Semilinear Elliptic Equations: Classical and Modern Theories 9783110556285, 9783110555356

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Takashi Suzuki Semilinear Elliptic Equations

De Gruyter Series in Nonlinear Analysis and Applications

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Editor-in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Mikio Kato, Tokyo, Japan Wojciech Kryszewski, Torun, Poland Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Simeon Reich, Haifa, Israel Vicenţiu D. Rădulescu, Krakow, Poland

Volume 35

Takashi Suzuki

Semilinear Elliptic Equations |

Classical and Modern Theories

Mathematics Subject Classification 2010 Primary: 35J61, 35B33; Secondary: 35K58, 35B40, 35B44 Author Prof. Dr. Takashi Suzuki Center for Mathematical Modeling and Data Science Osaka University 1-3 Machikaneyama-cho 560-8531 Osaka Japan [email protected]

ISBN 978-3-11-055535-6 e-ISBN (PDF) 978-3-11-055628-5 e-ISBN (EPUB) 978-3-11-055545-5 ISSN 0941-813X Library of Congress Control Number: 2020942754 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface Nonlinearity arises when one wishes to describe the complicated structure of a problem in a compact form, reducing the number of unknown parameters via conservative quantities, multiscaled simplifications, dualities, and so on. Then semilinear elliptic equations arise often when the reduced system is stationary. Since this model is independent of time, one may feel a sort of mildness to its solution, which, however, is far from my impression got from mathematical experience. Needless to say, an elliptic equation describes a transient process. Its nonlinearity inside is spatial, but it casts a time conductor to the outside. It is a silent driver, or writer of the unstable stationary state in the field variable. This book is a revised version of my old book published in 1994 [332]. As it was, I have not developed any general theory. On the contrary, this book is restricted to several special phenomena hidden in an event, which has an unexpected vertical hierarchy, gradually spreading to the horizon. In this new version, Part I is devoted to the fundamental theory, composed of four chapters. Chapter 1 is devoted to the calculus of variations to describe the existence and nonexistence of the solution for the power nonlinearity. It deals with Keller–Segel model as an introduction, Pohozaev identity [280], Lagrangian multiplier, critical exponent, and mountain pass lemma. The proof of Brezis–Kato’s theorem is included for completeness. Chapter 2 is concerned with symmetry and its breaking. It deals with an ignition model as an introduction, then the structure of its stationary state is revealed by several analytical methods. Here we added further results on the blowup of the nonstationary solution. The ODE approach is then justified by the symmetry result of Gidas, Ni, and Nirenberg theorem. The maximum principle is then applied to ensure the log-concavity [126] of the solution to the heat equation. Finally, symmetry breaking is approached by calculus of variations. Chapter 3 is a description of curvature equations in accordance with the surface theory [256]. Classical singular limit analysis is performed using complex variables. Then, Chapter 4 is concerned with classical and nonclassical results of the isoperimetric inequality, based on the method of rearrangement. Potential theory for spherically harmonic functions, capacity estimate of blowup set, extremal case of the Trudinger–Moser inequality are biproducts of these methods and results, and an analytic proof of Hamilton’s result on the normalized Ricci flow [181, 182] is provided. Part II is concerned with the applications oriented towards extremal structures of several specific models. First, Chapter 5 is concerned with the supplementary topics related to Part I, that is, Nehari principle, applications of moving planes, method of scaling, new comparison theorem associated with a differential inequality, Trudinger– Moser inequality in the original form, and topological arguments on the level set of variational functionals. Then Chapters 6 and 7 are devoted to the Gel’fand equation on an annulus and Hardy-BMO duality, respectively. The original version was composed of five chapters. The first two chapters were fundamental, concerning calculus of variations and maximum principles. All of them https://doi.org/10.1515/9783110556285-201

VI | Preface are included in this new edition in Part I. The following two chapters are mostly concerned with the Gel’fand equation in two space dimensions, that is, classification of singular limits of the classsical solution in accordance with the Green function and a uniqueness theorem. These topics are also kept in this edition in Part I, including later developments in Part II. Concerning the former, a paradigm shift is done from complex analysis to real analysis. Benefits are observed in the localization, refinement, extension, and the reverse theory – construction of the solution in accordance with the topological and variational methods. Some of the details are described in Part II. Progress on the uniqueness in Part I is seen for multiply-connected domains. Basically, a change of the study is derived with the movement from the Gel’fand equation to the Boltzmann–Poisson equation. Then the Hamiltonian control is reformulated as a recursive hierarchy. This static theory is presented in Part II of this book. A recent study, however, reveals a kinetic theory in the context of Liouville theory, the details of which will be presented in another monograph. Finally, Chapter 5 of the original version, concerned with the Euler–Lagrange equation of the Trudinger–Moser func2 tional, is put into a section in Chapter 5. The associated Euler equation is −Δu = λueu in two-space dimensions, where a quantized blowup mechanism was observed recently [105]. The original sections, Nehari Principle and Symmetric Criticality, are thus moved to the last chapter of Part I and the last section in Chapter 2, respectively. Symmetric Criticality, Gel’fand Equation on Annulus (2D and Higher Dimensional Cases), and Bifurcation from Symmetry are moved to Part II. The paragraphs on sinh-Poisson equation (quaternion, harmonic map), Neumann problem (of Boltzmann–Poisson equation), convex domains (for classification of singular limits), area functional (in singular perturbation), capacity estimates of the singular set (including parabolic case), Trudinger–Moser inequality, and normalized Ricci flow are added. New topics include an application of the moving plane method, new comparison theorem associated with differential inequalities, topological methods in the flavor of the Morse theory, regularity of the weak solution, blowup analysis inducing recursive hierarchy, Hardy spaces and BMO, bubble towers of harmonic map, and general form of Wente’s inequality. This book thus comes back to the system of chemotaxis, suggesting a quantized blowup mechanism and its recursive control of the Hamiltonian at the end. I have left some contents oriented for specialists, but most of the material is for beginners, graduate students, and general researchers interested in nonlinear mathematical analysis. As in the original version, each chapter is provided with introduction, description of the problem, methods, and conclusion. Thus Sections 7.1.1 and 7.1.2 may be used as an introduction to Chapter 1, dealing with calculus of variations. I have tried to make the description self-contained except for some cases. They are several fundamental results on the rearrangement (Propositions 4.18–4.22), those on the geometric measure theory, De Giorgi’s isoperimetric inequality (4.34) and Fleming– Rishel’s coarea formula (4.35), the asymptotics of a zero point of ODE equation for the proof of Carleson–Chang’ theorem, Lemma 5.64, quantization of collapsed masses

Preface | VII

in the sinh-Poisson equation emerged with the use of quaternion; the last part of Lemma 5.71, and several topological results used in §5.6. I conclude this preface with gratitude to Nikos Kavallaris for suggesting me to write this monograph. Thanks are also due to Yoshihisa Miyanishi for providing me with guidelines to the conformal geometry used in Sections 3.1.1, 3.1.2, and Futoshi Takahashi for permission of including a part of the previous article [342] in §7.4. June, 2020 Takashi Suzuki, Osaka

Acknowledgment This work was supported by JSPS Core to Core Program, International Research Network and JSPS Kakenhi 16H06576 and 26247013.

https://doi.org/10.1515/9783110556285-202

Preface for the Original Text It was around three years ago, but I was very confused at the request of writing a book about my work. The theory of semilinear elliptic equations had been advanced extensively, and my contribution was almost nothing. Besides, I am not a good story teller. However, it has been out! Therefore, I have to excuse myself. The book is mostly based on the lectures delivered at various places. I need five chapters. Each is provided with an introduction, preliminary material, proofs, and conclusion, and can be read independently, in spite of the fact that all are connected with each other significantly inside. I intended the book to be an invitation to the nonlinear problems. It is not self-contained, nor does it contains all of the celebrated work. But readers are supposed to be acquainted with some of the fundamental ideas. The equations are quite simple, but have a lot of suggestions. Maybe that is why many people have been attracted to them so much, and the study is still in progress. In the autumn of 1984, I learned a surprising result of H. Wente concerning the soap bubble. Before, I did not know the way to apply the complex function theory to nonlinear problems. That was the time I began the study such equations. Throughout these ten years, I have been engaged in a few problems, but most of them had been motivated by the personal experience. Here, I just express my gratitude to H. Ishii, S. Sakaguchi, and Y. Yamada for careful reading of the preliminary version. Even now, I am not sure that I had been a good writer, but I appreciate Professor N. Kenmochi very much for providing me with this opportunity. October, 1994 Takashi Suzuki

https://doi.org/10.1515/9783110556285-203

Contents Preface | V Acknowledgment | IX Preface for the Original Text | XI

Part I: Fundamentals 1 1.1 1.1.1 1.1.2 1.1.3 1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.4.1 1.4.2 1.4.3 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 1.5.6

Calculus of variations | 3 Reaction–diffusion systems | 3 Keller–Segel model | 3 Stationary problems | 5 Existence and nonexistence of a solution | 6 Pohozaev identity | 6 Basic identity | 6 Generalized identity | 9 Free boundaries and unique continuation | 14 Method of variations | 18 Lagrangian multiplier | 18 Sobolev and Morrey embeddings | 20 Differentiability and submersion | 27 Positivity and regularity | 32 Critical exponent | 33 Sobolev constant | 33 Lieb’s lemma | 36 Brezis–Kato’s theorem | 38 Mountain pass lemma | 40 Palais–Smale condition | 40 Deformation lemma | 43 Ekeland’s variational principle | 44 Legendre transformation | 46 Minimax principle | 49 Ghoussoub–Preiss version | 53

2 2.1 2.1.1 2.1.2 2.1.3

Maximum principles | 59 Parabolic dynamics | 59 Ignition model | 59 Stationary state: standard arguments | 62 Global-in-time dynamics | 66

XIV | Contents 2.1.4 2.1.5 2.1.6 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.3 2.3.1 2.3.2 2.3.3 2.4 2.4.1 2.4.2 2.4.3 2.4.4 3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.3.7 3.4

Branch of minimal solutions | 71 Unboundedness via the topological degree | 76 Nonminimal solutions | 78 Radial symmetry | 79 Gidas–Ni–Nirenberg’s theorem | 79 Maximum principles | 84 Method of the moving plane | 88 Some remarks | 91 Serrin’s corner point lemma | 94 Convexity of level sets | 95 Nonparametric convex surfaces | 95 Logarithmic concavity of the solution | 98 Identity for the Hessian | 100 Symmetric criticality | 103 Symmetric functionals | 103 Symmetric solutions | 105 Maximal symmetry | 105 Mountain pass approach | 108 Complex structure | 109 Theory of surfaces | 109 Curvatures | 109 Conformal geometry and soap bubbles | 112 Liouville integral and spherical derivatives | 115 Sinh-Poisson equation, quaternion, harmonic map | 118 Boltzmann–Poisson equation | 121 Asymptotic analysis | 121 Bol’s inequality | 123 Radial solutions | 124 Associated Legendre equations | 126 Laplace–Beltrami operator | 126 Potential theory | 127 Neumann problems | 128 Classification of the singular limit | 129 Summary | 129 Boundary estimate | 131 Obata’s relation | 134 Complex analysis | 137 Isolated singular points | 141 Simply-connected domains | 143 Convex domains | 145 Singular perturbation | 150

Contents | XV

3.4.1 3.4.2 3.4.3 3.4.4 4 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.3.6 4.3.7 4.3.8 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 4.4.6

Weston’s theory | 150 Moseley–Wente’s theory | 157 Convergence of the asymptotic expansion | 159 Area functional | 165 Rearrangement | 171 Elliptic eigenvalue problems | 171 Vibrating membrane | 171 Hartman–Wintner’s theorem | 175 Kuo’s lemma | 178 Nodal domains | 181 Faber–Krahn inequality | 183 Equimeasurable transformations | 186 Coarea formula | 186 Talenti’s comparison theorem | 188 Capacity estimates of the singular set | 191 Parabolic case | 194 Isoperimetric inequality on surfaces | 198 Bol’s inequality in analytic form | 198 Mean value theorems | 200 Reverse inequality | 205 Bandle’s rearrangement | 212 Global analysis | 216 A priori estimates and bending | 217 Connectivity via isoperimetric inequality | 219 Trudinger–Moser inequality | 226 Normalized Ricci flow | 230 Geometric motivation | 230 Analytic approach | 233 Benilan’s inequality | 236 Estimate from below | 238 Concentration of probability measures | 243 Compactness of the orbit | 247

Part II: Applications 5 5.1 5.1.1 5.1.2 5.1.3

Supplementary topics | 255 Nehari principle | 255 An alternative variation | 255 Equivalence to the mountain pass lemma | 257 Minimizing sequences | 259

XVI | Contents 5.1.4 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.5 5.5.1 5.5.2 5.5.3 5.5.4 5.5.5 5.5.6 5.6 5.6.1 5.6.2 5.6.3 5.6.4 5.6.5 5.6.6 6 6.1 6.1.1 6.1.2 6.1.3 6.1.4 6.2 6.2.1 6.2.2

Iterative sequences | 261 Moving plane revisited | 263 A priori estimate of the solution | 263 Positivity of the solution | 266 Uniqueness of the solution | 268 Lipschitz continuity of the nonlinearity | 271 Poincaré metric and symmetry | 272 Method of scaling | 276 Boltzmann–Poisson equation revisited | 276 Blow-up analysis | 278 Brezis–Merle’s inequality and related topics | 281 Vanishing residual | 284 Higher-dimensional case | 288 Differential inequality −Δu ≤ f (u) | 298 Blow-up profile | 298 A comparison theorem | 301 Mean value theorem in higher dimension | 305 Pattern formation | 307 Trudinger–Moser inequality revisited | 309 Threshold of asymptotics | 309 Fundamental theorem | 311 Nonuniformness | 317 Sharpness of Moser’s inequality | 320 Uniformness near the origin | 322 Carleson–Chang’s theorem | 327 Topological methods | 331 Total degree | 331 Sinh-Poisson equation revisited | 333 Topological deformation | 336 Blow-up analysis revisited | 339 Improved Trudinger–Moser inequality | 344 Lucia’s deformation lemma | 350 Equations on annulus | 357 2D case | 357 Radial solutions on annulus | 357 Liouville integral of radial solutions | 358 Generation of nonradial solutions | 363 Nonradial bifurcation | 365 Higher-dimensional case | 367 Radial solutions in higher dimensions | 367 Emden transformation | 370

Contents | XVII

6.2.3 6.2.4 6.3 6.3.1 6.3.2 6.4 6.4.1 6.4.2 6.4.3 6.4.4 7 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.1.5 7.1.6 7.2 7.2.1 7.2.2 7.2.3 7.3 7.3.1 7.3.2 7.3.3 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.4.5 7.5 7.5.1 7.5.2 7.5.3 7.5.4

Boundary pair | 374 Bending arbitrary many times | 377 Bifurcation from symmetry | 379 Morse indices | 379 Radial Morse indices | 384 Structure of radial solutions | 386 Summary | 386 Γ-curves | 389 Radial Morse indices revisited | 390 Zone property | 394 Hardy spaces and BMO | 395 Regularity of weak solutions | 395 Dirichlet principle | 395 Existence of weak solutions | 397 p-harmonic functions | 400 Local maximum principle | 402 Local minimum principle | 406 BMO, Harnack inequality, and C α regularity | 408 BMO | 412 Poincaré–Sobolev inequality | 412 Calderón–Zygmund decomposition | 415 John–Nirenberg’s inequality | 417 Hardy spaces | 419 Maximal theorem | 419 Div–Rot lemma | 421 Jacobian estimate | 423 Harmonic map revisited | 424 Quantized blow-up mechanism | 424 Monotonicity formula | 428 Hardy–BMO structure | 430 Energy concentration | 430 Bubble towers | 432 L∞ -estimates | 440 BMO estimate of the Green function | 440 Wente’s inequality | 443 Brezis–Merle’s inequality revisited | 445 System of chemotaxis | 446

Bibliography | 453 Index | 469

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Part I: Fundamentals

1 Calculus of variations The theme of the present chapter is the existence of a solution. The first issue is the notion of the critical exponent, which splits the situation into alternatives, existence and nonexistence. Existence is established through the method of variations, where the theory of real and abstract analysis comes in. Closely related to each other, three ways to tackle the problem are presented: Lagrangian multiplier, mountain pass, and the Nehari method. Actually, those are the variations taken on the restricted sets or the path spaces. Nonexistence, on the other hand, follows from an identity, discovered by several authors, and then the readers are referred to various applications. Finally, the method of variations is adopted in a qualitative study of the symmetry breaking. The text presents a system of equations describing some kind of living things of curious life style, having attracted many people.

1.1 Reaction–diffusion systems 1.1.1 Keller–Segel model Semilinear elliptic boundary value problems arise mostly when describing stationary states of various phenomena, so they are equations of balance. We start with a model of cellular slime mold, the life style of which changes regularly between animal and plant. In the animal phase, it produces some chemical material influencing itself. The density of chemotaxis and the material are denoted by u(x, t) and v(x, t), respectively, and are related as to satisfy a system of equations, called the Keller–Segel system, a kind of reaction–diffusion system, where x = (x1 , x2 , . . . , xn ) with n = 2, 3 and t stand for the space and time variables, respectively. Actually, the system is given in the following way, where Ω ⊂ Rn denotes the domain under consideration, ν is the outer unit normal vector on the boundary 𝜕Ω, Q = Ω × (0, T), and Γ = 𝜕Ω × (0, T): 𝜕u = D1 Δu + χ∇ ⋅ (u∇ψ(v)), u > 0, 𝜕t 𝜕v = D2 Δv + k(u, v), v > 0 in Q, 𝜕t with 𝜕 (u, v)|𝜕Ω = 0, (u, v)|t=0 = (u0 (x), v0 (x)) > 0, 𝜕ν where D1 , D2 , and χ are positive constants and Δ= https://doi.org/10.1515/9783110556285-001

𝜕2 𝜕2 𝜕2 + + ⋅ ⋅ ⋅ + 𝜕xn2 𝜕x12 𝜕x22

(1.1)

4 | 1 Calculus of variations denotes usual Laplacian. The functions ϕ(v) and k(u, v) have the following forms, and in particular are smooth in u, v > 0: ϕ(v) = log v,

k(u, v) = −av + bu,

where a and b are positive constants, too. We have followed the standard notation of vector analysis. Hence ∇=(

𝜕 𝜕 𝜕 , ,..., ) 𝜕x1 𝜕x2 𝜕xn

denotes the gradient operator, and ∇⋅a=

𝜕a 𝜕a1 𝜕a2 + + ⋅⋅⋅ + n 𝜕x1 𝜕x2 𝜕xn

the divergence of the vector field a = (a1 (x), a2 (x), . . . , an (x)). The second term on the right-hand side of (1.1), therefore, means 𝜕 𝜕 𝜕 𝜕 𝜕 𝜕 (u ϕ(v)) + (u ϕ(v)) + ⋅ ⋅ ⋅ + (u ϕ(v)). 𝜕x1 𝜕x1 𝜕x2 𝜕x2 𝜕xn 𝜕xn The above ϕ(v) is monotone increasing, while k(u, v) is monotone increasing and decreasing in u and v, respectively. Generally, an equation of the form 𝜕w − dΔw + ∇ ⋅ bw = f 𝜕t describes the convection–diffusion of w(x, t), which denotes the material density, subject to the convection velocity b(x, t).1 The term f (x, t) denotes the acting force, say gravitation, and constant d > 0 denotes the diffusion coefficient. Since b = χ∇φ in (1.1), this model describes the following situation: 1. Cellular slime molds u(x, t) diffuse and move in the direction where the chemical material v(x, t) increases the most, subject to the power proportional to the degree of its increasing. 2. The chemical material v(x, t) diffuses by itself. The magnitude of production, 𝜕v , 𝜕t increases as the chemotaxis u(x, t) increases, and decreases as v(x, t) itself increases. 3. Chemotaxis and the chemical material are never produced or absorbed on the boundary 𝜕Ω. Under these circumstances, it is natural to suppose that chemotaxis is slower than that of the chemical material. 1 See Section 7.1 of [335].

D1 D2

≪ 1, i. e., the diffusion of

1.1 Reaction–diffusion systems | 5

1.1.2 Stationary problems For the qualitative study of the Keller–Segel system, especially global-in-time behavior, it is useful to investigate its stationary states. In fact, stable stationary states keep the nonstationary solutions around them, while unstable ones exert serious influence on the dynamics locally and globally. Thus we pick up the following system: D1 Δu − χ∇ ⋅ (u∇ log v) = 0, D2 Δv − av + bu = 0, 𝜕 (u, v)|𝜕Ω = 0. 𝜕ν

v>0

u > 0, in Ω, (1.2)

For any constant u > 0, the pair (u, v) = (u,̄ a−1 bu)̄ solves the relation. This kind of solutions, called constant solutions, are not significant. Actually, for a nonstationary system we expect u(x, t) to have sharp peaks eventually blowing-up like a δ-function, which is regarded as a transition into the plant phase of the cellular slime molds. Related to the issue, it is proven that the stationary system itself has nontrivial solutions with spiky patterns.2 However, nonconstant stationary solutions arise only when the parameters are taken appropriately, as the results below indicate. In fact, we take w = log u − χD−1 1 log v to deduce ∇ ⋅ (D1 u∇w) = D1 ∇ ⋅ (u∇ log u) − χ∇ ⋅ (u∇ log u) = D1 Δu − χ∇ ⋅ (u∇ log v) = 0

in Ω

and 1 𝜕v 𝜕w 1 𝜕u = − χD−1 =0 1 𝜕ν u 𝜕ν v 𝜕ν

on 𝜕Ω.

Hence it holds by Green’s formula that 0 = ∫{∇ ⋅ (D1 u∇w)}w dx = ∫ (D1 u Ω

𝜕Ω

𝜕u )w ds − ∫ D1 u|∇w|2 dx 𝜕ν Ω

2

= − ∫ D1 u|∇w| dx, Ω

where ds denotes the surface element. Therefore, w is a constant denoted by log λ. In other words, u = λvχ/D1 which is to be substituted into (1.2). Putting p = χ/D1 , d = D2 /a, μ = (a−1 bλ)1/(p−1) , and w = μv, we arrive at dΔw − w + wp = 0,

w>0

2 Ni and Takagi [252], [214], [254]. See also [363].

in Ω,

𝜕w | = 0, 𝜕ν 𝜕Ω

(1.3)

6 | 1 Calculus of variations which is a boundary value problem for the single unknown function w(x) with the linear principal part dΔw − w. Such problems have been proposed in many areas, e. g., in physics, chemistry, and biology. The structure of the solution set, however, is not so simple as it looks [214, 253]. 1.1.3 Existence and nonexistence of a solution The boundary value problem (1.3) has the invertible linear part dΔw − w with the boundary condition taken into account, so that is equivalent to the fixed point equation in an appropriate function space, w = (−dΔ + 1)−1 wp . One may think that the existence of a solution will be established through the compactness of the operator defined by the right-hand side, and the construction of an appropriate invariant set. However, this method may produce only trivial (constant in this case) solutions.3 To illustrate the situation, we take a simpler problem − Δu = up ,

u>0

in Ω,

u|𝜕Ω = 0,

(1.4)

where 1 < p < ∞ denotes the given exponent. The trivial solution for this problem is u ≡ 0. We have the following theorems for this problem, where n∗ = {

+∞, n = 2, n+2 , n−2

n > 2.

(1.5)

Theorem 1.1. If 1 < p < n∗ , there exists a solution to (1.4). Theorem 1.2. If n > 2, p ≥ n∗ , and Ω is star-shaped, then there does not exist a solution to (1.4).4 Here and henceforth, the bounded domain Ω is assumed to have a smooth boundary 𝜕Ω, and the solution u ∈ C 2 (Ω) ∩ C 0 (Ω)̄ is the classical one unless otherwise stated.

1.2 Pohozaev identity 1.2.1 Basic identity Theorem 1.2 in the previous section is a consequence of the following lemma due to S. I. Pohozaev. 3 See Section 5.1 for an iterative scheme to avoid this triviality. 4 See, however, the celebrated work [17] for the existence of the solution with p = n∗ , when the domain Ω has a nontrivial topology.

1.2 Pohozaev identity |

7

Lemma 1.3 ([274]). For a continuous function f = f (u), let u = u(x) ∈ C 2 (Ω)̄ satisfy −Δu = f (u)

in Ω,

u|𝜕Ω = 0,

where Ω ⊂ Rn is a bounded domain. Then the identity N ∫ F(u) dx +

2

1 𝜕u 2−n ∫ f (u)u dx = ∫ ( ) (x ⋅ ν) ds 2 2 𝜕ν Ω

Ω

(1.6)

𝜕Ω

u

holds for F(u) = ∫0 f (u) du. Proof. We apply the divergence formula of Gauss to the vector field a(x) = [(x ⋅ ∇)u]∇u = (x1

𝜕u 𝜕u 𝜕u 𝜕u 𝜕u 𝜕u + x2 + ⋅ ⋅ ⋅ + xn )( , ,..., ), 𝜕x1 𝜕x2 𝜕xn 𝜕x1 𝜕x2 𝜕xn

i. e., ∫ ∇ ⋅ a dx = ∫ a ⋅ ν ds, Ω

(1.7)

𝜕Ω

where ⋅ denotes the usual inner dot-product in Rn . 2 𝜕u u In fact, writing uj = 𝜕x and uij = 𝜕x𝜕 𝜕x , we have j

i

∇⋅a=∑ j,i

j

𝜕u 𝜕u 𝜕 {x } 𝜕xj i 𝜕xi 𝜕xj

= [(x ⋅ ∇)u]Δu + ∑ xi uij uj + |∇u|2 , i,j

where [(x ⋅ ∇)u]Δu = − ∑ xi ui f (u) = − ∑ xi F(u)i = −x ⋅ ∇F(u), i

i

and hence ∫[(x ⋅ ∇)u]Δu dx = − ∫ x ⋅ ∇F(u) dx Ω

Ω

= − ∫ (x ⋅ ν)F(u) ds + n ∫ F(u) dx = n ∫ F(u) dx Ω

𝜕Ω

Ω

due to u|𝜕Ω = 0. Writing ν = t (ν1 , ν2 , . . . , νn ), we have Iij = ∫ xi uij uj dx = ∫ xi νi u2j ds − ∫ u2j dx − Iij , Ω

𝜕Ω

Ω

8 | 1 Calculus of variations and hence 1 1 ∫ (xi νi )u2j ds − ∫ u2j dx. 2 2

Iij =

Ω

𝜕Ω

Therefore, ∑ ∫ xi uij uj dx = i,j Ω

1 n ∫ (x ⋅ ν)|∇u|2 ds − ∫ |∇u|2 dx 2 2 𝜕Ω

=

Ω

2

𝜕u 1 n ∫ ( ) (x ⋅ ν) ds − ∫ |∇u|2 dx 2 𝜕ν 2 Ω

𝜕Ω

so that ∫ ∇ ⋅ a dx Ω

= n ∫ F(u) dx + (1 − Ω

2

𝜕u n 1 ) ∫ |∇u|2 dx + ∫ ( ) (x ⋅ ν) ds 2 2 𝜕ν Ω

= n ∫ F(u) dx + Ω

𝜕Ω

2

2−n 1 𝜕u ∫ f (u)u dx + ∫ ( ) (x ⋅ ν) ds, 2 2 𝜕ν Ω

(1.8)

𝜕Ω

while ∫ a ⋅ ν ds = ∫ [(x ⋅ ∇)u] 𝜕Ω

𝜕Ω

2

𝜕u 𝜕u ds = ∫ ( ) (x ⋅ ν) ds, 𝜕ν 𝜕ν

(1.9)

𝜕Ω

due to u|𝜕Ω = 0. The desired equality (1.6) follows from (1.7), (1.8), and (1.9). Proof of Theorem 1.2. Let Ω be star-shaped with respect to the origin and u(x) be a solution to (1.4). We apply the Pohozaev identity (1.6) for f (u) = up . Then it holds that F(u) =

1 p+1 u p+1

and hence 2

(

n 2−n 1 𝜕u + ) ∫ up+1 dx = ∫ ( ) (x ⋅ ν) ds. p+1 2 2 𝜕ν Ω

The left-hand side is negative if p > 𝜕Ω. When p = n+2 , it follows that n−2

n+2 , n−2

𝜕Ω

which is a contradiction because x ⋅ ν > 0 on

𝜕u =0 𝜕ν

on 𝜕Ω,

(1.10)

1.2 Pohozaev identity |

9

which is also impossible. In fact, u solves the linear problem − Δu = c(x)u,

u>0

in Ω,

u|𝜕Ω = 0,

(1.11)

for c(x) = up−1 , and hence Hopf lemma5 implies 𝜕u 0 satisfying ψ(r) = o(1) and |ψ󸀠 (r)| + |ψ󸀠󸀠 (r)| = O(r −n+δ ) as r ↓ 0 for some δ > 0. The boundary condition (1.19) is not necessary to deduce (1.15). In particular, the following identity can be proven by taking h(x) = x,

a(x) =

n−2 . 2

Corollary 1.7. Any function u ∈ C(B̄ R ) satisfying −Δu = f (u) in BR ≡ {|x| < R} ⊂ Rn admits the identity ∫ nF(u) + BR

2−n |∇u|2 dx 2 2

R 𝜕u = ∫ − |∇u|2 + RF(u) + R( ) ds. 2 𝜕r 𝜕BR

10 See the following chapter for the property u = u(r), r = |x|, that is, radial symmetry of the solution.

14 | 1 Calculus of variations Proof. Without the boundary condition, we cannot reduce the contribution of (1.21) in ∫Ω ⋅ dx to (1.22). Therefore, just the identity n − 2 𝜕u 𝜕u 1 u − (x ⋅ ∇u) ds ∫ |∇u|2 (x ⋅ ν) − F(u)(x ⋅ ν) − 2 2 𝜕ν 𝜕ν

𝜕Ω

= ∫ −nF(u) + Ω

n−2 uf (u) dx 2

(1.23)

follows from (1.15). Here, we have Ω = BR and x⋅∇=r

𝜕 , 𝜕r

x⋅ν =R

on 𝜕BR

so that the left-hand side of (1.23) is equal to 2

∫ 𝜕BR

n − 2 𝜕u 𝜕u R |∇u|2 − RF(u) − u − R( ) ds, 2 2 𝜕r 𝜕r

while ∫ uf (u) dx = ∫ u(−Δu) dx = ∫ |∇u|2 dx − ∫ u BR

BR

BR

𝜕BR

𝜕u ds 𝜕r

stands for the right-hand side of (1.23). Then the desired equality follows.

1.2.3 Free boundaries and unique continuation We present two applications of the Pohozaev identity, namely the free boundary problem and unique continuation theorem. The first requires the strong maximum principle for the Laplace operator. This principle is described in the following chapter for more general cases. Theorem 1.8. Let Ω ⊂ Rn be a bounded domain with C 3 boundary 𝜕Ω. Suppose the existence of a function u ∈ C 3 (Ω)̄ satisfying −Δu = 1 in Ω,

u|𝜕Ω = 0,

𝜕u =c 𝜕ν

on 𝜕Ω

where c is a constant. Then Ω is necessarily a ball. Proof. We takef (u) ≡ 1 in Lemma 1.3 to conclude ∫ nu + Ω

2−n c2 c2 |∇u|2 dx = ∫ (x ⋅ ν) ds = n|Ω| 2 2 2 𝜕Ω

(1.24)

1.2 Pohozaev identity |

15

where |Ω| denotes the volume of Ω. Here we have ∫ u dx = ∫(−Δu)u dx = ∫ |∇u|2 dx Ω

Ω

Ω

so that (n + 2) ∫ u dx = nc2 |Ω|.

(1.25)

Ω

Now, Schwarz’ inequality implies n

n

i=1

i,j=1

1 = (Δu)2 ≤ n ∑ u2ii ≤ n ∑ u2ij

(1.26)

and hence Δ(|∇u|2 +

2

2 2 𝜕2 𝜕u u) = ∑ 2 ( ) + Δu n 𝜕x n 𝜕x i i,j j = 2 ∑ u2ij + 2 ∑ ui uijj + i,j

ij

2 2 Δu = 2 ∑ u2ij − ≥ 0 n n i,j

since Δu = −1. Here, |∇u|2 +

2

𝜕u 2 u=( ) n 𝜕ν

on 𝜕Ω

so that either |∇u|2 + n2 u < c2 or |∇u|2 + n2 u = c2 in Ω because of the strong maximal principle for subharmonic functions. The former case does not occur since ∫ |∇u|2 +

2 2 u dx = ∫ u + u dx = c2 |Ω|, n n Ω

Ω

by (1.24) and (1.25). Therefore, Δ(|∇u|2 +

2 u) = 0 in Ω, n

1 and the equality holds in (1.26). Thus we obtain uij = − n1 δij so that u = 2n (A − r 2 ) for n r = |x − x0 |, with A being a constant and x0 some point in R . We again utilize the strong maximum principle to state that u > 0 in Ω and hence 𝜕Ω = {u = 0}. Then, Ω must be a ball.

16 | 1 Calculus of variations The above proof is due to H. F. Weinberger [365]. The original proof by J. Serrin is based on a sophisticated use of the maximum principle, and is exposed in the following chapter. The argument is applicable to more general equations (see, e. g., [351]), while the above proof has the advantage of using fewer assumptions on the regularity of the free boundary [127]. The second application is the unique continuation property for solutions of linear elliptic equations of second order, which states that two solutions coinciding within an open set must be identically equal to each other. We shall present the simplest version.11 Theorem 1.9. Let Ω ⊂ Rn be a domain and u ∈ C 2 (Ω) satisfy Δu = 0 in Ω. Let x0 ∈ Ω and suppose u2 dx = O(Rm ),

∫

R↓0

(1.27)

|x−x0 | 0 admits Cm > 0 such that m

∫ u2 dx ≤ C k ∫ u2 dx ≤ Cm C k (R0 2−k ) = Cm Rm 0(

BR0

BR

k

C ) 2m

02

−k

for k = 0, 1, 2, . . . Second, we take m ≫ 1 satisfying follows that

C 2m

< 1 and let k → +∞. Then, it

∫ u2 dx = 0 BR0

and hence u = 0 in BR0 . Therefore, the desired identity u ≡ 0 follows from a standard argument based on the connectivity of Ω.

1.3 Method of variations 1.3.1 Lagrangian multiplier A simple way to verify Theorem 1.1 is to adopt the variational method, particularly that based on the Lagrangian multiplier principle.12 The idea can be traced back to Lord Rayleigh’s principle for characterizing the first eigenvalue of the Laplacian, the constant λ1 = λ1 (Ω) introduced in the previous section and defined by − Δϕ1 = λ1 ϕ1 ,

ϕ1 > 0

in Ω,

ϕ1 |𝜕Ω = 0

(1.32)

with a function ϕ1 (x). This principle indicates that 1 1 λ1 = inf{ ‖∇v‖22 | v ∈ H01 , ‖v‖22 = 1}. 2 2 Here and henceforth, the set of p-integrable measurable functions on Ω is denoted by Lp = Lp (Ω) for 1 ≤ p ≤ ∞, Lp = {f : Ω → R, measurable | ‖f ‖p < +∞}, with the norm ‖ ⋅ ‖p defined by ‖f ‖p = {

(∫Ω |f |p dx)1/p ,

ess.sup |f |,

1 ≤ p < ∞, p = ∞.

Given m = 1, 2, . . . , W m,p = W m,p (Ω) denotes the Sobolev space defined by W m,p = {f ∈ Lp | Dα f ∈ Lp , |α| ≤ m}, 12 See [326] for applications of the variational method to differential equations.

1.3 Method of variations | 19

where Dα = (

α

α

αn

1 2 𝜕 𝜕 𝜕 ) ( ) ⋅⋅⋅( ) 𝜕x1 𝜕x2 𝜕xn

denotes the differentiation in the sense of distributions of L. Schwarz, and |α| = α1 + α2 +⋅ ⋅ ⋅+αn . The space W0m,p = W0m,p (Ω) denotes the closure in W m,p of C0∞ (Ω), the set of C ∞ functions with compact support13 in Ω. Finally, we put H01 = H01 (Ω) for W01,2 (Ω) [47]. The following facts can be proven: 1. The infimum λ1 is attained by a function ϕ1 (x) ≥ 0. 2. This ϕ1 (x) is a classical solution to (1.32) and ϕ1 (x) > 0 holds for any x ∈ Ω. 3. The constant λ1 is the minimum eigenvalue, i. e., − Δϕ = λϕ,

φ ≢ 0

in Ω,

ϕ|𝜕Ω = 0

(1.33)

implies λ ≥ λ1 . Furthermore, any solution ϕ to (1.33) with λ = λ1 is a constant multiplication of the above ϕ1 . The first fact is proven by the compactness argument. Let vk ∈ H01 , k = 1, 2, . . . , be a minimizing sequence: 1 ‖v ‖2 = 1, 2 k 2

1 ‖∇vk ‖22 → λ1 . 2

This sequence is bounded in H01 so that has a weak limit, say, v ∈ H01 , by passing to a subsequence. This property means lim ∫ ∇vk ⋅ ∇ψ dx = ∫ ∇v ⋅ ∇ψ dx

k→∞

for any ψ ∈

H01 ,

Ω

Ω

which implies 1 1 ‖∇v‖22 ≤ lim inf ‖∇vk ‖22 = λ1 , 2 2 k→∞

because the Poincaré inequality δ‖v‖22 ≤ ‖∇v‖22 ,

v ∈ H01

(1.34)

with δ > 0 assures ‖∇ ⋅ ‖2 to be a norm in H01 [26].14 The problem left, is therefore, whether 21 ‖v‖22 = 1 holds or not. This property actually follows from the Rellich– Kondrachov theorem which assures vk → v strongly in L2 if vk ⇀ v weakly in H01 so that 1 1 ‖v ‖2 = 1 → ‖v‖22 = 1. 2 k 2 2

The second fact is proven by the Lagrangian multiplier principle, which guarantees that the minimizer ϕ1 (x) of λ1 is a weak solution to (1.32). Here we suppose 13 f ∈ C0∞ (Ω) is a smooth function in Ω such that supp f = {x ∈ Ω | f (x) ≠ 0} ⊂ Ω. 14 We can actually take δ = λ1 in (1.34).

20 | 1 Calculus of variations ϕ1 (x) ≥ 0 by replacing |ϕ| for ϕ in the minimizing process. Then, the regularity theorem and strong maximum principle imply the smoothness and positivity of ϕ1 (x), respectively [91]. The third fact is proven by Jacobi’s method.15 Thus we take the above φ1 (x) > 0, x ∈ Ω and an arbitrary φ(x) satisfying (1.33) for λ = λ1 . We apply Hopf lemma, described ̄ Applying (1.33) to in the following chapter, to φ1 (x), to confirm ψ = φ/φ1 ∈ C 2 (Ω). φ = ψ ⋅ φ1 , we obtain 2 φ1 Δψ + 2∇φ1 ⋅ ∇ψ = φ1 (Δψ + ∇ log φ21 ⋅ ∇ψ) = φ1 [φ−2 1 ∇ ⋅ (φ1 ∇ψ)] = 0

and hence ∇ ⋅ (φ21 ∇ψ) = 0 in Ω. Then it follows that ∫ φ21 |∇ψ|2 dx = 0,

Ω

and therefore, ψ(x) is a constant. These procedures are applicable to the nonlinear problem except for the last part. In proving Theorem 1.1, we introduce the variational problem 1 1 ‖v‖p+1 = 1}. inf{ ‖∇v‖22 | v ∈ H01 , 2 p + 1 p+1 The Lagrangian multiplier λ will be reduced to 1, from homogeneous nonlinearity f (u) = up , 1 < p < ∞. The main difficulty of the problem is thus to verify the first procedure, that is, the compact embedding of H01 ⊂ Lp+1 . Hence the need to talk about real analysis. 1.3.2 Sobolev and Morrey embeddings 2n and is compact when The continuous embedding H01 ⊂ Lp+1 arises when 1 ≤ p+1 ≤ n−2 2n 1 ≤ p + 1 < n−2 . These facts are basic in proving Theorem 1.1. The key is the inequality

‖v‖ 2n ≤ Cn ‖∇v‖2 , n−2

v ∈ C0∞ (Rn ),

with a constant Cn > 0. No exponent q, other than ‖v‖q ≤ Cn ‖∇v‖2 ,

2n , n−2

admits

v ∈ C0∞ (Rn ).

(1.35)

To see this, we take a nonzero function v ∈ C0∞ (Rn ) and set vλ (x) = v(λx) for λ > 0. Then, it holds that −n

‖v‖q = λ q ‖v‖q ,

‖∇vλ ‖2 = λ

2−n 2

‖∇v‖2

15 Nodal domain theorem in § 4.1.4 provides another proof; see [268] for the argument valid to quasilinear problems.

1.3 Method of variations | 21

so that n 2−n ‖∇vλ ‖2 ‖∇v‖2 + = λq 2 ⋅ . ‖vλ ‖q ‖v‖q

Hence

n q

+

2−n 2

= 0 is necessary for (1.35) to hold.16 The next theorem implies the con2n

tinuous embedding H01 (Ω) ⊂ L n−2 (Ω) for any bounded domain Ω ⊂ Rn , n ≥ 3. Theorem 1.10 (Sobolev). If 1 ≤ q < n, the inequality v ∈ C0∞ (Rn )

‖v‖q∗ ≤ C(n, q)‖∇v‖q , holds with a constant C(n, q) > 0, where

1 q∗

=

1 q

(1.36)

− n1 .

For the proof we utilize the following lemma deduced by induction. Lemma 1.11. Assume n > 1 and f1 , f2 , . . . , fn ∈ Ln−1 (Rn−1 ). Let f (x) = f (x̃1 )f (x̃2 ) ⋅ ⋅ ⋅ f (x̃n ), where x̃i = (x1 , x2 , . . . , xi−1 , xi+1 , . . . , xn ) ∈ Rn−1 for 1 ≤ i ≤ n. Then, it holds that n

‖f ‖L1 (Rn ) ≤ ∏ ‖fi ‖Ln−1 (Rn−1 ) . i=1

Proof. The case n = 2 is obvious. For n = 3, we note 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∫󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx 3 = 󵄨󵄨󵄨f3 (x1 , x2 )󵄨󵄨󵄨 ∫󵄨󵄨󵄨f1 (x2 , x3 )󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨f2 (x1 , x3 )󵄨󵄨󵄨 dx3 R

R

1/2

1/2

󵄨 󵄨 󵄨 󵄨2 󵄨 󵄨2 ≤ 󵄨󵄨󵄨f3 (x1 , x2 )󵄨󵄨󵄨(∫󵄨󵄨󵄨f1 (x2 , x3 )󵄨󵄨󵄨 dx3 ) (∫󵄨󵄨󵄨f2 (x1 , x3 )󵄨󵄨󵄨 dx 3 ) , R

R

which implies 1/2

󵄨 󵄨 󵄨 󵄨2 ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx ≤ (∫ 󵄨󵄨󵄨f3 (x1 , x2 )󵄨󵄨󵄨 dx1 dx2 ) R3

R2

1/2

󵄨 󵄨2 󵄨 󵄨2 ⋅ (∫ dx1 dx2 ∫󵄨󵄨󵄨f1 (x2 , x3 )󵄨󵄨󵄨 dx3 ∫󵄨󵄨󵄨f2 (x1 , x3 )󵄨󵄨󵄨 dx3 ) R2

R

= ‖f1 ‖L2 (R2 ) ‖f2 ‖L2 (R2 ) ‖f3 ‖L2 (R2 ) . 16 See [255, 124] for the Gagliardo–Nirenberg inequality.

R

22 | 1 Calculus of variations We suppose the assertion up to n. To establish the assertion for the case of n + 1, we freeze the variable xn+1 of x = (x1 , . . . , xn , xn+1 ) to get 󵄨 󵄨 ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx1 dx2 ⋅ ⋅ ⋅ dxn Rn n

≤ ‖fn+1 ‖Ln (Rn ) (∫ |f1 f2 ⋅ ⋅ ⋅ fn | n−1 dx 1 ⋅ ⋅ ⋅ dx n )

n−1 n

Rn

since

1 n

+

1

n n−1

= 1. Here, by the induction assumption, we have n

∫ |f1 f2 ⋅ ⋅ ⋅ fn | n−1 dx1 dx2 ⋅ ⋅ ⋅ dxn Rn

n

n

i=1

i=1

n 󵄩 n 󵄩 ≤ ∏󵄩󵄩󵄩|fi | n−1 |󵄩󵄩󵄩Ln−1 (Rn−1 ) = ∏ ‖fi ‖Ln−1 n (Rn−1 )

and hence n

󵄨 󵄨 ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx1 dx2 ⋅ ⋅ ⋅ dxn ≤ ‖fn+1 ‖Ln (Rn ) ∏ ‖fi ‖Ln (Rn−1 ) . i=1

Rn

We apply ∫R ⋅ dxn+1 to the above inequality. Since n

1/n

n

∫ ∏ ‖fi ‖Ln (Rn−1 ) dx n+1 ≤ ∏(∫ ‖fi ‖nLn (Rn−1 ) dx n+1 )

R i=1

i=1

R

n times

as

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 1 1 + ⋅ ⋅ ⋅ + = 1, it holds that n n n

󵄨 󵄨 ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx1 dx2 ⋅ ⋅ ⋅ dx n+1 ≤ ‖fn+1 ‖Ln (Rn ) (∏ ‖fi ‖nLn (Rn ) ) Rn+1

n+1

i=1

= ∏ ‖fi ‖Ln+1 (Rn ) , i=1

and hence the assertion holds for n + 1. Proof of Theorem 1.10. For v ∈ C0∞ (Rn ), we have 󵄨󵄨 x1 󵄨󵄨 󵄨󵄨 𝜕v 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 (t, x2 , . . . , xn ) dt 󵄨󵄨󵄨 󵄨󵄨v(x1 , x2 , . . . , xn )󵄨󵄨 = 󵄨󵄨 ∫ 󵄨󵄨 󵄨󵄨 𝜕x 󵄨 −∞ 1 󵄨 ∞ 󵄨󵄨 󵄨󵄨󵄨 𝜕v 󵄨 ≤ ∫ 󵄨󵄨󵄨 (t, x2 , . . . , xn )󵄨󵄨󵄨 dt. 󵄨 𝜕x1 󵄨󵄨 󵄨 −∞

1/n

1.3 Method of variations | 23

Similarly, ∞ 󵄨󵄨 𝜕v 󵄨󵄨󵄨 󵄨 󵄨 󵄨󵄨 n−1 󵄨󵄨v(x)󵄨󵄨󵄨 ≤ fi (x̃i ) ≡ ∫ 󵄨󵄨󵄨 (x1 , x2 , . . . , xi−1 , t, xi+1 , . . . , xn )󵄨󵄨󵄨 dt 󵄨󵄨 𝜕xi 󵄨󵄨 −∞

for 1 ≤ i ≤ n. Hence n

󵄨 n 󵄨󵄨 󵄨󵄨v(x)󵄨󵄨󵄨 n−1 ≤ ∏ fi (x̃i ) ≡ f (x) i=1

so that 1

n n 󵄩 󵄩󵄩 𝜕v 󵄩󵄩󵄩 n−1 󵄨 󵄨 n ∫ 󵄨󵄨󵄨v(x)󵄨󵄨󵄨 n−1 dx ≤ ‖f ‖L1 (Rn ) ≤ ∏ ‖fi ‖Ln−1 (Rn−1 ) = ∏󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩 𝜕x 󵄩 1 n i=1 i=1 󵄩 i 󵄩L (R ) n

R

by Lemma 1.11. Thus, the case q = 1 has been established in the form of n

‖v‖

n L n−1

(Rn )

1

󵄩󵄩 𝜕v 󵄩󵄩 n−1 󵄩 󵄩 ≤ ∏󵄩󵄩󵄩 󵄩󵄩󵄩 . 󵄩 𝜕x 󵄩 1 n i=1 󵄩 i 󵄩L (R )

(1.37)

For t ≥ 1, we substitute |v|t−1 v for v in (1.37) and get ‖v‖t

tn L n−1

(Rn )

1/n n 󵄩 󵄩󵄩 𝜕v 󵄩󵄩󵄩󵄩 ≤ t ∏󵄩󵄩󵄩|v|t−1 󵄩󵄩 , 󵄩 𝜕xi 󵄩󵄩L1 i=1 󵄩

right-hand side of which is estimated above by n 󵄩 󵄩󵄩 𝜕v 󵄩󵄩󵄩1/n 󵄩󵄩 󵄩󵄩 t‖v‖t−1 ≤ ∏ 󸀠 Lq (t−1) 󵄩󵄩 𝜕x 󵄩󵄩 q i=1 󵄩 i 󵄩L

for given q ∈ [1, n), where to conclude

1 q

+

1 q󸀠

= 1. Here, we take

tn n−1

= q󸀠 (t − 1), i. e., t =

n−1 ∗ q n

≥ 1,

n 󵄩 󵄩󵄩 𝜕v 󵄩󵄩󵄩1/n ‖v‖Lq∗ ≤ t ∏󵄩󵄩󵄩 󵄩󵄩󵄩 ≤ t‖∇v‖Lq . 󵄩 𝜕x 󵄩 q i=1 󵄩 i 󵄩L

Assuming the extension operator [227], we can prove the following consequence. Corollary 1.12. If a bounded domain Ω ⊂ Rn has Lipschitz boundary, then the continuous inclusion W 1,q (Ω) ⊂ Lq∗ (Ω) holds for 1 ≤ q < n, where q1∗ = q1 − n1 . Proof. The extension operator Φ : W 1,q (Ω) → W 1,q (Rn ) has the property 󵄩󵄩 󵄩 󵄩󵄩Φ(v)󵄩󵄩󵄩W 1,q (Rn ) ≤ C‖v‖W 1,q (Ω) ,

Φ(v)|Ω = v,

where C is a constant. Then, inequality (1.36) is localized as 󵄩 󵄩 󵄩 󵄩 ‖v‖Lq∗ (Ω) ≤ 󵄩󵄩󵄩Φ(v)󵄩󵄩󵄩Lq∗ (Rn ) ≤ C(n, q)󵄩󵄩󵄩∇Φ(v)󵄩󵄩󵄩Lq (Rn ) ≤ C(n, q, Ω)‖v‖W 1,q (Ω) , with the constant C(n, q, Ω).

24 | 1 Calculus of variations To establish the compactness, we need the following fact deduced by Arzelà– Ascoli’s theorem. Lemma 1.13 (Fréchet–Kolmogorov). Let Ω ⊂ Rn be open, K ⊂ Ω compact, and 1 ≤ p < ∞. Assume that a bounded set ℱ ⊂ Lp satisfies the following property: (P) For any ε > 0 there exists δ > 0 with δ < dist(K, Ω∁ ) such that f ∈ ℱ , h ∈ ℛn , and |h| < δ imply ‖τh f − f ‖Lp (K) < ε,

(1.38)

where (τh f )(x) = f (x + h) denotes the translation operator. Then, ℱ |K is relatively compact in Lp (K). Proof. We may suppose that Ω is bounded. For f ∈ ℱ , set f ̄(x) = {

f (x), 0,

x ∈ Ω, x ∈ Rn \Ω,

and ℱ̄ = {f ̄ | f ∈ ℱ }. Since Ω is bounded, the set ℱ̄ is bounded in L1 (Rn ) and Lp (Rn ). We take 0 ≤ ρ ∈ C0∞ (Rn ) satisfying ∫Rn ρ dx = 1, supp ρ ⊂ {|x| ≤ 1} to define the molifier (ρj ∗ f ̄)(x) = ∫ f ̄(x − y)ρj (y) dy,

ρj (x) = jn ρ(jx).

Rn

Given ε > 0, we take δ > 0 as in (P) and let j > 1/δ. Then, for f ̄ ∈ ℱ it holds that ‖ρj ∗ f ̄ − f ̄‖Lp (K) < ε

(1.39)

by Young’s inequality. We have, on the other hand, that ℋ = ρj ∗ ℱ̄ |K is uniformly bounded and equicontinuous. In fact, with a constant Cj > 0, we have ‖ρj ∗ f ̄‖L∞ (Rn ) ≤ ‖ρj ‖∞ ‖f ̄‖1 ≤ Cj and 󵄨󵄨 󵄨 󵄨󵄨(ρj ∗ f ̄)(x1 ) − (ρj ∗ f ̄)(x2 )󵄨󵄨󵄨 ≤ |x1 − x2 | ⋅ ‖∇ρj ‖∞ ⋅ ‖f ̄‖1 ≤ Cj |x1 − x2 |,

x1 , x2 ∈ K,

for any f ̄ ∈ ℱ̄ . In particular, ℋ is precompact in Lp (K). Therefore, ℋ is covered by a finite number of ε-balls in Lp (K). From (1.39), the same is true for ℱ |K by using 2ε-balls. Corollary 1.14. Let Ω ⊂ Rn be open and ℱ ⊂ Lp (Ω) bounded, where 1 ≤ p < ∞. Suppose the following properties:

1.3 Method of variations | 25

1. 2.

For any ε > 0 and compact set K ⊂ Ω, there exists δ > 0 with δ < dist(K, Ω∁ ) satisfying (1.38) for |h| < δ and f ∈ ℱ . For any ε > 0, there exists a compact set K ⊂ Ω such that ‖f ‖Lp (Ω\K) < ε for f ∈ ℱ .

Then, ℱ is relatively compact in Lp (Ω). Proof. Given ε > 0, we take K as in the second condition. By Lemma 1.13, ℱ |K is covered by a finite number, say k, of ε-balls in Lp (K), denoted by Bε (gi ) ≡ {g ∈ Lp (K) | ‖g − gi ‖Lp (K) < ε} for some gi ∈ Lp (K), 1 ≤ i ≤ k. Therefore, putting g̃i (x) = {

gi (x), x ∈ K, 0, x ∈ Ω\K,

we obtain ℱ ⊂ ⋃ki=1 B̃ 2ε (g̃i ), where B̃ 2ε (g̃i ) = {g ∈ Lp (Ω) | ‖g − g̃i ‖Lp (Ω) < 2ε}. Theorem 1.15 (Rellich–Kondrachov). For 1 ≤ q < n and 1 ≤ p < q∗ , the embedding W01,q (Ω) ⊂ Lp (Ω) is compact, where Ω ⊂ Rn is a bounded domain. Proof. Given a bounded set ℱ ⊂ W01,q (Ω), we examine the conditions of Corollary 1.14 for 1 ≤ p < q∗ . First, from the assumption there exists α ∈ (0, 1] such that 1 α 1−α = + ∗ . p 1 q For K ⊂⊂ Ω, f ∈ ℱ and |h| < dist(K, Ω∁ ), we have ‖τh f − f ‖Lp (K) ≤ ‖τh f − f ‖αL1 (K) ‖τh f − f ‖1−α Lq∗ (K) α

≤ (|h|‖∇f ‖L1 (Ω) ) (2‖f ‖Lq∗ (Ω) )

1−α

≤ C|h|α .

Second, for f ∈ ℱ and K ⊂⊂ Ω, it holds that ‖f ‖Lp (Ω\K) ≤ ‖f ‖Lq∗ (Ω) |Ω\K|

1− qp∗

.

These inequalities assure the requirements in the previous corollary. We similarly have the compact embedding W 1,q (Ω) ⊂ Lp (Ω), 1 ≤ p < q∗ , if Ω has a Lipschitz boundary. The following theorem is a counterpart, where C α stands for the space of α-Hölder continuous functions [47]. Again, the inclusion W 1,q (Ω) ⊂ C α (Ω)̄ is valid when Ω has Lipschitz boundary. Theorem 1.16 (Morrey). If q > n, the continuous embedding W 1,q (Rn ) ⊂ C α (Rn ) ∩ L∞ (Rn ) holds with α = 1 − qn .

26 | 1 Calculus of variations Proof. It suffices to show 󵄨 󵄨󵄨 α 󵄨󵄨v(x) − v(y)󵄨󵄨󵄨 ≤ C|x − y| ‖∇v‖Lq (Rn ) ,

x, y ∈ Rn

(1.40)

and ‖v‖L∞ (Rn ) ≤ C‖v‖W 1,q (Rn )

(1.41)

for v ∈ C0∞ (Rn ). Let 0 ∈ Q be an open cube with side of length r. First, we have 󵄨󵄨 1 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 d 󵄨󵄨 ≤ ∫󵄨󵄨󵄨x ⋅ ∇v(tx)󵄨󵄨󵄨 dt v(x) − v(0) = v(tx) dt ∫ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 dt 󵄨󵄨 󵄨0 0 1

1 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕v 󵄨󵄨 𝜕v 󵄨󵄨 󵄨 󵄨󵄨 󵄨 ≤ ∑ |xi | ∫󵄨󵄨 (tx)󵄨󵄨 dt ≤ r ∑ ∫󵄨󵄨󵄨 (tx)󵄨󵄨󵄨 dt, 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕xi 󵄨󵄨 𝜕xi i i 0

x ∈ Q,

0

which implies 1 󵄨󵄨 𝜕v 󵄨󵄨 r 󵄨 󵄨 󵄨 󵄨󵄨 ∫ dx ∑ ∫󵄨󵄨󵄨 (tx)󵄨󵄨󵄨 dt 󵄨󵄨v − v(0)󵄨󵄨󵄨 ≤ 󵄨󵄨 𝜕xi 󵄨󵄨 |Q| i 0

Q

=

= where v = Since

1

r n−1 1

r n−1

1

󵄨󵄨 𝜕v 󵄨󵄨 󵄨 󵄨 ∫ dt ∫ ∑󵄨󵄨󵄨 (tx)󵄨󵄨󵄨 dx 󵄨 𝜕x 󵄨󵄨 i 󵄨 i 0

Q

1 󵄨󵄨 𝜕v 󵄨󵄨 dy 󵄨 󵄨 ∫ dt ∫ ∑󵄨󵄨󵄨 (y)󵄨󵄨󵄨 n 󵄨󵄨 𝜕xi 󵄨󵄨 t 0

tQ

1 ∫ v dx. |Q| Q

󵄨󵄨 𝜕v 󵄨󵄨 󸀠 󵄨 󵄨 ∫󵄨󵄨󵄨 (y)󵄨󵄨󵄨 dy ≤ ‖vi ‖Lq (Q) ⋅ |tQ|1/q , 󵄨󵄨 𝜕xi 󵄨󵄨

tQ

1 1 + =1 q q󸀠

holds for 0 < t < 1, we obtain 1

t n/q r 1−n/q 󵄨󵄨 󵄨 ‖∇v‖Lq (Q) n/q󸀠 ‖∇v‖Lq (Q) , ⋅ r ⋅ dt = ∫ 󵄨󵄨v − v(0)󵄨󵄨󵄨 ≤ n t 1 − n/q r n−1 󸀠

0

recalling q > n. From the same argument, it follows that r 1−n/q 󵄨󵄨 󵄨 ‖∇v‖Lq (Q) , 󵄨󵄨v − v(x)󵄨󵄨󵄨 ≤ 1 − n/q

x ∈ Q,

(1.42)

1.3 Method of variations | 27

and hence 1−n/q 󵄨 2r 󵄨󵄨 ‖∇v‖Lq (Q) , 󵄨󵄨v(x) − v(y)󵄨󵄨󵄨 ≤ 1 − n/q

x, y ∈ Q.

(1.43)

Given x, y ∈ Rn , we can take Q containing x, y for r = 2|x − y|. Then (1.43) guarantees (1.40) with α = 1 − n/q. Inequality (1.42) implies also 󵄨󵄨 󵄨 󵄨󵄨v(x)󵄨󵄨󵄨 ≤ |v| + C‖∇v‖Lq (Q) ≤ ‖v‖L1 (Q) + C‖∇v‖Lq (Q) ≤ C‖v‖W 1,q (Rn ) , x ∈ Q, and hence (1.41). The case q = n is called the limiting Sobolev exponent. The space W 1,n (Rn ) is embedded into another type of space, different from C α or Lp , called the Orlicz space.17

1.3.3 Differentiability and submersion The proof of Theorem 1.1 relies on the Lagrangian multiplier principle described below. Let X and Y be Banach spaces over R, and U ⊂ X an open set. For F ∈ C 1 (U, R) and G ∈ C 2 (U, Y), we consider the constrained variational problem C = inf{F(v) | v ∈ U, G(v) = 0}.

(1.44)

Here, the differentiation is taken in the Fréchet sense. Thus, G ∈ C 1 (U, Y) indicates the existence of the bounded linear operator G󸀠 (v) : X → Y for each v ∈ U satisfying G(v + w) = G(v) + G󸀠 (v)[w] + o(‖w‖X ),

‖w‖X → 0

in Y, where the mapping v ∈ U 󳨃→ G󸀠 (v) ∈ B(X, Y) is strongly continuous.18 Regarding G󸀠 : U → B(X, Y), we define the operator G󸀠󸀠 : X → B(X, Y) and the notion G ∈ C 2 (U, Y) similarly, where B(X, Y) = {T : X → Y | bounded linear} is equipped with the strong topology. 17 This case is described in Chapter 3 of [335]; see also [3, 222]. 18 The space B(X, Y) is provided with the uniform, strong, and weak topologies.

28 | 1 Calculus of variations We suppose the existence of a minimizer v0 ∈ U in (1.44) attaining the value C to deduce an equation. Henceforth, X ∗ and Y ∗ denote the dual spaces of X and Y, and ℛ(T) and 𝒩 (T) denote the range and the kernel of the linear operator T, respectively.19 The next theorem is proven later in this subsection. Theorem 1.17. If the condition 󸀠

ℛ(G (v0 )) = Y

(1.45)

is satisfied, there exists an element y∗ ∈ Y ∗ such that F 󸀠 (v0 ) + G󸀠 (v0 )∗ y∗ = 0

(1.46)

in X ∗ . and

To apply Theorem 1.17 for the proof of Theorem 1.1, we take X = U = H01 (Ω), Y = R, F(v) =

1 ∫ |∇v|2 dx, 2

G(v) =

Ω

1 ∫ |v|p+1 dx − 1. p+1 Ω

The condition F ∈ C 1 (X, Y) is obvious by F(v + w) − F(w) = F(v) + ∫ ∇v ⋅ ∇w dx,

v, w ∈ X

Ω

so that F 󸀠 (v)[w] = ∫Ω ∇v ⋅ ∇w dx. The condition G ∈ C 2 (X, Y) holds when the continuous embedding X = H01 (Ω) ⊂ Lp+1 (Ω) exists, that is, 1 ≤ p < ∞ for n = 2 and 1 ≤ p ≤ n+2 for n > 2. In fact, we have n−2 1 1 θ=1 {|v + w|p+1 − |v|p+1 } = [|θw + v|p+1 ]θ=0 p+1 p+1 1

= ∫ |θw + v|p−1 (θw + v) dθw 0

1

= |v|p−1 vw + ∫{|θw + v|p−1 (θw + v) − |v|p−1 v} dθw 0 1

θ

0

0

= |v|p−1 vw + p ∫ dθ ∫ dρ|ρw + v|p−1 w2 ,

19 Condition (1.45) and the element y∗ ∈ Y ∗ are regarded as a constraint qualification and Lagrangian multiplier, respectively.

1.3 Method of variations | 29

and therefore, G(v + w) − G(v) = ∫Ω |v|p−1 vw dx + R for 1

θ

0

0

R = ∫ p[∫ dθ ∫ dρ|ρw + v|p−1 w2 ] dx, Ω

which implies p−1 2 |R| ≤ Cp (‖w‖p+1 p+1 + ‖v‖p+1 ‖w‖p+1 )

by Hölder’s inequality, where Cp is a positive constant. Then, X ⊂ Lp+1 implies the Fréchet differentiability of G with the identity G󸀠 (v)[w] = ∫ |v|p−1 vw dx

(1.47)

Ω

and ‖G󸀠 (v)‖X ∗ ≤ C‖v‖pp+1 by

1

p+1 p

+

1 p+1

= 1. Then, the strong continuity of

v ∈ X 󳨃→ G󸀠 (v)[w] ∈ Y = R follows similarly from (1.47). We have also the twice differentiability of G, G󸀠󸀠 (v)[w1 , w2 ] = p ∫ |v|p−1 w1 w2 dx,

v, w1 , w2 ∈ X,

Ω

and the strong continuity of v 󳨃→ G󸀠󸀠 (v)[w1 , w2 ], using X ⊂ Lp+1 and

1

p+1 p−1

+

2 p+1

= 1.

The existence of the minimizer, on the other hand, requires the compact embedding. In fact, taking a minimizing sequence {vk } of 1 1 C = inf{ ∫ |∇v|2 dx | v ∈ H01 , ∫ |v|p+1 dx = 1}, 2 p+1 Ω

we have

1 ∫ |∇vk |2 dx → C, 2 Ω

(1.48)

Ω

1 ∫ |vk |p+1 dx = 1. p+1 Ω

Passing to a subsequence, we may suppose that {vk } converges weakly to an element v ∈ H01 . Then the lower semicontinuity of the norm implies 1 1 ∫ |∇v|2 dx ≤ lim inf ∫ |∇vk |2 dx = C k→∞ 2 2 Ω

as before. When the embedding H01 ⊂ Lp+1 is strong convergence vk → v in Lp+1 and hence

Ω

compact, furthermore, we obtain the

1 1 ∫ |v|p+1 dx = lim ∫ |vk |p+1 dx = 1. k→∞ p + 1 p+1 Ω

Ω

30 | 1 Calculus of variations This property means that v ∈ H01 is a minimizer of C in (1.48). We have also G󸀠 (v)[v] = ∫ |v|p+1 dx = p + 1 ≠ 0 Ω

and the constrained qualification (1.45) holds with Y = R. From the above theory, there exists a constant μ ∈ R such that ∫ ∇v ⋅ ∇w dx = μ ∫ |v|p−1 vw dx Ω

for any w ∈

H01 (Ω),

Ω

which implies that v ∈

H01 (Ω)

is a weak solution to

−Δv = μ|v|p−1 v. For the proof of Theorem 1.17, we require the following form of the implicit function theorem.20 Proposition 1.18. Let X, Y, Z be Banach spaces and V ⊂ Z × X be an open set. Given g ∈ C 1 (V, Y), we suppose the existence of (z0 , x0 ) ∈ V such that g(z0 , x0 ) = 0 and ℛ(gx (z0 , x0 )) = Y. Then there exist a neighborhood W of z0 ∈ Z and x ∈ C(W, Y) such that x(z0 ) = x0 and g(z, x(z)) = 0 for any z ∈ W. If 𝒩 (gx (z0 , x0 )) = {0}, furthermore, this x(z) is isolated as a zero of g(z, ⋅) for each z ∈ W, and the mapping z 󳨃→ x(z) is C 1 . Proof of Theorem 1.17. Recalling that v0 ∈ U is a minimizer of C in (1.44), we introduce the manifold ℳ = {v ∈ U | G(v) = 0} and its tangent space at v0 : Tv0 ℳ = {h ∈ X | v0 + th + v(t) ∈ ℳ, ∃v(t) = o(t) ∈ X, t → 0}. If h ∈ Tv0 ℳ, we have 0 = G(v0 + th + v(t)) = G(v0 ) + tG󸀠 (v0 )[h] + o(t), and hence h ∈ 𝒩 (G󸀠 (v0 )) due to G(v0 ) = 0. We shall show the converse property, referred to as the submersion. To this end, given h ∈ 𝒩 (G󸀠 (v0 )), we take Φ(t, w) = {

1 {G(v0 + t(h + w)) − G(v0 )}, t G󸀠 (v0 )[h + w] = G󸀠 (v0 )[w],

for w ∈ U and |t| ≪ 1. It holds that Φ(0, 0) = 0,

t ≠ 0, t = 0,

1 {G(v0 + t(h + w)) − G(v0 ) − tG󸀠 (v0 )[h + w]} t2 1 + {G󸀠 (v0 + t(h + w))[h + w] − G󸀠 (v0 )[h + w]} t 1 󸀠󸀠 → G (v0 )[h + w, h + w], t → 0, 2

Φt (t, w) = −

20 The proof is similar as in the finite-dimensional case; see [38].

1.3 Method of variations | 31

and Φw (t, w) = G󸀠 (v0 + t(h + w))[h + w]. Hence Φ ∈ C 1 (V, R), where V ⊂ R × X is a neighborhood of (0, 0). Since ℛ(Φw (0, 0)) = ℛ(G󸀠 (v0 )), the implicit function theorem, Proposition 1.18, is applicable with Z = R, and there are 0 < ε ≪ 1 and w = w(t) ∈ C((−ε, ε), X) such that w(0) = 0,

Φ(t, w(t)) = 0,

|t| < ε.

This property implies G(v0 + t(h + w(t))) = 0,

w(t) = o(1),

t → 0,

as G(v0 ) = 0, and hence h ∈ Tv0 ℳ. From this submersion, any h ∈ 𝒩 (G󸀠 (v0 )) admits v(t) = o(t) ∈ X with f (t) = F(v0 + th + v(t)) taking a local minimum at t = 0. Therefore, it holds that 0 = f 󸀠 (0) = F 󸀠 (v0 )[h], and hence ⊥

F 󸀠 (v0 ) ∈ 𝒩 (G󸀠 (v0 )) = ℛ(G󸀠 (v0 )∗ ) because ℛ(G󸀠 (v0 )) = Y is closed, and so is ℛ(G󸀠 (v0 )∗ ) by the closed range theorem. This property means (1.46) for some y∗ ∈ Y ∗ . The assumption G ∈ C 2 (U, Y) may be replaced by G ∈ C 1 (U, Y) and the splitting of the Banach space X into X1 = 𝒩 (G󸀠 (v0 )) and X2 = (1 − P)X, where P : X → X1 is a projection [374, 100]. In fact, this time we take Φ(v1 , v2 ) = G(v0 + v1 + v2 ) for v1 ∈ X1 and v2 ∈ X2 , to examine Φ(0, 0) = 0 and that Φv2 (0, 0) = G󸀠 (v0 ) : X2 → Y is an isomorphism. Then, there exists a C 1 mapping ψ : V1 → X2 with Φ(v1 , ψ(v1 )) = 0 and ψ(0) = 0, where V1 is a neighborhood of 0 in X1 . Since Φv1 (0, 0) = 0, we obtain ψ󸀠 (0) = 0 from 0 = Φv1 (0, 0) + Φv2 (0, 0)[ψ󸀠 (0)]. Therefore, for h ∈ 𝒩 (G󸀠 (v0 )) and |t| ≪ 1, it holds that w(t) ≡ ψ(th) = o(t), the submersion.

32 | 1 Calculus of variations 1.3.4 Positivity and regularity We complete the proof of Theorem 1.1. First, the minimizer v in (1.48) can be nonnegative, because ‖∇|w|‖2 = ‖∇w‖2 holds for w ∈ H01 : v ∈ H01 ,

‖v‖p+1 p+1 = p + 1 > 0.

v ≥ 0,

There exists also a Lagrangian multiplier μ ∈ R such that ∫ ∇v ⋅ ∇w dx = μ ∫ vp w dx, Ω

Ω

w ∈ H01 .

(1.49)

̄ to arrive at We shall prove the regularity, v ∈ C 2 (Ω) ∩ C 0 (Ω), − Δv = μvp ,

v ≥ 0,

v ≢ 0

in Ω,

v|𝜕Ω = 0.

(1.50)

Then, the strong maximum principle implies v > 0 in Ω and μ > 0. Hence, u = μ1/(p−1) v is a solution to (1.4). To show this property of v, we require two different types of regularity valid for the solution to linear elliptic boundary value problems [139]. Both results are expressed in the form of inequality with C independent of v.21 The finiteness of the right-hand side implies that of the left-hand side, and the boundary is supposed to be C 2 and C 2+θ , respectively, in the first and the second case, where |Dα v(x) − Dα v(y)| 󵄨 󵄨 ‖v‖Cm+θ (Ω) = ∑ sup󵄨󵄨󵄨Dα v󵄨󵄨󵄨 + ∑ sup |x − y|θ |α|≤m x∈Ω |α|=m x =y̸ for m ∈ N and 0 < θ ≤ 1: 1. Lp estimate for 1 < p < ∞: ‖v‖W 2,p (Ω) ≤ C{‖Δv‖Lp (Ω) + ‖v‖W 2−1/p,p (𝜕Ω) }. 2.

Schauder estimate for 0 < θ < 1: ‖v‖C2+θ (Ω) ≤ C{‖Δv‖Cθ (Ω) + ‖v‖C2+θ (𝜕Ω) }.

Both estimates are consequences of their local versions, the estimate near and off the boundary 𝜕Ω. More precisely, for any ω ⊂⊂ Ω the norms ‖v‖W 2,p (ω) and ‖v‖C2+θ (ω) are estimated above by ‖Δv‖Lp (ω1 ) + ‖v‖Lp (ω1 ) and ‖Δv‖Cθ (ω1 ) + ‖v‖Cθ (ω1 ) , respectively, where ω ⊂⊂ ω1 ⊂⊂ Ω. Similarly, for any C 2 and C 2+θ domain ω ⊂ Rn with ω ∩ Ω ≠ 0, the norms ‖v‖W 2,p (ω∩Ω) and ‖v‖C2+θ (ω∩Ω)̄ are estimated above by ‖Δv‖Lp (ω1 ∩Ω) + ‖v‖W 2−1/p,p (𝜕Ω∩ω1 ) and ‖Δv‖Cθ (ω1 ∩Ω) + ‖v‖C2+θ (𝜕Ω∩ω1 ) , respectively, if ω ⊂⊂ ω1 .

21 The first inequality represents also the existence of the trace operator v 󳨃→ v|𝜕Ω .

1.4 Critical exponent |

33

The relation (1.49) implies (1.50) in the distributional sense. If n > 2, it holds that −Δv = μvp ∈ L(n∗ +1)/p (Ω), with the exponent n∗ defined by (1.5). If n = 2, we obtain v ∈ W 2,q for any 1 < q < ∞ by the Lp estimate, which implies v ∈ C 1+θ (Ω)̄ with some 0 < θ < 1 by Morrey’s theorem. Then v ∈ C 2+θ (Ω)̄ follows from the Schauder estimate, because of ̄ −Δv = μvp ∈ C θ (Ω),

v|𝜕Ω = 0.

2, n∗p+1

(Ω) from the Lp estimate. If n∗p+1 > n2 , then Getting back to n > 2, we have v ∈ W v ∈ C θ (Ω)̄ follows from Morrey’s theorem for some 0 < θ < 1. Therefore, v ∈ C 2+θ (Ω)̄ holds similarly. If this inequality is not true, we use Sobolev’s embedding, noting the inclusion 2n

2n

n+2

W 2, n+2 (1−α) ⊂ L n−2 (1− n−2 α) , valid for 0 < α
1, and hence v ∈ W 2,r for any r > 1 by the Lp estimate. Then it similarly follows ̄ Such a scheme of iteration for deriving the desired regularity of the that v ∈ C 2+θ (Ω). solution has been called the bootstrap argument. In the Lp elliptic estimate, p = 1 is critical, but we still have the following form due to H. Brezis and W. Strauss [56]:22 3. L1 estimate: ‖v‖W 1,q (Ω) ≤ C{‖Δv‖L1 (Ω) + ‖v‖L1 (𝜕Ω) }, where 1 ≤ q
2, equation (1.4) is still reduced to the variational n−2 problem (1.48), which, however, may lose the minimizer because of the lack of compactness. We have actually this situation when Ω is star-shaped, as any classical solution is not admitted in (1.4). Critical nonlinearity appears occasionally in differential geometry and has been studied in detail [199]. 22 This inequality arises as a dual form of the Stampacchia estimate derived from the truncation method [320].

34 | 1 Calculus of variations The problem which we shall consider here is n+2

− Δu = u n−2 + λu,

u>0

in Ω,

u|𝜕Ω = 0

(1.51)

where Ω ⊂ Rn is a bounded domain with Lipschitz boundary.23 Let λ1 > 0 and ϕ1 (x) be the first eigenvalue and eigenfunction, respectively: −Δϕ1 = λ1 ϕ1 ,

ϕ1 > 0

in Ω,

ϕ1 |𝜕Ω = 0,

and the solution is taken in the classical sense. Theorem 1.19 ([54]). When n ≥ 4 and 0 < λ < λ1 , problem (1.51) has a solution. Theorem 1.20 ([54]). If Ω is a three-dimensional ball, problem (1.51) admits a solution if and only if λ41 < λ < λ1 . First, the identity n+2

(λ1 − λ) ∫ uϕ1 dx = ∫ u n−2 ϕ1 dx > 0 Ω

Ω

is valid for any solution u to (1.51) so that the condition λ1 ≤ λ admits no solution. Taking L2 inner product with the first eigenfunction is used for the parabolic problem, referred to as Kaplan’s method [178]. The above solution is constructed by the variational method, so that we introduce the constants: S0 = inf{∫ |∇v|2 dx | v ∈ H01 , ‖v‖ 2n = 1}, Ω

n−2

Sλ = inf{∫ |∇v|2 − λv2 dx | v ∈ H01 (Ω), ‖v‖ 2n = 1}, Ω

n−2

λ < λ1 .

We shall prove the existence part of these theorems in the following procedures, while the nonexistence will follow from the generalized Pohozaev identity described in § 1.2.2: 1. A minimizer v ≥ 0 of Sλ exists whenever Sλ < S0 . 2. Under the assumption of the theorems, the condition Sλ < S0 actually holds. 3. If λ < λ1 , the strong maximum principle and elliptic regularity yield the positivity of the minimizer v and then a solution to (1.51) arises. The second part is significantly influenced by the following facts24 where 2∗ = n∗ + 1 = 2n : n−2 23 See also [221] for a similar phenomenon in a free-boundary problem. 24 They are sublimated to the concentration compactness principle [215, 326].

1.4 Critical exponent |

1. 2.

35

The constant S0 , referred to as the Sobolev constant, depends only on the dimension n and is independent of Ω. For Ω = Rn , S0 is attained by the function U(x) =

C , (1 + |x|2 )(n−2)/2

where C > 0 is a constant defined by ‖U‖2∗ = 1. Therefore, the same is true for its modifications Uε (x) = 3.

Cε , (ε + |x|2 )(n−2)/2

Cε = ε

n−2 4

C,

and Uε (x − x0 ) defined for x0 ∈ Rn . No function attains S0 if Ω ≠ Rn .

To establish Sλ < S0 , a minimizer v of S0 would be useful to calculate Qλ (v) =

∫Ω |∇v|2 − λv2 dx ‖v‖22∗

.

Since it does not exist, we adopt the above function Uε (x) as an approximation. Thus, taking x0 ∈ Ω and ζ ∈ C0∞ (Ω) with ζ ≡ 1 near x0 , we substitute vε = ζUε in Qλ . This vε concentrates on x = x0 , and after an elementary calculation we have n−2

S − λKε + O(ε 2 ), Qλ (vε ) = { 0 S0 − λKε log ε1 + O(ε),

n ≥ 5, n = 4,

as ε ↓ 0 with K = K(n) > 0 independent of ζ . Thus we obtain Sλ < S0 if n ≥ 4 and λ > 0. In the case of n = 3, the asymptotics is slightly different. We have positive constants A and B depending on ζ satisfying Qλ (vε ) = S0 + (A − Bλ)ε1/2 + O(ε). Therefore, Sλ < S0 follows from λ > A/B. Efforts are made to minimize A/B by chang2 ing ζ . In the unit ball case, AB = π4 = 41 λ1 holds when ζ (x) = cos(π|x|/2), which is the best possible. To show that λ ≤ 41 λ1 implies the nonexistence of the solution in this case, we recall the identity given in Corollary 1.6 when n = 3, F(u) = f (u) = u5 + λu. Assuming a solution u, we have ∫ u2 (λψ󸀠 + B

1 6 u 6

+ λ2 u2 , and hence

1 󸀠󸀠󸀠 2 1 ψ ) dx = ∫ u6 (rψ − r 2 ψ󸀠 ) 2 dx + 2πu2r (1)ψ(1), 4 3 r B

and take ψ(r) = sin √4πλr. It holds that λψ󸀠 + 41 ψ󸀠󸀠󸀠 = 0 and rψ − r 2 ψ󸀠 > 0 for 0 < r < 1, and, therefore, having ψ(1) ≥ 0, which is equivalent to λ ≤

π2 , implies a contradiction. 4

36 | 1 Calculus of variations 1.4.2 Lieb’s lemma We notice that the lack of compactness is recovered when Sλ < S0 , the idea which can be traced back to T. Aubin [14]. Here, we utilize the following lemma. Lemma 1.21 ([203]). Let (Ω, ℬ, μ) a measure space and {fk } a sequence of measurable functions converging to zero almost everywhere with the property that supk ‖fk ‖p < ∞, where 0 < p < ∞. Then, it holds that lim {‖fk + g‖pp − ‖fk ‖pp } = ‖g‖pp

k→+∞

for any g ∈ Lp . The lemma is obvious for p = 2 because of the structure of the Hilbert space of L2 . Actually, Lemma 1.21 is an immediate consequence of the following proposition applied to j(t) = |t|p , ϕε (t) = |t|p , and ψε (t) = Cε |t|p . Proposition 1.22 ([51]). Let j ∈ C(R, R), j(0) = 0, satisfy the following conditions: Given ε > 0, there exist continuous functions ϕε , ψε : R → [0, +∞) such that 󵄨󵄨 󵄨 󵄨󵄨j(a + b) − j(a)󵄨󵄨󵄨 ≤ εϕε (a) + ψε (b),

a, b ∈ R.

Let (Ω, ℬ, μ) be a measure space, {fk } a sequence of measurable functions, and g a measurable function with the following properties: fk → 0,

a. e., j(g) ∈ L1 ,

ψε (g) ∈ L1 ,

C = sup ∫ ϕε (fk )dμ < +∞. ε,k

Ω

Then, it holds that 󵄨 󵄨 lim ∫󵄨󵄨󵄨j(fk − g) − j(fk ) − j(g)󵄨󵄨󵄨 dμ = 0.

k→∞

Ω

Proof. Given ε > 0, we put 󵄨 󵄨 Wε,k = [󵄨󵄨󵄨j(fk + g) − j(fk ) − j(g)󵄨󵄨󵄨 − εϕε (fk )]+ which converges to zero almost everywhere. We have 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨j(fk + g) − j(fk ) − j(g)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨j(fk + g) − j(fk )󵄨󵄨󵄨 + 󵄨󵄨󵄨j(g)󵄨󵄨󵄨 󵄨 󵄨 ≤ εϕε (fk ) + ψε (g) + 󵄨󵄨󵄨j(g)󵄨󵄨󵄨 so that 0 ≤ Wε,k ≤ ψε (g) + |j(g)| ∈ L1 and hence ∫Ω Wε,k dμ → 0 by the dominated convergence theorem. Here we use 󵄨󵄨 󵄨 󵄨󵄨j(fk + g) − j(fk ) − j(g)󵄨󵄨󵄨 ≤ Wε,k + εϕε (fk )

1.4 Critical exponent |

37

to deduce 󵄨 󵄨 lim sup ∫󵄨󵄨󵄨j(fk + g) − j(fk ) − j(g)󵄨󵄨󵄨 dμ ≤ εC, k→+∞

Ω

and hence the conclusion. Now we perform the first procedure described in the previous subsection. Suppose Sλ < S0 for 0 < λ < λ1 , and take a minimizing sequence {vk } of Sλ : ∫ |∇vk |2 − λvk2 dx = Sλ + o(1),

Ω

∫ |vk |2 dx = 1. ∗

(1.52)

Ω

The boundedness of {vk } in X = H01 (Ω) follows from λ < λ1 because of Poincaré inequality λ1 ‖v‖22 ≤ ‖∇v‖22 ,

v ∈ X.

Therefore, it has a weak limit v, passing to a subsequence. Taking a subsequence once more, if necessary, we may suppose that wk = vk − v converges to zero weakly in H01 , strongly in Lq for 1 ≤ q < 2∗ , and hence almost everywhere in Ω. Then relation (1.52) is reduced to ∫ |∇v|2 + |∇wk |2 − λv2 dx = Sλ + o(1)

(1.53)

Ω

because ‖wk ‖2 → 0 since 2∗ > 2. The sequence {wk }, on the other hand, is bounded ∗ in L2 . Hence 1 − ‖wk ‖22∗ = ∫ |wk + v|2 − |wk |2 dx = ∫ |v|2 dx + o(1) ∗

∗

∗

Ω

∗

Ω

holds by Proposition 1.22, which means ‖v‖22∗ + ‖wk ‖22∗ = 1 + o(1). ∗

∗

Since 2∗ > 2, this property implies 1 ≤ ‖v‖22∗ + ‖wk ‖22∗ + o(1). Relations (1.53) and (1.54) imply ∫ |∇v|2 − λv2 + |∇wk |2 dx ≤ Sλ {‖v‖22∗ + ‖wk ‖22∗ } + o(1).

Ω

(1.54)

38 | 1 Calculus of variations In terms of the Sobolev constant S0 , the right-hand side is estimated above by Sλ ‖v‖22∗ +

Sλ ‖∇wk ‖22 + o(1), S0

and hence it follows that Sλ ‖v‖22∗ ≤ ∫ |∇v|2 − λv2 dx ≤ ( Ω

Sλ − 1)‖∇wk ‖22 + Sλ ‖v‖22∗ + o(1). S0

Therefore, Sλ < S0 implies wk → 0, or vk → v strongly in H01 (Ω), which indicates that v is a minimizer of Sλ , and hence a weak solution to (1.51). 1.4.3 Brezis–Kato’s theorem To establish the existence of the classical solution, we have to work on the regularity of the weak solution [354]. This is rather more delicate compared with the subcritical case, based on the method of truncation [154]. Here we use the following theorem 2n 2n where au ∈ L1loc (Ω), because n+2 < n2 is the conjugate exponent of n−2 . Theorem 1.23 ([50]). Let Ω ⊂ Rn , n > 2, be an open set and u ∈ H01 (Ω) satisfy −Δu = au n in Ω as a distribution, where a(x) ∈ L 2 (Ω). Then, u ∈ Lq (Ω) holds for any 1 ≤ q < ∞. 4

n

2n

Let u ∈ H01 be a solution to (1.51), then a = λ + u n−2 ∈ L 2 since u ∈ H01 ⊂ L n−2 . Then, Theorem 1.23 implies u ∈ Lq for any 1 ≤ q < ∞, and hence n+2

−Δu = λu + u n−2 ∈ Lp ,

1 < p < ∞.

The desired regularity u ∈ C 2 (Ω) ∩ C 0 (Ω)̄ now follows as before. We use the following lemma to prove the above theorem. Lemma 1.24. Under the assumption of the previous theorem, any ε > 0 admits Cε > 0 such that 󵄨 󵄨 ∫󵄨󵄨󵄨a(x)󵄨󵄨󵄨v2 dx ≤ ε‖∇v‖22 + Cε ‖v‖22 ,

Ω

v ∈ H01 .

Proof. By Sobolev’s inequality, we have 󵄨 󵄨 󵄨 󵄨 ∫󵄨󵄨󵄨a(x)󵄨󵄨󵄨v2 dx ≤ k ∫ v2 dx + ∫ 󵄨󵄨󵄨a(x)󵄨󵄨󵄨v2 dx Ω

{|a|≤k}

≤ ≤

k‖v‖22 k‖v‖22

{|a|>k}

+ ‖a‖Ln/2 ({|a|>k}) ‖v‖22∗

+ ‖a‖Ln/2 ({|a|>k}) S0−1 ‖∇v‖22

1.4 Critical exponent |

for any k > 0, recalling 2∗ =

2n . n−2

39

Then we obtain the result because lim ‖a‖Ln/2 ({|a|>k}) = 0

k→∞

follows from a ∈ Ln/2 (Ω). Proof of Theorem 1.23. By Lemma 1.24, the bilinear form u, v ∈ H01

𝒜(u, v) = ∫ ∇u ⋅ ∇v − a(x)uv + μuv dx, Ω

provides an inner product on H01 for μ > 0 sufficiently large. By the representation theorem of Riesz, any f ∈ H −1 = (H01 )∗ admits a unique u ∈ H01 satisfying −Δu−au+μu = f in the sense of distributions, which means 𝒜(u, v) = ⟨v, f ⟩,

v ∈ H01

and it holds that ‖∇u‖2 ≤ C‖f ‖H −1 . Writing f = μu, we obtain u ∈ H01 ,

2n

−Δu − au + μu = f ∈ H01 󳨅→ L n−2 󳨅→ H −1 .

Letting ak = {

a, ±k,

|a| ≤ k, ±a > k,

we have uk satisfying uk ∈ H01 ,

−Δuk − ak uk + μuk = f .

(1.55)

It holds also that ‖∇uk ‖2 ≤ C‖f ‖H −1 , independent of k, since |ak | ≤ |a|. Passing to a subsequence, uk converges weakly to an element in H01 , denoted by u,̃ and hence weakly ∗ in L2 . From ∫ ∇uk ⋅ ∇v − ak uk v + μuk v dx = ⟨v, f ⟩, Ω

v ∈ H01 ,

we have the strong and weak convergence of ak to a and of uk to ũ in Ln/2 and L2 , respectively. Since Ω is bounded, this property implies ∗

𝒜(u,̃ v) = ⟨v, f ⟩,

v ∈ H01 ,

and hence ũ = u. This uniqueness of the limit function assures the convergence of the original sequence. Hence we have the weak convergence of uk to u in H01 .

40 | 1 Calculus of variations Next, we truncate uk as ukℓ = {

uk , |uk | ≤ ℓ, ±ℓ, ±uk ≥ ℓ.

Given p > 1, we multiply |ukℓ |p−2 ukℓ to (1.55). Since uk uℓk ≥ 0, it holds that ∫ ∇uk ⋅ ∇(|ukℓ |p−2 ukℓ ) dx =

p 2 4(p − 1) 󵄨󵄨 󵄨 ∫󵄨󵄨∇|ukℓ | 2 󵄨󵄨󵄨 dx 2 p

Ω

Ω

≤ ∫ ak (x)uk |ukℓ |p−2 ukℓ dx + ⟨|ukℓ |p−2 ukℓ , f ⟩ Ω

󵄨 󵄨 ≤ ∫󵄨󵄨󵄨ak (x)󵄨󵄨󵄨 |ukℓ |p dx + Ω

kℓp−1 |uk | dx + ‖f ‖p ⋅ ‖ukℓ ‖p−1 p

∫ {|ukℓ |>ℓ}

󵄨 󵄨 ≤ ∫󵄨󵄨󵄨ak (x)󵄨󵄨󵄨 |ukℓ |p dx + k‖uk ‖pLp ({|u |>ℓ}) + ‖f ‖p ‖ukℓ ‖p−1 p , k Ω

p

provided that f ∈ L . We apply Lemma 1.24 to estimate the first term on the right-hand side above by p 2(p − 1) 󵄩󵄩 󵄩2 p 󵄩󵄩∇|ukℓ | 2 |󵄩󵄩󵄩2 + Cp ‖ukℓ ‖p . 2 p

Then Sobolev’s inequality implies ‖ukℓ ‖p 2∗ ≤ Cp (‖ukℓ ‖pp + ‖u‖p ‖ukℓ ‖p−1 p + k‖uk ‖Lp ({|uk |>ℓ}) ) 2

as f = μu. Letting ℓ ↑ +∞, we obtain ‖uk ‖p⋅ 2∗ ≤ Cp (‖uk ‖pp + ‖u‖p ‖uk ‖p−1 p ), 2

p

provided that uk , u ∈ L . ∗ Let γ = 22 > 1. From the above relation, we see that uk , u ∈ Lp implies uk ∈ Lp and also the weak convergence of uk to u as k → ∞ in Lγp , in particular, u ∈ Lγp . We repeat this procedure, starting from p = 2∗ , to get the result.

1.5 Mountain pass lemma 1.5.1 Palais–Smale condition The Lagrange multiplier cannot be reduced to one except for the homogeneous nonlinearity. Instead, introducing the functional 1 1 p+1 v dx, J(v) = ∫ |∇v|2 − 2 p+1 + Ω

v ∈ X = H01 ,

we see the following facts for 1 < p < n∗ , where v+ = max{v, 0}: (a) J ∈ C 1 (X, R). (b) J 󸀠 (u) = 0 holds for u ∈ X\{0} if and only if u is a solution to (1.4).

(1.56)

1.5 Mountain pass lemma

| 41

These facts are proven by the arguments in § 1.3.3 and § 1.3.4. Seeking a critical point u of J, we find a difficulty again in confirming its nontriviality u ≠ 0. Actually, u = 0 is an isolated stable critical point of J as we shall see. We can prove the following facts for BR = {v ∈ X | ‖v‖X < R}, R > 0, where ‖v‖X = ‖∇v‖2 stands for the norm in X = H01 : (c) (mountain pass situation) It holds that J|𝜕BR ≥ ρ and J(e) ≤ J(0) for R, ρ > 0 and e ∈ X\B̄ R . (d) (Palais–Smale condition) The sequence {vk } ∈ X is precompact if J(vk ) = O(1) and ‖J 󸀠 (vk )‖X ∗ = o(1). Lemma 1.25 ([12]). Let X be a Banach space over R and J ∈ C 1 (X → R) satisfy (c)–(d). Introduce the set of paths Γ = {γ ∈ C([0, 1], X) | γ(0) = 0, γ(1) = e}.

(1.57)

j = inf max J(γ(t))

(1.58)

Then γ∈Γ 0≤t≤1

is a critical value of J, i. e., there is u0 ∈ X such that J(u0 ) = j,

J 󸀠 (u0 ) = 0.

Each path γ ∈ Γ must meet 𝜕BR so that the inequality j ≥ ρ > 0 holds. Hence the nontriviality u0 ≠ 0 is assured. We call this principle mountain pass lemma. We shall examine conditions (c) and (d) for the functional J defined by (1.56). Proof of (c). First, the inequality ‖v‖p+1 ≤ C‖∇v‖2 ,

v ∈ H01

holds with a constant C > 0, because p+1 < 2∗ and Ω is bounded. Therefore, we obtain C p+1 p+1 1 R , J(v) ≥ R2 − 2 p+1

v ∈ 𝜕BR ,

the right-handed side of which is positive if R > 0 is sufficiently small. Second, we take an element v ∈ X which is positive in Ω. Then it holds that 1 λp+1 J(λv) = λ2 ‖∇v‖22 − ‖v‖p+1 , 2 p + 1 p+1 the right-handed side of which is negative if λ > 0 is sufficiently large. Thus, e = λv satisfies ‖e‖X > R and J(e) ≤ 0. Proof of (d). Given such a sequence, first, we establish its boundedness. In fact, we have J 󸀠 (v)[w] = ∫ ∇v ⋅ ∇w − v+p w dx, Ω

42 | 1 Calculus of variations and hence J(v) −

1 1 1 󸀠 J (v)[v] = ( − ) ∫ |∇v|2 dx, p+1 2 p+1

v∈X

Ω

with

1 2

−

1 p+1

> 0. Then it holds that 1 1 1 󵄩󵄩 󸀠 󵄨 󵄨 󵄩 ( − )‖vk ‖2X ≤ 󵄩J (vk )󵄩󵄩󵄩X ∗ ‖vk ‖X + 󵄨󵄨󵄨J(vk )󵄨󵄨󵄨, 2 p+1 p + 1󵄩

and hence ‖vk ‖X = O(1). Second, the differential operator −Δ in Ω which is equipped with the null Dirichlet boundary condition is realized as a positive self-adjoint operator in L2 (Ω) denoted by A. The relation D(A1/2 ) = X holds with the identity a(v, w) ≡ ∫ ∇v ⋅ ∇w dx = (A1/2 v, A1/2 w),

v, w ∈ X = H01

Ω

where (⋅, ⋅) denotes the L2 -inner product. Then, a(⋅, ⋅) casts the inner product in X (see [180, 349]). If we identify X ∗ with X through Riesz’ representation theorem, namely the element J 󸀠 (v) ∈ X ∗ is identified with f = v − A−1 v+p ∈ X. Hence ‖J 󸀠 (vk )‖X ∗ = o(1) means p fk ≡ vk − A−1 vk+ →0

in X.

(1.59)

Since 1 < p < n∗ , the mapping v ∈ X 󳨃→ v+p ∈ L

p+1 p

is bounded, that is, any bounded set is mapped onto a bounded set. Also, the linear p+1

2, p+1

1, p+1

operator A−1 : L p → W p ∩ W0 p is bounded by the Lp -estimate, and the Sobolev ∗ inequality implies W 2,r ⊂ W 1,r for r = p+1 , where r ∗ = 1 1 1 > 2 since p r

−n

1 1 1 1 1 + > + ≥ . p + 1 n n∗ n 2 Then, Rellich–Kondrachov’s theorem assures the compact embedding W 2,r ∩ W01,r ⊂ H01 = X, and hence the mapping v ∈ X 󳨃→ A−1 (v+p ) ∈ X is compact, so that the image of a bounded sequence contains a strongly converging subsequence. p Now, the sequence {vk } ⊂ X is bounded and hence {A−1 (vk+ )} has a subsequence converging strongly in X. The same is true for {vk } by (1.59).

1.5 Mountain pass lemma

| 43

The properties (a)–(d) hold for more general nonlinearities. To show the existence of the solution to − Δu = f (u),

u>0

in Ω,

u|𝜕Ω = 0,

(1.60)

we introduce the functional 1 J(v) = ∫ |∇v|2 − F(v+ ) dx, 2

v ∈ X,

(1.61)

Ω

u

where F(u) = ∫0 f (u) du. Let f = f (u) be C 1 in u ∈ [0, +∞) and satisfy the following conditions: (f1) (superlinearity) f (0) = 0 ≥ f 󸀠 (0), f (+∞) = f 󸀠 (+∞) = +∞, and 1 F(u) ≤ ( − ε)uf (u), 2

u≫1

for some ε > 0. (f2) (subcriticality) limu↑+∞ f (u)/up = 0 for some p < n∗ . (f3) (positivity) f (u) > 0 for u ≫ 1. Then, this functional J in (1.61) satisfies conditions (a)–(d) [288]. By the theory of Orlicz spaces, we can replace the subcriticality (f2) by lim (log f )(u)/uα = 0

u↑+∞

with some α < 2 in the case of n = 2. Via the mountain pass lemma, we are able to prove the following theorem. Theorem 1.26. The existence of a solution to (1.60) is assured if the nonlinearity f (u) satisfies (f1)–(f3). For the proof of the positivity of the critical function u of J in this theorem, we take the open set ω = {u < 0} ⊂ Ω to get −Δu = 0

in ω,

u|𝜕ω = 0,

since f (0) = 0, which implies u = 0 in ω, or equivalently, ω = 0.

1.5.2 Deformation lemma Deformation lemma is a standard tool to prove Lemma 1.25. In fact, if j in (1.58) is not a critical value, then j must be smaller by the following proposition, and we get a contradiction [288].

44 | 1 Calculus of variations Proposition 1.27. Let X be a Banach space over R and J ∈ C 1 (X, ℛ) satisfy the Palais– Smale condition. Given j ∈ R, let Kj = {v ∈ X | J(v) = j, J 󸀠 (v) = 0},

Aj = {v ∈ X | J(v) ≤ j}.

If Kj = 0, then for any ε > 0 there exist δ ∈ (0, ε) and η ∈ C(X, X) such that 󵄨 󵄨󵄨 󵄨󵄨J(x) − j󵄨󵄨󵄨 > ε 󳨐⇒ η(x) = x

(1.62)

and η(Aj+δ ) ⊂ Aj−δ . ρ

First, we assume Kj = 0 for j ≥ ρ defined by (1.58). Second, for ε = 2 > 0, we take δ ∈ (0, ε) and the function η ∈ C(X, X) as in Proposition 1.27. Then there exists a path γ ∈ Γ such that max J(γ(t)) ≤ j + δ

0≤t≤1

(1.63)

by the definition of j. Let h = η∘γ ∈ C([0, 1], X). Then we have h(0) = η(γ(0)) = η(0) = 0. Furthermore, J(e) ≤ 0 < j − ε implies η(e) = e by (1.62), and hence h(1) = η(e) = e. Thus we obtain γ ∈ Γ so that j ≤ max J(h(t)). 0≤t≤1

However, (1.63) gives γ[0, 1] ⊂ Aj+δ , h[0, 1] = η(γ[0, 1]) ⊂ Aj−δ , and max J(γ(t)) ≤ j − δ

0≤t≤1

in turn, a contradiction. A different method to approach the mountain pass lemma from the deformation lemma is based on Ekeland’s variational principle [15, 131]. It localizes the Palais– Smale condition and simultaneously restricts the location of critical points.25 1.5.3 Ekeland’s variational principle Describing the principle, we note that the path space Γ = {γ ∈ C([0, 1], X) | γ(0) = 0, γ(1) = e} forms a complete metric space with the metric 󵄩 󵄩 d(γ1 , γ2 ) = max󵄩󵄩󵄩γ1 (t) − γ2 (t)󵄩󵄩󵄩, 0≤t≤1

25 See also [346, 161, 162, 280, 279, 282, 19, 198, 360] for the other structures of the mountain pass critical points.

1.5 Mountain pass lemma

| 45

recalling the norm ‖ ⋅ ‖ in X. The functional 𝒥 : Γ → (−∞, +∞] defined by 𝒥 (γ) = max J(γ(t)) 0≤t≤1

enjoys the following properties, provided that J ∈ C(X, R) and (c) holds: (𝒥 1) (proper) 𝒥 ≢ +∞; (𝒥 2) (lower semicontinuous) d(γk , γ) → 0 󳨐⇒ 𝒥 (γ) ≤ lim inf 𝒥 (γk ); (𝒥 3) (bounded below) j ≡ infΓ 𝒥 > −∞. Under these circumstances, one can prove the following lemma [106]. Lemma 1.28 (Ekeland). Let ε > 0 and γ0 ∈ Γ satisfy 𝒥 (γ0 ) < j + ε. Then, given N > 0, there exists γ∗ ∈ Γ such that 𝒥 (γ∗ ) + Nεd(γ∗ , γ0 ) ≤ 𝒥 (γ0 ), 𝒥 (γ) > 𝒥 (γ∗ ) − Nεd(γ, γ∗ ),

(1.64) ∀γ ≠ γ∗ .

(1.65)

We note that inequality (1.64) implies 𝒥 (γ∗ ) ≤ 𝒥 (γ0 ) and Nεd(γ∗ , γ0 ) ≤ 𝒥 (γ0 ) − 𝒥 (γ∗ ) ≤ 𝒥 (γ0 ) − j ≤ ε, that is, d(γ∗ , γ0 ) ≤ 1/N. Proof of Lemma 1.28. We define the sequence {γk }∞ k=1 inductively. Thus, given γ0 , γ1 , . . . , γk , we put Sk = {γ ∈ Γ \ {γk } | 𝒥 (γ) ≤ 𝒥 (γk ) − Nεd(γk , γ)}. 1. If Sk = 0, we take γk+1 = γk . 2. If α ∈ Sk it holds that 0 < Nεd(γk , α) ≤ 𝒥 (γk ) − 𝒥 (α) ≤ 𝒥 (γk ) − infSk 𝒥 . Therefore, there is γ 󸀠 ∈ Sk such that 𝒥 (γ 󸀠 ) ≤ infSk 𝒥 + 21 {𝒥 (γk )−infSk 𝒥 }. Then we put γk+1 = γ 󸀠 . By this definition, we have 1 2

𝒥 (γk+1 ) ≤ inf 𝒥 + {𝒥 (γk ) − inf 𝒥 }, Sk

Sk

k = 1, 2, . . .

(1.66)

We have also either γk+1 ∈ Sk or γk+1 = γk , so that 0 ≤ Nεd(γk , γk+1 ) ≤ 𝒥 (γk ) − 𝒥 (γk+1 ),

(1.67)

Nεd(γl , γk ) ≤ 𝒥 (γk ) − 𝒥 (γl ),

(1.68)

and in particular, ∀l > k.

As is observed in (1.67), the sequence {𝒥 (γk )} is monotone decreasing and has a limit by (𝒥 3). Therefore, {γk } is a Cauchy sequence in Γ by (1.68), so that has a limit denoted by γ∗ ∈ Γ. Since inequality (1.68) implies 𝒥 (γk ) + Nεd(γ0 , γk ) ≤ 𝒥 (γ0 ),

k = 1, 2, . . . ,

46 | 1 Calculus of variations we obtain (1.64) by (𝒥 2). If (1.65) were not true, on the other hand, there would exist γ ≠ γ∗ such that 𝒥 (γ) ≤ 𝒥 (γ∗ ) − Nεd(γ, γ∗ ). We now let l → ∞ in (1.68), to obtain Nεd(γk , γ∗ ) ≤ 𝒥 (γk ) − lim 𝒥 (γl ). l→∞

These two inequalities imply 𝒥 (γ) ≤ {𝒥 (γ∗ ) − Nεd(γ, γ∗ )} + {𝒥 (γk ) − lim 𝒥 (γl ) − Nεd(γk , γ∗ )} l→∞

≤ 𝒥 (γk ) − Nεd(γ, γk ),

(1.69)

by (𝒥 2). Since we assumed γ ≠ γ∗ , it holds that γ ≠ γk for k large, because of d(γk , γ∗ ) → 0. This property, together with (1.69), implies γ ∈ Sk , k ≫ 1, and hence 1 2

𝒥 (γk+1 ) ≤ inf 𝒥 + {𝒥 (γk ) − inf 𝒥 } Sk

Sk

1 1 = {𝒥 (γk ) + inf 𝒥 } ≤ {𝒥 (γ) + 𝒥 (γk )} Sk 2 2

(1.70)

by (1.66). Inequalities (1.69) and (1.70) imply 𝒥 (γk+1 ) − 𝒥 (γk ) ≤ −

Nε d(γ, γk ), 2

and hence d(γ, γ∗ ) ≤ 0, or γ = γ∗ , by letting k → ∞, a contradiction. To establish the proof of the mountain pass lemma via Ekeland’s variational principle, we need two more tools, namely Fenchel–Moreau’s duality theorem and von Neumann’s minimax theorem.

1.5.4 Legendre transformation Duality principle of Fenchel–Moreau is stated in the context of convex analysis. Let X be a Banach space over R equipped with the norm ‖ ⋅ ‖ and V : X → (−∞, +∞] a mapping satisfying the following properties: (V1) (proper) D(V) ≡ {x ∈ X | V(x) ≠ +∞} ≠ 0; (V2) (convex) x, y ∈ X, 0 < λ < 1 󳨐⇒ V(λx + (1 − λ)y) ≤ λV(x) + (1 − λ)V(y); (V3) (lower semicontinuous) xk → x 󳨐⇒ V(x) ≤ lim infk V(xk ). The set epi V = {(x, λ) ∈ X × R | V(x) ≤ λ} is called the epigraph of V. The following properties are verified: 1. V lower semicontinuous ⇐⇒ epi V closed ⇐⇒ {x ∈ X | V(x) ≤ λ} closed for any λ ∈ R;

1.5 Mountain pass lemma

2. 3. 4. 5. 6. 7.

| 47

V1 , V2 lower semicontinuous 󳨐⇒ V1 + V2 lower semicontinuous; Vi , i ∈ I lower semicontinuous 󳨐⇒ V = supi∈I Vi lower semicontinuous; A lower semicontinuous function attains its minimum on every compact set; V convex ⇐⇒ epi V convex; V1 , V2 convex 󳨐⇒ V1 + V2 convex; Vi , i ∈ I convex 󳨐⇒ V = supi∈I Vi convex.

For V : X → (−∞, +∞], the mapping V ∗ (p) = sup{⟨x, p⟩ − V(p)}, x∈X

p ∈ X∗

is called its conjugate or Legendre transformation, where ⟨x, p⟩ = p(x) stands for the paring between p ∈ X ∗ and x ∈ X. The mapping V ∗ : X → (−∞, +∞] is always convex and lower semicontinuous. Proposition 1.29. If V : X → (−∞, +∞] is proper, convex, and lower semicontinuous, then V ∗ : X ∗ → (−∞, +∞] is proper. Then we define the second Legendre transformation for such V by V ∗∗ (p) = sup {⟨x, p⟩ − V ∗ (p)}. p∈X ∗

Lemma 1.30 (Fenchel–Moreau). If V : X → (−∞, +∞] is proper, convex, and lower semicontinuous, then it holds that V ∗∗ = V. These properties are standard results in convex analysis.26 Here we turn to the following fact. Corollary 1.31. If V : X → (−∞, +∞] is proper, convex, lower semicontinuous, and (V4) (homogeneous) x ∈ X, λ ≥ 0 󳨐⇒ V(λx) = λV(x), then it is a supporting function of K = {p ∈ X ∗ | ⟨x, p⟩ ≤ V(x), ∀x ∈ X}, which means V(x) = σK (x) ≡ sup⟨x, p⟩. p∈K

Proof. We show that V ∗ = IK , the indicator function of K defined by IK (p) = {

0, p ∈ K, +∞, p ∈ ̸ K.

Then it follows that IK∗ (x) = sup {⟨x, p⟩ − IK (p)} = sup⟨x, p⟩ = σK (x) p∈X ∗

and hence V = V

∗∗

= σK follows.

26 See [15, 47] for the proof.

p∈K

(1.71)

48 | 1 Calculus of variations To show V ∗ = IK , first, we take p ∈ K. By the definition K, it holds that V ∗ (p) = sup{⟨x, p⟩ − V(x)} ≤ 0, x∈X

while V(0) = 0 follows from (V4) so that V ∗ (p) ≥ 0. Hence we obtain V ∗ (p) = 0. Second, for p ∈ ̸ K, we have ⟨x, p⟩ − V(x) > 0 for some x ∈ X. Letting λ ↑ +∞ in ⟨λx, p⟩ − V(λx) = λ{⟨x, p⟩ − V(x)}, we see V ∗ (p) = +∞. Hence it follows that V ∗ = IK . Now we turn to the notion of subdifferentials [46]. Let V : X → (−∞, +∞] be a proper, convex, lower semicontinuous functional defined on a Banach space X over R. Then, given p ∈ X ∗ and x ∈ X, we define p ∈ 𝜕V(x) by V(y) ≥ V(x) + ⟨y − x, p⟩,

∀y ∈ X.

If 𝜕V(x) ≠ 0, therefore, V(x) ≠ +∞, and hence x ∈ D(V). In the following, the first proposition is a direct consequence of the definition. We ∗ have 𝜕‖ ⋅ ‖(x) ≠ 0 by Hanh–Banach’s theorem,27 and x ∈ X 󳨃→ 𝜕‖ ⋅ ‖(x) ∈ 2X is called the duality map. The second proposition, on the other hand, follows from the first and the representation theorem of F. Riesz, where ℳ[0, 1], μ = μ+ − μ− with 0 ≤ μ± ∈ ℳ[0, 1], and 𝒫 [0, 1] denote the set of measures on [0, 1], Jordan decomposition of μ ∈ ℳ[0, 1], and the set of probability measures on [0, 1], respectively [118]. Proposition 1.32. It holds that 𝜕‖ ⋅ ‖(x) = {p ∈ X ∗ | ⟨x, p⟩ = ‖x‖, ‖p‖X ∗ = 1}. Proposition 1.33. Let X = C[0, 1] be the Banach space over R provided with the maximum norm ‖α‖ = max0≤t≤1 |α(t)| for α ∈ X. Then it holds that 𝜕‖ ⋅ ‖(α) = {μ ∈ ℳ[0, 1] | |μ| ∈ 𝒫 [0, 1], supp μ± ⊂ Mα± }, where |μ| = μ+ + μ− is the total variation of μ ∈ ℳ[0, 1] and Mα± = {t ∈ [0, 1] | ±α(t) = ‖α‖}. The mapping h ∈ (0, ∞) 󳨃→ h1 {V(x0 + hx) − V(x0 )} is nondecreasing for x0 ∈ D(V). It is, furthermore, bounded below if 𝜕V(x0 ) ≠ 0. Hence we can define the superdifferential of V at x0 by 1 1 D+ V(x0 )[x] = inf {V(x0 + hx) − V(x0 )} = lim {V(x0 + hx) − V(x0 )}. h>0 h h↓0 h 27 See [47] for the proof.

1.5 Mountain pass lemma

| 49

Writing 1 D+ V(x0 )[x] = lim sup {V(x0 + hx) − V(x0 )}, h h↓0 we see that x ∈ X 󳨃→ D+ (x0 )[x] ∈ R is proper, convex, and lower semicontinuous. Proposition 1.34. If V : X → (−∞, +∞] is proper, convex, and lower semicontinuous, it holds that D+ V(x0 ) = σ𝜕V (x0 ) for any x0 ∈ X in 𝜕V(x0 ) ≠ 0. Proof. Regarding the homogeneity D+ V(x0 )[λx] = λD+ V(x0 )[x],

x ∈ X,

λ ≥ 0,

we have D+ V(x0 )[x] = σK (x) = supp∈K ⟨x, p⟩ by Corollary 1.31, where K = {p ∈ X ∗ | ⟨x, p⟩ ≤ D+ V(x0 )[x], ∀x ∈ X}. Hence it suffices to derive K = 𝜕V(x0 ). First, given p ∈ K, we have ⟨x, p⟩ ≤ D+ V(x0 )[x] ≤

1 {V(x0 + hx) − V(x0 )}, h

x ∈ X,

h > 0,

and in particular, V(x0 ) < +∞. Writing y = x + hx0 for h = 1, we obtain V(y) ≥ V(x0 ) + ⟨y − x0 , p⟩, and hence p ∈ 𝜕V(x0 ). Second, if p ∈ 𝜕V(x0 ), it holds that V(x0 ) < +∞ and furthermore, ⟨x, p⟩ ≤

1 {V(x0 + hx) − V(x0 )}, h

x ∈ X,

h > 0,

and hence ⟨x, p⟩ ≤ D+ V(x0 )[x]. 1.5.5 Minimax principle Here we describe the minimax principle of von Neumann [15]. Let X0 and Y0 be topological vector spaces and X ⊂ X0 and Y ⊂ Y0 be their subsets. Given f = f (x, y) : X × Y → R, we have α ≡ inf sup f (x, y) ≥ β ≡ sup inf f (x, y). x∈X y∈Y

Now we study the condition for α = β to hold.

y∈Y x∈X

(1.72)

50 | 1 Calculus of variations In the context of game theory, the function f (x, y) stands for cost. A person on the X-side tries to make it decrease, while his opponent on the Y-side does the converse. The equality α = β describes the situation when this game is even. An element (x,̄ y)̄ ∈ X × Y satisfying f (x,̄ y) ≤ f (x,̄ y)̄ ≤ f (x, y)̄ for (x, y) ∈ X × Y is called a saddle point or core of the game. If such (x,̄ y)̄ exists, then the inequality α ≤ sup f (x,̄ ⋅) ≤ f (x,̄ y)̄ ≤ inf f (⋅, y)̄ ≤ β Y

Y

actually follows. To ensure the existence of a core, we introduce the family ℱ = {K ⊂ Y | #K < +∞}, to introduce v = sup inf sup f (x, y). K∈ℱ x∈X y∈K

The following inequality is obvious: β = sup inf f (x, y) = sup inf max f (x, y) ≤ v. y∈Y α∈X

y

x∈X y∈{y}

(1.73)

For K ∈ ℱ and x ∈ X, it holds that maxy∈K f (x, y) ≤ supy∈Y f (x, y), and hence inf max f (x, y) ≤ inf sup f (x, y) = α

x∈X y∈K

x∈X y∈Y

or v ≤ α. Proposition 1.35. If X is compact and x ∈ X 󳨃→ f (x, y) ∈ R is lower semicontinuous for each y ∈ Y, there exists x̄ ∈ X such that sup f (x,̄ ⋅) ≤ v. Y

(1.74)

Proof. For each y ∈ Y, the set Sy = {x ∈ X | f (x, y) ≤ v} is closed by the assumption. First, we shall examine the finite intersection property of {Sy }y∈Y . To this end, we take a finite set K = {y1 , y2 , . . . , ym } ⊂ ℱ . The mapping x ∈ X 󳨃→ maxy∈K f (x, y) is lower semicontinuous so it attains the minimum on the compact space X. Hence there is x̄ ∈ X such that max f (x,̄ y) = inf max f (x, y) ≤ v, y∈K

x∈X y∈K

and hence x̄ ∈ Syi , 1 ≤ i ≤ m. Thus we obtain ⋂m i=1 Syi ≠ 0. X Since {Sy }y∈Y ∈ 2 has the finite intersection property and X is compact, it follows that ⋂y∈Y Sy ≠ 0, i. e., f (x,̄ y) ≤ v for any y ∈ Y with some x̄ ∈ X, or, equivalently, (1.74) holds.

1.5 Mountain pass lemma

| 51

Under the assumption of Proposition 1.35, it holds that α ≡ inf sup f ≤ sup f (x,̄ ⋅) ≤ v ≤ α, X

Y

Y

(1.75)

and hence v = α. Given K = {y1 , y2 , . . . , ym } ⊂ ℱ , we put αK = inf max f (x, y) x∈X y∈K

m

Mm = {t (λ1 , λ2 , . . . , λm ) ∈ Rm | ∑ λi = 1, λi ≥ 0, 1 ≤ ∀i ≤ m} i=1

m

βK = sup inf ∑ λi f (x, yi ). λ∈Mm x∈X i=1

It holds that v = supK∈ℱ αK by the definition. Proposition 1.36. If Y is convex and y ∈ Y 󳨃→ f (x, y) ∈ R is concave for each x ∈ X, then it holds that βK ≤ β ≡ supY infX f for each K ∈ ℱ . Proposition 1.37. If X is convex and x ∈ X 󳨃→ f (x, y) ∈ R convex for each y ∈ Y, then it holds that αK = βK for each K ∈ ℱ . Admitting these propositions, we obtain v ≡ sup αK = sup βK ≤ β ≤ v K∈ℱ

K∈ℱ

(1.76)

by (1.73), and hence the following lemma. Lemma 1.38. Let X be convex and compact, and the mapping x ∈ X 󳨃→ f (x, y) ∈ R convex and lower semicontinuous for each y ∈ Y. Furthermore, let Y be convex, and the mapping y ∈ Y 󳨃→ f (x, y) ∈ R concave for each x ∈ X. Then, there exists some x̄ ∈ X such that sup f (x,̄ ⋅) = β = α. Y

(1.77)

Proof. In fact, equality (1.77) follows from (1.76) and (1.75). Lemma 1.38 implies the minimax principle in the following form: Theorem 1.39 (von Neumann). Let X, Y be convex, compact, and assume that x 󳨃→ f (x, y), y ∈ Y, and y 󳨃→ f (x, y), x ∈ X, are convex, lower semicontinuous, and concave, upper semicontinuous, respectively. Then there exists (x,̄ y)̄ ∈ X × Y such that ̄ f (x,̄ y) ≤ f (x,̄ y)̄ ≤ f (x, y), and hence it holds that infX supY f = supY infX f .

∀(x, y) ∈ X × Y

52 | 1 Calculus of variations Proof. Applying Lemma 1.38 for ±f (x, y), we have (x,̄ y)̄ ∈ X × Y such that sup f (x,̄ ⋅) = inf f (⋅, y)̄ = α = β, X

Y

where α, β are defined by (1.72). Then it holds that ̄ f (x,̄ y) ≤ sup f (x,̄ ⋅) = inf f (⋅, y)̄ ≤ f (x,̄ y),

∀y ∈ Y,

̄ f (x, y)̄ ≥ inf f (⋅, y)̄ = sup f (x,̄ ⋅) ≥ f (x,̄ y),

∀x ∈ X.

X

Y

X

Y

We conclude this section with the proof of Propositions 1.36 and 1.37. Proof of Proposition 1.36. For λ = t (λ1 , λ2 , . . . , λm ) ∈ Mm and x ∈ X, it holds that m

m

i=1

i=1

∑ λi f (x, yi ) ≤ f (x, ∑ λi yi ), and hence m

m

inf ∑ λi f (x, yi ) ≤ inf f (x, ∑ λi yi ) ≤ sup inf f (x, y) = β.

x∈X

i=1

x∈X

y∈Y x∈X

i=1

Thus, βK ≤ β follows. Proof of Proposition 1.36. Given K = {y1 , y2 , . . . , ym } ⊂ Y, let m

f ̃ = f ̃(x, λ) ≡ ∑ λi f (x, yi ) : X × Mm → ℛ. i=1

It is obvious that supλ∈Mm f ̃(x, λ) = max1≤i≤m f (x, yi ) for each x ∈ X so that αK ≡ inf max f (x, y) = inf sup f ̃(x, λ). x∈X y∈K

x∈X λ∈M

m

Since βK = supλ∈Mm infx∈X f ̃(x, λ), the original problem α = β has been reduced to the finite-dimensional case (4.92). In particular, βK ≤ αK is obvious similarly as β ≤ α in (1.72). To show the reverse inequality, αK ≤ βK , we use the following fact proven by a separation theorem of Mazur type: Claim 1.40. For any a > βK , there exists some xa ∈ X such that sup f ̃(xa , ⋅) ≤ a. Mm

This claim implies a ≥ αK and hence βK ≥ αK , the desired inequality.

(1.78)

1.5 Mountain pass lemma

| 53

Proof of Claim 1.40. We note that inequality (1.78) is equivalent to aθ ∈ Φ(X) + Rm +

(1.79)

where Φ = Φ(x) ≡ (

1 . θ = ( .. ) ∈ Rm +

f (x, y1 ) .. m . ):X→R ,

1

f (x, ym )

t m for Rm + ≡ { (λ1 , λ2 , . . . , λm ) ∈ R | λi ≥ 0, 1 ≤ i ≤ m}. In fact, (1.79) means

a = f (xa , yi ) + ui ,

ui ≥ 0,

1 ≤ i ≤ m,

and hence (1.78) as m

m

i=1

i=1

f ̃(xa , λ) = ∑ λi f (xa , yi ) ≤ ∑ λi a = a,

∀λ ∈ Mm .

m To confirm (1.79), first, we note that Φ(X) + Rm + ⊂ R is a convex set. In case of m aθ ∈ ̸ Φ(X) + R+ for some α > βK , therefore, there exists some λ = t (λ1 , . . . , λm ) ∈ Rm + \ {0} such that

λ ⋅ aθ ≤

inf

u∈Φ(x)+Rm +

λ ⋅ u ≤ inf λ ⋅ Φ(x) + infm λ ⋅ v, x∈X

v∈R+

where ⋅ denotes the inner dot-product in Rm . Hence infv∈Rm+ λ ⋅ v > −∞ holds. This property implies infv∈Rm+ λ ⋅ v = 0, and hence λ ⋅ aθ ≤ inf λ ⋅ Φ(x).

(1.80)

x∈X

̄ We have also λ ⋅ θ = ∑i λi > 0 by λ ∈ Rm + \{0} so that λ ≡ λ/(λ ⋅ θ) ∈ Mm . Then (1.80) implies a ≤ inf λ̄ ⋅ Φ(x) = inf f ̃(x, λ) ≤ sup f ̃ = βK , x∈X

x

Mm

which contradicts the assumption a > βK . 1.5.6 Ghoussoub–Preiss version Given a Banach space X over R equipped with the norm ‖ ⋅ ‖, let K ⊂ X be a closed set and x̄ ∈ X\{0}. We introduce the path space Γ = {γ ∈ C([0, 1], X) | γ(0) = 0, γ(1) = x}̄

54 | 1 Calculus of variations with the d(γ1 , γ2 ) = max0≤t≤1 ‖γ1 (t) − γ2 (t)‖ for γ1 , γ2 ∈ Γ. The set K is supposed to separate 0 and x,̄ i. e., 0, x̄ ∈ ̸ K, and γ[0, 1] ∩ K ≠ 0 for any γ ∈ Γ. Given J ∈ 𝒞 1 (X, R), we set 𝒥 (γ) = max J(γ(t)), 0≤t≤1

γ ∈ Γ.

The following theorem can be regarded as a refined version of Lemma 1.25. Theorem 1.41 ([132]). If infK J ≥ j ≡ infΓ 𝒥 > −∞ there exists {xk } ⊂ X such that J(xk ) → j, ‖J 󸀠 (xk )‖X ∗ → 0, and dist(xk , K) ≡ infx∈K ‖xk − x‖ → 0. In fact, assuming the mountain pass situation, (c) in § 1.5.1, we take x̄ = e and K = {x ∈ X | J(x) ≥ j}. Then, the required conditions 0, x̄ ∈ ̸ K, γ[0, 1] ∩ K ≠ 0 for any γ ∈ Γ, and infK J ≥ j are satisfied, and therefore, there exists {xk } ∈ X satisfying the conditions in Theorem 1.41. If the Palais–Smale condition (d) in § 1.5.1 holds then this {xk } is precompact, and hence j is a critical value of J. It, however, sometimes happens that the Palais–Smale condition does not hold just for the special range of j, referred to as the local Palais–Smale condition. Even in this case, j is a critical value of J. Another outcome of this theorem is the specification of the location of critical point. For instance, if one can find BR such that inf𝜕BR J = j, then the above K is taken to be 𝜕BR . Hence we have the existence of a critical point x ∈ 𝜕BR of J. Proof of Theorem 1.39. We slightly modify J. Since X is locally connected, there exists a pair of open sets (U, V) such that 0 ∈ U, x̄ ∈ V, U ∩ V = 0, and U ∪ V ∪ K = X. For 0 < ε ≪ 1, we take the ε-neighborhood of K denoted by Kε = {x ∈ X | dist(x, K) < ε}. First, there is γ0 ∈ Γ such that 𝒥 (γ0 ) < j +

ε2 . 4

(1.81)

Second, putting a = sup{t ∈ [0, 1] | γ0 (t) ∈ U\Kε } and b = inf{t ∈ [a, 1] | γ0 (t) ∈ V\Kε }, we have 0 < a < b < 1 and γ0 (t) ∈ Kε for a < t < b. Then the path space Γ is modified as Γ̃ = {γ ∈ C([a, b], X) | γ(a) = γ0 = a, γ(b) = γ0 (b)}, which forms also a complete metric space. Since ψ(x) = max{0, ε2 − ε ⋅ dist(x, K)} satisfies 0 ≤ ψ ≤ ε2 and ‖ψ‖Lip = sup x =y̸

‖ψ(x) − ψ(y)‖ ≤ ε, ‖x − y‖

̃ the functional 𝒥 ̃ (γ) = maxa≤t≤b J(γ(t)) defined for γ ∈ Γ̃ and J ̃ = J + ψ is lower semicontinuous and bounded below. Each path γ ∈ Γ̃ satisfies γ(a) = γ0 (a) ∈ U and γ(b) = γ0 (b) ∈ V, and hence γ[a, b] ∩ K ≠ 0. There is t ∗ ∈ [a, b] such that γ(t ∗ ) ∈ K, which implies ∗ ̃ J(γ(t )) = J(γ(t ∗ )) + ψ(γ(t ∗ )) ≥ j + ε2

1.5 Mountain pass lemma

| 55

and, in particular, inf 𝒥 ̃ ≥ j + ε2 .

(1.82)

Γ̃

For γ0 ∈ Γ in (1.81), on the other hand, we have γ1 ≡ γ0 |[a,b] ∈ Γ.̃ It holds that max J(γ1 (t)) ≤ 𝒥 (γ0 ) < j +

a≤t≤b

ε2 , 4

and, therefore, ψ ≤ ε2 implies 𝒥 ̃ (γ1 ) < j +

ε2 5 2 ε ≤ inf 𝒥 ̃ + 4 4 Γ̃

by (1.82). Then we apply Lemma 1.28, taking ε2 /4 and 2/ε for ε and N, respectively. Thus we have γ∗ ∈ Γ̃ satisfying 𝒥 ̃ (γ∗ ) ≤ 𝒥 ̃ (γ1 ), d(γ∗ , γ1 ) ≤ ε2 , and ε 2

𝒥 ̃ (γ) ≥ 𝒥 ̃ (γ∗ ) − d(γ, γ∗ ),

∀γ ∈ Γ.̃

(1.83)

̃ ∗ (t)) = 𝒥 ̃ (γ∗ )}, we have the following fact. Putting M = {t ∈ [a, b] | J(γ Claim 1.42. There exists t∗ ∈ M such that 3 󵄩󵄩 󸀠 󵄩 󵄩󵄩J (γ∗ (t∗ ))󵄩󵄩󵄩X ∗ ≤ ε. 2

(1.84)

̃ ∗ (t∗ )) = 𝒥 ̃ (γ∗ ) ≤ 𝒥 ̃ (γ1 ) < j + 5 ε2 . Hence 0 ≤ ψ ≤ ε2 This t∗ ∈ M satisfies j + ε2 ≤ J(γ 4 implies j ≤ J(γ∗ (t∗ )) < j +

5 2 ε. 4

Recalling γ1 = γ0 |[a,b] , a ≤ t∗ ≤ b, and γ1 (t∗ ) ∈ Kε , on the other hand, we obtain 1 3 dist(γ∗ (t∗ ), K) ≤ dist(γ1 (t∗ ), K) + d(γ1 , γ∗ ) ≤ ε + ε = ε. 2 2 We thus end up with 3 󵄩󵄩 󸀠 󵄩 󵄩󵄩J (xε )󵄩󵄩󵄩X ∗ ≤ ε, 2

j ≤ J(xε ) < j +

5 2 ε, 4

dist(xε , K) ≤

3 ε 2

for x∗ = γ1 (t∗ ) ∈ X, and then the desired sequence {xk } is obtained by taking ε = k = 1, 2, . . . .

1 k

Proof of Claim 1.40. First, we note a, b ∈ ̸ M. In fact, by (1.82) and (1.81), we obtain 2

𝒥 ̃ (γ∗ ) ≥ j + ε >

3 2 3 ε + 𝒥 (γ0 ) ≥ ε2 + max{J(γ0 (a)), J(γ0 (b))}. 4 4

for

56 | 1 Calculus of variations ̃ 0 (a)) = J(γ ̃ ∗ (a)) since dist(γ0 (a), K) = ε and γ∗ ∈ Γ,̃ and therefore, Here, J(γ0 (a)) = J(γ ̃ ∗ (a)) is impossible, so that a ∈ ̸ M. Similarly, it holds that b ∈ ̸ M. Second, 𝒥 ̃ (γ∗ ) = J(γ since j > −∞, we may assume J(γ(t)) ≥ 0 for any γ ∈ Γ and 0 ≤ t ≤ 1, adding a constant ̃ in γ ∈ Γ.̃ to J. Hence it follows that a, b ∈ ̸ Mα± for any α = J(γ) Letting C0 ([a, b], X) = {γ ∈ C([a, b], X) | γ(a) = γ(b) = 0}, we have γ∗ + hγ ∈ Γ̃ for each h > 0 and γ ∈ C0 ([a, b], X). By (1.83), it holds that 1 ̃ ε 󵄩 󵄩 {𝒥 (γ∗ + hγ) − 𝒥 ̃ (γ∗ )} ≥ − max 󵄩󵄩󵄩γ(t)󵄩󵄩󵄩. h 2 a≤t≤b

(1.85)

Here, we have ̃ ∗ (t) + hγ(t)) = J(γ ̃ ∗ (t)) + hJ 󸀠̃ (γ∗ (t))[γ(t)] + o(h) J(γ ̃ ∗ (t)) + hJ 󸀠 (γ∗ (t))[γ(t)] + h‖ψ‖Lip 󵄩󵄩󵄩γ(t)󵄩󵄩󵄩 + o(h) ≤ J(γ 󵄩 󵄩 ̃ ∗ (t)) + hJ 󸀠 (γ∗ (t))[γ(t)] + εh max 󵄩󵄩󵄩γ(t)󵄩󵄩󵄩 + o(h), ≤ J(γ 󵄩 󵄩 a≤t≤b

h↓0

uniformly in t ∈ [a, b], and hence 1 ̃ {𝒥 (γ∗ + hγ) − 𝒥 ̃ (γ∗ )} h 1 ̃ (t) + hγ(t)) − max J(γ ̃ ∗ (t))} = {max J(γ h a≤t≤b ∗ a≤t≤b 1 ̃ ∗ (t)) + hJ 󸀠 (γ∗ (t))[γ(t)]) − max J(γ ̃ ∗ (t))} ≤ {max (J(γ h a≤t≤b a≤t≤b 󵄩 󵄩 + ε max 󵄩󵄩󵄩γ(t)󵄩󵄩󵄩 + o(1). a≤t≤b

(1.86)

We take the Banach space Y = C[a, b] over R equipped with the norm 󵄨 󵄨 N(κ) ≡ max 󵄨󵄨󵄨κ(t)󵄨󵄨󵄨, a≤t≤b

κ∈Y

̃ ∗ (t)) and βγ (t) ≡ J 󸀠 (γ∗ (t))[γ(t)]. It holds that and put α(t) ≡ J(γ 3 1 lim inf {N(α + hβγ ) − N(α)} ≥ − εN(γ) h↓0 h 2

(1.87)

by (1.85) and (1.86). Then the left-hand side is equal to D+ N(α)[βγ ] = σ𝜕N(α) (βγ ) = sup ⟨μ, βγ ⟩, μ∈𝜕N(α)

by Proposition 1.34, with ⟨⋅, ⋅⟩ standing for the pairing between Y ∗ and Y. Here, Proposition 1.33 assures 𝜕N(α) = {μ ∈ ℳ[0, 1] | |μ| ∈ 𝒫 [0, 1], supp μ± ⊂ Mα± },

1.5 Mountain pass lemma

| 57

for Mα± = {t ∈ [a, b] | ±α(t) = N(α)}. Hence inequality (1.87) implies 3 − ε ≤ inf sup ⟨μ, βγ ⟩, γ∈Γ1 μ∈𝜕N(α) 2 for Γ1 = {γ ∈ C([a, b], X) | γ(a) = γ(b) = 0, maxa≤t≤b ‖γ(t)‖ ≤ 1}. Here we use Theorem 1.39. In fact, the space of measures ℳ[0, 1] = C[0, 1]∗ is ∗-weakly compact, the unit ball Γ1 in C0 ([a, b], X) is compact, and the bilinear mapping (μ, γ) ∈ ℳ0 × Γ1 󳨃→ ⟨μ, βγ ⟩ ∈ R,

ℳ0 = {μ ∈ ℳ[0, 1] | |μ| ∈ 𝒫 [0, 1]}

is continuous with respect to the above mentioned topologies. Hence, b

3 − ε ≤ sup inf ∫ J 󸀠 (γ∗ (t))[γ(t)]dμ(t). 2 μ∈𝜕N(α) γ∈Γ1

(1.88)

a

The right-hand side is equal to b

󵄩 󵄩 sup {− ∫󵄩󵄩󵄩J 󸀠 (γ∗ (t))󵄩󵄩󵄩X ∗ d|μ|(t)}

μ∈𝜕N(α)

a

̃ ∗ ) ≥ 0, this because a, b ∈ ̸ Mα± so that suppμ ⊂ (a, b) for μ ∈ 𝜕N(α). Since α = J(γ quantity is equal to b

󵄩 󵄩 󵄩 󵄩 − inf ∫󵄩󵄩󵄩J 󸀠 (γ∗ (t))󵄩󵄩󵄩X ∗ d|μ|(t) = − min󵄩󵄩󵄩J 󸀠 (γ∗ (t))󵄩󵄩󵄩X ∗ . μ∈𝜕N(α)

a

Hence there is t∗ ∈ M satisfying (1.84) by (1.88).

t∈M

2 Maximum principles Learning the previous chapter, the reader may have wondered about the problem of uniqueness. It is, in fact, crucial in the theory of nonlinear elliptic equations, in connection with the qualitative study. Here, two methods are presented: introducing parameters and utilizing the symmetry. Both of them require careful uses of the maximum principle. Once the topology of the domain is changed, however, the structure of the total set of solutions takes a variety of profiles. We shall see this situation using calculus of variations. The chapter is divided into four sections. We describe the theory of global bifurcation and radial symmetry. Then the strong maximum principle is applied to guarantee the convexity of level sets of the solution to the heat equation. The last section is devoted to the symmetric criticality, where an efficient use of the mountain pass lemma of Ghouss–Preiss is applied to confirm the emergence of nonradial solutions in any mode. The Gel’fand equation will then appear.

2.1 Parabolic dynamics 2.1.1 Ignition model A typical equation we have examined in the previous chapter is the form of − Δu = up ,

u>0

in Ω,

u|𝜕Ω = 0

(2.1)

for 1 < p < ∞, where Ω ⊂ Rn is a bounded domain with smooth boundary 𝜕Ω. Here we introduce a parameter λ ∈ R to study the problem − Δu = λf (u),

u>0

in Ω,

u|𝜕Ω = 0

(2.2)

for f ∈ C 1 (R, R). This problem is reduced to (2.1) when f (u) = up with p ≠ 1, but the solution set 2

0

̄ | u solves (2.1)} 𝒮λ = {u ∈ C (Ω) ∩ C (Ω) will change as λ varies in the other cases. For instance, if f (u) = u problem (2.1) admits a solution if and only if λ = λ1 the first eigenvalue of −Δ with Dirichlet boundary condition, and the solution set becomes a one-dimensional manifold, namely, {cϕ1 (x) | c > 0} for a suitably normalized eigenfunction ϕ1 (x) > 0. Thus, we have to take the position of finding (λ, u) simultaneously, and in this context problem (2.2) is called the nonlinear eigenvalue problem. Then, the total set of solutions, C = ⋃ {λ} × 𝒮λ , λ>0

will experience a serious effect from the nonlinearity f (u). https://doi.org/10.1515/9783110556285-002

60 | 2 Maximum principles Problem (2.2) has already arisen in the previous chapter with the appearance of the Lagrangian multiplier. Moreover, in dimensionless process of physical constants, the parameter λ appears inevitably. Below we see the situation in the theory of ignition of uniform complexity of gases [225, 33]. In fact, according to the law of Arrhenius, the velocity of reaction w(x, t) and the temperature T(x, t) are related as w = Am exp(−

E ) RT

where R, E, A, and m denote the ignition constant, activity energy, concentration of products, and order of reaction, respectively. This equation is coupled with the heat equation ρc

𝜕T − ∇ ⋅ (κ∇T) = qρw 𝜕t

where p, c, κ, and q are the mass density, specific heat, conductivity, and reaction quotient, respectively. Given Te (x, t), we may adopt T = Te as the boundary condition. Starting the process dimensionless, we use the diameter of the molecules of gases reacting, d, as a scaling parameter: ξ = x/d, τ = d2κρc t. Then the dimensionless quantity v=

E (T − Te ) RTe2

is used for the temperature T. It holds that E E v = − , RT RTe 1 + ϵv

ϵ=

RTe , E

and hence 𝜕v v − Δξ v = λ exp{ } in Q, 𝜕τ 1 + ϵv

v|𝜕Ω = 0

(2.3)

for Q ≡ Ω × (0, T) ⊂ Rn+1 , where λ=

qρAm d2 E E exp(− ) κ RTe RTe2

is the Frank–Kamenetskii constant. For the physical reasons, ϵ > 0 is small and equation (2.2) is reduced to 𝜕v − Δξ v = λev 𝜕τ

in Q,

v|𝜕Ω = 0.

(2.4)

2.1 Parabolic dynamics | 61

This nonlinearity and its modifications appear in many areas, although the total sets of stationary problems, say, −Δu = λ exp{

u }, 1 + ϵu

u>0

in Ω,

u|𝜕Ω = 0

and − Δu = λeu

in Ω,

u|𝜕Ω = 0

(2.5)

are rather different.1 This difference is reflected to the global-in-time behavior of the solution to nonstationary problems, for example, blowup of the solution to (2.4), in contrast of always global-in-time existence of the solution to (2.3). These solutions are, however, rather similar in the fixed time interval [0, T], provided that ϵ > 0 is sufficiently small. Here we mention also that (2.5) arises also as a rigorous description of the self-dual part in gauge theory, mean field of stationary many vortex points, matrix and its deformation in differential geometry, and so forth. Local-in-time unique existence of the classical solution to the nonstationary problem vt − Δv = λf (v)

in Q,

v|𝜕Ω = 0,

v|t=0 = v0 (x) ≥ 0

(2.6)

holds if v0 ∈ X = C(Ω) and f ∈ C 1 (R, R). This property is a consequence of the linear theory [120, 349]. In fact, we have the fundamental solution {U(x, y; t) | x, y ∈ Ω, t > 0} to the initial-boundary value problem vt − Δv = 0

in Q,

v|𝜕Ω = 0,

v|t=0 = v0 (x),

and hence v(x, t) = ∫ U(x, y; t)v0 (y) dy,

(x, t) ∈ Q.

Ω

Now we introduce an analytic contraction semigroup {Tt }t≥0 on X by v(⋅, t) = Tt v0 . Then, by the Duhamel principle, the inhomogeneous problem vt − Δv = λf (x, t)

in Q,

v|𝜕Ω = 0,

v|t=0 = v0 (x) ∈ X

is reduced to the integral equation in X for v = v(⋅, t), t

v(t) = Tt v0 + λ ∫(Tt−s f )(s) ds,

(2.7)

0

provided that t ∈ [0, T] 󳨃→ f (⋅, t) ∈ X is Hölder continuous. 1 See [312] and [98] for the former and the classical treatment for the latter, respectively. Since λeu > 0 for any u ∈ R, the solution u(x) to (2.5) is always positive in Ω.

62 | 2 Maximum principles Regarding this linear theory, we take the integral equation in X, t

v(t) = Tt v0 + λ ∫ Tt−s (f (v(s))) ds.

(2.8)

0

The iteration t

vk+1 = Tt v0 + λ ∫ Tt−s (f (vk (s))) ds,

k = 0, 1, 2, . . .

0

with v0 (t) ≡ v0 converges in X̂ = C([0, T], X), provided that T > 0 is sufficiently small, to produce the solution to (2.8). This v = v(x, t) is sufficiently smooth for t > 0, and becomes a classical solution to (2.6).2 Uniqueness of classical solution, on the other hand, is proven by taking w = v1 −v2 , with vi , i = 1, 2 being solutions. Actually, this w turns out to be identically zero from the linear theory because it satisfies 1

wt − Δw = λ ∫ f 󸀠 (θv1 + (1 − θ)v2 )dθ ⋅ w

in Q

0

with w|𝜕Ω = 0 and w|t=0 = 0. 2.1.2 Stationary state: standard arguments To examine global-in-time behavior of the nonstationary state, we study the stationary problem − Δu = λeu

in Ω,

u|𝜕Ω = 0

(2.9)

where Ω ⊂ Rn is a bounded domain with a smooth boundary 𝜕Ω. Given λ ∈ R+ = (0, +∞), we introduce the total set of solutions, 𝒮λ = {u ∈ C 2 (Ω)∩C 0 (Ω) | u solves (2.9)}, which is rather different from the case of f (u) = up . Theorem 2.1. There exists λ > 0 determined by Ω such that 𝒮λ ≠ 0 and 𝒮λ = 0 according to 0 < λ < λ and λ > λ, respectively. The proof is divided into three parts and several arguments are known for each step. Step 1. 𝒮λ ≠ 0 if 0 < λ ≪ 1. (a) (Fixed point theorems) The linear problem − Δu = f (x) in Ω,

u|𝜕Ω = 0

2 See [166] for H 1 theory and the smoothing effect of the solution.

(2.10)

2.1 Parabolic dynamics | 63

admits a Green’s function G(x, y) > 0 with the singularity at x = y, that is, u(x) = ∫ G(x, y)f (y) dy. Ω

Problem (2.9) is reduced to the fixed point equation u = λΦ(u),

u ∈ C0 (Ω) ≡ {v ∈ C 0 (Ω) | v|𝜕Ω = 0}

for Φ(u)(x) = ∫ G(x, y)eu(y) dy.

(2.11)

Ω

Due to the elliptic estimate in § 1.3.4, we can apply the fixed point theorem of Schauder, or the contraction mapping if 0 < λ ≪ 1, to create a solution to (2.9). (b) (Implicit function theorem) Problem (2.11) is reduced finding a zero of Φ(λ, u) ≡ −Δu − λeu : R̄ + × C02+θ (Ω) → C θ (Ω),

0 < θ < 1,

where C02+θ (Ω) = {v ∈ C 2+θ (Ω) | v|𝜕Ω = 0}. Since Φ(0, 0) = 0 and the linearized operator Φu (0, 0) = −Δ : C02+θ (Ω) → C θ (Ω) is invertible, Proposition 1.18 is applicable to Φ(λ, u) = 0. Therefore, there is a continuous mapping λ ∈ [0, λ0 ) 󳨃→ uλ ∈ C02+θ (Ω), 0 < λ0 ≪ 1, with uλ solving (2.9) and u0 ≡ 0. Furthermore, local uniqueness of the solution is valid. Hence no other solution to (2.9) exists in a neighborhood of uλ in C02+θ (Ω) if 0 < λ ≪ 1. Step 2. 𝒮λ = 0 for λ ≫ 1. (a) (Kaplan’s method) We take the first eigenvalue μ1 > 0 and the eigenfunction ϕ1 (x) for −Δ equipped with the Dirichlet boundary condition: −Δϕ1 = μ1 ϕ1 ,

ϕ1 > 0

in Ω,

ϕ1 |𝜕Ω = 0.

Here we adopt the normalization of ∫Ω ϕ1 dx = 1 and let u = u(x) be a solution to (2.9). Since u > 0 in Ω, it holds that J = ∫Ω uϕ1 dx > 0 and μ1 J = ∫(−Δu)ϕ1 dx = λ ∫ eu ϕ1 dx ≥ λeJ Ω

Ω

by Jensen’s inequality. Then we obtain λ ≤ μ1 supJ>0 Je−J < +∞ whenever there is a solution to (2.9). (b) (Isoperimetric inequality) The idea is to take a comparison between the radial case Ω = B ≡ {|x| < 1} ⊂ Rn and u = u(|x|). If n = 2, there is a sharp estimate of λ below by Bol’s inequality. See a later chapter.

64 | 2 Maximum principles Step 3. λ1 > λ2 > 0, 𝒮λ1 ≠ 0 󳨐⇒ 𝒮λ2 ≠ 0. (a) (Method of super/subsolutions) Given − Δu = λf (u) in Ω,

u|𝜕Ω = 0,

(2.12)

we call u or u a sub- or supersolution if −Δu ≤ λf (u) in Ω, u|𝜕Ω ≤ 0 or −Δu ≥ λf (u) in Ω, u|𝜕Ω ≥ 0, respectively. The assertion follows from the following lemma because any element in 𝒮λ1 casts a supersolution to (2.9) for λ = λ2 < λ1 , while u ≡ 0 is always a subsolution to (2.9). Lemma 2.2. If there is a pair of super- and subsolutions, denoted by u and u, respectively, satisfying u ≤ u in Ω, then there is a solution u to (2.12) such that u ≤ u ≤ u in Ω. This lemma holds in a general setting [6, 10, 11, 16].3 If the nonlinearity is monotone as in (2.9), a simple scheme of iteration works. Thus we obtain u ≡ u0 ≤ u1 ≤ ⋅ ⋅ ⋅ ≤ uk ≤ uk+1 ≤ ⋅ ⋅ ⋅ ≤ uk+1 ≤ uk ≤ ⋅ ⋅ ⋅ ≤ u1 ≤ u0 ≡ u for uk , uk , k = 1, 2, . . ., defined by −Δuk+1 = λf (uk ) −Δuk+1 = λf (uk )

in Ω, in Ω,

uk+1 |𝜕Ω = 0,

uk+1 |𝜕Ω = 0,

provided that f (u) is monotone nondecreasing in u. Then the monotone convergence theorem assures the assertion. The following theorem holds by the same argument [186]. Theorem 2.3. Any 𝒮λ ≠ 0 admits a minimal element uλ , which means that uλ ∈ 𝒮λ and v ≥ uλ in Ω, for any v ∈ 𝒮λ . Proof. We reduce (2.9) to (2.10) in X = C 0 (Ω). The nonlinear operator Φ : X → X defined by (2.11) is order-preserving so that v ≤ w implies Φ(v) ≤ Φ(w), where v ≤ w means v(x) ≤ w(x) for any x ∈ Ω. We define {uk } by u0 ≡ 0,

uk+1 = λΦ(uk ),

k = 0, 1, 2, . . .

Due to u1 ≥ 0 = u0 , we obtain the monotonicity of {uk }, uk+1 ≥ uk by the orderpreservation property of Φ. Given v ∈ 𝒮λ , we have v = λΦ(v) ≥ 0 = u0 and hence v ≥ (λΦ)k (u0 ) = uk similarly. Therefore, the function u ∈ L∞ (Ω) is defined by u(x) = lim uk (x) k→∞

3 See, however, [123] for an example of its breaking down.

2.1 Parabolic dynamics | 65

for each x ∈ Ω with the property u ≤ v. Since G(x, ⋅)eu(⋅) ∈ L1y (Ω), the monotone convergence theorem is applicable to uk+1 (x) = λ ∫ G(x, y)euk (y) dy, Ω

which confirms u(x) = λ ∫ G(x, y)eu(y) dy,

x ∈ Ω.

Ω

Hence u ∈ X is smooth and becomes a classical solution to (2.9). Here v ∈ 𝒮λ is arbitrary, and hence this u is the minimal element of 𝒮λ . This minimal element, called the minimal solution, is obviously unique. The following property is valid to general convex nonlinearity, and controls global-in-time behavior of the stationary solution [122, 197].4 Theorem 2.4. No triple of solutions {u1 , u2 , u3 } ⊂ 𝒮λ to (2.9) admits the relation u1 ≤ u2 ≤ u3 with u1 ≠ u2 ≠ u3 . Proof ([185]). Suppose the existence of ui ∈ C 2 (Ω) ∩ C 0 (Ω), i = 1, 2, 3, satisfying − Δui = λf (ui )

in Ω,

ui |𝜕Ω = 0

(2.13)

for f (u) = eu , and ϕ ≡ u3 − u2 ≥ 0,

ϕ ≢ 0

in Ω;

ψ ≡ u2 − u1 ≥ 0,

ψ ≢ 0

in Ω.

(2.14)

It holds that − Δϕ = c(x)ϕ

in Ω,

ϕ|𝜕Ω = 0;

−Δψ = d(x)ψ

in Ω,

ψ|𝜕Ω = 0

(2.15)

with c = λf 󸀠 ((1 − θ)u3 + θu2 ) ≥ 0, d = λf ((1 − γ)u2 + γu1 ) ≥ 0, 󸀠

c ≢ 0, d ≢ 0,

(2.16)

for 0 < θ = θ(x) < 1 and 0 < γ = γ(x) < 1. The strong maximum principle described in the next section implies ϕ(x) > 0 for x ∈ Ω. Then, 0 is the first eigenvalue of −Δ − c(x) under the Dirichlet boundary 4 By this theorem, a criterion for blowup in finite or infinite time follows, that is, an initial value larger than the nonminimal stationary state.

66 | 2 Maximum principles condition, which is proven by Jacobi’s method in § 1.3.1. Then the Rayleigh principle guarantees 0 = inf{R(v) | v ∈ H01 \ {0}} for R(v) =

I(v) , ‖v‖22

I(v) ≡ ∫{|∇v|2 − c(x)v2 } dx Ω

H01 (Ω). 󸀠

and hence I(v) ≥ 0 for any v ∈ Since f (u) = eu is convex, f is monotone nondecreasing, which guarantees c ≥ d and c ≢ d by (2.14) and (2.16). Therefore, for v ∈ H01 in v ≢ 0, we have J(v) ≡ ∫{|∇v|2 − d(x)v2 } dx > I(v) ≥ 0. Ω

However, ψ ≢ 0 solves (2.15) so that J(ψ) = 0, a contradiction. 2.1.3 Global-in-time dynamics The nonstationary problem vt − Δv = λf (v)

in Q,

v|𝜕Ω = 0,

v|t=0 ≡ v0 (x) ≥ 0

(2.17)

is reduced to the integral equation (2.8). The solution exists uniquely locally-in-time, and its existence time is estimated below by ‖v0 ‖X . From this fact, we can prove the following result concerning continuation of the solution in time because (2.17) is autonomous. Proposition 2.5. The classical solution v(x, t) to (2.17) is extendable beyond t = T < +∞ if lim inft↑T ‖v(⋅, t)‖L∞ < +∞. The following comparison principle is useful to study global-in-time behavior of the nonstationary solution. It is obtained by the linear theory and taking the difference of two solutions. Proposition 2.6. Let vi (x, t) be classical solutions to (2.6) for the initial values v0 = vi (x), i = 1, 2, satisfying v1 ≥ v2 in Ω. Then it holds that v1 (⋅, t) ≥ v2 (⋅, t) in Ω for any t ∈ [0, T). The method of super/subsolutions arises from this proposition. The following theorems are valid for the general convex nonlinearity f (v) with some additional assumptions. We assume, however, always f (v) = ev for simplicity. Henceforth, [0, T) with T ∈ (0, +∞] denotes the maximal time interval for the existence of the classical solution v(x, t) to (2.6).5 Theorem 2.7 ([35]). If λ > λ, it holds that T < +∞ and hence 󵄩 󵄩 lim󵄩󵄩󵄩v(⋅, t)󵄩󵄩󵄩L∞ = +∞. t↑T 5 The requirement for f (v) in Theorem 2.7 is reduced to the convexity of f (v) and M < +∞. This fact is proven by the use of weak steady state [58]. A related result is shown in [190].

2.1 Parabolic dynamics | 67

Lemma 2.8. Assume on the contrary, T = +∞, in the above theorem for λ = λ + δ with δ > 0, and put a=

1/2

δ 4λ + δ 4λ + δ ⋅ (1 − ( ) ), M 4(2λ + δ) 2(2λ + δ)

∞

M=∫ 0

ds = 1, f (s)

and Q1 = Ω × (0, T1 ) for T1 = a1 . Then we can define w(x, t) by wt − Δw = λg(t)f (w) in Q1 ,

w|𝜕Ω = 0,

w|t=0 = 0,

(2.18)

and it holds that (

wt ) ≥ 0, f (w) t

0 ≤ t < T1 .

(2.19)

Proof. Since 0 ≤ g(t) ≤ 1 for 0 ≤ t ≤ T1 , the global-in-time solution v(x, t) ≥ 0 to (2.6) casts a supersolution to (2.18). Since w ≡ 0 is always a subsolution, there is a solution w(x, t) to (2.6). Therefore, we can define w = w(x, t) in Q1 by (2.18). 󸀠 s Regarding M(s) ≡ ∫0 fds = 1 − e−s , we take u ≡ M(w) = 1 − e−w to obtain (s󸀠 ) ut − Δu = |∇u|2 f 󸀠 (w) + δg(t),

u|𝜕Ω = 0,

u|t=0 = 0.

w

t )t = utt satisfies Hence z = ( f (w)

zt − Δz = (|∇w|2 f 󸀠 (w))tt + δg 󸀠󸀠 (t), It holds that ut =

wt , f (w)

z|𝜕Ω = 0,

z|t=0 = 0.

and therefore,

(|∇u|2 f 󸀠 (w))tt = (2(∇u ⋅ ∇ut ) ⋅ f 󸀠 (w) + |∇u|2 ⋅ [f 󸀠󸀠 f ](w) ⋅ ut )t

= 2(∇utt ⋅ ∇u)f 󸀠 (w) + 2|∇ut |2 f 󸀠 (w) + 4(∇u ⋅ ∇ut ) ⋅ [f 󸀠󸀠 f ](w) ⋅ ut + |∇u|2 utt ⋅ [f 󸀠󸀠 f ](w) + |∇u|2 u2t ⋅ [(f 󸀠󸀠 f ) f ](w), 󸀠

which implies L[z] ≡ zt − Δz − 2f 󸀠 (w)∇z ⋅ ∇u − [f 󸀠󸀠 f ](w) ⋅ |∇u|2 z

= δg 󸀠󸀠 (t) + |∇u|2 u2t ⋅ [(f 󸀠󸀠 f ) f ](w) + 2|ut |2 f 󸀠 (w) 󸀠

+ 4[(∇ut ⋅ ∇u)ut ] ⋅ [f 󸀠󸀠 f ](w).

Here, since 1 󵄨󵄨 󵄨 1 2 2 2 󵄨󵄨(∇ut ⋅ ∇u)ut 󵄨󵄨󵄨 ≤ {ε|∇u| ut + |∇u| }, 2 ε

ε=(

f 󸀠󸀠 f )(w) > 0, f󸀠

68 | 2 Maximum principles it holds that L[z] ≥ δg 󸀠󸀠 (t) + |∇u|2 u2t {[(f 󸀠󸀠 f ) f ](w) − 2ε[f 󸀠󸀠 f ](w)} 1 + 2|∇ut |2 (f 󸀠 (w) − [f 󸀠󸀠 f ](w)) ε (f 󸀠󸀠 f )2 󸀠 󸀠󸀠 2 2 = δg (t) + |∇u| ut {[(f 󸀠󸀠 f ) f ](w) − 2 󸀠 (w)} f 󸀠󸀠 f = δg 󸀠󸀠 (t) − |∇u|2 ⋅ u2t ⋅ [f 󸀠 2 f ](w) ⋅ ( 󸀠 ) (w) ≥ δg 󸀠󸀠 (t) > 0. f 󸀠

Using z|𝜕Ω = z|t=0 = 0, we then obtain z ≥ 0. Proof of Theorem 2.7. Let λ = λ + δ for δ > 0 and assume T = +∞. Since it follows that v ≥ w in Q1 = Ω × (0, T1 ), it suffices to show lim w(x0 , t) = +∞, t↑T1

∃x0 ∈ Ω

(2.20)

for w = w(x, t) in Lemma 2.8 to get a contradiction. From the definition of λ, the stationary solution to (2.17) does not admit a positive supersolution. Therefore, there is x0 ∈ Ω such that Δw + (λ +

δ )f (w) > 0, 4

x = x0 ,

t = t0 ≡

1/2

1 4λ + δ ) ( a 2(2λ + δ)

Here, (2.18) implies wt Δw = + (λ + δ)g(t0 ) f (w) f (w) δ δ > (−(λ + ) + λ + δ)g(t0 ) = g(t0 ) 4 2 at (x, t) = (x0 , t0 ), and hence wt δ ≥ g(t0 ), f (w) 2

t ∈ [t0 , T1 ),

x = x0

by (2.19). This inequality implies v(x0 ,t)

lim ∫ t↑T1

0

δ δ 1 ds ≥ g(t )(T − t ) = a2 t02 ( − t0 ) f (s) 2 0 1 0 2 a 1/2

=

δ 4λ + δ 4λ + δ 1 ⋅ ⋅ ⋅ (1 − ( ) ) 2 2(2λ + δ) a 2(2λ + δ) ∞

=M =∫ 0

and hence (2.20).

ds , f (s)

< T1 .

2.1 Parabolic dynamics | 69

Dini’s theorem used in the proof of the following theorem says that a sequence of continuous functions on a compact metric space converges uniformly if it is monotone and the limit function is continuous. Theorem 2.9 ([122]). If 𝒮λ ≠ 0 and 0 ≤ v0 ≤ uλ , the classical solution v(x, t) to (2.17) exists global-in-time and satisfies 󵄩 󵄩 lim 󵄩󵄩󵄩v(⋅, t) − uλ 󵄩󵄩󵄩∞ = 0,

(2.21)

t↑+∞

where uλ denotes the minimal solution. Proof. Let w(x, t) be the solution to (2.17) for v0 ≡ 0. Then it holds that 0 ≤ w(⋅, t) ≤ v(⋅, t) ≤ uλ

(2.22)

by Proposition 2.6, and therefore, we may suppose v0 ≡ 0 for the proof of this theorem, and also obtain T = +∞ by Proposition 2.5. Then w = vt satisfies wt − Δw = λf 󸀠 (v)w

in Q,

w|𝜕Ω = 0,

w|t=0 = w0 (x)

(2.23)

for w0 = Δv0 + λf (v0 ) ≥ 0, and hence w = vt ≥ 0 in Q = Ω × (0, ∞) which implies the convergence u(x) = limt↑+∞ v(x, t) for each x ∈ Ω. It holds that ∞

G(x, y) = ∫ U(x, y; t) dt 0

since (−Δ)−1 = ∫0 etΔ dt, while the monotone convergence theorem applied to ∞

t

v(x, t) = λ ∫ ds ∫ U(x, y; s)f (v(y, t − s)) dy 0

(2.24)

Ω

implies ∞

u(x) = λ ∫ ds ∫ U(x, y; s)f (u(y)) dy = λ ∫ G(x, y)f (u(y)) dy 0

Ω

Ω

for each x ∈ Ω. Then u ∈ 𝒮λ follows from the proof of Theorem 2.3.6 Since u ≤ uλ by (2.22), it holds that u = uλ by the minimality of uλ ∈ 𝒮λ . Then Dini’s theorem assures (2.21). 6 Another argument uses Lyapunov function for u ∈ 𝒮λ to prove [341].

70 | 2 Maximum principles If 𝒮λ \{uλ } ≠ 0, any u∗ ∈ 𝒮λ \{uλ } satisfies u∗ > uλ in Ω by the strong maximum principle. Furthermore, the first eigenvalue of the linearized operator −Δ − λf 󸀠 (u∗ ) under the Dirichlet boundary condition, denoted by μ1 , is nonpositive, μ1 ≤ 0. This fact, Lemma 2.11, is proven in the following subsection and is used in the proof of the second case of the following theorem.7 In the first case, on the other hand, it is obvious that v(⋅, t) = u∗ if v0 = u∗ . Theorem 2.10 ([196]). If v0 ≤ u∗ , v0 ≢ u∗ , then it holds that T = +∞ and (2.21) is satisfied. If ∫Ω v0 φ∗1 dx > ∫Ω u∗ φ∗1 dx, on the contrary, we have T < +∞, where φ∗1 (x) > 0, x ∈ Ω, denotes the first eigenfunction of −Δ − λf 󸀠 (u∗ ) under the Dirichlet boundary condition. Proof. In the first case, we may assume v0 = (1 − α)uλ + αu∗ ,

0 < α < 1,

by Proposition 2.6. Then w = vt satisfies (2.23) with w0 = Δv + λf (v)|t=0

= −λ{(1 − α)f (uλ ) + αf (u∗ )} + λf ((1 − α)uλ + αu∗ ) ≤ 0.

We thus obtain w = vt ≤ 0, which implies T = +∞ and the convergence u(x) = limt↑+∞ v(x, t) for each x ∈ Ω. It holds that u ∈ 𝒮λ by the proof of Theorem with (2.24) replaced by t

v(x, t) = ∫ U(x, y; t)v0 (y) dy + λ ∫ ds ∫ U(x, y; s)f (v(y, t − s)) dy. Ω

0

Ω

In fact, here, the first term on the right-hand side converges uniformly to zero because the first eigenvalue −Δ with the Dirichlet boundary condition is positive. Since uλ ≤ u ≤ u∗ and u ≢ u∗ , we obtain u ≡ uλ by Theorem 2.4. The uniform convergence (2.21), finally, follows from Dini’s theorem. In the second case, we may suppose the existence of μ1 ≤ 0 and φ∗1 = φ∗1 (x) such that −Δφ∗1 = (μ1 + λf 󸀠 (u∗ ))φ∗1 ,

φ∗1 > 0

in Ω,

φ∗1 |Ω = 0,

∫ φ∗1 dx = 1.

Ω

Since w = v − u∗ satisfies wt − Δw = λ(f (w + u∗ ) − f (u∗ )) in Q,

w|𝜕Ω = 0,

7 The assumption in the second case of this theorem follows if v0 ≥ u∗ , v0 ≢ u∗ . This case is treated by [122], using a similar argument as in the first case, although the possibility of blowup in infinite time is not excluded.

2.1 Parabolic dynamics |

it holds that dJ + μ1 J = λ ∫(f (w + u∗ ) − f (u∗ ) − wf 󸀠 (u∗ ))φ∗1 dx, dt Ω

71

J = ∫ wφ∗1 dx. Ω

We have also f 󸀠󸀠 (v) ≥ δ > 0 for v ≥ 0 with δ = 1. This property means that h(v) = f (v) − δ2 v2 is convex in v ≥ 0. Then, inequality h(w + u∗ ) − h(u∗ ) − wh󸀠 (u∗ ) ≥ 0 follows, which implies f (w + u∗ ) − f (u∗ ) − wf 󸀠 (u∗ ) ≥ We thus obtain

δ 2 w ≡ g(w). 2

dJ + μ1 J ≥ λg(J), J(0) > 0 dt by Jensen’s inequality and the assumption, which guarantees J(t) > 0 for 0 ≤ t < T and hence dJ ≥ λg(J) > 0 by μ1 ≤ 0. We thus obtain dt J(t)

∫ J(0)

and therefore, T < +∞ by

dJ ≥ λt, g(J)

+∞ dJ ∫ g(J)

0 ≤ t < T,

< +∞.

2.1.4 Branch of minimal solutions The parabolic dynamics, Theorems 2.7, 2.9, and 2.10, hold for more general nonlinearities in (2.17), as well as the description of the stationary state, Theorems 2.1, 2.3, and 2.4. Here we take f ∈ C 2 (R, R) satisfying f (0) > 0,

f 󸀠 (0) ≥ 0,

f 󸀠󸀠 (u) > 0,

u > 0.

(2.25)

The case f (0) = 0 is admitted in Theorem 2.4 if the trivial solution u ≡ 0 is included in the set of stationary states denoted by 𝒮λ . In fact, in this case the trivial solution u ≡ 0 casts the minimal solution, but henceforth, we shall concentrate on the case (2.25). As in the proof of Theorem 2.10, the role of minimal solutions can be understood from the spectrum of the linearized operator. Given the solution (λ, u) to the stationary state − Δu = λf (u),

u>0

in Ω,

u|𝜕Ω = 0,

(2.26)

the linearized operator Lλ,u denotes the differentiation −Δ − λf 󸀠 (u) equipped with the Dirichlet boundary condition. It spectrum consists of eigenvalues {μj (λ, u)}∞ j=1 with the property −∞ < μ1 (λ, u) < μ2 (λ, u) ≤ ⋅ ⋅ ⋅ → ∞. First, we note the following fact for u ∈ 𝒮λ (see [186]).

72 | 2 Maximum principles Lemma 2.11. If u ∈ 𝒮λ satisfies μ1 (λ, u) > 0, then it holds that u = uλ . For the proof, we require the following form of the maximum principle. Henceforth, we say that u ∈ 𝒮λ is strictly minimal if μ1 (λ, u) > 0. Proposition 2.12. Let c = c(x) ∈ C(Ω)̄ and take A = −Δ − c(x) equipped with the Dirichlet boundary condition. Suppose that the first eigenvalue μ1 of A is positive. Then, the condition −Δv ≤ c(x)v in Ω with v ≤ 0 on 𝜕Ω implies either v < 0 in Ω or v ≡ 0. to

This proposition follows from the existence of the Green’s function G = G(x, y) > 0 (−Δ − c(x))v = f (x),

v|𝜕Ω = 0,

derived from that of the fundamental solution U(x, y, t) > 0 to vt − Δv = c(x)v

in Q = Ω × (0, T),

v|𝜕Ω = 0,

v|t=0 = v0 (x)

and the Laplace transformation justified since μ1 > 0, A

−1

∞

= ∫ e−tA dt. 0

Proof of Lemma 2.11. Suppose u ∈ 𝒮λ and put v = u − uλ , which satisfies v ≥ 0, v ≢ 0 in Ω and v = 0 on 𝜕Ω. Then, regarding the monotonicity of f 󸀠 , we obtain −Δv = λf 󸀠 (θu + (1 − θ)uλ )v ≤ λf 󸀠 (u)v

in Ω,

v|𝜕Ω = 0

with 0 < θ = θ(x) < 1. Then μ1 (λ, u) > 0 yields that v < 0 in Ω, a contradiction. The implicit function theorem implies 𝒮λ ≠ 0 for 0 < λ ≪ 1 in the proof of Theorem 2.1. In particular, there exists a branch, that is, a one-dimensional manifold, 𝒞∗󸀠 ⊂ 𝒞 ≡ ⋃λ>0 {λ} × 𝒮λ starting from (λ, u) = (0, 0) toward λ > 0. This is proven using the fact that μ1 (0, 0) > 0 and hence the branch 𝒞∗󸀠 is composed of strict minimal solutions. This branch extends as far as μ1 (λ, u) > 0 is kept, to create a maximal branch denoted by 𝒞∗ . In particular, we have μ1 (λ, u) > 0 for each (λ, u) ∈ 𝒞∗ and 𝒞∗ = {(λ, uλ ) | 0 < λ < λ∗ }

(2.27)

for some λ∗ ≤ λ by Lemma 2.11. Step 3 in Theorem 2.1 is proven by the method of super/subsolutions. Hence u = uλ󸀠 and u = 0 are super- and subsolutions to (2.12) for λ < λ󸀠 , and therefore, λ < λ󸀠 ,

𝒮λ󸀠 ≠ 0 󳨐⇒ 𝒮λ ≠ 0,

Then, two cases arise by the Schauder estimate.

uλ ≤ uλ󸀠

in Ω.

2.1 Parabolic dynamics | 73

Case 1. limλ↑λ∗ ‖uλ ‖L∞ = +∞; Case 2. 𝒮λ∗ ≠ 0, uλ∗ = limλ↑λ∗ uλ uniformly on Ω, and μ1 (λ∗ , uλ∗ ) = 0. In the first case, we say that the branch blows-up at λ = λ∗ . In the second case, a sharp description of the total set of solutions 𝒞 is obtained around (λ∗ , uλ∗ ) ∈ 𝒞 . For this purpose we apply the implicit function theorem again, utilizing the positivity of the first eigenfunction of the linearized operator, and the convexity of f (u) [94]. Since λ(s) < λ∗ for |s| ≪ 1, the total set of solutions 𝒞 bends at (λ, u) = (λ∗ , u∗ ) in the following lemma.8 Lemma 2.13. If (λ∗ , u∗ ) ∈ 𝒞 satisfies μ1 (λ∗ , u∗ ) = 0, there exists a branch 𝒞 ∗ = {(λ(s), u(s)) | |s| ≪ 1} ⊂ 𝒞 such that (λ(0), u(0)) = (λ∗ , u∗ ). Furthermore, the maṗ = 0 and ping s 󳨃→ (λ(s), u(s)) ∈ R+ × C0 (Ω) is a C 2 diffeomorphism and satisfies λ(0) ̈ λ(0) < 0. Proof. Let ϕ1 (x) be the eigenfunction corresponding to μ1 (λ∗ , u∗ ) = 0: − Δϕ1 = λ∗ f 󸀠 (u∗ )ϕ1 ,

ϕ1 > 0

in Ω,

ϕ1 |𝜕Ω = 0.

(2.28)

The first eigenvalue is simple, and we obtain ker(Lλ∗ ,u∗ ) = ⟨ϕ1 ⟩ ≡ {cϕ1 | c ∈ R}, recalling that Lλ∗ ,u∗ denotes the linearized operator, −Δ − λ∗ f 󸀠 (u∗ ), with the Dirichlet boundary condition. Letting Y ≡ {v ∈ C02+α (Ω) | ∫Ω vϕ1 dx = 0} with 0 < α < 1, we define the nonlinear operator Φ : R × R × Y → C α (Ω)̄ by Φ(s, σ, v) = −Δ(u∗ + sϕ1 + v) − (λ∗ + σ)f (u∗ + sϕ1 + v). It is obvious that Φ(0, 0, 0) = 0. Furthermore, since f > 0, we have ∫ f (u∗ )ϕ1 dx > 0,

Ω

and hence the linearized operator Φ(σ,v) (0, 0, 0) = (

R −f (u∗ ) ) : × → C α (Ω) Lλ∗ ,u∗ Y

is an isomorphism. The implicit function theorem assures the C 2 mapping s 󳨃→ (σ(s), v(s)) ∈ R × Y defined for |s| ≪ 1 such that Φ(s, σ(s), v(s)) = 0,

|s| ≪ 1,

λ(0) = λ∗ ,

u(0) = u∗ .

8 Using the perturbation theory, from the simple eigenvalue we obtain the differentiability in s of d μ1 (λ(s), u(s)), and also ds μ1 (λ(s), u(s))|s=0 ≠ 0.

74 | 2 Maximum principles Then we define 𝒞 ∗ = {(λ(s), u(s)) | |s| ≪ 1} by σ(s) = λ(s) − λ∗ , Henceforth, we write ̇ for

𝜕 . 𝜕s

v(s) = u(s) − u∗ − sϕ1 .

First, Φ(s, σ(s), v(s)) = 0 and

Φs (0, 0, 0) = −Δϕ1 − λ∗ f 󸀠 (u∗ )ϕ1 = 0 ̇ ̇ implies Φσ (0, 0, 0)[σ(0)] + Φv (0, 0, 0)[v(0)] = 0. Hence it holds that ̇ ̇ σ(0) = λ(0) = 0,

̇ ̇ v(0) = u(0) − ϕ1 = 0

(2.29)

because Φσ,v (0, 0, 0) is an isomorphism. Second, −Δu(s) = λ(s)f (u(s)) in Ω,

u(s)|𝜕Ω = 0

implies −Δü = λf̈ (u) + 2λḟ 󸀠 (u)u̇ + λf 󸀠󸀠 (u)u̇ 2 + λf 󸀠 (u)ü in Ω,

u|̈ 𝜕Ω = 0,

and then we obtain ̈ ∫ f (u )ϕ dx + λ ∫ f 󸀠󸀠 (u )ϕ3 dx = 0, λ(0) ∗ 1 ∗ ∗ 1 Ω

Ω

̈ by (2.28) and (2.29). Since f 󸀠󸀠 > 0, it follows that λ(0) < 0. The following result is a consequence of Lemma 2.13, and, in particular, any minimal solution uλ ∈ Sλ has the property μ1 (λ, uλ ) ≥ 0. We call uλ extremal minimal solution if it exists. Theorem 2.14. Under the assumption of (2.25), the set of strictly minimal solutions forms a branch, 𝒞∗ = {(λ, uλ ) | 0 < λ < λ}. If 𝒮λ = 0, this branch blows-up as λ ↑ λ. Otherwise, it holds that 𝒮λ = {uλ }, μ1 (λ, uλ ) = 0, and the total set of solutions 𝒞 bends at (λ, u) = (λ, uλ ). Proof. It suffices to show #𝒮λ ≤ 1. In fact, if there is a u ∈ 𝒮λ \ {uλ }, we obtain a contradiction by the proof of the second case of Theorem 2.10, due to μ1 (λ, uλ ) = 0. The following conclusion is shown to be the best possible concerning the dimension n in later sections. Theorem 2.15 ([95]). If f (u) = eu and n ≤ 9, the set 𝒞 bends at the extremal minimal solution.

2.1 Parabolic dynamics |

75

Proof. We show (2.30)

lim sup ‖uλ ‖∞ < +∞ λ↑λ

under the assumption. In fact, the minimal solution u = uλ ∈ 𝒮λ satisfies ∫ |∇w|2 − λf 󸀠 (u)w2 dx ≥ 0, Ω

w ∈ H01

since μ1 (λ, uλ ) ≥ 0. Given ϕ ∈ C 1 (R, R) with ϕ(0) = 0, we have w = ϕ(u) ∈ H01 , and hence ∫ ϕ󸀠 (u)2 |∇u|2 − λf 󸀠 (u)ϕ(u)2 dx ≥ 0. Ω

Given ψ ∈ C 1 (R, R) with ψ(0) = 0, on the other hand, we multiply by ψ(u) equation (2.26) to obtain ∫ ψ󸀠 (u)|∇u|2 − λf (u)ψ(u) dx = 0. Ω

Therefore, it holds that 2

ψ󸀠 ≥ (ϕ󸀠 ) 󳨐⇒ ∫ f (u)ψ(u) dx ≥ ∫ f 󸀠 (u)ϕ(u)2 dx. Ω

(2.31)

Ω

Given f (u) = eu , we take ϕ(u) = emu − 1 and u

ψ(u) = ∫ ϕ󸀠 (u)2 du = 0

m 2mu {e − 1}, 2

with m > 0 to be defined later. Then, (2.31) is reduced to m ∫ e(2m+1)u − eu dx ≥ ∫ e(2m+1)u − 2e(m+1)u + eu dx, 2 Ω

Ω

which implies (1 −

m ) ∫ e(2m+1)u dx ≤ 2 ∫ e(m+1)u dx 2 Ω

Ω

≤ 2|Ω|

m 2m+l

(∫ e Ω

(2m+1)u

dx)

m+1 2m+1

.

76 | 2 Maximum principles We thus obtain ‖eu ‖2m+1 = O(1) for m < 2 and hence ‖λf (u)‖p = O(1) for 1 < p < 5 since λ ≤ λ. Then the Lp estimate implies ‖u‖W 2,p = O(1), so that ‖u‖∞ = O(1) in the case of n < p by Sobolev’s embedding. Thus we obtain (2.30) for n < 10. 2 2.1.5 Unboundedness via the topological degree For the nonlinear eigenvalue problem (2.26) with nonlinearity f ∈ C 2 (R, R) satisfying (2.25), there is a branch of strictly minimal solutions denoted by 𝒞 = {(λ, uλ ) | 0 < λ < λ},

where λ stands for the upper bound of λ for the existence of the classical solution. Let

𝒞 ̂ be the connected component of 𝒞 , the total set of solutions containing 𝒞 .

Theorem 2.16 ([286, 287]). The set 𝒞 ̂ is unbounded in R+ × C 0 (Ω). We use topological degree for the proof. Let 𝒪 ≠ 0 be a nonempty bounded open set in a Banach space X and Φ : 𝒪 → X a continuous compact mapping, so that Φ(𝒪) is a compact set in X. For Ψ = I − Φ and p ∈ X with p ∈ ̸ Ψ(𝜕𝒪), the integer called topological or Leray– Schauder degree d(Ψ, p, 𝒪) is constructed with the following properties, where I denotes the identity mapping Iv = v for v ∈ X. 1. (homotopy invariance) This integer is invariant under continuous deformation. Namely, if the mapping (t, x) ∈ [0, 1] × 𝒪 󳨃→ Φt (x) ∈ X is continuous, Φt (⋅) : 𝒪 → X is compact, and p ∈ ̸ Ψi (𝜕𝒪), Ψt = I − Φt , for any t ∈ [0, 1], then it holds that d(Ψ0 , p, 𝒪) = d(Ψ1 , p, 𝒪). 2. 3.

(boundary valued dependence) The integer d(Ψ, p, 𝒪) depends only on Ψ|𝜕𝒪 . (domain decomposition) For a finite family {𝒪j } of disjoint open sets in 𝒪, it holds that p ∈ ̸ Ψ(𝒪\ ⋃ 𝒪j ) 󳨐⇒ d(Ψ, p, 𝒪) = ∑ d(Ψ, p, 𝒪j ). j

j

4. (product formula) If Xj , 𝒪j , pj ∈ Xj , and Φj : 𝒪j → Xj are as above, that is, 𝒪j ≠ 0 is a bounded open set in a Banach space Xj , Φj : 𝒪 → Xj is a continuous compact mapping, and pj ∈ ̸ Ψj (𝜕𝒪j ), Ψj = I − Φj , for j = 1, 2, then it holds that d(Ψ1 × Ψ2 , p1 × p2 , 𝒪1 × 𝒪2 ) = d(Ψ1 , p1 , 𝒪1 ) ⋅ d(Ψ2 , p2 , 𝒪2 ). If X is a finite-dimensional vector space and p ∈ X is a regular value of Ψ, this topological degree is equal to d(Ψ, p, 𝒪) =

∑ v∈Ψ−1 (p)

sign JΨ (v)

2.1 Parabolic dynamics | 77

with the agreement of Σ0 = 0, where JΨ (v) denotes the Jacobian of Ψ at v, and sign(s) is equal to −1, 0, and +1, according to s < 0, s = 0, and s > 0, respectively. The number d(Ψ, p, 𝒪) in the general case is defined by extending this number, following the above properties [38, 100]. Then we obtain the following proposition. Proposition 2.17. If d(Ψ, p, 𝒪) ≠ 0, then {v ∈ 𝒪 | Ψ(v) = p} ≠ 0. Let X, 𝒪, p, and Ψ = I − Φ be as above and put p = 0 for simplicity. If v0 ∈ 𝒪 is an isolated zero of Ψ, the value d(Ψ, 0, Bϵ (v0 )), denoted by i(Ψ, v0 ), is independent of ϵ > 0 sufficiently small, and is called the fixed point index. If Φ ∈ C 1 (𝒪, X), furthermore, then T = Φ󸀠 (v0 ) ∈ B(X, X) is a compact bounded operator by the assumption, and therefore, its zero eigenvalue μ ≠ 0 enjoys the property ker(μI − T) ⫋ ker(μI − T)2 ⫋ ⋅ ⋅ ⋅ ⫋ ker(μI − T)p = ker(μI − T)p+1 = ⋅ ⋅ ⋅ with a positive integer p. Then the number nμ = dim ker(μI − T)p is finite, and called the algebraic multiplicity of μ. The following fact is obvious from the definition if dim X < +∞. Lemma 2.18. If T = Φ󸀠 (v0 ) is invertible, then it holds that i(Ψ, v0 ) = (−1)∑μ>1 nμ . Admitting these fundamental properties of the topological degree, we show Theorem 2.16. Proof of Theorem 2.16. Assume on the contrary, that is, 𝒞 ̂ is bounded in X̂ = R+ ×C 0 (Ω). Then it is compact from the Schauder estimate and the existence of the upper bound λ < +∞ of 𝒞 ̂ on λ. Since 𝒞 ∩ {λ = 0} = {(0, 0)}, we can take an open set 𝒪 ⊂ X̂ containing 𝒞 ̂ satisfying 𝜕𝒪λ ∩ 𝒞 = 0, 𝒪λ = 0 for λ ≫ 1, and 𝒪λ ∩ 𝒞 = {uλ } for 0 < λ ≪ 1, where 𝒪λ = {v ∈ C 0 (Ω) | (λ, v) ∈ 𝒪}. In the Banach space X = C 0 (Ω), topological degree d(Ψλ , 0, 𝒪λ ) is taken for any λ ∈ R+ , where Ψλ = I − Φλ with Φλ (v) = (−Δ)−1 λf (v). From the homotopy invariance, d(Ψλ , 0, 𝒪λ ) is independent of λ ∈ R+ . However, Proposition 2.17 and Lemma 2.18 imply d(Ψλ , 0, 𝒪λ ) = 0,

λ ≫ 1,

d(Ψλ , 0, 𝒪λ ) = 1,

0 < λ ≪ 1,

a contradiction. If f (0) = 0, f 󸀠 (0) = 1, f 󸀠󸀠 (0) ≠ 0 in (2.26), the first eigenvalue of −Δ provided with the Dirichlet condition, denoted by λ1 , is a bifurcation point from the trivial solution u = 0. Then, bifurcated positive solutions form a branch, and the connected component of the total set of solutions containing this branch is denoted by 𝒞 .̂ It is either unbounded in R+ × C 0 (Ω) or crosses with the branch of trivial solutions {(λ, 0) | λ ≥ 0} in (λ1 , +∞). The latter case is excluded because the strong maximum principle assures that 𝒞 ̂ contains only positive solutions [286].

78 | 2 Maximum principles 2.1.6 Nonminimal solutions Continuing the study on the nonlinear eigenvalue problem (2.26) for C 2 nonlinearity f (u) satisfying (2.25), we apply the variational method, restricting its growth rate as in § 1.5.1: lim f (u)/up = 0,

n > 2,

u↑+∞

lim (log f )(u)/uα = 0,

u↑+∞

∃p < n∗ =

n = 2,

n+2 , n−2

∃α < 2.

Then the functional 1 J(v) = ∫ |∇v|2 − λF(v) dx v ∈ X = H01 (Ω), 2

(2.32)

u

F(u) = ∫ f (u)du 0

Ω

1

is introduced, satisfying J ∈ C (X, R) and the Palais–Smale condition on X. The critical point of this functional corresponds to the solution to (2.26) and the existence of e ∈ X with ‖e‖X sufficiently large such that J(e) ≤ 0 is verified if we impose the superlinearlity at infinity: lim f (u)/u = +∞.

(2.33)

u↑+∞

Since f (0) > 0, the strict minimal solution uλ for 0 < λ < λ casts the trivial solution to create a mountain pass situation. In fact, above properties established for J are inherited by I(v) = J(v + uλ ) − J(uλ ),

v∈X

besides I(0) = 0. To assure a ball BR = BR (0) ⊂ X with 0 < R ≪ 1 satisfying inf𝜕BR I > 0, we use 1 I(v) = ∫ |∇v|2 + ∇v ⋅ ∇uλ − λF(v + uλ ) + λF(uλ ) dx 2 Ω

=

1 ∫ |∇v|2 dx − λ ∫ F(v + uλ ) − F(uλ ) − f (uλ )v dx, 2 Ω

Ω

noting that uλ is a solution. It follows that I(v) =

1 λ ∫ |∇v|2 dx − ∫ f 󸀠 (uλ )v2 dx + o(||v||2L2 ) 2 2 Ω

Ω

from the mean value theorem, and then Poincaré inequality implies ∫ |∇v|2 − λf 󸀠 (uλ )v2 dx ≥ δ‖∇v‖22 ,

Ω

v ∈ H01

with a constant δ > 0 because uλ is strictly minimal. This property assures the mountain pass situation, and thus, we obtain the following theorem [95].

2.2 Radial symmetry |

79

Theorem 2.19. If f (u) satisfies (2.32)–(2.33) besides (2.25), problem (2.26) admits a nonminimal solution u∗λ for each 0 < λ < λ. This theorem is applicable for f (u) = eu if n = 2. This result is also the best possible with respect to the dimension n.

2.2 Radial symmetry 2.2.1 Gidas–Ni–Nirenberg’s theorem Optimality of n ≤ 9 and n ≤ 2 in Theorems 2.15 and 2.19 is an entrance of to a sharp description of the total set of solutions 𝒞 . Assuming simple shape of the domain is the first step to touch the truth [96, 97]. If the domain Ω is the unit ball B = B1 (0) ≡ {|x| < 1} ⊂ Rn and the solution u(x) is a function of r = |x|, the problem − Δu = f (u),

u>0

in Ω,

u|𝜕Ω = 0

(2.34)

is reduced to the two-point boundary value problem urr +

n−1 u + f (u) = 0, r r

0 < r = |x| ≤ 1,

ur (0) = u(1) = 0.

(2.35)

We call such a solution radially symmetric, or radial in short. The following theorem ensures that no solution u ∈ C 2 (B) to (2.34) is missed by reducing it to (2.35), provided that f (u) is a C 1 function [137].9 Theorem 2.20 (Gidas–Ni–Nirenberg). If u ∈ C 2 (B) and f ∈ C 1 (R, R) in (2.34), then it holds that u = u(r) and ur < 0 for 0 < r = |x| ≤ 1. If f (u) = up , 1 < p < ∞, problem (2.35) means urr +

n−1 u + up = 0, r r

u > 0,

0 < r ≤ 1,

ur (0) = 0,

(2.36)

with u(1) = 0.

(2.37)

Let u(r) and v(r) be solutions to (2.36), and put λ = {u(0)/v(0)}

p−1 2

,

2

w(r) = λ p−1 v(λr).

9 This assumption is reduced to f = f1 +f2 with f1 being Lipschitz continuous and f2 monotone increasing. The problem in the entire space Rn is treated first by [138]. See also [250] for later developments.

80 | 2 Maximum principles Then, it holds that wr (0) = 0 = ur (0), w(0) = u(0), and wrr +

2p n−1 n−1 wr + wp = λ p−1 (vρρ + v + vp ) = 0 r ρ ρ

for ρ = λr. Using this property, we see that any order of derivative of u and w must coincide at x = 0, that is, Dα u(0) = Dα v(0) for any multiindex α. Here, analytic regularity to elliptic problems is available [92]. In fact, since u and w solve −Δu = up , they are real-analytic at x = 0, and therefore, u ≡ w follows. Regarding the boundary condition (2.37), we conclude λ = 1, and hence the uniqueness of the problem, u ≡ v. Combining this uniqueness and the existence result in the previous chapter, we reach the following theorem. Theorem 2.21. For 1 < p < ∞, the problem −Δu = up , u > 0 in B, u|𝜕B = 0 admits a unique classical solution if and only if p < n∗ . The eigenvalue problem − Δu = λeu

in B,

u|𝜕B = 0

(2.38)

is reduced to urr +

n−1 u + λeu = 0, r r

r > 0,

ur (0) = 0,

(2.39)

with u(1) = 0.

(2.40)

For this nonlinearity, the solution set of (2.39) is invariant under the transformation u0 (r) 󳨃→ u0 (eα/2 r) + α, α ∈ R. Therefore, for a suitable solution u0 (r) to (2.39), we have only to determine the parameter α ∈ R to adjust the boundary condition (2.40): u0 (eα/2 ) + α = 0,

(2.41)

which creates the solution to (2.38) by u(x) = u0 (eα/2 |x|) + α. Then we find a significant dependence on the dimension n of the total set of solutions, 𝒞 . Case 1. n = 1. First, we seek a function u0 (r), −1 < r < 1, satisfying u󸀠󸀠 + λeu = 0,

u󸀠 (0) = 0,

λeu(0) = 2.

We have 0 = u󸀠󸀠󸀠 + λeu u󸀠 = {u󸀠󸀠 − 21 (u󸀠 )2 }󸀠 and hence u󸀠󸀠 − 21 (u󸀠 )2 = −2, or, 2 − u󸀠 2 − u󸀠 2 − u󸀠 ) = 2(1 − ) . 4 4 4 󸀠

(

(2.42)

2.2 Radial symmetry | 81

Therefore, l(r) = 41 {2 − u󸀠 (r/2)} solves the logistic equation l󸀠 = (1 − l)l,

1 l(0) = . 2

Writing this equation as ( 1l )󸀠 = − 1l + 1, we see l(x) = 21 (1 + tanh x2 ), or, u󸀠 (r) = −2 tanh r. Thus it holds that u󸀠󸀠 (r) = −2/ cosh2 r and hence u0 (r) = −2 log cosh r + log

2 λ

by (2.42). Second, equation (2.40) means e−α/2 cosh(eα/2 ) = ( λ2 )1/2 . Here, g(t) = 1t cosh t defined for t > 0 has a minimum at some t ∈ (0, ∞), decreases and increases in (0, t) and (t, +∞), respectively, and furthermore, lim g(t) = lim g(t) = +∞. t↓0

t↑∞

Therefore, we have two branches, denoted by t± (g), of the root to g(t) = g for g ≥ g0 ≡ inft>0 g(t) > 0 such that limg↑+∞ t− (g) = 0 and limg↑+∞ t+ (g) = +∞. Then the solution to (2.38) is given by u(x) = u0 (eα/2 |x|) + α = −2 log cosh(eα/2 |x|) + log = 2 log(

2 +α λ

cosh t± (g) cosh eα/2 ) = ug± (x), ) = 2 log( cosh(t± (g)|x|) cosh eα/2 |x|

and, in particular, it holds that limg↑+∞ ug− (x) = 0 and limg↑+∞ ug+ (x) = +∞ for any x ∈ B. Proposition 2.22. If n = 1, the total set of solutions to (2.38), denoted by 𝒞 , forms a branch, which is a one-dimensional manifold starting from (λ, u) = (0, 0), bending at (λ, uλ ), and absorbed into the hyperplane λ = 0. In particular, the number of solutions to (2.38) is 0, 1, and 2, according to λ > λ, λ = λ, and 0 < λ < λ, respectively. Furthermore, along the nonminimal branch as λ ↓ 0 it holds that u(x) → +∞,

x ∈ B.

(2.43)

We call (2.43) the entire blow-up. Case 2. n = 2. Regarding (2.39), that is, u󸀠󸀠 + 1r u󸀠 +λeu = 0, 0 < r < ∞, u󸀠 (0) = 0, we apply the Liouville transformation s = log r to obtain 1 u + λeu = 0, r 2 ss

82 | 2 Maximum principles d 2 ) (u+2s)+λeu+2s = 0, which takes the same form as in the case of n = 1. Therefore, or ( ds the function u defined by

u + 2s = −2 log cosh(eβ/2 s) + log

2 + β, λ

β∈R

solves the above equation. Taking β = 0, we have u0 (r) = log{

8/λ }, (r 2 + 1)2

u0r (0) = 0.

This time, equation (2.40) is reduced to the algebraic equation 8 2 = (eα + 1) /eα , λ and hence the solution to (2.39) for n = 2 is explicitly given by uλ± (x) = log{

8β ± /λ }, (1 + β± |x|2 )2

β± =

1/2

4 λ λ {1 − ± (1 − ) }. λ 4 2

Thus we obtain λ = 2, and the profile of the total set of solutions 𝒞 is similar to that in the case n = 1. This time, however, we have lim uλ+ (x) = 4 log λ↓0

1 , |x|

locally uniformly in x ∈ B\{0}

(2.44)

and limλ↓0 uλ+ (0) = +∞. We say that the upper branch {(λ, uλ+ ) | 0 < λ < 2} experi1 ences a one-point blow-up as λ ↓ 0, and call the function 4 log |x| in (2.44) a singular limit.10

Proposition 2.23. If n = 2, the total set of solutions to (2.38) has the same profile as that in the case n = 1. The upper branch, however, experiences a one point blow-up as λ ↓ 0 1 with the singular limit 4 log |x| . Case 3. n > 2. This case is quite different from the previous ones. First, any solution to (2.39) does not take an explicit form. Second, the nonlinearity is stronger than the diffusion, that is, we are in the supercritical case where the method of variations does not work. Here, we make use of the algebraic property of f (u) = eu more systematically. From (2.39) we deduce (r n−1 u󸀠 ) + λr n−1 eu = 0, 󸀠

r > 0,

u(0) = A > 0,

u󸀠 (0) = 0.

10 An important observation to (2.44) is that uλ+ is locally uniformly bounded in B\{0} as λ ↓ 0. Hence the blow-up set is equal to {0}.

2.2 Radial symmetry |

83

Then we adopt Emden’s transformation, u(r) = w(t) − 2t + A,

r = Bet ,

B={

1/2

2(n − 2) A e } λ

to arrive at ẅ + (n − 2)ẇ + 2(n − 2)(ew − 1) = 0, ̇ − 2} = 0. lim {w(t) − 2t} = lim e−t {w(t)

t↓−∞

t↓−∞

(2.45)

This system (2.45) is autonomous, and does not contain A > 0 or λ > 0. Actually, it admits a unique solution which is global-in-time. Now we define the orbit 𝒪 = ̇ {(w(t), w(t)) | −∞ < t < ∞} to pick up the boundary condition u(1) = 0. In fact, each point (w0 , ẇ 0 ) ∈ 𝒪 corresponds to the time t = t0 . In terms of the variables w and t, the requirement u = 0 at r = 1 means A = −w0 + 2t0 ,

et0 = B−1 .

(2.46)

Given t0 , conversely, the constants A and B defined by (2.46) produce the solution (λ, u) to (2.38) for λ = 2(n − 2)e−A B−2 = 2(n − 2)ew0 . Therefore, the total set of solutions to (2.39), 𝒞 , is homeomorphic to 𝒪 through the relation (2.46). Here we study the shape of 𝒪. We write (2.45) in the form of w ẇ w d ( )=( )≡F( ) w −2(n − 2)(e − 1) − (n − 2)ẇ ẇ dt ẇ where t (w, w)̇ = t (0, 0) is the unique zero of F. The linearized equation around this point is given by h 0 1 h d ( )=( )( ̇ ) −2(n − 2) −(n − 2) h dt ḣ and the linearlized eigenvalues are equal to μ± = 21 {(2 − n) ± √(n − 2)(n − 10)}. Hence the stationary point t (w, w)̇ = t (0, 0) of this system is a spiral attractor if and only if 2 < n < 10. In the other case when n ≥ 10, the orbit 𝒪 is absorbed into the origin exponentially. ̇ By comparison argument, we can prove limt↑+∞ (w(t), w(t)) = (0, 0) which indicates infinitely many and no bendings of 𝒞 for 2 < n < 10 and n ≥ 10, respectively. The stationary point (w, w)̇ = (0, 0) corresponds to λ = 2(n − 2),

u(x) = 2 log

1 , |x|

(2.47)

which is a weak solution to (2.38), since −Δ(2 log

1 2(n − 2) )= ∈ L1 (B) |x| |x|2

holds in the sense of distributions if n > 2. Thus, the dimensions n ≤ 9 and n ≤ 2 in Theorems 2.15 and 2.19 are optimal.

84 | 2 Maximum principles Proposition 2.24 ([130, 170]). If n > 2, the total set of solutions 𝒞 forms a branch staring at (λ, u) = (0, 0) and blowing-up at the weak solution (2.47). If 2 < n < 10, this 𝒞 bends in λ infinitely many times, while no bending occurs to 𝒞 if n ≥ 10. 2.2.2 Maximum principles The proof of Theorem 2.20 is based on the strong maximum principle and Hopf lemma, which we have already utilized for the Laplace operator. We need now, however, more delicate and general forms [278, 139]. Let Ω ⊂ Rn be a bounded domain and n

L = ∑ aij (x) i,j=1

n 𝜕 𝜕2 + ∑ bj (x) + c(x) 𝜕xi 𝜕xj j=1 𝜕xj

a second order partial differential operator with real-valued coefficients aij = aji . We suppose the ellipticity, 0 < λ(x)|ξ |2 ≤ ∑ aij (x)ξi ξj ≤ Λ(x)|ξ |2 , i,j

(2.48)

valid for x ∈ Ω and ξ = (ξ1 , ξ2 , . . . , ξn ) ∈ Rn \ {0}. Theorem 2.25 (Weak maximum principle I). If Ω is bounded, c ≡ 0, and 󵄨 󵄨 Bj ≡ sup󵄨󵄨󵄨bj (x)󵄨󵄨󵄨/λ(x) < +∞, x∈Ω

1 ≤ ∃j ≤ n,

(2.49)

the function u ∈ C 2 (Ω) ∩ C 0 (Ω) satisfying Lu ≥ 0

in Ω

(2.50)

admits supΩ u = max𝜕Ω u. Proof. Step 1. The function u never assumes an interior maximum if Lu > 0 in Ω. In fact, if u(x0 ) = maxΩ u with some x0 ∈ Ω, we have uxj (x0 ) = 0, 1 ≤ j ≤ n, and H(x0 ) = (uxi ,xj (x0 ))1≤i,j≤n ≤ 0 since ϕ󸀠 (0) = 0 ≥ ϕ󸀠󸀠 (0) for ϕ(t) = u(x0 + tξ ), |t| ≪ 1, and ξ ∈ Rn . We introduce the matrix A(x) = (aij (x)) ≥ λ(x)I > 0, where I denotes the unit matrix. Then, it holds that Lu(x0 ) = ∑ aij (x0 )uxi ,xj (x0 ) = tr{A(x0 )H(x0 )} i,j

= tr{A(x0 )1/2 H(x0 )A(x0 )∗1/2 } ≤ 0, a contradiction to (2.49), where A(x0 )∗ denotes the transposed matrix of A(x0 ).

2.2 Radial symmetry | 85

Step 2. Assume j = 1 in (2.49) for the general case of (2.50). Taking ξ = (1, 0, . . . , 0) in (2.48), we see a11 (x) = ξ t A(x)ξ ≥ λ(x), and hence L(eγx1 ) = (a11 (x)γ 2 + b1 (x)γ)eγx1 ≥ λ(x)(γ 2 − B1 γ)eγx1 > 0 for γ ≫ 1. Therefore, for ϵ > 0 the function vϵ (x) = u(x) + ϵeγx1 satisfies Lvϵ > 0 in Ω, so that supΩ vϵ = max𝜕Ω vϵ from the first step. Letting ϵ ↓ 0, we obtain the result. Theorem 2.26 (Weak maximum principle II). If Ω is bounded, c ≤ 0, and (2.48) and (2.49) hold, then sup u ≤ max{max(u, 0)} Ω

𝜕Ω

(2.51)

for u ∈ C 2 (Ω) ∩ C 0 (Ω) satisfying (2.50). Proof. The conclusion is obvious when u ≤ 0 in Ω. If Ω+ ≡ {x ∈ Ω | u(x) > 0} ≠ 0, then it holds that {L − c(x)}(u) ≥ 0 in Ω+ . We apply the previous theorem in each connected component of Ω+ , to obtain sup u = sup u = sup u. Ω

Ω+

𝜕Ω+

(2.52)

Here, 𝜕Ω+ ⊂ 𝜕Ω ∪ {u = 0}, and hence sup u ≤ max{max u, 0} = max{max(u, 0)}. 𝜕Ω

𝜕Ω+

𝜕Ω

(2.53)

Then we obtain (2.51) by (2.52) and (2.53). Proceeding to the Hopf lemma, we say that an operator L is uniformly elliptic if (2.49) holds for any j = 1, 2, . . . , n, and also (2.48) is satisfied with λ0 = inf λ(x) > 0, x∈Ω

a = sup Λ(x)/λ(x) < +∞. x∈Ω

Let b = supj Bj . Theorem 2.27 (Hopf lemma I). Let L be uniformly elliptic in Ω, c ≡ 0, and u ∈ C 2 (Ω) satisfy (2.50). Suppose, furthermore, the following: 1. u(x) is continuous at x = ξ ∈ 𝜕Ω. 2. u(ξ ) > u(x) for any x ∈ Ω. 3. There is a ball B = BR (y) ⊂ Ω satisfying B ∩ 𝜕Ω = {ξ }. 4. There exists 𝜕u , where ν denotes the unit outer normal vector at ξ ∈ 𝜕Ω. 𝜕ν Then it holds that

𝜕u 𝜕ν

> 0 at ξ .

86 | 2 Maximum principles ≥ 0 holds at ξ . Subject to the ball BR (y) in the third condiProof. It is obvious that 𝜕u 𝜕ν tion, we introduce the auxiliary function 2

2

v(x) = e−αr − e−αR > 0,

r = |x − y| < R

(2.54)

with α > 0 to be specified later. Since 2

vxj = −2α(xj − yj )e−ar , 2

2

vxi xj = −2αδij e−σr + 4α2 (xi − yj )(xj − yj )e−αr , it holds that 2

Lv(x) = e−ar (4α2 ∑ aij (xj − yi )(xj − yj ) − 2α ∑(ajj + bj (xj − yj ))) i,j

≥e

−αr 2

≥e

−ar 2

2

j

2

(4α λ(x)r − 2α[Tr(A(x)) + n sup |bj | ⋅ r]) j

2 2

λ0 {4α r − 2α(na + nbr)}.

(2.55)

Taking the annulus A = BR (y)\Bρ (y), 0 < ρ < R, we make α ≫ 1 in (2.55), to achieve Lv > 0

in A.

(2.56)

By the second condition, we have 0 < ϵ ≪ 1 so that w(x) ≡ u(x) − u(ξ ) + ϵv(x) ≤ 0

on 𝜕Bρ (y),

while w(x) = u(x) − u(ξ ) ≤ 0

on 𝜕BR (y),

holds due to v = 0 on 𝜕BR (y). Since (2.56) implies Lw = Lu + ϵLv > 0

in A,

(2.57)

we thus obtain w(x) = u(x) − u(ξ ) + ϵv(x) ≤ 0 by Theorem 2.25. Regarding w(ξ ) = v(ξ ) = 0, we have (2.58), which implies 𝜕u > 0 there, because 𝜕ν

in A, 𝜕w 𝜕ν

2 𝜕v 𝜕v = = −2aRe−aR < 0. 𝜕ν 𝜕r

=

𝜕u 𝜕ν

(2.58) 𝜕v + ϵ 𝜕ν ≥ 0 at ξ ∈ 𝜕Ω by

2.2 Radial symmetry |

87

Theorem 2.28 (Hopf lemma II). The condition c ≡ 0 can be replaced by c ≤ 0, u(ξ ) ≥ 0, and 󵄨 󵄨 C ≡ sup󵄨󵄨󵄨c(x)󵄨󵄨󵄨/λ(x) < +∞

(2.59)

x∈Ω

in Theorem 2.27. Proof. In this case the function v(x) defined by (2.54) is estimated by 2

Lv(x) ≥ e−ar λ0 {4α2 r 2 − 2α(na + nbr) − C}. Hence inequality (2.56) follows from α ≫ 1. The function w(x) defined by (2.58) satisfies (2.57), as Lw(x) = Lu(x) − c(x)u(ξ ) + ϵLv(x) ≥ Lu(x) + eLv(x) > 0

in A.

Then, Theorem 2.26 implies supA w ≤ max𝜕A {max w, 0} = 0, where 0 < ϵ ≪ 1, and the proof is the same from this stage. Theorem 2.29 (Hopf lemma III). The condition c ≡ 0 can be replaced by u(ξ ) = 0 and (2.59) in Theorem 2.27. Proof. Putting v = e−αx1 u, we have 0 ≤ Lu = L(eαx1 v) = eαx1 {Lv + a11 (α2 v + 2α

𝜕v 𝜕v ) + α(∑ a1j + b1 v)}, 𝜕x1 𝜕xj j=1̸

or [L + 2αa11

𝜕 𝜕 + α ∑ a1j + (α2 a11 + αb1 )]v ≥ 0 𝜕x1 𝜕x j j=1̸

in Ω.

Then the second condition of the theorem implies L󸀠 v ≥ 0 in Ω for α ≫ 1, where L󸀠 ≡ ∑ aij (x) i,j

𝜕2 𝜕 𝜕 𝜕 + ∑ bj (x) + 2αa11 + α ∑ a1j . 𝜕xi 𝜕xj 𝜕x 𝜕x 𝜕x j 1 j j j=1̸

By Theorem 2.27, we obtain 𝜕v 𝜕 𝜕u 𝜕u = ( e−αx1 )u + e−αx1 = e−σx1 > 0, 𝜕ν 𝜕ν 𝜕ν 𝜕ν and hence

𝜕u 𝜕ν

> 0 at ξ ∈ 𝜕Ω.

Theorem 2.30 (Strong maximum principle I). Let L be uniformly elliptic with c ≡ 0, and u ∈ C 2 (Ω) satisfy inequality (2.50). Then, u is a constant whenever it assumes an interior maximum.

88 | 2 Maximum principles Proof. From the assumption we have M ≡ supΩ u < +∞ and u(x0󸀠 ) = M for some x0󸀠 ∈ Ω. In case u ≢ M we have Ω1 ≡ {x ∈ Ω | u(x) < M} ≠ ϕ and 𝜕Ω1 ∩ Ω ≠ 0. Then we take x0 ∈ 𝜕Ω1 ∩ Ω, Bρ (x0 ) ⊂ Ω, and z ∈ Bρ/2 (x0 ) ∩ Ω1 ≠ 0. It holds that z ≠ x0 , Bd (z) ⊂ Ω1 ∩ B(x0 , ρ), and 𝜕Bd (z) ∩ 𝜕Ω1 ≠ 0 for d = dist(z, 𝜕Ω1 ∩ Ω) ∈ (0, ρ/2]. Reducing the radius, if necessary, we may assume B ⊂ Ω1 and #(B ∩ 𝜕Ω1 ) = 1, for B = Bd (z). Then we put B ∩ 𝜕Ω1 = {ξ } to apply Theorem 2.27. It follows that 𝜕u > 0 at ξ ∈ Ω1 , 𝜕ν which contradicts M = supΩ u < +∞ for ξ ∈ Ω. Replacing Theorem 2.29 by Theorem 2.27 in the proof, we obtain the following result. Theorem 2.31 (Strong maximum principle II). If L is uniformly elliptic, provided with (2.59), and 0 ≤ u ∈ C 2 (Ω) satisfies (2.50), then u is identically zero if it assumes an interior zero. 2.2.3 Method of the moving plane The proof of Theorem 2.20 relies on a sophisticated use of the maximum principle, called Alexandroff’s moving plane method [8]. Let Ω ⊂ Rn be a bounded domain with C 2 boundary 𝜕Ω, and γ ∈ Rn be a unit vector. We introduce the hyperplane Tλ = {x ∈ Rn | x ⋅ γ = λ} perpendicular to γ, where λ ∈ R and ⋅ denotes the inner product in Rn . We have Tλ ∩ Ω = 0 for λ ≫ 1, and a portion cut by Tλ appears as λ decreases, denoted by Σ(λ). Let its reflection with respect to Tλ be Σ󸀠 (λ). This reflection remains in Ω until one of the following situations happens: 1. Σ󸀠 (λ) touches 𝜕Ω inside Ω at some p ∈ ̸ Tλ . 2. Tλ is perpendicular to 𝜕Ω at some q ∈ 𝜕Ω. In such a case we put Tλ = Tγ and call Σ(λ) the maximal cap. There arises the case that Σ(λ) remains in Ω even after the maximal cap. Putting λ∗ = inf{λ | Σ(λ) ⊂ Ω}, we call Σ(λ∗ ) the optimal cap. For the moment we consider the equation Δu + b1 (x)ux1 + f (u) = 0,

u>0

in Ω,

u|𝜕Ω = 0

(2.60)

for b1 ∈ C(Ω) with b1 (x) ≥ 0, f ∈ C 1 (R, R), and u ∈ C 2 (Ω). Let Σ = Σ(λ1 ) be the maximal cap, Σγ , for γ = (1, 0, . . . , 0). It holds that λ1 < λ0 ≡ maxx∈Ω̄ x1 . We set Σ󸀠 = Σ󸀠 (λ1 ) ⊂ Ω. Then we have the following theorem which implies Theorem 2.20. Theorem 2.32. Any λ ∈ (λ1 , λ0 ) admits ux1 (x) < 0,

u(x) < u(xλ ),

∀x ∈ Σ(λ),

(2.61)

where x λ denotes the reflection of x with respect to Tλ . We have, in particular, ux1 < 0 in Σ, but the condition ux1 (q) = 0,

∃q ∈ Tλ1 ∩ Ω

2.2 Radial symmetry |

89

implies the symmetry of the domain, differential operator, and solution, in the sense that Ω = Σ ∪ Σ󸀠 ∪ (Tλ1 ∩ Ω),

b1 (x) ≡ 0,

u(x) = u(xλ1 ),

x ∈ Ω.

(2.62)

Proof of Theorem 2.20. From the first part of Theorem 2.32, it follows that ux1 < 0 for x1 > 0. Taking the transformation x1 󳨃→ −x1 , we realize ux1 > 0 for x1 < 0, and hence ux1 = 0 on x1 = 0. Using the second part of this theorem, we see the symmetry of u with respect to the hyperplane {x1 = 0}. The axis x1 can be taken to be an arbitrary direction which ensures u = u(|x|), as well as ur < 0 for 0 < r = |x| ≤ R. The proof of Theorem 2.32 is executed by continuation argument. Step 1. (2.61) holds for 0 < λ0 − λ ≪ 1. Step 2. If (2.61) is satisfied at λ = λ∗ ∈ (λ1 , λ0 ), then it holds for any λ in 0 < λ∗ − λ ≪ 1. Step 3. If (2.61) holds for λ = λj satisfying λj ↓ λ∗ ∈ (λ0 , λ1 ), then it is kept at λ = λ∗ . Thus, we shall show that the set of λ ∈ (λ1 , λ0 ) satisfying (2.61) contains a leftneighborhood of λ0 , and is simultaneously left-open and left-closed. We show the following lemmas to complete the proof. Up to the end of this subsection, we take the situation described in Theorem 2.32. Henceforth, ν(x) = (ν1 (x), . . . , νn (x)) denotes the outer unit normal vector at x ∈ 𝜕Ω. Lemma 2.33. If x0 ∈ 𝜕Ω satisfies ν1 (x0 ) > 0, there is δ > 0 such that ux1 < 0 in Ω∩Bδ (x0 ). Lemma 2.34. If λ1 < λ < λ0 and ux1 (x) ≤ 0, u(x) ≤ u(x λ ), and u(x) ≢ u(xλ ) for any x ∈ Σ(λ), then these inequalities can be improved to (2.61). Lemma 2.34 is associated with the exclusion of the equality everywhere. Such a principle is prepared in linear theory, strong maximum principle, and Hopf lemma. Thus, the proof relies on the linearization, justified because uλ (x) ≡ u(xλ ) satisfies the same equation as that of u(x), at the cost of f ∈ C 1 . The simply-looking Lemma 2.33, on the other hand, is involved in a delicate profile of the solution, as the conclusion ux1 < 0 is asserted in an open set. There arises actually the case that this quantity takes zero on the boundary. The proof, therefore, is divided into two cases. In the first, f (0) ≥ 0, this property is excluded, while in the second, f (0) < 0, it actually happens. Proof of Lemma 2.33. We write uxi = ui , uxi xj = uij , etc., for simplicity. Given 0 < ε ≪ 1, we put Ωε = Ω ∩ Bε (x0 ). It holds that u ∈ C 2 (Ωε ), u > 0 in Ωε , and u = 0 on S ≡ 𝜕Ω ∩ Ωε . Since 0 < ε ≪ 1, we have ν1 (x) > 0 for x ∈ S, and hence u1 ≤ 0

on S.

(2.63)

If the conclusion is false, we have a sequence {xk } ⊂ Ω converging to x0 such that u1 (xk ) ≥ 0,

(2.64)

u1 (x0 ) = 0.

(2.65)

and hence

90 | 2 Maximum principles If f (0) ≥ 0, we have Δu + b1 (x)u1 + f (u) − f (0) ≤ 0 by (2.60), and the mean value theorem admits c = c(x) ∈ L∞ (Ω) such that Δu + b1 (x)u1 + c1 (x)u ≤ 0

in Ωε .

Theorem 2.29 then implies uν (x0 ) < 0, and hence u1 (x0 ) < 0, a contradiction to (2.65). Turning to the case f (0) < 0, we use the fact that the segment starting from x k for k ≫ 1, parallel to x1 -axis, and with the right-direction, crosses S at one point, denoted by yk . The inequality u1 (yk ) ≤ 0 holds by (2.63), and therefore, the inequality u11 > 0 on [xk , yk ] is impossible by (2.64). Hence we find z k ∈ [xk , yk ] satisfying u11 (z k ) ≤ 0, which implies u11 (x0 ) ≤ 0. We have, on the other hand, u(x0 ) = 0, u1 (x0 ) = 0, and u > 0 in Ωε , so that the function ϕ(x1 ) = u(x1 , x20 , . . . , xn0 ) satisfies ϕ󸀠󸀠 (x10 ) ≥ 0, where x0 = (x10 , x20 , . . . , xn0 ). This property means u11 (x0 ) ≥ 0, and hence u11 (x0 ) = 0.

(2.66)

For any tangential vector τ, we next obtain u|S = 0 󳨐⇒ uττ (x0 ) = 0.

(2.67)

uν (x0 ) = 0

(2.68)

We have also

by (2.65), u|S = 0, and ν1 (x0 ) > 0, and therefore, uν |S attains the maximum at x0 because uν ≤ 0 holds on S, which implies uτν (x0 ) = 0.

(2.69)

Δu(x0 ) = uνν (x0 ) + (n − 1)κuν (x0 ) = uνν (x0 ) = −f (0)

(2.70)

Finally, u|S = 0 implies by (2.68), where κ stands for the mean curvature.11 Equalities (2.67), (2.69), and (2.70) are summarized as uij (x0 ) = −f (0)νi νj , which implies u11 (x0 ) = −f (0)ν12 > 0, a contradiction to (2.66). Proof of Lemma 2.34. Recall that Σ󸀠 (λ) denotes the reflection of Σ(λ) with respect to Tλ . We take the function v(x) = u(xλ ), which satisfies Δv − b1 (xλ )v1 + f (v) = 0, 11 See §2.2.3 of [335].

v1 ≥ 0

in Σ󸀠 (λ)

2.2 Radial symmetry | 91

by (2.60) and (2.60). Since b1 (x) ≥ 0, it holds that Δ(v − u) + b1 (x)(v − u)1 + f (v) − f (u) = (b1 (xλ ) + b1 (x))v1 ≥ 0.

(2.71)

Hence w(x) = v(x) − u(x) satisfies Δw + b1 (x)w1 + c(x)w ≥ 0,

w ≤ 0,

w ≢ 0

in Σ󸀠 (λ),

where c(x) is a bounded function. Then Theorem 2.31 implies w < 0 in Σ󸀠 (λ), or u(x) < u(x λ ) for x ∈ Σ(λ). We have, on the other hand, w = 0 on Tλ ∩ Ω, and therefore, Theorem 2.29 assures w1 > 0 on Tλ ∩ Ω, or equivalently, ux1 (x) < 0 for x ∈ Tλ ∩ Ω. We are ready to give the following proof. Proof of Theorem 2.32. The first step, (2.61) for 0 < λ0 − λ ≪ 1, is a direct consequence of Lemma 2.33. The third step, on the other hand, follows from Lemma 2.34. In fact, we have uλ∗ ≢ u in Σ(λ∗ ) because u is positive in Ω and takes a zero on 𝜕Ω. The second step, however, is delicate in controlling the solution near the boundary as in the proof of Lemma 2.33. Assuming on the contrary, we lose one of the inequalities (2.61) in 0 < λ∗ − λ ≪ 1. The first inequality, however, is valid inside Ω for such λ because of that for λ = λ∗ . Near the boundary, on the other hand, Lemma 2.33 is applicable so that there remains the possibility of a breakdown of the second inequality. Thus, there arise λ1 < λk ↑ λ∗ and xk ∈ Σ(λk ) satisfying λ

u(xk ) ≥ u(xkk ).

(2.72)

We may assume xk → x ∈ Σ(λ∗ ), which implies u(x) ≥ u(x λ∗ ). Since (2.61) is assumed for λ = λ∗ , there is no x ∈ Σ(λ∗ ) such that x ∈ Tλ∗ ∩ Ω. It is also excluded that x ∈ 𝜕Ω by Lemma 2.33, and hence x ∈ Tλ ∩ Ω. This last possibility is also a contradiction to the first inequality of (2.61), valid for λ = λ∗ . Turning to the second part, we suppose u1 = 0 at some q ∈ Tλ1 ∩ Ω. Then Lemma 2.34 implies u(x) = u(x λ ) for any x ∈ Σ(λ1 ), and hence Ω = Σ ∪ Σ󸀠 ∪ (Tλ1 ∩ Ω) since u > 0 in Ω and u|𝜕Ω = 0. Finally, we use (2.71) valid for λ = λ1 and v ≡ u. Since v1 ≠ 0 on Ω \ Tλ1 , it follows that b1 (xλ1 ) + b1 (x) = 0 in Ω, and hence b1 ≡ 0 by b1 (x) ≥ 0. 2.2.4 Some remarks There are several applications of Theorem 2.32. Theorem 2.35. If Ω = A ≡ {a < |x| < 1} with 0 < a < 1 in (2.34), then it holds that for a+1 ≤ r = |x| ≤ 1. 2

𝜕u 𝜕r

u(1) for 0 < x < 1, then it holds that ̇ u(x) < 0,

1 < x < 1. 2

(2.73)

If u(̇ 21 ) = 0, furthermore, we have b(x) = 0 and u(1 − x) = u(x) for 0 < x < 1. Proof. This theorem is a direct consequence of Theorem 2.32 if u(x) is differentiable at x = 1. Without this property, we notice u(x) > u(1) to ensure the existence of the sequences ϵk , δk ↓ 0 such that u(x) > u(1 − ϵk ),

δk < ∀x < 1 − ϵk .

Applying Theorem 2.32 to u(x) − u(1 − ϵk ), we get ̇ u(x) 0

in Ω,

u|𝜕Ω = 0

are contained in Ω\ω, where f (u) is an arbitrary C 1 nonlinearity. Proof. In fact, the assertion follows for ω = ⋃|γ|=1 Σγ , where Σγ denotes the maximal cap. Theorem 2.38. Let Ω, Σ(λ), and λ ∈ (λ1 , λ0 ) be as in Theorem 2.32, and u ∈ C 2 (Ω) solve the fully nonlinear equation F(x, u, u1 , . . . , un , u11 , . . . , unn ) = 0

in Ω,

where F(x, u, pi , pjk ) is a continuous function with continuous first derivatives with respect to u, pi , and pjk . Suppose, furthermore, the ellipticity m|ξ |2 ≤ Fpjk ξj ξk ≤ M|ξ |2 with the constants m, M > 0, and the following conditions:

2.2 Radial symmetry |

1. 2.

93

The function g(x) = F(x, 0, . . . , 0) satisfies exclusively g(x) ≥ 0 or g(x) < 0 for x ∈ 𝜕Ω ∩ {x1 > λ1 }. For λ ∈ [λ1 , λ0 ), x ∈ Σ(λ), u > 0, and p1 < 0, it holds that F(xλ , u, −p1 , p2 , pα , p11 , −p1α , pβγ ) ≥ F(x, u, p1 , pα , pjk ), where 1 ≤ j, k ≤ n and 2 ≤ α, β, γ ≤ n.

Then (2.61) holds, and furthermore, Ω = Σ(λ1 ) ∪ Σ(λ1 )󸀠 ∪ (Tλ1 ∩ Ω), provided that (2.62) is satisfied. The proof is similar, and Theorems 2.20, 2.35, and 2.36 are also generalized. Theorem 2.39. The conclusions of Theorems 2.20 and 2.35 are valid even if the nonlinearity is inhomogeneous as in f = f (r, u), if f and fu are continuous and f is monotone nonincreasing in r. Theorem 2.40. The conclusion of Theorem 2.36 is valid even if ü + f (x, u, u)̇ = 0,

u(x) > u(1),

0 < x < 1,

provided that f , fu , and fp are continuous, and f (y, u, −p) ≥ f (x, u, p),

p ≤ 0,

y + x > 1,

y < x.

The following theorem may be regarded as a refinement of Theorem 2.37. Here we use the Kelvin transformation, v(y) = |x|n−2 u(x), y = x/|x|2 , which induces Δy v = |x|n+2 Δx u. Theorem 2.41. The conclusion of Theorem 2.37 holds for the general domain with C 2 boundary, if n = 2 and f (u) ≥ 0. Proof. We show that each x0 ∈ 𝜕Ω admits a neighborhood U of x0 determined by Ω such that no stationary point of u is located in Ω ∩ U. For this purpose we take a ball B ⊂ Ωc so that 𝜕B ∩ Ω = {x0 }. Without loss of generality, we suppose that B is a unit disc with the center origin and x0 = (1, 0), to take the Kelvin transformation T : x 󳨃→ y = x/|x|2 and v(y) = |x|n−2 u(x). Since n = 2, it follows that − Δv = |y|−4 f (v)

in Ω̃ = TΩ ⊂ B.

(2.74)

It holds that Ω̃ ∩ 𝜕B = {x0 } and hence Ω̃ is strictly convex around x0 . Therefore, the maximal cap Σ for γ̃ with |γ|̃ = 1 and |γ̃ − γ| ≪ 1 forms an Ω̃ neighborhood of x0 . Since f (u) ≥ 0, Theorem 2.38 is applicable to (2.74), which implies ∇v ≠ 0 in Σ, or equivalently, ∇u ≠ 0 in T −1 Σ, as desired.

94 | 2 Maximum principles 2.2.5 Serrin’s corner point lemma Based on the method of the moving plane, we can give an alternative proof of Theorem 1.8. Theorem 2.42. If Ω ⊂ Rn is a bounded domain with C 2 boundary 𝜕Ω and there is a u ∈ C 2 (Ω) satisfying 𝜕u 󵄨󵄨󵄨󵄨 (2.75) − Δu = 1 in Ω, u|𝜕Ω = 0, 󵄨 = constant ≡ c, 𝜕ν 󵄨󵄨󵄨𝜕Ω then Ω is a ball. In this proof, we take a unit vector γ ∈ Rn arbitrarily, and show Ω = Σ∪Σ󸀠 ∪(Tλ1 ∩Ω), where Σ = Σ(λ1 ) is the maximal cap and Σ󸀠 = Σ󸀠 (λ1 ). Let v(x) = u(xλ1 ). Since u > 0 in Ω, the strong maximum principle reduces the desired conclusion to w ≡u−v =0

in Σ󸀠 .

(2.76)

Here we note Δw = 0

in Σ󸀠 ,

w|Tλ

1

∩𝜕Σ󸀠

= 0,

w|𝜕Σ󸀠 \Tλ ≥ 0. 1

(2.77)

If (2.76) does not hold, then w > 0 in Σ󸀠 by the strong maximum principle. Since Σ = Σ(λ1 ) is the maximal cap, two cases are to be considered. Case 1. The set Σ󸀠 touches 𝜕Ω at some p ∈ ̸ Tλ1 from inside. Case 2. The hyperplane Tλ1 is perpendicular to 𝜕Ω at some q ∈ 𝜕Ω. In the first case we have w = u − v = 0 at p ∈ 𝜕Σ󸀠 and hence rem 2.27. This property contradicts the third identity of (2.75). In the second case, on the other hand, we have Dα w|x=q = 0,

𝜕w 𝜕ν

∀|α| ≤ 2.

< 0 there by Theo-

(2.78)

In fact, w = 0 at q is obvious. The equalities wν = wτν = 0 at q follow from the third condition of (2.75). Also, since uτ = 0 on 𝜕Ω, it follows that wτ = wττ = 0 at q because of the location of q. Finally, we have −1 = Δu = uνν +(n−1)κuν on 𝜕Ω, and hence wνν = 0 at q, where κ denotes the mean curvature. Here, the following condition implies w ≡ 0 in Σ by (2.77), (2.78), and w ≥ 0, a contradiction. Lemma 2.43 (Serrin [308]). Let D∗ ⊂ Rn be a domain with C 2 boundary 𝜕D∗ and q ∈ 𝜕D∗ . Let T and D be a hyperplane containing q and one of the portions of D∗ cut by T, respectively. Let, furthermore, s be a nontangential unit vector from q towards D. Then, any w ∈ C 2 (D) such that Δw ≤ 0, satisfies either

𝜕w | 𝜕s x=q

> 0 or

w ≥ 0, 2

𝜕w | 𝜕s2 x=q

> 0.

w ≢ 0

in D,

w|x=q = 0

2.3 Convexity of level sets | 95

Proof. We can take a ball K1 = Br1 (y) ⊂ D∗ satisfying K1 ∩ 𝜕D∗ = {q}. Let K2 = Br1 /2 (q) and K 󸀠 = K1 ∩K2 ∩D. We may suppose that q is the origin, T = {x1 = 0}, and D ⊂ {x1 > 0}. Take the auxiliary function 2

2

z(x) = x1 (e−αr − e−αr1 ),

r = |x − y|

with α ≫ 1. It holds that 2

Δz = 2αx1 e−αr {2αr 2 − (n + 2)} > 0,

z > 0 in K 󸀠 ,

z|𝜕K1 ∪T = 0

for α sufficiently large. From Theorem 2.30 it follows that w > 0 in D, and there is η > 0 such that w ≥ ηx1

on 𝜕K2 ∩ 𝜕K 󸀠 ,

by Theorem 2.27. The boundary 𝜕K 󸀠 is divided into three parts, Γ1 = 𝜕K 󸀠 ∩ 𝜕K1 , Γ2 = 𝜕K 󸀠 ∩ 𝜕K2 , and Γ3 = 𝜕K 󸀠 ∩ T. Since w|Γ1 ∪Γ2 ≥ 0, z|Γ1 ∪Γ3 = 0, and z|Γ2 ≤ constant × x1 , there is 0 < ε ≪ 1 such that v ≡ w − εz ≥ 0

on 𝜕K 󸀠 .

Furthermore, we have Δv = Δw − εΔz < 0 in K 󸀠 , and therefore, Theorem 2.30 implies 2 | > 0 or 𝜕𝜕sv2 |x=q ≥ 0. Then v = w − ϵz > 0 in K 󸀠 , and hence v|x=q = 0 implies either 𝜕v 𝜕s x=q

we obtain the result by

𝜕z | 𝜕s x=q

𝜕2 z | 𝜕s2 x=q

= 0 and

> 0.

Using Lemma 2.43, Theorem 2.32 is extended to optimal caps and to special domains with corners [137]. For rectangular domains, however, a different argument is necessary.12

2.3 Convexity of level sets 2.3.1 Nonparametric convex surfaces Maximum principles are applicable to induce geometric properties of the graph of the solution to elliptic and parabolic equations. If Ω ⊂ R2 is a (strictly) convex bounded domain with smooth boundary 𝜕Ω then the first eigenfunction φ1 = φ1 (x) of −Δ under the Dirichlet condition − Δφ1 = λ1 φ1 ,

φ1 > 0

in Ω,

φ1 |𝜕Ω = 0

(2.79)

is (strictly) logarithmically concave, which means that − log ϕ1 is (strictly) convex in Ω [2]. The method for the proof is applicable to nonlinear equations, which implies 12 See Chapter 5.

96 | 2 Maximum principles the uniqueness of the critical point of the Robin function for such domains [61]. See § 3.3.1. This property is valid to the heat equation in Rn with (strictly) convex potential V(x), under the use of Trotter–Lie’s product formula and the Brunn–Minkowski type inequality [44]. Applying the argument used for (2.79), however, we obtain a similar fact for ut = Δu

in Ω × (0, +∞),

u|𝜕Ω = 0,

u|t=0 = u0 (x) > 0,

(2.80)

where Ω ⊂ R2 is a bounded domain with smooth boundary 𝜕Ω.13 Recall that the domain Ω ⊂ R2 is said to be convex (resp. strictly convex) if λz1 + (1 − λ)z2 ∈ Ω for any z1 , z2 ∈ Ω (resp. z1 , z2 ∈ Ω), where 0 < λ < 1. The function h = h(z) is said to be convex (resp. strictly convex) if h(λz1 + (1 − λ)z2 ) ≤ λh(z1 ) + (1 − λ)h(z2 )

(resp. h(λz1 + (1 − λ)z2 )) < λh(z1 ) + (1 − λ)h(z2 )

for any z1 , z2 ∈ Ω in z1 ≠ z2 and 0 < λ < 1. The nonparametric C 2 surface ℳ : ζ = h(z),

z = (x, y) ∈ Ω

on the convex (resp. strictly convex) domain Ω ⊂ R2 , on the other hand, is convex (resp. strictly convex) if and only if its Hessian matrix ℋ = ℋ(x, y) ≡ (

hxx hxy

hxy ) hyy

is nonnegative (resp. positive) definite in Ω (see [20, Theorem 1.3.1]). If s is a regular value of h and Ωs ⊂ Ω for Ωs = {z ∈ Ω | h(z) > s}, furthermore, the curvature of the level line Γs = {h(z) | z ∈ 𝜕Ωs } ⊂ ℳ is given by κs = ∑( i

where hi =

𝜕h 𝜕xi

and hij =

hi 1 ) = {∑ |∇h|2 Δh − hi hj hij }, |∇h| i |∇h|3 i,j

𝜕2 h . 𝜕xi 𝜕xj

(2.81)

Here, we have

|∇h|2 Δh − ∑ hi hj hij = hxx h2y + hyy h2x − 2hx hy hxy = ℬ[∇h, ∇h] i,j

for ℬ = ℬ(x, y) ≡ (

hyy −hxy

−hxy ). hxx

13 This result is extended to higher dimension n ≥ 3 by a slightly different argument, called the concavity maximum principle [191].

2.3 Convexity of level sets | 97

Since the eigenvalues of ℋ and ℬ coincide, it holds that κs ≥ 0 (resp. κs > 0) everywhere on Γs , provided that ℳ is convex (resp. strictly convex). This section is devoted to the following theorem. Theorem 2.44. Let Ω ⊂ R2 be (strictly) convex and − log u0 be (strictly) convex in Ω. Then − log u(⋅, t) is also (strictly) convex for each t > 0. Via the limiting process, Theorem 2.44 is reduced to the strictly convex category. Assuming that Ω is strictly convex, we show that ℋ = ℋ(⋅, t) ≡ (

vxx vxy

vxy ), vyy

v = − log u

stays positive definite in 0 < t < +∞, provided that ℋ(⋅, 0) is positive definite. Putting ψ = − log φ1 in (2.79), we have Δψ = |∇ψ|2 + λ1

in Ω,

v|𝜕Ω = +∞,

which implies Δψ > 0 everywhere in Ω. Putting v = − log u in (2.80), on the other hand, induces vt = Δv − |∇v|2

in Ω × (0, +∞),

v|𝜕Ω = +∞,

(2.82)

which has no such property. We use K = tr ℋ ≡ Δv,

2 H = det ℋ ≡ vxx vyy − vxy

(2.83)

to approach this technical problem. Lemma 2.45. If Ω ⊂ R2 is strictly convex and − log u0 is strictly convex in Ω, it holds that K > 0 in Ω × [0, T] as long as H ≥ 0 in Ω × [0, T]. Proof. By (2.82), we have vxxt = Δvxx − 2|∇vx |2 − 2∇v ⋅ ∇vxx , and hence Kt = ΔK − 2(|∇vx |2 + |∇vy |2 ) − 2∇v ⋅ ∇K. From |∇vx |2 + |∇vy |2 = (vxx + vxy )2 + (vxy + vyy )2

2 2 = vxx + vyy + 2vxy (vxy + vxx + vyy )

2 = (vxx + vyy )2 − 2vxx vyy + 2vxy + 2vxy K

= K 2 − 2H + 2vxy K,

98 | 2 Maximum principles it follows that Kt = ΔK + aK + 4H + b ⋅ ∇K

in Ω × (0, +∞),

(2.84)

with continuous scalar and vector fields denoted by a and b, respectively. It holds also that K = Δv = u−2 |∇u|2 − u−1 ut , and hence (uK)|𝜕Ω = +∞ by the Hopf lemma applied to (2.80), which implies K|𝜕Ω = +∞,

(2.85)

in particular. Finally, K(⋅, 0) > 0 in Ω follows because ℋ(⋅, 0) is positive definite everywhere in Ω. We thus obtain the result by the strong maximum principle for parabolic equation, applied to (2.84)–(2.85). 2.3.2 Logarithmic concavity of the solution By Lemma 2.45, Theorem 2.44 is reduced to H>0

in Ω × (0, +∞).

(2.86)

For this purpose, first, we control the boundary behavior of this H. Second, we derive a differential inequality which guarantees the positivity of H by the strong maximum principle. For the first part, we have the following lemma. Lemma 2.46. If Ω is strictly convex, it holds that H|𝜕Ω = +∞. Proof. A direct calculation implies 2 H ≡ vxx vyy − vxy 1 1 = 2 (uxx uyy − u2xy ) + 3 (2ux uy uxy − uxx u2y − uyy u2x ), u u

and hence φ ≡ e−2v H = u2 H = (uxx uyy − u2xy ) +

1 {∑ u u u − |∇u|2 Δu}. u i,j i j ij

(2.87)

2.3 Convexity of level sets | 99

The strong maximum principle, on the other hand, guarantees that Γ = 𝜕Ω is the level line of u = u(⋅, ⋅), Γ = {z ∈ Ω | u(z, t) = 0}, where z = (x, y). Fix t, and let Γs = {z ∈ Ω | u(z, t) = s}. It is a regular curve for 0 < s ≪ 1 by the Hopf lemma applied to (2.80). We thus obtain φ = (uxx uyy − u2xy ) +

|∇u|3 κ s s

on Γs ,

0 < s ≪ 1,

by (2.81), where κs stands for the curvature of Γs . From the assumption, we have κs → κ > 0 on 𝜕Ω as s ↓ 0, while |∇u| ≥ δ on Γs , 0 < s ≪ 1 with δ > 0, holds by the Hopf lemma. Hence it follows that φ|Γs → +∞,

s ↓ 0,

which implies φ|𝜕Ω = +∞. Then (2.87) follows from φ = u2 H. The second part is described by the following lemma. Lemma 2.47. Let u = u(x, t) be the solution to (2.80) and define v and H by (2.83). Then it holds that Ht −

2Δv 1 ℬ(∇vt , ∇vt ) = ΔH + (B ⋅ ∇H + CH) θ θ

in {θ > 0},

(2.88)

where B and C are continuous vector and scalar fields on Ω × [0, +∞), respectively, and ℬ = ℬ(⋅, t) ≡ (

vyy −vxy

−vxy ), vxx

2 θ = 4vxy + (vxx − vyy )2 ≥ 0.

Admitting this lemma, we give the following proof. Proof of Theorem 2.44. Assume the contrary to (2.86). We recall H(⋅, 0) > 0 and put T∗ ≡ sup{t > 0 | min H(⋅, t) > 0} < +∞. Ω

By Lemma 2.45, it holds that ℋ = ℋ(⋅, t) ≡ (

vxx vxy

vxy )>0 vyy

in Ω × [0, T∗ ),

which implies K = tr ℋ > 0,

ℋ

−1

= H −1 ℬ > 0

there. We thus obtain 1 Ht ≥ ΔH + (B ⋅ ∇H + CH) θ

in {θ > 0} ∩ (Ω × [0, T∗ ]).

(2.89)

100 | 2 Maximum principles Since H(⋅, 0) > 0 in Ω and by Lemma 2.46, therefore, there is x0 ∈ Ω such that H=θ=0

at P = (x0 , T∗ )

by (2.89). Then it holds that vxy = 0,

1 vxx = vyy = Δv, 2

H=

1 (Δv)2 = 0 4

at P,

which contradicts K = Δv > 0 in Ω × [0, T∗ ] derived from Lemma 2.45. 2.3.3 Identity for the Hessian Here we show Lemma 2.47 to complete the proof of Theorem 2.44. Confirm that equalities (2.82) and (2.83) imply vxt = Δvx − 2∇v ⋅ ∇vx ,

vxxt = Δvxx − 2|∇vx |2 − 2∇v ⋅ ∇vxx ,

∇H = vxx ∇vyy − 2vxy ∇vxy + vyy ∇vxx . Then we obtain ΔH = 2I + 2II for 1 I = (vxx Δvyy − 2vxy Δvxy + vyy Δvxx ) 2 and II = ∇vxx ⋅ ∇vyy − |∇vxy |2 . The second inequality of (2.90) implies 1 I = vxx (vyyt + 2|∇vy |2 + 2∇v ⋅ ∇vxx ) − vxy (vxyt + 2∇vy ⋅ ∇vx 2 1 + 2∇v ⋅ ∇vxy ) + vyy (vxxt + 2|∇vx |2 + 2∇v ⋅ ∇vxx ) 2 1 = (vxx vyyt − 2vxy vxyt + vyy vxxt ) 2 + (vxx |∇vy |2 − 2vxy ∇vx ⋅ ∇vy + vyy |∇vx |2 ) + ∇v ⋅ (vxx ∇vyy − 2vxy ∇vxy + vyy ∇vxx ),

with vxx vyyt − 2vxy vxyt + vyy vxxt = Ht ,

∇v ⋅ (vxx ∇vyy − 2vxy ∇vxy + vyy ∇vxx ) = ∇v ⋅ ∇H

(2.90)

2.3 Convexity of level sets | 101

and vxx |∇vy |2 − 2vxy ∇vx ⋅ ∇vy + vyy |∇vx |2

2 2 2 2 = vxx (vxy + vyy ) − 2vxy (vxx vyy + vxy vyy ) + vyy (vxx + vxy ) 2 = vxx vyy Δv − vxy Δv = HΔv.

Hence it holds that 2I = Ht + 2(HΔv + ∇v ⋅ ∇H). Next, one has 2 2 II = ∇vxx ⋅ ∇vyy − |∇vxy |2 = vxxx vxyy + vxxy vyyy − vxxy − vxyy 2 2 = (Δvx − vxyy )vxyy + vxxy (Δvy − vxxy ) − vxxy − vxyy 2 2 = (Δvx )vxyy + (Δvy )vxxy − 2(vxyy + vxxy ).

Here we write the third equality of (2.90) as (

vxx − vyy −2vxy

−2vxy v H − vyy Δvx ) ( xyy ) = ( x ) −vxx + vyy vxxy Hy − vxx Δvy

(2.91)

and put A=(

vxx − vyy −2vxy

−2vxy ). −vxx + vyy

We have 2 det A = −(vxx − vyy )2 − 4vxy = −θ,

and it holds that t

A = A,

A2 = θI,

A−1 = θ−1 A.

(2.92)

Then (2.91) implies (

vxyy H − vyy Δvx ), ) = θ−1 A ( x vxxy Hy − vxx Δvy

and hence 2 2 vxyy + vxxy = θ−1 {(Hx − vyy Δvx )2 + (Hy − vxx Δvy )2 }

= θ−1 (|∇H|2 − 2{(vyy Δvx )Hx + (vxx Δvy )Hy } + (vyy Δvx )2 + (vxx Δvy )2 )

(2.93)

102 | 2 Maximum principles by (2.92). Equality (2.93) implies also (Δvx )vxyy + (Δvy )vxxy = θ−1 A (

Hx − vyy Δvx Δvx )⋅( Hy − vxx Δvy Δvy

= θ−1 A (

Δvx v Δv Δvx ) ⋅ ∇H − θ−1 A ( yy x ) ⋅ ( ). Δvy vxx Δvy Δvy

)

We thus end up with II =

1 B ⋅ ∇H − θ−1 III 2θ

for Δvx v Δv B = A( ) − 2∇H + 4 ( yy x ) , Δvy vxx Δvy 2 III = A (

vyy Δvx Δvx )⋅( ) + 2(vyy Δvx )2 + 2(vxx Δvy )2 . vxx Δvy Δvy

The term III is equal to III = {(vxx − vyy )vyy Δvx − 2vxy vxx Δvy }Δvx

+ {−2vxy vyy Δvx + (−vxx + vyy )vxx Δvy }Δvy + 2{(vyy Δvx )2 + (vxx Δvy )2 }

2 = (Δvx )2 {(vxx − vyy )vyy + 2vyy } + 2Δvx Δvy (−vxy vxx − vxy vyy ) 2 + (Δvy )2 {(−vxx + vyy )vxx + 2vxx }

= Δv{vyy (Δvx )2 − 2vxy Δvx Δvy + vxx (Δvy )2 }. Then the first equality of (2.90) implies vyy (Δvx )2 − 2vxy Δvx Δvy + vxx (Δvy )2 = IV + V for IV = 4{vyy (∇v ⋅ ∇vx )2 − 2vxy (∇v ⋅ ∇vx )(∇v ⋅ ∇vy ) + vxx (∇v ⋅ ∇vy )2 } and 2 V = vyy (vxt + 4vxt ∇v ⋅ ∇vx ) − 2vxy (vxt vyt

2 + 2vxt ∇v ⋅ ∇vy + 2vyt ∇v ⋅ ∇vx ) + vxx (vyt + 4vyt ∇v ⋅ ∇vy ).

It holds that

IV 2 2 = vx2 (vyy vxx − vxy vxx vxy + vxx vxy ) 4 3 + 2vx vy (vyy vxx vxy − vxy vxx vyy − vxy + vxx vxy vyy )

2.4 Symmetric criticality | 103

2 2 2 + vy2 (vyy vxy − 2vxy vyy + vxx vyy )

2 2 2 = vx2 vxx (vxx vyy − vxy ) + 2vx vy vxy (vxx vyy − vxy ) + vy2 vyy (vxx vyy − vxy )

= H(vx2 vxx + 2vx vy vxy + vy2 vyy ),

and finally, 2 2 2 V = vyy vxt − 2vxy vxt vyt + vxx vyt + vxx vyt

+ 4vxt (vyy ∇v ⋅ ∇vx − vxy ∇v ⋅ ∇vy ) + 4vyt (vxx ∇v ⋅ ∇vy − vxy ∇v ⋅ ∇vx )

= ℬ[∇vt , ∇vt ] + 4vxt vx H + 4vyt vy H. We thus complete the proof.

2.4 Symmetric criticality 2.4.1 Symmetric functionals In Chapter 1 we have examined the method of variations to find a solution to − Δu = f (u),

u>0

in Ω,

u|𝜕Ω = 0,

(2.94)

where Ω ⊂ Rn is a bounded domain with smooth boundary 𝜕Ω. If the nonlinearity f (u) satisfies f(1)–f(3) in § 1.5.1, this problem is reduced to finding a critical point u of the functional 1 J(v) = ∫ |∇v|2 − F(v+ ) dx, 2 Ω

v ∈ X = H01 (Ω)

u

for F(u) = ∫0 f (u)du. For a domain with symmetry, the variational method approaches solutions with and without symmetry as follows. Let G ⊂ O(n) be a group indicating the symmetry of the domain: TΩ = Ω for any T ∈ G. Then the Banach space XG = {v ∈ X | T ∗ v = v, ∀T ∈ G} is defined, where T ∗ v = v ∘ T. The method of variations for J on X is applicable to J on XG , to create a critical point uG ∈ XG . Here arise three questions. 1. Is uG a critical point of J on X? 2. Does the original critical point u differ from uG , or more precisely, u ∈ ̸ XG ? 3. Is G the maximal symmetry of uG , i. e., uG ∈ ̸ XH for any group H with G ⊂ H ⊂ O(n) and H ≠ G? We illustrate the situation for the case of f (u) = up , 1 < p < ∞, and Ω = {R1 < |x| < R2 } ⊂ Rn . In fact, we take G = O(n) and set X∞ = XO(n) to define j = inf{

‖∇v‖22 | v ∈ X\{0}}, ‖v‖2p+1

j∞ = inf{

‖∇v‖22 | v ∈ X∞ \{0}}. ‖v‖2p+1

104 | 2 Maximum principles As is seen, if p < n∗ , there exists a minimizer u of j which solves the problem. Similarly, we have a minimizer u∞ ∈ X∞ of j∞ , a critical value of J|X∞ , which solves R2

󸀠 − up∞ v)r n−1 dr = 0, ∫ (u󸀠∞ v∞

∀v ∈ X∞ .

R1

Hence it follows that −

1

r n−1

d n−1 du∞ {r } = up∞ , dr dr

R1 < r < R2 ,

u∞ |r=R1 ,R2 = 0,

and therefore, u∞ is a solution to (2.94) for f (u) = up . Since the variational problem associated with j∞ is essentially one-dimensional, the solution exists even for the supercritical nonlinearity p ≥ n∗ . Thus we obtain a counterpart of Theorem 1.2 [184]. Proposition 2.48. If Ω = {R1 < |x| < R2 } ⊂ Rn and f (u) = up , equation (2.94) possesses a radially symmetric solution for any 1 < p < ∞. n+2

The same situation of the annular domain arises for f (u) = u n−2 + λu with n > 2. The value Sλ∞ = inf{∫ |∇v|2 − λv2 dx | v ∈ X∞ , ‖v‖

2n

L n−2

Ω

= 1}

is attained and creates a radially symmetric solution whenever λ < λ1 . The value S0 = inf{∫ |∇v|2 dx | v ∈ X, ‖v‖ Ω

2n

L n−2

= 1},

however, is not attained so that we have S0 < S0∞ . This property implies Sλ < Sλ∞ ,

0 < λ ≪ 1.

(2.95)

In fact, since Sλ = inf{∫ |∇v|2 − λv2 dx | v ∈ X, ‖v‖ Ω

2n

L n−2

= 1}

is upper semicontinuous in λ, we obtain limλ↓0 sup Sλ ≤ S0 , while Sλ∞ is continuous in λ because of the existence of the minimizer, which implies S0∞ = limλ↓0 Sλ∞ . Therefore, (2.95) follows from S0 < S0∞ . If n ≥ 4 and 0 < λ < λ1 , on the other hand, the value Sλ is attained to create a n+2 solution u to (2.94) for f (u) = u n−2 + λu, as we have seen in § 1.4. Then inequality (2.95) assures u ∈ ̸ X∞ , and we thus obtain the following result. n+2

Proposition 2.49 ([54]). If Ω = {R1 < |x| < R2 } ⊂ Rn , n ≥ 4, and f (u) = u n−2 + λu, equation (2.94) admits both radial and nonradial solutions for 0 < λ ≪ 1.

2.4 Symmetric criticality | 105

2.4.2 Symmetric solutions The first question in the previous subsection is affirmative, and is called symmetric criticality [270]. Theorem 2.50. Any critical point u ∈ XG of J|XG is a critical point of J on X. Here we give a proof in the context of convex analysis [340]. Lemma 2.51. Let X be a Banach space over R and D : X → (−∞, +∞] be a proper, convex, and lower semicontinuous mapping. Let K ⊂ X be a closed convex set, and suppose that u, f ∈ X satisfy f ∈ 𝜕(D + 1K )(u), where 1K denotes the indicator function of K defined by (1.71). Assume, furthermore, the existence of w ∈ K such that f ∈ 𝜕D(w). Then it follows that f ∈ 𝜕D(u). Proof. The relation f ∈ 𝜕(D + 1K )(u) means u ∈ K,

D(ζ ) ≥ D(u) + ⟨f , ζ − u⟩,

∀ζ ∈ K,

(2.96)

where ⟨⋅, ⋅⟩ denotes the pairing between X and X ∗ . Since f ∈ 𝜕D(w), we have D(ζ ) ≥ D(w) + ⟨f , ζ − w⟩,

∀ζ ∈ X.

(2.97)

Adding (2.96) with ζ = w ∈ K and (2.97), we obtain D(ζ ) ≥ D(u) + ⟨f , ζ − u⟩,

∀ζ ∈ X,

which means f ∈ 𝜕D(u). Proof of Theorem 2.50. We take K = XG and D(v) = 21 ∫Ω |∇v|2 dx. Then the criticality of u ∈ K of J|K and that in X are represented by f ∈ 𝜕(D+1K )(u) and f ∈ 𝜕D(u), respectively, where f = f (u). The existence of w ∈ K satisfying f ∈ 𝜕D(w) is a consequence of the unique solvability of the Poisson equation because f = f (u) ∈ K = XG holds by u ∈ K. Hence the result follows. 2.4.3 Maximal symmetry We take the two-dimensional annulus A = {a < |x| < a + 1} ⊂ R2 and consider the equation − Δu = up ,

u>0

in A,

u|𝜕A = 0

(2.98)

for 1 < p < ∞. Let Tk (reiθ ) = rei(θ+2π/k) be the rotation by 2π for x = reiθ and Gk = {Tkl | k 1 l = 0, 1, . . . , k − 1}. We set X = H0 (A), Xk = XGk for k = 1, 2, . . ., and X∞ = XO(2) , to take the variation jk = inf{

‖∇v‖22 | v ∈ Xk \{0}}. ‖v‖2p+1

(2.99)

106 | 2 Maximum principles From Theorem 2.50, each jk is attained by a solution uk ∈ Xk of (2.98). Then we have the generation of nonradial solutions. Proposition 2.52. It holds that j∞ → +∞ and jk = O(1) for k finite as a ↑ +∞. Proof. To show jk = O(1) for a finite k, we take the fundamental region Ak = {reiθ | a < r < a + 1, 0 < θ
0 such that lima↑+∞ i(k) = ki∗ . See [89, 200, 208, 160, 42] including the higher-dimensional case, and also later results [173, 174, 175] for variable coefficients and polyhedric holes.

108 | 2 Maximum principles Proof. The first part is a consequence of Proposition 2.48. For each finite k, the relation jk < j∞ arises as a ↑ +∞. Then the minimizer uk of jk in (2.99) solving (2.98) has mode k. In fact, if uk ∈ Xk󸀠 is the case for some k 󸀠 > k, we get jk󸀠 = jk in (2.100), and hence jk󸀠 = j∞ by Proposition 2.48. This property implies jk = j∞ , which contradicts jk < j∞ . 2.4.4 Mountain pass approach Here we take the mountain pass approach to the third question of § 2.4.1, the maximal symmetry. For this purpose, given a Banach space X over R, let the functionals I, K ∈ C 1 (X, R) satisfy K(0) = I(0) = 0 and K ≤ I. Suppose the conditions K|𝜕B ≥ ρ and E = {e ∈ X\B | I(te) ≤ 0, t ≥ 1} ≠ 0 for a ball B = BR (0) ⊂ X and a constant ρ > 0. The lower semicontinuous functionals 𝒦, ℐ : Γ → R are defined by 𝒦(γ) = max K(γ(t)),

ℐ (γ) = max I(γ(t))

0≤t≤1

0≤t≤1

where Γ = {γ ∈ C([0, 1], X) | γ(0) = 0, γ(1) = e} denotes the path space connecting 0 and e ∈ E. We put k ≡ infΓ 𝒦 ≥ ρ and i ≡ infΓ ℐ ≥ k. The following theorem is a consequence of Theorem 1.41, the refined version of the mountain pass lemma. This theorem is available to separate the mountain pass critical values based on the symmetric criticality as is argued above [333]. Theorem 2.55. Suppose (PS)i , the Palais–Smale condition at the value i for the functional I, which means that any {xk } ⊂ X is precompact, provided that I(vk ) → i and ‖I 󸀠 (vk )‖X ∗ → 0. Then, the condition i = k implies the existence of v ∈ X such that I 󸀠 (v) = 0 and I(v) = K(v) = i. Proof. The closed set DK = {v ∈ X | K(v) ≥ k} separates 0 and e. Since i = k and K ≤ I, it holds that DK ⊂ DI = {v ∈ X | I(v) ≥ i}, and hence inf I ≥ inf I ≥ i. DK

DI

By Theorem 1.41, we obtain a sequence {vk } ⊂ X satisfying 󵄩󵄩 󸀠 󵄩 󵄩󵄩I (vk )󵄩󵄩󵄩X ∗ → 0,

I(vk ) → i,

dist(vk , DK ) → 0.

Then (PS)i implies the existence of v such that I 󸀠 (v) = 0,

i = I(v),

K(v) ≥ k.

Therefore, the assumption i = k implies I(v) = K(v) by k ≤ K(v) ≤ I(v) = i = k.

3 Complex structure Combination of two-dimensional Laplacian and exponential nonlinearity is observed in physics and geometry, statistical mechanics for many point vortices, self-dual gauge theory, as well as in the theory of surfaces. There are interactions between nonlinear elliptic problems, complex function theory, differential geometry, special functions, potential theory, statistical mechanics, and field theory. We begin with the theory of surfaces associated with the complex structure, and observe remarkable structures of the Boltzmann–Poisson equation. Then we apply the complex function theory to classify the singular limits of the Boltzmann–Poisson equation and to construct onepoint blow-up solutions. This final part, referred to as the singular perturbation, has developed in the context of real and functional analysis recently. Here we describe the classical approach done by V. H. Weston and J. L. Moseley, however. The other sections deal with the applications of the complex structure. The developed asymptotic analysis is a driving force to reach a deep principle behind the equation, that is, the elliptic uniformization theory toward the dynamics close to equilibrium.

3.1 Theory of surfaces 3.1.1 Curvatures In spite of the significant relation between the surface theory and complex structure, the connection to elliptic theory was recognized rather recently in accordance with the development in the theory of nonlinear partial differential equations and nonlinear functional analysis. Let x1 (u, v)

3

ℳ : x = ( x2 (u, v) ) ∈ R ,

x3 (u, v)

(u, v) ∈ D ⊂ R2

(3.1)

be a parametric surface. Writing dx = xu du + xv dv, we define the first fundamental form I = ds2 by I = dx ⋅ dx = Edu2 + 2Fdudv + Gdv2 , where E = |xu |2 , F = xu ⋅ xv , and G = |xv |2 stand for the first fundamental quantities. Fix P ∈ ℳ, and let n be the unit normal vector on ℳ at P. Then it holds that n=

xu × xv √EG − F 2

where × denotes the outer product in R3 . https://doi.org/10.1515/9783110556285-003

110 | 3 Complex structure Choosing a unit tangent vector t on ℳ at P, we take a plane π made by n and t, and curve 𝒞 on ℳ cut by π. Parametrize this 𝒞 by the arc-length as in x = x(s) ∈ 𝒞 . If ρC and nC denote the curvature radius and the principal unit normal vector of 𝒞 , it dt = ρ1 nC , and the normal curvature of ℳ at P in the direction t is defined holds that ds C by 1 cos ψC = , R ρC where ψC denotes the angle made by n and nC , that is, (ψC , ρC ) = (0, R) or (π, −R). The principal curvatures stand for the minimum and maximum of this R1 when t ∈ S2 , denoted by R1 and R1 , respectively. Then the mean curvature and the Gauss curvature 1

2

of the surface ℳ at P are defined by 2H =

1 1 + , R1 R2

K=

1 1 ⋅ . R1 R2

(3.2)

Since 𝒞 : x = x(u(s), v(s)) ∈ ℳ, it holds that 2

2

cos ψC dt du du dv dv =n⋅ = L( ) + 2M + N( ) ρC ds ds ds ds ds II Ldu2 + 2Mdudv + Ndv2 , = I Edu2 + 2Fdudv + Gdv2

= where

L = xuu ⋅ n = −xu ⋅ nu ,

M = xuv ⋅ n = −xu ⋅ nv = −xv ⋅ nu , N = xvv ⋅ n = −xv ⋅ nv

(3.3)

stand for the second fundamental quantities, and II = Ldu2 + 2Mdudv + Ndv2 denotes the second fundamental form. Therefore, R1 and R1 are the two solutions to 2

(EG − F 2 )

2

1 1 − (GL + EN − 2FM) + LN − M 2 = 0 R R2

(3.4)

by definition, which guarantees 2H =

GL + EN − 2FM = tr A, EG − F 2

K=

LN − M 2 = det A, EG − F 2

(3.5)

where A=(

E F

F L ) ( G M −1

A11 M )≡( 2 N A1

A12 A22

)

(3.6)

3.1 Theory of surfaces |

111

is called the Weingarten matrix. Then (3.3) implies the Weingarten formula nu = −A11 xu − A21 xv ,

nv = −A12 xu − A22 xv .

(3.7)

Given a Riemannian metric ds2 = gij dxi dxj , we recall the Christoffel symbol 1 Γikℓ = g im (gmk,ℓ + gmℓ,k − gkℓ,m ) 2 where (g jk ) = (gjk )−1 , gmk,ℓ =

𝜕gmx , 𝜕xℓ

and so forth. For the parametric surface we obtain

Γ111 E + Γ211 F = xuu ⋅ xu ,

Γ111 F + Γ211 G = xuu ⋅ xv ,

Γ111 E + Γ211 F = xvv ⋅ xu ,

Γ122 F + Γ222 G = xuv ⋅ xu

Γ112 E + Γ212 F = xuv ⋅ xu ,

Γ112 F + Γ212 G = xuv ⋅ xv ,

by this definition, which ensures the Gauss formula xuu = Γ111 xu + Γ211 xv + Ln,

xuv = Γ112 xu + Γ212 xv + Mn, xvv = Γ122 xu + Γ222 xv + Nn.

(3.8)

A direct calculation, on the other hand, produces GEu − 2FFu + FEv GEv − FGu , Γ112 = Γ121 = , 2(EG − G2 ) 2(EG − F 2 ) 2GFv − GGu − FGv 2EFu − EEv − FEu Γ122 = , Γ211 = , 2(EG − F 2 ) 2(EG − F 2 ) EGu − FEv EGv − 2FFv + FGu Γ212 = Γ221 = , Γ222 = . 2 2(EG − F ) 2(EG − F 2 ) Γ111 =

(3.9)

Equalities (3.7) and (3.8) are summarized as the fundamental equation in surface theory, Fu = FΩ,

Fv = FΛ,

(3.10)

for F = [xu xv n] and Γ111

Ω = ( Γ211 L

Γ211

Γ212 M

−A11

−A21 ) , 0

Γ121

Λ = ( Γ221 M

Γ122 Γ222 N

−A12

−A21 ) . 0

(3.11)

Theorem 3.1. The coefficient matrices Ω and Λ in (3.11) satisfy the Lie structure relation Ωv − Λu = ΩΛ − ΛΩ.

(3.12)

112 | 3 Complex structure Proof. Equality (3.12) arises as a compatibility condition to (3.10). In fact, we have Fuv = (FΩ)v = Fv Ω + FΩv = F(ΛΩ + Ωv ), and similarly, Fvu = F(ΩΛ + Λu ), from which the result follows because matrix F is nonsingular. Equality (3.11) is composed of the Theorem Egregium, K=

E(Ev Gv − 2Fu Gv + Gu )2 4(EG − F 2 )2 F(Eu Gv − Ev Gu − 2Ev Fv − 2Fu Gu + 4Fu Fv ) + 4(EG − F 2 ) +

G(Eu Gu − 2Eu Fv + Ev2 )2 Evv − 2Fuv + Guu − , 4(EG − F 2 ) 2(EG − F 2 )

and the Codazzi–Mainardi equation, Lv − Mu = LΓ112 + M(Γ212 − Γ111 ) − NΓ211 ,

Mv − Nu = LΓ122 + M(Γ222 − Γ112 ) − NΓ212 .

3.1.2 Conformal geometry and soap bubbles A soap bubble is a closed surface of constant mean curvature. Are there such objects other than the round sphere? The question is affirmative and is shown by the nonlinear eigenvalue problem − Δu = λ sinh u,

u>0

in Ω,

u|𝜕Ω = 0

(3.13)

where Ω ⊂ R2 is a rectangle [368]. Previously we knew the fact of the nonexistence of such a surface with genus 0 or embedded in R3 [165]. The moving plane method described in §2.2.3 was applied first to prove the latter fact. We note that the Weingarten matrix A in (3.6) satisfies (AX, Y)I = (X, Y)II ,

X, Y ∈ R2

for (X, Y)I = ∑ Iij Xi Yj ,

(X, Y)II = ∑ Jij Xi Yj ,

I11 = E,

J11 = L,

I12 = I21 = F,

J12 = J21 = M,

I22 = G,

J22 = N.

3.1 Theory of surfaces | 113

1 and R1 are the solutions to (3.4), this equation is equivalent to det |λI − A| = 0 R1 2 = R1 by (3.2) and (3.5). If |xu | = |xv | = 1 and xu ⋅ xv = 0 at P ∈ ℳ, then it holds that

Since for λ

Axu = k1 xu , where k1 =

1 R1

and k2 =

1 R2

Axv = k2 xv ,

denote the principal curvatures. We obtain S−1 AS = (

k1 0

0 ), k2

S = [xu xv ],

which implies (L, M, N) = (k1 , 0, k2 ) by (E, F, G) = (1, 0, 1) and (3.6). Then it holds that nu = −k1 xu and nv = −k2 xv by (3.7). The parametrization in (3.1) is called conformal if |xu |2 = |xv |2 ≡ E = E(u, v),

xu ⋅ xv = 0

(3.14)

everywhere in D. Then it holds that A=

1 L ( E M

M ), N

2H = tr A =

L+N , E

K = det A =

LN − M 2 , E2

and E = G = e2σ ,

F=0

for σ = σ(u, v). Then we have 2σu e4σ = σu , 2e4σ −2σu e4σ Γ122 = = −σu , 2e4σ 2σ e4σ Γ212 = Γ221 = u 4σ = σu , 2e Γ111 =

2σv e4σ = σv , 2e4σ −2σv e4σ Γ211 = = −σv , 2e4σ 2σ e4σ Γ222 = v 4σ = σv 2e Γ112 = Γ121 =

by (3.9). It holds also that A = e−2σ (

L M

M A1 ) = ( 21 N A1

A12 ), A22

and hence σu Ω = ( −σv L

σv σu M

−e−2σ L −e−2σ L ) , 0

σv Λ = ( σu M

−σu σv N

by (3.11). Equation (3.12) now implies σuu + σvv + e−2σ (LN − M 2 ) = 0,

−e−2σ M −e−2σ N ) 0

114 | 3 Complex structure Lv − Mu = σv (L + N),

Nu − Mv = σu (L + N),

(3.15)

and, in particular, Gauss equation ensures K=

Δ log E LN − M 2 = −e−2σ (σuu + σvv ) = − 2E E2

(3.16)

since E = e2σ . Due to Coddazi–Mainardi equation, on the other hand, one can write Ev H = 2e2σ σv H = 2Eσv H = (L + N)σv = Lv − Mu ,

Eu H = 2e2σ σu H = 2Eσu H = (L + N)σu = Nu − Mv . Then we obtain 1 1 1 Mv + (L − N)u = Nu − Eu H + (L − N)u = −Eu H + (L + N)u 2 2 2 = −Eu H + (EH)u = EHu ,

(3.17)

and similarly, 1 1 1 Mu − (L − N)v = Lv − Ev H − (L − N)v = −Ev H + (L + N)v 2 2 2 = −Ev H + (EH)v = EHv .

(3.18)

Now we use the complex variable z = u + 𝚤v. Since 𝜕 1 𝜕 𝜕 = ( − 𝚤 ), 𝜕z 2 𝜕u 𝜕v

𝜕 1 𝜕 𝜕 = ( + 𝚤 ), 𝜕z 2 𝜕u 𝜕v

it holds that 1 EHz = E(Hu − 𝚤Hv ) 2 1 1 1 = {Mv + (L − N)u − 𝚤(Mu − (L − N)v )} 2 2 2 1 1 = {(L − N − 2𝚤M)u + 𝚤(L − N − 2𝚤M)v } = ϕz , 4 2 where ϕ = L − N − 2𝚤M. Let a closed surface ℳ of genus 1 with constant mean curvature H be realized by a doubly periodic conformal mapping x = x(u, v) : R2 → R3 . Then the above ϕ = L − N − 2𝚤M is holomorphic due to ϕz = 2EHz = 0, and therefore, a constant by Liouville’s theorem. Since |ϕ|2 = (L − N)2 + 4M 2 = (L + N)2 + 4(M 2 − LN) = 4E 2 (H 2 − K),

3.1 Theory of surfaces |

115

one gets Δ log E λ2 = H2 − K = H2 + , 2E 4E 2

λ = |ϕ|,

so that Δ log E + 2EH 2 − Therefore, putting E =

λ u e , 2H

λ2 = 0. 2E

we obtain

−Δu = 2EH 2 −

λ2 = λHe − v − λHe−2 = 2λH sinh u 2E

or − Δu = λ sinh u in R2 , doubly periodic

(3.19)

if H = 21 . So using differential geometry, we have created a doubly periodic conformal immersion x : R2 → R3 from the solution u to (3.19), and conversely. Equation (3.13) was used for this purpose, where Ω ⊂ R2 stands for a rectangle, where the theory of partial differential equations comes in.

3.1.3 Liouville integral and spherical derivatives Gauss curvature is given by (3.16) under the conformal local chart (3.14). If this K is a positive constant, we can write − Δu = eu

in Ω = D ⊂ R2

(3.20)

for u = log E + log(2K). An integral for this equation was found by J. Liouville [216]. Here we take the complex variable z = x1 + 𝚤x2 and z = x1 − 𝚤x2 for x = (x1 , x2 ) ∈ Ω, to introduce the function 1 s = uzz − u2z . 2 From (3.20) it follows that uzz = − 41 eu and hence 1 1 sz = uzzz − uz uzz = − eu uz + eu uz = 0, 4 4 which means that s = s(z) is a holomorphic function of z ∈ Ω ⊂ C.

(3.21)

116 | 3 Complex structure Regarding (3.21) as a Riccati equation, we see that ϕ = e−u/2 satisfies 1 ϕzz + sϕ = 0. 2

(3.22)

Taking a point x∗ = (x1∗ , x2∗ ) ∈ Ω, we introduce the fundamental system of solutions {ϕ1 , ϕ2 } to the linear ordinary equation (3.22) in z by (ϕ1 ,

𝜕ϕ1 󵄨󵄨󵄨󵄨 = (1, 0), )󵄨 𝜕z 󵄨󵄨󵄨z=z ∗

(ϕ2 ,

𝜕ϕ2 󵄨󵄨󵄨󵄨 = (0, 1) )󵄨 𝜕z 󵄨󵄨󵄨z=z ∗

(3.23)

for z ∗ = x1∗ + 𝚤x2∗ . These ϕ1 = ϕ1 (z) and ϕ2 = ϕ2 (z) are analytic functions in z ∈ Ω and the relation ϕ ≡ e−u/2 = f 1 (z)ϕ1 (z) + f 2 (z)ϕ2 (z) holds for some functions f 1 and f 2 of z. We can write f1 (z) = C1 ϕ1 (z),

f2 (z) = C2 ϕ2 (z)

(3.24)

󵄨 C1 = e−u/2 󵄨󵄨󵄨x=x∗ ,

C2 =

λ u/2 󵄨󵄨 e 󵄨󵄨x=x∗ 8

(3.25)

for

if x∗ = (x1∗ , x2∗ ) ∈ Ω is a critical point of u = u(x), where f1 (z) = f 1 (z) and f2 (z) = f 2 (z). In fact, we have W(ϕ1 , ϕ2 ) ≡ ϕ1 ϕ2z − ϕ1z ϕ2 = 1, and hence f 1 (z) = W(ϕ, ϕ2 ) = ϕϕ2z − ϕz ϕ2 ,

f 2 (z) = W(ϕ1 , ϕ) = ϕ1 ϕz − ϕ1z ϕ.

Here we confirm that these functions are independent of z. Then, putting z = z ∗ , we get f 1 (z) = ϕ(z ∗ , z),

f 2 (z) = ϕz (z ∗ , z).

Since ϕ = e−u/2 is real-valued, it also solves 1 ϕzz + sϕ = 0 2

(3.26)

for s(z) = s(z), and so do f 1 (z) = ϕ(z ∗ , z) and f 2 (z) = ϕz (z ∗ , z). A fundamental system of solutions to (3.26) is given by {ϕ1 , ϕ2 } for ϕ1 (z) = ϕ1 (z) and ϕ2 (z) = ϕ1 (z), which satisfies (ϕ1 ,

𝜕ϕ1 󵄨󵄨󵄨󵄨 )󵄨 = (1, 0), 𝜕z 󵄨󵄨󵄨z=z ∗

(ϕ2 ,

𝜕ϕ2 󵄨󵄨󵄨󵄨 )󵄨 = (0, 1). 𝜕z 󵄨󵄨󵄨z=z ∗

3.1 Theory of surfaces | 117

Then we obtain 󵄨 f 1 (z ∗ ) = ϕ(z ∗ , z ∗ ) = e−u/2 󵄨󵄨󵄨x=x∗ = C1 , 𝜕 𝜕 −u/2 󵄨󵄨 f (z ∗ ) = ϕz (z ∗ , z ∗ ) = e 󵄨󵄨x=x∗ = 0, 𝜕z 1 𝜕z 𝜕 −u/2 󵄨󵄨 e f 2 (z ∗ ) = ϕz (z ∗ , z ∗ ) = 󵄨󵄨x=x∗ = 0, 𝜕z and 𝜕 1 󵄨 f (z ∗ ) = ϕzz (z ∗ , z ∗ ) = Δe−u/2 󵄨󵄨󵄨x=x∗ 𝜕z 2 4 λ 1 = − e−u/2 Δu|x=x∗ = eu/2 |x=x∗ = C2 , 8 8 using ∇u(x∗ ) = 0. These relations imply (3.24), which results in e−u/2 = C1 |ϕ1 |2 + C2 |ϕ2 |2

(3.27)

for C1 and C2 defined by (3.25). Put ψ1 = C11/2 8−1/4 ϕ1 , ψ2 = C1−1/2 81/4 ϕ2 , and F = ψ2 /ψ1 . Since W(ψ1 , ψ2 ) = W(ϕ1 , ϕ2 ) = 1, it holds that 1/2

1 1 |F 󸀠 | ( ) eu/2 = = ≡ ρ(F). 2 2 8 |ψ1 | + |ψ2 | 1 + |F|2

(3.28)

Here, F is a meromorphic function of z ∈ Ω ⊂ C as a quotient of two linearly independent solutions to (3.22), so that it satisfies {F; z} = − 21 s, where {F; z} =

2

3 F 󸀠󸀠 1 F 󸀠󸀠󸀠 ( 󸀠) − 4 F 2 F󸀠

is the Schwarzian derivative. This ρ(F) stands for the spherical derivative of the conformal mapping F : Ω → S2 , where S2 ⊂ R3 denotes the Gauss sphere, that is, a round sphere with diameter 1. This property means ρ(F) = dσ , where the left-hand side stand ds for the ratio of the standard line element in Ω and that in S2 mapped by F. The Gel’fand equation, − Δv = λev

in Ω ⊂ R2 ,

v|𝜕Ω = 0,

(3.29)

similarly, has the integral ( 8λ )1/2 ev/2 = ρ(F). Therefore, this boundary value problem is reduced to finding a conformal mapping F : Ω → S2 such that 1/2

λ dσ 󵄨󵄨󵄨󵄨 󵄨 =( ) . ds 󵄨󵄨󵄨𝜕Ω 8

118 | 3 Complex structure Since |S2 | = π, this expression induces the quantized blow-up mechanism, which exhibits ∫ ρ(F)2 dx = Ω

1 ∫ λev dx → πm, 8

m∈N

Ω

as the singular limit of the solution to (3.29) is approached.

3.1.4 Sinh-Poisson equation, quaternion, harmonic map Wente surface indicates a closed surface with genus 1 and mean curvature H = 21 . To construct such an object, we use (3.13) for the rectangular domain Ω = (0, a) × (0, b): − Δz = λ sinh z,

z>0

in Ω,

z|𝜕Ω = 0.

(3.30)

Take a doubly periodic function z̃ by the odd extension of z = z(u, v), to put ω(u, v) = 1 z(̃ u , v ) which satisfies 2 √ √ λ

λ

Δω + sinh ω cosh ω = 0

in R2 .

(3.31)

Given such ω, we construct a conformal mapping x : R2 → R3 satisfying (3.14) for E = e2ω . In fact, this condition is represented by an overdetermined system of (xu , xv , n), xuu = ωu xu − ωv xv + eω sinh ωn, xuv = ωv xu + ωu xv ,

xvv = −ωu xu + ωv xv + eω cosh ωn, nu = −e−ω sinh ωxu ,

nv = −e−ω cosh ωxv , which is solvable by noting (3.31). The problem left is to make the surface ℳ = x(R2 ) closed-up, which is possible by taking the parameters (λ, b/a) appropriately based on the method of singular perturbation.1 As the Gel’fand equation admits the integral in complex variables, the sinhGordon equation is associated with quaternions. Here we study the general case, −Δu = 2λ1 eu − 2λ2 e−u

in Ω ⊂ R2 ,

1 Solutions to (3.30) for rectangular Ω are classified in terms of elliptic functions [1]. See [369, 43] and the references therein for latest development.

3.1 Theory of surfaces | 119

where λi > 0, i = 1, 2, are constants. Then we write z = x1 + 𝚤x2 and z = x1 − 𝚤x2 for x = (x1 , x2 ) ∈ Ω as before. Confirm 𝜕 1 𝜕 𝜕 = ( −𝚤 ), 𝜕z 2 𝜕x1 𝜕x2

𝜕 1 𝜕 𝜕 = ( +𝚤 ) 𝜕z 2 𝜕x1 𝜕x2

to deduce uzz = −

λ1 u λ2 −u e + e . 2 2

Letting U=

uz 1 ( 2 √λ1 eu/2

−√λ2 e−u/2 ), 0

V=

−√λ1 eu/2 ), uz

0 1 ( 2 √λ2 e−u/2

we get 1 Uz = ( 2

uzz √λ1 u e 2 uz 2

0 1 Vz = ( λ2 − u2 2 − 2 e uz

√λ2 − u e 2 2

0 −

)=(

√λ1 u e 2 uz 2

uzz

u

−λ1 eu +λ2 e−u 4

√λ2 e− 2 uz 4

√λ1 e 2 uz 4

0

u

) = −(

), u

√λ1 e 2 uz 4

0 u

λ1 eu −λ2 e−u 4

√λ2 e− 2 uz 4

),

which results in Uz − Vz =

−λ1 eu + λ2 e−u 1 ( 4 √λ1 eu/2 uz + √λ2 e−u/2 uz

√λ2 e−u/2 uz + √λ1 eu/2 uz λ1 eu − λ2 e−u

).

It holds also that −√λ1 eu/2 uz − √λ2 e−u/2 uz

UV =

−λ2 e−u 1 ( 4 0

VU =

−λ1 eu 1 ( 4 √λ2 e−u/2 uz + √λ1 eu/2 uz

),

−λ1 eu

0

−λ2 e−u

),

and hence Uz − Vz + [U, V] = 0,

[U, V] = UV − VU.

(3.32)

We recall that the set of quaternions constitutes an algebra ℋ, generated by {1, i, j, k} satisfying ij = k, jk = i, ki = j. This algebra has a representation, σ1 = 𝚤i, σ2 = 𝚤j, σ3 = 𝚤k, where 1=(

1 0 ), 0 1

σ1 = (

0 1 ), 1 0

σ2 = (

0 𝚤

−𝚤 ), 0

σ3 = (

1 0

0 ). −1

120 | 3 Complex structure Thus we regard ℋ = ⟨1, σ1 , σ2 , σ3 ⟩ ⊂ M2 (C). Then the vector A = (a1 , a2 , a3 ) ∈ R3 is identified with A = −𝚤 ∑3i=1 ai σi ∈ Im ℋ, while any ϕ ∈ ℋ∗ = ℋ \ {0} is represented as ϕ=(

a −b

b ), a

|a|2 + |b|2 ≠ 0,

a, b ∈ C.

(3.33)

Hence α ≡ Udz + Vdz uz ds 1 = ( u/2 2 √λ1 e dz + √λ2 e−u/2 dz

−√λ2 e−u/2 dz − √λ1 eu/2 dz ) uz dz

is a one-form in Ω with values in ℋ. If Ω is simply-connected then there is ϕ : Ω → ℋ∗ such that dϕ ⋅ ϕ−1 = α.

(3.34)

In fact, this equation is reduced to dϕ = αϕ, which is equivalent to ϕz = Uϕ, ϕz = Vϕ. Then, the Frobenius–Poincaré theorem assures its solvability under the compatibility (Uϕ)z = (Vϕ)z , which is actually the case due to (3.32) and (Uϕ)z = Uz + Uϕz = (Uz + UV)ϕ,

(Vϕ)z = Vz ϕ + Vϕz = (Vz + VU)ϕ. It is unique under the constraint ϕ(x∗ ) = eu(x∗ )/2 1 for fixed x∗ ∈ Ω, and then, ϕ = ϕ(z, z) ≠ 0 follows from the componentwise analyticity. Hence we obtain ϕ ∈ ℋ∗ , which induces (3.34). By (3.33) we obtain det ϕ ≠ 0, which may be positive without loss of generality. Then h = log(det ϕ) satisfies 1 hz = tr(ϕ−1 ϕz ) = tr(ϕz ϕ−1 ) = tr U = uz , 2

1 hz = tr V = uz 2

by (3.34), and hence h − u2 = constant in Ω. Recalling h(x∗ ) = det(eu(x∗ )/4 1) = eu(x∗ )/2 , we obtain h = u2 , or det ϕ = eu/2 . Now we take Ω = B ≡ B1 (0) and show that N ≡ ϕ−1 kϕ : B → S2 . In fact, using (3.33) for ϕ = ϕ(x) ∈ ℋ∗ , x ∈ B, we obtain ϕ−1 =

a −b 1 ( ). det ϕ b a

Then it follows that N = ϕ−1 ⋅ (−𝚤)σ3 ⋅ ϕ = =

a −b a −𝚤 ( )( det ϕ b a −b

|a|2 − |b|2 −𝚤 ( det ϕ 2ab

b ) a

3 2ab 2 ) = −𝚤 ∑ αi σi |b| − |a| i=1 2

3.2 Boltzmann–Poisson equation

| 121

for α1 =

2(a1 b1 + a2 b2 ) , |a|2 + |b|2

α2 =

2(a2 b1 − a1 b2 ) , |a|2 + |b|2

α3 =

|a|2 − |b|2 |a|2 + |b|2

from det ϕ = |a|2 + |b|2 , where a = a1 + 𝚤a2 and b = b1 + 𝚤b2 . We thus have N : Ω → R3 = Im ℋ. Writing 2(a1 b1 + a2 b2 ) α1 1 ( N(x) = ( α2 ) = 2 2(a2 b1 − a1 b2 ) ) , |a| + |b|2 |a|2 − |b|2 α3 we see that |N(x)| = 1 for any x ∈ B. We have thus proven that N : B → S2 is a Gauss map on B, a surface with constant mean curvature, and hence is a harmonic map [326].2 Then it follows that − ΔN = |∇N|2 N,

|N| = 1

in Ω.

(3.35)

3.2 Boltzmann–Poisson equation 3.2.1 Asymptotic analysis The Gel’fand equation, or Emden–Fowler equation with the exponential nonlinearity, − Δu = λeu

in Ω,

u|𝜕Ω = 0

(3.36)

arises in the theories of thermionic emission and isothermal gas sphere other than the gas combustion [130, 70, 335]. Here, λ is a positive constant and Ω ⊂ Rn is a bounded domain with smooth boundary 𝜕Ω. The study of this equation has clarified that the structure of the solution set is very sensitive to the domain Ω, its dimension, topology, and geometry. In the case of two dimensions, this equation is associated with complex function theory and differential geometry, as it describes a parametric surface with constant Gauss curvature. There is also a physical background, as in the self-dual gauge theory and turbulence caused by point vortices. In the latter case it is called the Boltzmann– Poisson equation, usually represented in the form of nonlocal term, − Δu =

λeu ∫Ω eu dx

equivalent to (3.36) mathematically.3 2 See [172], for a direct proof of (3.35). 3 See [335, Chapter 9] and [336, Chapter 2].

in Ω,

u|𝜕Ω = 0,

(3.37)

122 | 3 Complex structure A local analysis is to pick up mild solutions, having character close to the trivial solution u = 0 for λ = 0. The tools for this purpose are the implicit function theory, comparison principle, and linearization, and we have confirmed the following facts in § 2.1.4, where 𝒮λ denotes the set of classical solutions to (3.36): 1. There exists a λ ∈ (0, +∞) such that 𝒮λ = 0 and 𝒮λ ≠ 0 for λ > λ and 0 < λ < λ, respectively. 2. The set 𝒮λ has a minimal element u = uλ whenever it is not void. 3. There exists no triple {u1 , u2 , u3 } ⊂ 𝒮λ satisfying u1 ≤ u2 ≤ u3 and u1 ≠ u2 ≠ u3 . 4. Minimal solutions {(λ, uλ ) | 0 < λ < λ} form a branch 𝒮 λ , the one-dimensional manifold starting from (λ, u) = (0, 0) in the λ–u plane. 5. The branch 𝒮 λ continues up to λ = λ and then bends back. 6. The minimal solution uλ for λ ∈ (0, λ) is linearly stable. The first eigenvalue μ1 = μ1 (p, Ω) of the linearized operator Ap ≡ −ΔD − p is thus positive for 0, λ < λ, where −ΔD denotes the Laplacian provided with the Dirichlet condition ⋅|𝜕Ω = 0 and p = λeuλ . It holds that μ1 (p, Ω) = 0, on the other hand, for p = λeuλ with λ = λ. Topological arguments developed in § 2.1.5 assure the unboundedness of the connected component of the total set of solutions 𝒞 = ⋃λ>0 {λ} × 𝒮λ , while the existence of the second solution follows from the variational method in § 2.1.6. To proceed further, an a priori estimate stated below may be useful. Actually, it suggests striking character as λ ↓ 0 of the nonminimal solutions: 1. It holds that 𝒮 λ \ {uλ } ≠ 0 for λ ∈ (0, λ). 2. Each ϵ > 0 admits Cϵ > 0 satisfying ‖u‖∞ ≤ Cϵ for any u ∈ 𝒮λ in λ ≥ ϵ. The asymptotic analysis is the first step to study nonminimal solutions, and is divided into two steps. The first is to classify the singular limit, that is, the limiting functions of classical solutions as λ ↓ 0. We shall see that any classical solution converges to the trivial solution u = 0, produces a finite point blow-up, or even the entire blow-up. The quantized blow-up mechanism and control of the Hamiltonian, called recursive hierarchy in the context of statistical mechanics, are the striking profiles of this singular limit. The second is to construct classical solutions close to these singular limits. This procedure is sometimes called the singular perturbation.4 Finally, global analysis will reveal how those singular limits are connected in the λ–u plane. That is the theme of the later chapter. Below we show several mathematical structures hidden in (3.36) for two dimensions.5 4 This idea was developed satisfactory after the publication of the original version [332]. At that time, only one-point blow-up solutions on simply-connected domains were constructed. 5 Chapter 12 of [335] deals with sup + inf inequality of Shafrir [310], prescaled analysis of Brezis–Merle [53], classification of entire solutions by Chen–Li [78], local blow-up analysis of Li–Shafrir [202], and local uniform estimate of Li [201]. Chapter 2 of [336] contains also further progress on the convex domain done by Grossi–Takahashi [144].

3.2 Boltzmann–Poisson equation

| 123

3.2.2 Bol’s inequality Let C be the Riemann sphere with (0, 0, 0) and (0, 0, 1) denoting the south and north poles, respectively, and let dσ1 be its canonical metric. The meromorphic function F in the previous subsection induces the conformal mapping F = τ ∘ F from Ω into C through the inverse stereographic mapping τ : C ∪ {∞} → C. Under F, the relation dσ1 = ρ(F) ds

(3.38)

is verified, where ds2 = dx12 + dx22 denotes the Euclidean metric on Ω. From this fact, the quantity ρ(F) has been called the spherical derivative of F in the complex function theory. In particular, ρ(F) is invariant under the O(3)-action on C. Thus for each subdomain ω ⊂ Ω, the quantities ℓ1 (𝜕w) ≡ ∫ ρ(F)1/2 ds,

m1 (ω) ≡ ∫ ρ(F) dx

𝜕ω

ω

indicate the length of F(𝜕ω) and the area of F(ω) in C, respectively, as immersions that enjoy the isoperimetric inequality on the round sphere [267], ℓ1 (𝜕w)2 ≥ 4m1 (ω)(π − m1 (ω)). This inequality is nothing but a special form of Bol’s inequality for surfaces with Gaussian curvatures dominated by a constant above [59]. For instance, if Ω ⊂ R2 is simply-connected and p = p(x) > 0 satisfies − Δ log p ≤ p in Ω,

(3.39)

then it holds that 1 ℓ(𝜕ω)2 ≥ m(ω)(8π − m(ω)) 2 for each subdomain ω ⊂ Ω, where ℓ(𝜕ω) = ∫𝜕ω p1/2 ds and m(ω) = ∫ω pdx [24].6 As a

result of this property, isoperimetric inequalities for λ, the least upper bound for the existence of classical solution to (3.21), are given, that is, we have some estimates of λ above and below in terms of the geometric features of Ω, which coincide if Ω is a disc [22, 23]. A modification of the Schwarz symmetrization is also possible by this inequality. Then we obtain an isoperimetric inequality for the first eigenvalue μ1 (p, Ω) of the linearized operator Ap ≡ −ΔD − p, where p = λeu with (λ, u) being a solution to (3.36). 6 See also [29] for multiply-connected Ω.

124 | 3 Complex structure Putting ν1 = inf{∫ |∇v|2 dx | v ∈ H01 (Ω), Ω

∫ v2 p dx = 1}, Ω

we obtain ν1 (p, Ω) ≥ ν1 (p∗ , Ω∗ ),

(3.40)

if p satisfies (3.39) and Σ ≡ ∫Ω pdx < 8π. Here, Ω∗ = {|x| < 1} ⊂ R2 denotes the unit disk, and p∗ = λ∗ eu with (λ∗ , u∗ ) satisfying ∗

− Δu∗ = λ∗ eu

∗

in Ω∗ ,

∫ λ∗ eu dx = Σ. ∗

u∗ |𝜕Ω∗ = 0,

(3.41)

Ω∗

Below we show that such a pair (λ∗ , u∗ ) exists uniquely for each Σ ∈ (0, 8π). Inequality (3.40) is derived by Faber–Krahn’s method. Given a nonnegative function v = v(x) in x ∈ Ω and t > 0, we put Dt = {v > t} to take open concentric disc D∗t of Ω∗ satisfying ∫D∗ p∗ dx = ∫D p dx ∈ (0, 8π). Then Bandle’s spherically decreasing reart

t

rangement of v, denoted by v∗ = v∗ (x), x ∈ Ω∗ , is defined by v∗ (x) = sup{t | x ∈ D∗t }. It is an equimeasurable rearrangement, and it holds that ∫ v2 p dx = ∫ v∗2 p∗ dx. Ω

(3.42)

Ω∗

A decrease of the Dirichlet integral, 󵄨 󵄨2 ∫ |∇v|2 dx ≥ ∫ 󵄨󵄨󵄨∇v∗ 󵄨󵄨󵄨 dx,

Ω

(3.43)

Ω∗

on the other hand, is valid from the coarea formula and Bol’s inequality, if, for example, v is a nonnegative C 1 function with the value zero on the boundary 𝜕Ω, as is described in later chapters.

3.2.3 Radial solutions By Theorem 2.20, if Ω = B ≡ {|x| < 1} ⊂ R2 is a disc, every solution to (3.36) is radially symmetric. Using explicit form of this solution in (2.43), we see λ = 2. Furthermore, according to 0 < λ < 2 or λ = 2, the number of solutions is two or one, respectively, and it holds that lim u− (x) = 0, λ↓0

lim u+ (x) = 4 log λ↓0

1 , |x|

x ∈ B.

3.2 Boltzmann–Poisson equation

| 125

In the λ–u plane, the solutions form a branch 𝒞 ∗ starting from (λ, u) = (0, 0), bending 1 1 at (λ, u) = (2, 2 log 1+|x| 2 ), and growing up to the singular limit u0 (x) = 4 log |x| as λ ↓ 0. This radially symmetric solution takes the form 1/2

μ1/2 λ ( ) eu± /2 = 2 ± , 8 |x| + μ±

λ μ1/2 ± =( ) 2

−1/2

λ {1 ∓ √1 − }. 2

(3.44)

The Liouville integral ( 8λ )1/2 eu/2 = ρ(F) is now realized by 1/2

1 C± = { {4 − λ ± 2√4 − 2λ}} , λ

F(z) = C± z,

(3.45)

and then the quantities 1/2

λ ℓ1 (𝜕B) = ∫ ( eu ) 8 𝜕B

1/2

λ ds = 2π( ) , 8

λ 1 m1 (B) = ∫ eu dx = Σ 8 8 B

denote the immersed length and area of F(𝜕B) and F(B), respectively. Therefore, Σ grows from 0 to 8π monotonously along the branch 𝒞 ∗ with the bending point λ = 2 corresponding to Σ = 4π, while λ increases first, up to the point where Σ = 4π, that is, λ = 2, and then decreases until λ = 0. The latter fact describes the exact features of bending. Theorem 3.2. If Ω = B, there is a continuous, one-to-one correspondence (λ, u) ∈ 𝒞 ∗ 󳨃→ Σ = ∫B eu dx ∈ (0, 8π). Writing this correspondence as (λ, u) = (λΣ , uΣ ), we have λΣ =

1 Σ(8π − Σ), 8π 2

m(Σ) ≡ ∫ euΣ dx = B

Σ 8π 2 = , λ 8π − Σ

(3.46)

and D(Σ) ≡ ∫ |∇uΣ |2 dx = −16π log(8π − Σ) ⋅ {1 + o(1)},

Σ ↑ 8π.

(3.47)

B

Furthermore, the bending point (λ, uλ ) corresponds to Σ = 4π. Proof. First, by (3.44), it holds that Σ = 2π(2± √4 − 2λ) and hence (3.46) is true. Second, we have ∫ |∇u± |2 dx = λ ∫ u± eu± dx = 8π{log

Ω

Ω

8C± C± − }, λ C± + 1

while Σ ∼ 8π corresponds to C = C+ ∼ 8λ . Hence we obtain 2

8 D(Σ) = 8π log( ) + o(1) = −16π log(8λ − Σ) ⋅ (1 + o(1)) λ by (3.46).

126 | 3 Complex structure 3.2.4 Associated Legendre equations From what has been described, the condition Σ = ∫B λ∗ eu dx < 4π implies the strict minimum of the radial solution u∗ . As indicated in § 3.2.1, this property means the positivity of the first eigenvalue μ1 (p∗ , Ω∗ ) of the linearized operator A∗p ≡ −ΔD − p∗ ∗

on Ω∗ , where p∗ = λ∗ eu . Combining this with the isoperimetric inequality (3.40), we obtain the following fact [22, 25]. ∗

Theorem 3.3. If a C 2 function p = p(x) > 0 satisfies (3.39) and Σ = ∫Ω p dx < 4π, then it holds that μ1 (p, Ω) > 0. We note that μ1 (p∗ , Ω∗ ) > 0 is equivalent to ν1 (p∗ , Ω∗ ) ≡ { ∫ |∇v|2 dx | v ∈ H01 (Ω∗ ), ∫ v2 p∗ dx = 1} > 1. Ω∗

(3.48)

Ω∗

The analytic proof of (3.48) under the condition Σ = ∫Ω∗ p∗ dx < 4π is reduced to the study of the eigenvalue problem Δϕ +

1 ζ (x)ϕ = 0 Λ 0

ζ0 (x) = p∗ = λ∗ eu = ∗

in B,

ϕ|𝜕B = 0,

8ρ , + ρ)2

(|x|2

ρ > 0.

(3.49)

After the separation of variables ϕ = Φ(r)eιmθ , we use the transformation ξ = (ρ − r 2 )/(ρ + r 2 )

(3.50)

to induce a two point boundary value problem for the associated Legendre equation [(1 − ξ 2 )Φξ ]ξ + [2/Λ − m2 /(1 − ξ 2 )]Φ = 0, Φ(1) = 1,

Φ(ξρ ) = 0,

ξρ = (ρ − 1)/(ρ + 1).

ξρ < ξ < 1,

(3.51)

Hence (3.48) is reduced to Φ(0) > 0 for the solution Φ to (3.51) with Λ = 1 and m = 0, that is, Φ(ξ ) = ξ . Actually, ∫B p∗ dx < 4π is equivalent to ρ > 1, or ξρ > 0, and therefore, Σ < 4π implies (3.48). 3.2.5 Laplace–Beltrami operator If (λ∗ , u∗ ) satisfies (3.41), it is associated with the conformal mapping F = τ∘F : Ω∗ → C through F(z) in (3.45), where τ : C ∪ {∞} → C denotes the inverse stereographic mapping. Under this transformation, the value ν1 (p∗ , Ω∗ ) is equal to the first eigenvalue μ̂ 1 (ω) of the Laplace–Beltrami operator under the Dirichlet condition in ω ⊂ C, where

3.2 Boltzmann–Poisson equation

| 127

ω is a disc of the area Σ/8. This observation explains the reason why the associated Legendre equation has arisen in the study of (3.49). In fact, this equation appears with the separation of variables of the three-dimensional Laplacian, or the Laplace–Beltrami operator on the two-dimensional round sphere. Then the transformation (3.50) indicates the inversion of the stereographic projection j : C → C ∪ {∞}. Then, inequality (3.48) means that if the area of ω is less than that of the hemiball of C, then μ̂ 1 (ω) > 1 holds. Spherically decreasing rearrangement of Bandle is a Schwarz symmetrization on a round sphere in this context. Let 8C be the two-dimensional round sphere of the area 8π, and dσ be its canonical metric. Given p = p(x) > 0 in a domain Ω ⊂ R2 satisfying (3.39) and Σ = ∫Ω p dx < 8π, we take a disc ω in 8C such that ∫ω dV = Σ. Then, for each nonnegative function v(x) in Ω, we define v∗ (x) in x ∈ ω by v∗ (x) = sup{t | x ∈ ωt }, where ωt denotes the open concentric disc of ω satisfying ∫ω dV = ∫{v>t} p dx. Then t the relations (3.42) and (3.43) mean that ∫ v2 p dx = ∫ v∗2 dV Ω

and

ω

󵄨 󵄨2 ∫ |∇v|2 dx ≥ ∫󵄨󵄨󵄨dv∗ 󵄨󵄨󵄨 dV,

Ω

ω

respectively, where d denotes the differentiation with respect to the canonical metric on 8C. Proposition 3.3 thus means the following: Given a domain ω ⊂ 8C, the first eigenvalue μ1 (ω) of the Laplace–Beltrami operator in ω under the Dirichlet condition is greater than one, provided that ∫ω dV < 4π. 3.2.6 Potential theory In connection with the description in § 3.2.1, asymptotic analysis reveals the following situation: classical solutions {u(x)} to (3.36) make a finite or entire blow-up as λ ↓ 0, unless they converge uniformly to zero, where the complex structure of (3.20), − Δu = λeu

in Ω ⊂ R2 ,

(3.52)

is used [235]. Thus u ∈ C 2 (Ω) solves (3.52) if and only if there is an analytic function

F(z) in Ω with single-valued ρ(F) =

|F 󸀠 | 1+|F|2

1/2

> 0 such that

λ ( ) eu/2 = ρ(F). 8

(3.53)

Equality (3.53) suggests an analogy of (3.52) to −Δu = 0 in Ω ⊂ R2 . In fact, this linear equation has the integral u = Re F, where F(z) is an analytic function in Ω. Here, we notice that the Harnack principle for harmonic functions, i. e., solutions to −Δu = 0, has a similarity to the asymptotics of the solution to (3.36) described above. In fact, the former assures us of the alternatives for monotone harmonic functions

128 | 3 Complex structure between the uniform convergence on every compact set and the entire blow-up, while the latter indicates the third possibility, the finite point blow-up for solutions {(λ, u)} satisfying (3.52) with u = 0 on the boundary. The latter phenomenon is actually regarded as a natural extension of the Harnack principle for subharmonic functions and to those to (3.52). Thus we can derive a mean value theorem for subsolutions to (3.52). Then, a variant of Harnack’s inequality follows, which establishes a Harnack principle [329].

3.2.7 Neumann problems The Neumann problem − Δv = λ(

1 ev − ) in Ω, v |Ω| ∫Ω e

𝜕v 󵄨󵄨󵄨󵄨 󵄨 =0 𝜕ν 󵄨󵄨󵄨𝜕Ω

∫ v = 0, Ω

(3.54)

arises as a stationary state of the simplified system of chemotaxis [168]. By the procedure of doubling, it is reduced to the mean field equation on a closed compact Riemannian surface ℳ, − Δv = λ(

1 ev − ) ∫ℳ ev |ℳ|

in ℳ,

∫ v = 0.

(3.55)

ℳ

Blow-up of the family of solutions occurs only for λ ∈ 8πN in (3.55), while these quantized values are reduced to 4πN in (3.54) [304, 260, 261]. If v = v(x) is a solution to (3.54) and its linearized operator is degenerate, then −Δψ = λ(

∫Ω ev ψ v ev ψ − ⋅ e ) in Ω, ∫Ω ev (∫Ω ev )2

has a nontrivial solution. Using p =

λev ∫Ω ev

and φ = ψ −

− Δφ = νpφ in Ω,

𝜕φ =0 𝜕ν

𝜕ψ 󵄨󵄨󵄨󵄨 󵄨 =0 𝜕ν 󵄨󵄨󵄨𝜕Ω

∫Ω ev ψ ∫Ω ev

, this property means that

on 𝜕Ω

(3.56)

has the eigenvalue ν = 1 with nonconstant eigenfunction. Actually, the first eigenvalue of (3.56) is ν = 0 with constant eigenfunction, and the other eigenvalue ν > 0 does not have a constant eigenfunction. Here, we note that the method of symmetrization does not work to evaluate μ2 (Ω) in Theorem 4.17 of the next chapter. Hence [357] adopts a direct calculation, while [345] uses the method of conformal transplantation. The latter is valid only to the case when Ω ⊂ R2 is simply-connected, but is applicable to (3.56). Using Theorem 4.36 in the next chapter, we obtain the following theorem.

3.3 Classification of the singular limit | 129

Theorem 3.4 ([25]). If Ω ⊂ R2 is a bounded simply-connected domain with a smooth boundary 𝜕Ω, p = p(x) > 0 is continuous on Ω, C 2 in Ω, and satisfies (3.39), and λ = ∫Ω p ≤ 4π, it holds that 1 1 λ + ≥ , ν2 ν3 2π

(3.57)

where νj , j = 2, 3, denotes the jth eigenvalue of (3.56). Since we obtain −Δ log p < p in Ω for p =

λev ∫Ω ev

with v = v(x) satisfying (3.55), the

equality is excluded in (3.57), and therefore, ν2 = ν2 (v, λ) < 1 for any solution v = v(x) to (3.54) with λ = 4π. Since v = 0 is a trivial solution, an immediate consequence is that ν2 (0, λ1 ) = 0 for some λ1 < 4π. If this ν2 = ν2 (0, λ1 ) is simple, then we obtain the bifurcated branch 𝒞 [93, 94]. This 𝒞 can reach λ = 4π only if it loses stability caused by secondary bifurcation, bending, and so forth. Thus, in contrast with (3.37), we obtain multiple existence of the solution to (3.54) in some cases for 0 < λ < 4π and simplyconnected Ω.7

3.3 Classification of the singular limit 3.3.1 Summary Our aim is to study the asymptotic behavior of solutions for nonlinear elliptic eigenvalue problems on bounded domains in R2 with exponentially dominated nonlinearities. Let Ω ⊂ R2 be a bounded domain with a smooth boundary 𝜕Ω, and f : R → R a smooth function. We study − Δu = λf (u),

u>0

in Ω,

u|𝜕Ω ,

(3.58)

where λ is a positive constant. As mentioned in § 3.2.1, we expect striking patterns of the solution as λ ↓ 0. If f (u) = eu + c(x)e−u and Ω is simply-connected, there arises a one-point blow-up family of solutions {u} as λ ↓ 0 via the singular perturbation method, where c(x) is a real-analytic function [228, 229]. Thus, there is κ ∈ Ω such that this {u} has a limit for every x ∈ Ω \ {κ}, while u(κ) tends to infinity as λ ↓ 0. On the contrary, for f (u) = up with 1 < p < +∞, we have a family of solutions {u = λ−1/(ρ−1) v} satisfying (3.58), where v solves it for λ = 1 and f (u) = up . Such a family of solutions experiences entire blow-up as λ ↓ 0: they tend to infinity at any interior point x ∈ Ω. By the implicit function theory, on the other hand, there exist {u} converging uniformly to the trivial solution u = 0 as λ ↓ 0 for the general nonlinearity f (u). 7 If Ω is a disc, ν2 (0, λ1 ) is double. In this case the two-dimensional manifold bifurcates from there. An open question in this case is the nonexistence of nontrivial solution for 4π < λ < 8π.

130 | 3 Complex structure We take the case f (u) = eu + g(u),

󵄨 󵄨󵄨 u 󵄨󵄨g(u)󵄨󵄨󵄨 = o(e ),

u ↑ +∞.

(3.59)

For the moment we assume |g(u) − g 󸀠 (u)| ≤ G(u) and 󵄨 󵄨 G(u) + 󵄨󵄨󵄨G󸀠 (u)󵄨󵄨󵄨 ≤ Ceγu ,

∀u ≥ 0

(3.60)

for some γ < 1/4. We also impose that either Ω is convex or f (u) ≥ 0 for any u ≥ 0.8 Let (λ, u) be a solution to (3.58) and put Σ = λ ∫Ω f (u) dx. Theorem 3.5 ([235]). As λ ↓ 0, the family {Σ} accumulates to 8πm, where m = 0, 1, . . . , +∞. According to the sequence converging to this value, the solutions {u} behave as follows: 1. m = 0. Uniform convergence to 0, i. e., ‖u‖∞ → 0. 2. 0 < m < +∞. Finite point blow-up, i. e., there exists a set composed of m pints denoted by 𝒮 = {κ1 , κ2 , . . . , κm } ⊂ Ω such that {u(x)} has a limit u0 (x) for each x ∈ Ω \ 𝒮 , while 𝒮 is the blow-up set of u.9 3. m = +∞. Entire blow-up, i. e., u(x) → +∞ for every x ∈ Ω.10 Theorem 3.6 ([235]). In the second case of Theorem 3.5, it holds that m

u0 (x) = 8π ∑ G(x, κj ),

(3.61)

j=1

where G(x, y) denotes the Green function of −Δ under the Dirichlet condition: −Δx G(x, y) = δ(x − y),

x ∈ Ω,

G|x∈𝜕Ω = 0

defined for each y ∈ Ω. The blow-up points κj ∈ Ω, 1 ≤ j ≤ m, satisfy the relation 1 ∇ R(κj ) + ∑ ∇x G(κj , κℓ ) = 0, 2 x ℓ=j̸ where R(x) = [G(x, y) +

1 2π

1 ≤ j ≤ m,

(3.62)

log |x − y|]y=x denotes the Robin function.

Equality (3.62) means that x∗ = (κ1 , . . . , κm ) is a critical point of H(x1 , . . . , xm ) =

1 ∑ R(xi ) + ∑ G(xi , xj ). 2 i i 0. We write ‖u‖L1 (Ω) = O(1) if any compact loc set K ⊂ Ω admits C > 0 such that ‖u‖L1 (K) ≤ C for 0 < λ ≪ 1. The following assertion may be called a boundary estimate. Here we assume that either Ω is convex or twodimensional and f (u) ≥ 0. Lemma 3.9 ([99]). If ‖u‖L1 (Ω) = O(1), each m = 0, 1, 2, . . . admits an Ω-neighborhood ω loc of 𝜕Ω such that 󵄩󵄩 α 󵄩󵄩 󵄩󵄩D u󵄩󵄩L∞ (ω) = O(1),

|α| ≤ m.

(3.63)

11 See also [290, 152] for higher-dimensional problems. 12 See [149, 148] and for a complex function theoretical proof and PDE theoretical proof, respectively. The related topic is the concavity maximum principle [2, 121]. See also [65] for the higher-dimensional case. 13 See Part II. The case of a general domain is also known. Hence if Ω is multiply-connected, then each m = 1, 2, . . . admits a sequence of solutions which experience an m-point blow-up as λ ↓ 0 [101].

132 | 3 Complex structure Proof. From the proof of Theorem 2.40, there are α > 0 and t0 > 0 such that d u(x + tξ ) < 0, dt

−t0 < t < 0

for any ξ ∈ R2 in |ξ | = 1, ξ ⋅ ν(x) ≥ α, and x ∈ 𝜕Ω, where ν = ν(x) denotes the outer unit normal vector on 𝜕Ω. Then we obtain an Ω-neighborhood ω = ω0 of 𝜕Ω and a positive constant γ such that each x ∈ ω0 admits a measurable set Ix ⊂⊂ Ω \ ω0 satisfying |Ix | ≥ γ and u|Ix ≥ u(x). Setting Ω0 = ⋃x∈ω0 Ix ⊂⊂ Ω, we obtain u(x) ≤

1 1 ∫ u(y) dy ≤ ‖u‖L1 (Ω0 ) , γ γ

∀x ∈ ω0 ,

Ix

and hence (3.63) for m = 0. From equation (3.58) and the boundary elliptic Lp -estimate in § 1.3.4, each 1 < p < ∞ admits an Ω-neighborhood ω1 ⊂ ω0 of 𝜕Ω such that ‖u‖W 2,p (ω1 ) = O(1). Taking p > 2 = n, we have 0 < β < 1 such that ‖u‖C1+β (ω1 ) = O(1) by Theorem 1.10, and hence (3.63) for m = 1. Now the assertion holds for m = 2, 3, . . . due to the boundary Schauder estimate and the bootstrap argument in § 1.3.4. Proposition 3.10. If Σ → +∞, then {u} experiences an entire blow-up. Proof. We employ Kaplan’s argument introduced in § 1.4.1. Let λ1 > 0 and ϕ1 (x) > 0 be the first eigenvalue and eigenfunction of −Δ under the Dirichlet boundary condition: −Δϕ1 = λ1 ϕ1 ,

ϕ1 > 0

in Ω,

ϕ1 |𝜕Ω = 0.

It holds that λ1 ∫ uϕ1 dx = J,

J = λ ∫ f (u)ϕ1 dx.

Ω

(3.64)

Ω

Therefore, if J = O(1) we have ‖u‖L1 (Ω) = O(1) and hence the boundary estimate loc holds by Lemma 3.9. It holds also that ‖λf (u)‖L1 (Ω) = O(1), which implies Σ = O(1). loc Hence Σ → +∞ implies J → +∞. Each K ⊂⊂ Ω admits a constant γK > 0 satisfying G(x, y) ≥ γK ϕ1 (y),

(x, y) ∈ K × Ω

by the Hopf lemma, where G = G(x, y) denotes the Green function for −Δ under the Dirichlet boundary condition. Since 1󵄨 󵄨 f (u) ≥ 󵄨󵄨󵄨f (u)󵄨󵄨󵄨 − C, 2

u≥0

3.3 Classification of the singular limit | 133

holds with a constant C from the assumption, we obtain 1 inf u(x) = inf ∫ G(x, y)λf (u(y)) dy ≥ γK J − C 󸀠 λ → +∞, x∈K 2

x∈K

Ω

and hence the assertion follows. If ‖u‖∞ = O(1), then ‖λf (u)‖∞ → 0 as λ ↓ 0, so that the first case occurs from the elliptic estimate, ‖u‖∞ → 0. Theorem 3.5 has thus been reduced to the following theorem. Claim 3.11. If ‖u‖∞ → +∞ and Σ = O(1), the second case occurs in Theorem 3.5. It is obvious that Σ = O(1) implies ‖Δu‖1 = O(1), and hence ‖u‖W 1,p (Ω) = O(1),

1 0 in Ω ⊂ Rn solves Δw =

n |∇w|2 + F(w) 2 w

(3.65)

with C 1 nonlinearity F(w), then the equality ∇⋅V =J +

n−1 |∇w|2 w−n {F(w) + wF 󸀠 (w)} n

(3.66)

holds, where V = (V j ),

V j = w−n+1 {∇( n

J = w−n+1 { ∑ ( i,j=1

2

1 𝜕w 𝜕w ) ⋅ ∇w − Δw}, 𝜕xj n 𝜕xj

𝜕2 w 1 ) − (Δw)2 } ≥ 0. 𝜕xi 𝜕xj n

The semilinear equation −Δu = λf (u),

u > 0 in Ω ⊂ Rn

is transformed into (3.65) by taking w = ϕ(u) for ϕ(u) = {

e−αu , n = 2, 2 u− n−2 , n > 2,

and α ∈ R. Then it holds that F(w) = −λϕ󸀠 (u)f (u), and the second term of (3.66) takes the form n−1 |∇w|2 w−n (F(w) + wF 󸀠 (w)) n α2 e−αu |∇u|2 {2αf (u) − f 󸀠 (u)}, n = 2, n = { n−1 2 2 n+2 2 − n−2 󸀠 ( ) |∇u| u { n−2 f (u) − f (u)}, n > 2. n n−2 For n = 2, equality (3.66) implies 1 ∇ ⋅ V ≥ α2 λe−(αu) |∇u|2 {2αf (u) − f 󸀠 (u)} 2 and then ∫ ∇ ⋅ V dx = ∫ V ⋅ ν ds = O(1) Ω

15 This idea is due to [318].

𝜕Ω

3.3 Classification of the singular limit | 135

follows from the boundary estimate (3.63). Hence each α ≠ 0 admits λ ∫ e−αu |∇u|2 {2αf (u) − f 󸀠 (u)} dx ≤ Cα . Ω

Using f 󸀠 (u) − f (u) ≤ Ceγu for 0 ≤ u < +∞, we obtain (2α − 1)λ ∫ e−αu |∇u|2 f (u) dx ≤ Cα {1 + λ ∫ e(γ−α)u |∇u|2 dx}, Ω

(3.67)

Ω

while (3.58) implies λ ∫ e−βu f (u) dx = − ∫ Ω

𝜕Ω

𝜕u ds − β ∫ e−βu |∇u|2 dx, 𝜕ν Ω

and hence ∫ e−βu |∇u|2 dx = O(1),

β>0

(3.68)

Ω

due to u > 0, (3.59), Σ = 0(1), and (3.63). From (3.67), (3.68), and (3.59), it follows that 󵄨 󵄨 λ ∫ e−αu |∇u|2 󵄨󵄨󵄨f (u)󵄨󵄨󵄨 dx = O(1),

α > 1/2,

Ω

because γ − α < 0 holds whenever γ < 1/4. This inequality means

u

󵄩󵄩 󵄩 −1/2 󵄩󵄩∇Gα (u)󵄩󵄩󵄩2 = O(λ )

(3.69)

for Gα (u) = ∫0 e(−α/2)u |f (u)|1/2 du.

We reduce (3.69) to ‖∇Gα (u)‖p = O(λ−1/2 ) for 1 < p < 2, and apply Sobolev’s inequality to get 󵄩󵄩 󵄩 −1/2 󵄩󵄩Gα (u)󵄩󵄩󵄩p∗ = O(λ ),

1 1 1 = − . p∗ p 2

(3.70)

1 < 2, 1−α

(3.71)

Henceforth we fix α > 1/2 such that (γ + 1/2)σ < 3/2,

σ=

recalling γ < 1/4. Since these exponents admit 󵄨󵄨 󵄨1/2σ ≤ C(Gα (u) + 1), 󵄨󵄨f (u)󵄨󵄨󵄨

0 ≤ u < +∞

inequality (3.70) implies ‖f (u)‖r = O(λ−σ ) for r = p∗ /2σ ∈ (1/σ, ∞). Since Σ = O(1) implies ‖f (u)‖1 = O(λ−1 ), we thus obtain 󵄩󵄩 󵄩p 󵄩 󵄩1/q 󵄩 󵄩p−1/q −1/q−(p−1/q)σ ), 󵄩󵄩f (u)󵄩󵄩󵄩p ≤ 󵄩󵄩󵄩f (u)󵄩󵄩󵄩1 󵄩󵄩󵄩f (u)󵄩󵄩󵄩(p−1/q)q󸀠 = O(λ

136 | 3 Complex structure where 1 < q < +∞ and 1/q󸀠 + 1/q = 1. Letting q ↓ 1, we get 󵄩󵄩 󵄩 −σ+(σ−1)/p−ϵ ), 󵄩󵄩f (u)󵄩󵄩󵄩p = O(λ

1 2. Here we have 󵄨󵄨 󵄨 󵄨 󸀠 󵄨 󵄨 󵄨γ 󵄨󵄨G(u)󵄨󵄨󵄨 = 󵄨󵄨󵄨H (u)󵄨󵄨󵄨 ≤ C(󵄨󵄨󵄨f (u)󵄨󵄨󵄨 + 1),

0 ≤ u < ∞,

and then (3.72) implies 󵄩 󵄩 󵄩 󵄩 λ󵄩󵄩󵄩H 󸀠 (u)f (u)󵄩󵄩󵄩p ≤ Cλ(󵄩󵄩󵄩f (u)γ+1 󵄩󵄩󵄩p + 1)

󵄩 󵄩γ+1 = Cλ(󵄩󵄩󵄩f (u)󵄩󵄩󵄩p(γ+1) + 1) = O(λρ−ϵ )

for any ϵ > 0, where ρ = 1 + (γ + 1){

σ−1 σ − σ} = 1 + − σ(γ + 1). p(γ + 1) p

(3.74)

3.3 Classification of the singular limit | 137

Letting p ↓ 2, we have 1 ρ ↑ 1 + (σ − 1) − σ(γ + 1) = −(γ + 1/2)σ + 1/2 > −1 2 by (3.71), and hence 󵄩 󵄩 λ󵄩󵄩󵄩H 󸀠 (u)f (u)󵄩󵄩󵄩p = o(λ−1 ).

(3.75)

Next, we take q ∈ (1, ∞) in qγ > 1/2, to note 󵄩󵄩 󸀠󸀠 󵄩 󸀠󸀠 󵄩 2󵄩 2 󵄩󵄩H (u)|∇u| 󵄩󵄩󵄩p ≤ 󵄩󵄩󵄩H (u)󵄩󵄩󵄩pq ‖∇u‖2pq󸀠 ,

1/q󸀠 + 1/q = 1.

Since 2pq󸀠 > 1 is obvious, it holds that ‖∇u‖22pq󸀠 = O(λ(−1+1/pq )(σ−1)−ϵ ), 󸀠

ϵ>0

by (3.73). One also has that 󵄨󵄨 󸀠󸀠 󵄨󵄨 󵄨󵄨 󸀠 󵄨󵄨 󵄨 󵄨γ 󵄨󵄨H (u)󵄨󵄨 = 󵄨󵄨G (u)󵄨󵄨 ≤ (󵄨󵄨󵄨f (u)󵄨󵄨󵄨 + 1),

0 ≤ u < +∞,

and hence 󵄩󵄩 󸀠󸀠 󵄩󵄩 󵄩 󵄩γ 󵄩󵄩H (u)󵄩󵄩pq ≤ C(󵄩󵄩󵄩f (u)󵄩󵄩󵄩pqγ + 1). Since qγ > 1/2 and p > 1 imply pqγ > 1, inequality (3.72) is valid. Hence we obtain 󵄩󵄩 󸀠󸀠 󵄩󵄩 −γσ+(σ−1)/pq−ϵ ), 󵄩󵄩H (u)󵄩󵄩pq = O(λ

ε > 0,

to end up with 󵄩󵄩 󸀠󸀠 2󵄩 ρ󸀠 −ϵ ), 󵄩󵄩H (u)|∇u| 󵄩󵄩󵄩p = O(λ

ϵ>0

where ρ󸀠 = −γσ + (−1 + 1/p)(σ − 1). Letting p ↓ 2, we obtain ρ󸀠 ↑ − 21 (σ − 1) − γσ > −1 by (3.71). Hence it follows that 󵄩󵄩 󸀠󸀠 2󵄩 −1 󵄩󵄩H (u)|∇u| 󵄩󵄩󵄩p = o(λ ).

(3.76)

We get the conclusion λ‖∇H(u)‖∞ = o(1), combining (3.74), (3.75), and (3.76). 3.3.4 Complex analysis Here we show that the family of solutions {u} experiences a finite point blow-up if ‖u‖∞ → +∞ and Σ = O(1), using complex function theory. For this purpose, we take

138 | 3 Complex structure complex variables z = x1 + 𝚤x2 and z = x1 − 𝚤x2 for x = (x1 , x2 ) ∈ Ω, to introduce the function 1 s = s(z, z) = uzz − u2z . 2 Regarding this equality as a Riccati equation in u, we see that ϕ = e−u/2 satisfies 1 ϕzz + sϕ = 0, 2

(3.77)

󵄩󵄩 𝜕s 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 → 0 󵄩󵄩 𝜕z 󵄩󵄩∞

(3.78)

while

follows from sz = − 4λ uz (f 󸀠 (u) − f (u)) and Lemma 3.13. Since Σ = O(1) implies ‖Δu‖1 = O(1), we obtain ‖u‖W 1,p = O(1) for 1 ≤ p < 2 by the L1 -elliptic estimate in § 1.3.4. Then the boundary estimate (3.63) holds so that {u} accumulates weakly in W 1,p , 1 < p < 2, and strongly in C 2 (ω) to a function u0 ∈ W 1,p (Ω) ∩ C 2 (ω), where ω denotes an Ω-neighborhood of 𝜕Ω. By (3.78) the function s0̃ (z) = u0zz − 21 u20zz is holomorphic in ω, and it holds that ‖s − s0̃ ‖L∞ (ω) → 0.

(3.79)

Setting f0 = Re s0̃ and h0 = Im s0̃ , we put U0 (x) = ∫ 𝜕Ω

𝜕 G(x, y)f0 (y) dsy , 𝜕νy

V0 (x) = ∫ 𝜕Ω

𝜕 G(x, y)h0 (y) dsy 𝜕νy

to define the smooth function s0 = U0 + 𝚤V0 in Ω, where G(x, y) denotes the Green 𝜕 function. Then (3.79) implies the following lemma, and in particular, 𝜕z s0 = 0 as a distribution by (3.78) which means that s0 (z) is holomorphic. Lemma 3.14. It holds that ‖s − s0 ‖L∞ (Ω) = o(1). loc

Proof. Letting U = Re s, V = Im s, R = Ux1 − Vx2 , and Q = Ux2 + Vx1 , we obtain |sz |2 =

1 (|R|2 + |Q|2 ), 4

and hence ‖R‖∞ , ‖Q‖∞ → 0. By (3.79) it holds that ‖f − f0 ‖L∞ (𝜕Ω) → 0 for f = U|𝜕Ω . On the other hand, ΔU = Rx1 + Qx2 implies, for x ∈ Ω, that U(x) − U0 (x) = − ∫ G(x, y)(Rx1 (y) + Qx2 (y)) dy Ω

+∫ 𝜕Ω

𝜕G 𝜕 𝜕G (x, y)f (y) − f (y) dsy = ∫ G(x, y)R(x, y) 𝜕νy 𝜕νy 0 𝜕y1 Ω

𝜕 𝜕 + G(x, y)Q(y) dy + ∫ G(x, y)(f (y) − f0 (y)) dsy . 𝜕y2 𝜕νy 𝜕Ω

3.3 Classification of the singular limit | 139

Since G(x, y) = we have

1 2π

1 log |x−y| + R(x, y) with a smooth function R = R(x, y) on Ω × Ω,

󵄩󵄩 𝜕G 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 < +∞ sup󵄩󵄩󵄩∇y G(x, ⋅)󵄩󵄩󵄩1 + sup󵄩󵄩󵄩 (x, ⋅)󵄩󵄩󵄩 󵄩󵄩L1 (𝜕Ω) 󵄩 𝜕νy x∈K 󵄩 x∈K for any compact set K ⊂ Ω, which implies ‖U − U0 ‖L∞ (K) → 0. Similarly, we get ‖V − V0 ‖L∞ (K) → 0, and the proof is complete. Let u(x) solve (3.58) for some λ > 0 and x∗ = (x1∗ , x2∗ ) ∈ Ω be its maximal point. For some Ω-neighborhood ω of 𝜕Ω, the relation x∗ ∈ Ω\ω holds by the argument of the boundary estimate. Therefore, {x∗ } accumulates to a point x0∗ ∈ Ω as λ ↓ 0, and then we take the fundamental system {ϕ1 , ϕ2 } of the linear ordinary differential equation (3.77) in z satisfying ϕ1 |z=z ∗ =

𝜕 ϕ | ∗ = 1, 𝜕z 2 z=z

𝜕 ϕ | ∗ = ϕ2 |z=z ∗ = 0 𝜕z 1 z=z

for z ∗ = x1∗ + ιx2∗ , which implies Wz (ϕ1 , ϕ2 ) ≡ ϕ1 ϕ2z − ϕ1z ϕ2 = 1. The coefficient s in (3.77) depends also on z so that we may write ϕ1 = ϕ1 (z, z) and ϕ2 = ϕ2 (z, z). By the following lemma, the Liouville integral holds asymptotically in (3.58) if the nonlinearity f (u) is exponentially dominated. Let 󵄨 C1 = e−u/2 󵄨󵄨󵄨x=x∗ ,

C2 =

λ −u/2 e f (u)|x=x∗ . 8

Lemma 3.15. It holds that e−u/2 = C1 ϕ1 (z, z ∗ )ϕ1 (z, z) + C2 ϕ2 (z, z)ϕ2 (z, z) + o(1)

(3.80)

uniformly in K as λ ↓ 0, where K is an arbitrary simply-connected compact set in Ω containing x0∗ in its interior. Proof. We have x∗ → x0∗ ∈ int K and ϕj (z, z), j = 1, 2, are single-valued on K. Since ϕ = e−u/2 satisfies (3.77), the relation ϕ ≡ e−u/2 = f 1 (z)ϕ1 (z, z) + f 2 (z)ϕ2 (z, z)

(3.81)

holds for f j (z), j = 1, 2, independent of z. Putting z = z ∗ , we obtain f 1 (z) = Wz (ϕ, ϕ2 ) = ϕ(z ∗ , z),

f 2 (z) = −Wz (ϕ, ϕ1 ) = ϕz (z ∗ , z).

(3.82)

Since ϕ(z, z) = e−u/2 is real-valued, it holds that ϕzz + 21 sϕ = 0 for s(z, z) = s(z, z). Hence (3.82) implies 1 1 f 2zz + s∗ f 2 + s∗z ϕ = 0, 2 2

s∗ (z) = s(z, z ∗ ),

s∗z (z) = sz (z, z ∗ ).

(3.83)

140 | 3 Complex structure In other words, f2 (z) = f 2 (z) solves 1 1 f2zz + s∗ f2 + s∗z ϕ = 0, 2 2

s∗ (z) = s(z, z ∗ ),

s∗z (z) = sz (z, z ∗ )

on K. Using ∇u(x ∗ ) = 0, we obtain 1 f 2 |z=z ∗ = ϕz (z ∗ , z ∗ ) = − e−u/2 uz |x=x∗ = 0, 2 while 𝜕 1 f 2 |zz ∗ = ϕzz (z ∗ , z ∗ ) = − e−u/2 uzz |x=x∗ 𝜕z 2 λ −u/2 = e f (u)|x=x∗ = C2 ∈ R. 8 We thus end up with f2 (z) = C2 ϕ2 (z, z ∗ ) + o(1) in L∞ (K),

(3.84)

by ‖s‖L∞ (Ω) = O(1), x∗ → x0∗ , and ‖sz∗ ϕ‖L∞ (K) ≤ ‖sz ‖∞ ‖ϕ‖L∞ (K) → 0. loc Similarly, we have f 1 |z=z ∗ = ϕ(z z , z ∗ ) = e−u/2 |x=x∗ = C1 , 1 𝜕 f | ∗ = ϕz (z ∗ , z ∗ ) = − e−u/2 uz |x=x∗ = 0. 𝜕z 1 z=z 2 Since f1zz + 21 s∗ f1 = 0 is valid by (3.82) and (3.83), here we obtain f1 (z) = C1 ϕ1 (z, z ∗ ).

(3.85)

Equalities (3.81), (3.84), and (3.85) now imply (3.80), because ‖ϕ2 ‖L∞ (K) = O(1) holds by Lemma 3.14. Since u = u(x) > 0 has a maximum at x = x ∗ , it holds that f (u)|x=x∗ ≥ 0, and hence C2 = 8λ e−u/2 f (u)|x=x∗ ≥ 0, while C1 = e−u/2 |x=x∗ > 0 is obvious. Proposition 3.16. If ‖u‖L∞(Ω) → +∞ and Σ = O(1), the family {u} experiences a finite point blow-up. ∗ ∗ Proof. We recall that x0∗ = (x10 , x20 ) ∈ Ω is an accumulation point of {x∗ }. From Lemma 3.15 we may suppose that, for any simply-connected compact set K ⊂ Ω such that x0 ∈ int K, we have ‖ϕj − ϕj0 ‖L∞ (K) → 0 for j = 1, 2, where {ϕ10 , ϕ20 } denotes the fundamental system of solutions for

1 ϕzz + s0 ϕ = 0, 2

3.3 Classification of the singular limit | 141

𝜕 𝜕 ∗ ∗ ϕ20 |z=z0∗ = 1 and 𝜕z ϕ10 |z=z0∗ = ϕ20 |z=z0∗ = 0 for z0∗ = x10 + ιx20 . satisfying ϕ10 |z=z0∗ = 𝜕z The function s0 (z) is holomorphic in Ω and hence ϕj0 ≢ 0, j = 1, 2, are analytic there. Hence the set 𝒮 = {z ∈ Ω | ϕ20 (z) = 0} is discrete. We have assumed that ‖u‖∞ → +∞ and hence C1 = e−‖u‖∞ /2 → 0. Let C20 ∈ [0, +∞] be an accumulation point of {C2 }. We take limit in (3.80), noting that the blowup points of {u} correspond to zeros of C20 |ϕ20 |2 . If C20 = 0, we have an entire blow-up u(x) → +∞ for x ∈ Ω, which is excluded by the boundary estimate. In the other case of C20 ∈ (0, +∞] it holds that ‖u‖L∞ (Ω\𝒮) = O(1), and then we see that {u} experiences loc a finite-point blow-up, using the boundary estimate.

3.3.5 Isolated singular points Suppose that {u} = {uk } experiences an m-point blow-up with the blow-up set 𝒮 = {κ1 , κ2 , . . . , κm } ⊂ Ω, that is, ‖u‖L∞ (Ω\𝒮) = O(1), loc

j

∃xk → κj ,

j

uk (xk ) → +∞,

1≤j≤m

with λ = λk ↓ 0. It follows that 𝒮 ⊂ Ω due to the boundary estimate. Here we show that the limit function u0 (x) takes the form (3.63). First, through a bootstrap argument described in § 1.3.4, we see that u0 ∈ W 1,p (Ω),

1≤p 2, n=2

denotes the fundamental solution, −ΔE = δ. Proof. Since the support of the distribution Δv ∈ 𝒟󸀠 is contained in {0}, it holds that Δv = ∑ Cα Dα δ,

∃k, ∃Cα ∈ R.

|α|≤k

We take ζ ∈ C0∞ (B) satisfying (−1)|α| Dα ζ (0) = Cα for |α| ≤ k, to put ζϵ (x) = ζ (x/ϵ) for 0 < ϵ ≪ 1. Then it holds that − ∫ ∇v ⋅ ∇ζϵ dx = ⟨ζ , Δv⟩𝒟,𝒟󸀠 = ∑ Cα2 /ϵ|α| . B

|α|≤k

Since ‖∇ζϵ ‖Lp󸀠 (B) = o(ϵ−1 ) and ∇v ∈ Lp (B) for

1 p󸀠

+

1 p

∫ ∇v ⋅ ∇ζϵ dx = o(ϵ−1 ),

= 1, we have ϵ↓0

B

which implies Cα = 0 for |α| ≥ 1. Hence the assertion follows. Proof of Lemma 3.17. Since aj ≠ 0 means that κj ∈ Ω is not a blow-up point of {u}, we assume aj ≠ 0, to derive aj = 4 and ∇Hj (κj ) = 0 in Proposition 3.18, where Hj (x) = u0 (x) + aj log |x − κj |, harmonic at x = κj . In fact, letting wj (x) = −aj log |x − κj |, we have 1 1 s0 ≡ u0zz − u0z = (wjzz − wjz2 ) + (Hjzz − Hjz2 ) − wjz Hjz 2 2 =

aj /2 − a2j /8 (z − κj )2

aj 1 + (Hjzz − Hjz2 ) + H , 2 2(z − κj ) jz

16 We use an argument in [52]. See also [307, 309].

3.3 Classification of the singular limit | 143

𝜕 log |z| = z2 . This function s0 (z) is holomorphic at z = κj , and hence it follows due to 𝜕z that aj = 4 and Hjz (κj ) = 0. However, Hj (x) is real-valued and hence the latter relation is equivalent to (3.88).

Proof of Theorem 3.6 (completion). By virtue of the boundary condition u0 |𝜕Ω = 0, the harmonic function H = H(x) in (3.89) takes the form m

H(x) = 4 ∑ ∫

j=1 𝜕Ω

The function K(x, y) = G(x, y) + Δx K(x, y) = 0,

𝜕G (x, ξ ) log |ξ − κj |dsξ , 𝜕νξ

1 2π

x ∈ Ω.

log |x − y| for y ∈ Ω, on the other hand, solves

x ∈ Ω,

K(x, y) =

1 log |x − y|, 2π

x ∈ 𝜕Ω,

and therefore, K(x, y) =

1 𝜕G (x, ξ ) log |ξ − y| dsξ , ∫ 2π 𝜕νξ

x, y ∈ Ω.

𝜕Ω

We thus obtain H(x) = 8π ∑m j=1 K(x, κj ) for x ∈ Ω, so that m

m

j=1

j=1

u0 (x) = H(x) − 4 ∑ log |x − κj | = 8π ∑ G(x, κj ) which means that (3.61) holds. The function Hj (x) in (3.87) is now given by Hj (x) = H(x) − 4 ∑ log |x − κℓ | = 8π{∑ G(x, κℓ ) + K(x, κj )}. ℓ=j̸

ℓ=j̸

Hence (3.88) means ∇x K(κj , κj ) + ∑ ∇x G(κj , κℓ ) = 0,

1 ≤ j ≤ m,

ℓ=j̸

or that (3.62) is true. 3.3.6 Simply-connected domains Corollary 3.8 is a consequence of Corollary 3.7 and the properties of the Hamiltonian H = H(x1 , . . . , xm ) =

1 ∑ R(xj ) + ∑ G(xi , xj ) 2 j i 0. The function U = H ∗ u ≡ u ∘ H solves the equation −ΔU = λk(x)f (U)

in B,

U|𝜕B = 0.

Then we apply generalized Pohozaev identity, using ∫ ∇ ⋅ P dx = ∫ P ⋅ ν ds, B

P(x) = [(x ⋅ ∇)U]∇U,

𝜕B

that is, ∫ λ(2k(x) + (x ⋅ ∇k))F(U) dx = B

2

1 𝜕U ∫ ( ) x ⋅ ν ds, 2 𝜕ν 𝜕B

u

where F(u) = ∫0 f (u) du. Here, we use ∫𝜕B ∫( 𝜕B

ds x⋅ν

= 2π and Schwarz inequality to deduce

2 2 (∫ 𝜕U ds)2 𝜕U 1 ) x ⋅ ν ds ≥ 𝜕B 𝜕ν ds = (∫ λk(x)f (U) dx) 𝜕ν 2π ∫𝜕B x⋅ν 2

=

B

1 (∫ λf (u) dx) . 2π Ω

Since 󵄨 󵄨 ∫ λ(2k(x) + (x ⋅ ∇k))F(U) dx ≤ C ∫ λ󵄨󵄨󵄨F(U)󵄨󵄨󵄨 dx B

B

for C = maxB |2k(x) + x ⋅ ∇k|, it follows that 2

1 󵄨 󵄨 (∫ λf (u) dx) ≤ C ∫ λ󵄨󵄨󵄨F(u)󵄨󵄨󵄨 dx, 4π Ω

Ω

which implies Σ = O(1) because the nonlinearity f = f (u) is exponentially dominated.

3.3 Classification of the singular limit | 145

3.3.7 Convex domains If the family of solutions experiences an ℓ-point blow-up, the blow-up points {κ1 , . . . , κℓ } satisfy (3.62), and hence (κ1 , . . . , κℓ ) is a critical point of H(x1 , . . . , xℓ ) =

1 ℓ ∑ R(xj ) + ∑ G(xi , xj ). 2 j=1 1≤i 0 to realize, however, we have to suppose an additional condition (C2) below. After passing this process, we can solve any order approximate equation without additional assumptions. (C2) P(ζ ) has no zero in |ζ | < 1. In terms of U = U(ζ ) defined by 1/2

λ |F 󸀠 | 󵄨 󵄨 ( ) 󵄨󵄨󵄨g 󸀠 󵄨󵄨󵄨eU/2 = , 8 1 + |F|2 we conclude the following proposition. Proposition 3.22 ([370]). Under the assumptions of (3.116), (C1), and (C2), there exists a family of solutions {(λ, Ũ n )} with 0 < λ ≪ 1 satisfying 󵄨 󵄨2 ̃ − ΔŨ n = λ󵄨󵄨󵄨g 󸀠 󵄨󵄨󵄨 eUn

in B,

‖Ũ n ‖L∞ (𝜕B) = O(λn )

(3.117)

and Ũ n (δ) → +∞ for n = 1, 2, . . . Then, quasi-Newton method is utilized to reach genuine solutions. First, the Green function on B is given by G(x, y) =

󵄨󵄨 z − w 󵄨󵄨 1 󵄨 󵄨󵄨 log󵄨󵄨󵄨 󵄨, 󵄨󵄨 1 − zw 󵄨󵄨󵄨 4π

where x = (x1 , x2 ),

y = (y1 , y2 ),

z = x1 + 𝚤x2 ,

w = y1 + 𝚤y2 .

(3.118)

Let 󵄨 󵄨2 (KU)(x) = ∫ G(x, y)(λ󵄨󵄨󵄨g 󸀠 󵄨󵄨󵄨 eU )(y) dy, B

(3.119)

3.4 Singular perturbation

| 153

and put S(U) = (1 − KU󸀠 0 ) (K(U) − KU󸀠 0 (U)), −1

where KU󸀠 0 ∈ B(X, X) denotes the Fréchet derivative for X = C 0 (B). We adopt the iteration scheme Uk+1 = S(Uk ),

k = 0, 1, 2, . . .

(3.120)

If the sequence {Uk }∞ k=0 converges in X, the relation U = S(U) is achieved. It is equivalent to U = K(U), and then the solution U to (3.112) is obtained. Proposition 3.23. If the initial guess U0 ∈ X satisfies 1+Γ 󵄩󵄩 󵄩 − (1 + Γ)−1 󵄩󵄩U0 − K(U0 )󵄩󵄩󵄩X ≤ log Γ

(3.121)

for −1 󵄩 󵄩 Γ ≥ 󵄩󵄩󵄩(1 − KU󸀠 0 ) KU󸀠 0 󵄩󵄩󵄩X,X ,

the sequence {Uk }∞ k=0 in (3.120) converges to some U∗ in X, satisfying ‖U∗ − U0 ‖X ≤ log

1+Γ . Γ

Assumption (3.121) is related to the accuracy of U0 . To achieve this property, we take the nth asymptotic solution Ũ n in Proposition 3.22 as U0 . In fact, since 󵄨 󵄨2 −Δ(KU0 ) = λ󵄨󵄨󵄨g 󸀠 󵄨󵄨󵄨 eU0

in B,

KU0 |𝜕B = 0,

we obtain −Δ(U0 − KU0 ) = 0

in B,

‖U0 − KU0 ‖L∞ (𝜕B) = O(λn ),

which ensures ‖U0 − KU0 ‖X = O(λn ) by the maximum principle. For this scheme to realize, spectral analysis of the linearized operator KU󸀠 0 is essential. The following estimate is the most crucial part of Weston’s theory, where a fine property of the associated Legendre equation is used. Lemma 3.24. ([370]) Under an additional generic condition concerning δ ∈ B, denoted by (C3), any Ũ n = U0 admits 󵄩󵄩 󸀠 −1 󵄩 󵄩󵄩(1 − KU0 ) 󵄩󵄩󵄩X,X ≤ C/λ, where C = Cn > 0 is a constant.

0 < λ ≪ 1,

(3.122)

154 | 3 Complex structure Since condition (C3) is rather hard to write, we postpone this to the next subsection, but it is expressed in the form of |α|2 ≠ β2

(3.123)

for α ∈ C and β ∈ R determined by Ω ⊂ R2 and κ ∈ Ω. Since 󵄩 󵄩󵄩 󵄩 󸀠 −1 󵄩 󸀠 −1 󵄩󵄩(1 − KU0 ) KU0 󵄩󵄩󵄩X,X ≤ 1 + 󵄩󵄩󵄩(1 − KU0 ) 󵄩󵄩󵄩X,X , we can take Γ = λ−1−ℓ for ℓ ≥ 0 and 0 < λ ≪ 1 in Proposition 3.23. Then the right-hand side of (3.121) becomes O(

1 ) = O(λ2ℓ+2 ). Γ2

Therefore, if n = 2ℓ + 3 for U0 = Ũ n , inequality (3.121) holds for 0 < λ ≪ 1. We thus obtain the following theorem. Theorem 3.25. ([370]) If δ ∈ B satisfies (3.116) with generic conditions (C1), (C2), and (C3), scheme (3.120) with U0 taken as the nth asymptotic solution for n ≥ 3 converges to the genuine solution U∗ to (3.112). Furthermore, it holds that ‖U∗ − U0 ‖X ≤ Cλ(n−1)/2 ,

0 < λ ≪ 1.

We conclude this section with the proof of Proposition 3.23.22 Lemma 3.26. Let X be a Banach space with the norm ‖ ⋅ ‖, and S : X → X a C 1 mapping in the sense of Fréchet. Given ϕ ∈ C 1 ([0, ∞), R) with a fixed point t ∗ > 0, we suppose 󵄩󵄩 󸀠 󵄩󵄩 󸀠 󵄩󵄩S (u)󵄩󵄩 ≤ ϕ (t),

‖u − u0 ‖ ≤ t,

t > 0,

(3.124)

and 󵄩󵄩 󵄩 󵄩󵄩S(u0 ) − u0 󵄩󵄩󵄩 ≤ ϕ(0) with some u0 ∈ X. Then the iteration sequence {un }∞ n=0 ⊂ X defined by un+1 = S(un ) converges to a fixed point u∗ ∈ X, satisfying ‖u∗ − u0 ‖ ≤ t ∗ . Proof. For each n ≥ 0, we have the identity 1

un+2 − un+1 = S(un+1 ) − S(un ) = ∫ 1

0

d S(tun+1 + (1 − t)un ) dt dt

= ∫ S󸀠 (tun+1 + (1 − t)un )[un+1 − un ] dt. 0

22 See [355] for the other applications.

(3.125)

3.4 Singular perturbation

| 155

Since ‖u1 − u0 ‖ ≤ ϕ(0) by assumption, it holds that 󵄩 󵄩󵄩 󵄩󵄩(tu1 + (1 − t)u0 ) − u0 󵄩󵄩󵄩 = t‖u1 − u0 ‖ ≤ tϕ(0), and hence (3.125) implies 1

t=1

‖u2 − u1 ‖ ≤ ∫ ϕ󸀠 (tϕ(0))ϕ(0) dt = [ϕ(tϕ(0))]t=0 = ϕ2 (0) − ϕ(0). 0

We thus obtain the first step, n = 1, of ‖un − u0 ‖ ≤ ϕn (0),

‖un+1 − un ‖ ≤ ϕn+1 (0) − ϕn (0).

(3.126)

Assuming (3.126) up to n, we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩(tun+1 + (1 − t)un ) − u0 󵄩󵄩󵄩 = 󵄩󵄩󵄩t(un+1 − u0 ) + (un − u0 )󵄩󵄩󵄩 ≤ t‖un+1 − u0 ‖ + ‖un − u0 ‖ ≤ t(ϕn+1 (0) − ϕn (0)) + ϕn (0), and hence ‖un+2 − un+1 ‖ 1

󵄩 󵄩 ≤ ∫󵄩󵄩󵄩S󸀠 (tun+1 + (1 − t)un )󵄩󵄩󵄩‖un+1 − un ‖ dt 0

1

≤ ∫ ϕ󸀠 (t(ϕn+1 (0) − ϕn (0)) + ϕn (0))(ϕn+1 (0) − ϕn (0)) dt 0

t=1

= [ϕ(t(ϕn+1 (0) − ϕn (0)) + ϕn (0))]t=0 = ϕn+2 (0) − ϕn+1 (0). Then it follows that ‖un+2 − u0 ‖ ≤ ‖un+2 − un+1 ‖ + ‖un+1 − un ‖ + ⋅ ⋅ ⋅ + ‖u1 − u0 ‖ ≤ ϕn+1 (0)

because the second inequality of (3.126) is available up to n + 1. Thus we obtain (3.126) for any n. Then (3.126) implies N

∑ ‖un+1 − un ‖ ≤ ϕN+1 (0) ≤ ϕN+1 (t ∗ ) = t ∗ < +∞,

n=0

and hence ∞

∑ ‖un+1 − un ‖ ≤ t ∗ .

n=0

156 | 3 Complex structure Thus the convergence un → u∗ follows for some u∗ ∈ X, satisfying S(u∗ ) = u∗ . Finally, we have ‖un − u0 ‖ ≤ ϕn (0) ≤ ϕn (t ∗ ) = t ∗ , and hence ‖u∗ − u0 ‖ ≤ t ∗ . Proof of Proposition 3.23. We check the assumptions of Lemma 3.26 in X = C 0 (B) for the operator S defined by (3.119) and ϕ(t) = ϕ(0) + Γ(eℓ − t − 1), First, we have

󵄩 󵄩 ϕ(0) = (1 + Γ)󵄩󵄩󵄩U0 − K(U0 )󵄩󵄩󵄩.

S(U0 ) − U0 = −(1 − KU󸀠 0 ) (U0 − K(U0 )), −1

and hence 󵄩󵄩 󵄩 󵄩 󵄩 󸀠 −1 󵄩 󵄩 󸀠 󵄩󵄩S(U0 ) − U0 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩(1 − KU0 ) 󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩U0 − Is (U0 )󵄩󵄩󵄩 󵄩 󵄩 ≤ (1 + Γ)󵄩󵄩󵄩U0 − K(U0 )󵄩󵄩󵄩 = ϕ(0).

To verify (3.124), we make use of the integral kernel of the linear operator K 󸀠 (U0 ), 󵄨 󵄨2 ρ(x, y) = λG(z, w)󵄨󵄨󵄨g 󸀠 (w)󵄨󵄨󵄨 eU0 (w) > 0,

(3.127)

recalling (3.118). Then it holds that

(KU󸀠−1 KU󸀠 )[h] = e−U0 +U ⋅ h, 0 where ⋅ denotes multiplication. Since S󸀠 (U) = (1 − KU󸀠 0 ) KU󸀠 0 (KU󸀠 −1 KU󸀠 − 1), 0 −1

inequality ‖U − U0 ‖X ≤ t implies 󵄩󵄩 󸀠 󵄩󵄩 󵄩󵄩 󵄩 󸀠 −1 󸀠 󵄩 󵄩 󸀠 −1 󸀠 󵄩󵄩S (U)󵄩󵄩 ≤ 󵄩󵄩(1 − KU0 ) KU0 󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩KU0 KU − 1󵄩󵄩󵄩 󵄩 󵄩 ≤ Γ󵄩󵄩󵄩e−U0 +U − 1󵄩󵄩󵄩X ≤ Γ exp(‖U0 − U‖X − 1) ≤ ϕ󸀠 (t).

Seeking the value t ∗ > 0 with ϕ(t ∗ ) = t ∗ , we put ψ(t) = ϕ(t) − t. Since ψ󸀠 (t) = , which is denoted by t0 . If Γ(e − 1) − 1, one has ψ󸀠 (t) = 0 if and only if t = log 1+Γ Γ ψ(t0 ) ≤ 0, the function ϕ admits a fixed point t ∗ ≤ t0 . Since t

ψ(t0 ) = ϕ(0) − (1 + Γ) log inequality ψ(t0 ) ≤ 0 is equivalent to

that is, (3.121).

1+Γ , Γ

1+Γ 󵄩󵄩 󵄩 − (1 + Γ)−1 , 󵄩󵄩U0 − K(U0 )󵄩󵄩󵄩 ≤ log Γ

3.4 Singular perturbation

| 157

3.4.2 Moseley–Wente’s theory Moseley’s refinements of the above theory are listed as follows [228, 229]: 1. Equation (3.116) means that δ ∈ B is a critical point of R̃ = (1 − |ζ |2 )2 |g 󸀠 (ζ )|2 , and hence it actually exists. 2. Using the freedom of g : B → Ω up to the Möbius transformation, the above δ is reduced to 0 ∈ B and in this case the conditions (3.116) and D(δ) ≠ 0 in (C) are equivalent to g 󸀠󸀠 (0) = 0 and σ ≡ |g 󸀠󸀠󸀠 (0)/g 󸀠 (0)| ≠ 2, respectively. We may suppose also g 󸀠 (0) > 0. 3. In the form of (3.27) of the Liouville integral, the requirements for the construction of the third asymptotic solution that P(ζ ) has at most one zero on |ζ | in (C1) and (C2) are not necessary. In fact, by this formulation the condition ρ(F) > 0 follows always from 0 < λ ≪ 1. Conditions g 󸀠󸀠 (0) = 0 and σ ≠ 2 are thus sufficient to construct asymptotic solutions {(λ, Ũ n )} for any n. 4. If Ω ⊂ R2 is convex, the solution δ ∈ B of (3.116) is unique. Furthermore, after transforming it to 0, it holds that σ < 2. There is a complex function theoretic proof of this fact. 5. Weston’s method works even for − Δu = λ(eu + a(x)e−u )

in Ω,

u|𝜕Ω = 0

(3.128)

if Ω ⊂ R2 is simply-connected and a(x) is real-analytic. The latter fact was used by Wente to construct soap bubbles of genus 1, where spectral properties of the linearized operator are examined in details. Although Wente’s surface is represented by an elliptic function, Wente contributed to the theory of partial differential equations in the following directions: 1. Weston’s condition (C3) is reduced to (3.123) for α and β defined by b23 , b32 = μα + o(μ),

b22 , b33 = μβ + o(μ)

∞ ∞ for bij = (1 − Λ−1 i )δij + (Lϕj , ϕi )ν0 + (LMϕj , ϕi )ν0 and {Λj }j=1 and {ϕj }j=1 denoting the eigenvalues and the eigenfunctions of K0 in X = C 0 (B) with the kernel G(x, y)ν0 (y), where

ν0 (y) =

8μ , (|y|2 + μ)2

μ=

λ 󸀠 2 g (0) > 0. 8

Furthermore, (⋅, ⋅)ν0 denotes the inner product in Y = L2 (B, ν0 (x)dx), (f , g)ν0 = ∫ f (x)g(x)ν0 (x) dx, B

158 | 3 Complex structure the eigenfunctions are normalized in Y, and finally, the operators M, L are defined by KU󸀠 0 = K0 + L, M = (1 − N)−1 L, K0 = P + N, and 4

P = ∑(⋅, ϕj )ν0 ϕj . j=2

2.

If Ω is symmetric with respect to both x1 and x2 axes, and σ < 2, Weston’s condition (C3) actually holds. In the case of a(x) ≡ −1 in (3.129), in particular, the quasi-Newton iteration (3.119) converges in X = C 0 (B), if U0 is the fourth order asymptotic solution and 0 < λ ≪ 1.

Condition (C3) is related to the spectrum of a linear operator. First, using the kernel ρ(x, y) of KU󸀠 0 in (3.127), we deduce ν(ζ ) ≡ λ|g 󸀠 (ζ )|2 eU0 (ζ ) ∼ ν0 (ζ ) in the sense that 1 ‖L‖ = O(λ1/2 log ). λ Second, to study the principal part K0 of KU󸀠 0 , we take the eigenvalue problem Δϕ +

1 ν (x)ϕ = 0 Λ 0

in B,

ϕ|B = 0.

Using the separation of variables, ϕ = Φ(r)e𝚤mθ , and the inverse stereographic trans-

formation ξ =

μ−r 2 , μ+r 2

we reach the associated Legendre equation (3.51). Then its kth

eigenvalue is denoted by Λm k for m = 0, ±1, ±2, . . . Among these eigenvalues, we ana±1 0 lyze those which satisfy Λm k → 1 as μ ↓ 0. They are actually Λ1 and Λ2 , and then we −1 −1 −1 obtain ‖(1 − N) ‖ = O(1). Using M = (1 − N) L and (1 − N) P = P, we arrive at (1 − KU󸀠 0 )

−1

= (1 − M)−1 P(1 − KU󸀠 0 )

−1

+ (1 − M)−1 (1 − N)−1 ,

and then apply the Lyapunov–Schmidt procedure. Then we end up with the finitedimensional problem Pv = (1 − M)−1 Pv + P(1 − M)−1 (1 − N)−1 w for a given w. Performing a fine asymptotic analysis, we see that the problem is reduced to the invertibility of the 3 × 3 matrix A=(

μ(

β α 0

α ) β

0 −1

− 32 (log μ1 )

)

where condition (C3) comes in. To verify it, however, some more analysis is needed, based on the orthogonality of {ϕij } in L2 (B, ν0 (x)dx), which is not illustrated here. Singular perturbation requires rather hard analysis. Concerning the exact problem of a(x) ≡ 0, however, we have a simple criterion for the construction of one-point blow-up solutions [331].

3.4 Singular perturbation

1. 2.

| 159

If a(x) ≡ 0, Moseley’s asymptotic solutions {Ũ n }∞ n=1 converge if 0 < λ ≪ 1. The condition g 󸀠󸀠 (0) = 0 with σ ≠ 2 is equivalent to saying that κ = g(0) ∈ Ω is a 1 log |x−y|]y=x . nondegenerate critical point of the Robin function R(x) = [G(x, y)+ 2π

Once the solution is constructed by the singular perturbation, any property concerning it can be taken, up to an arbitrary order of λ.23 Theorem 3.27. Each nondegenerate critical point κ ∈ Ω of the Robin function R(x) admits a one-point blow-up family of solutions {(λ, u)} to −Δu = λeu

in Ω,

u|𝜕Ω = 0

with the blow-up point κ. 3.4.3 Convergence of the asymptotic expansion Here we show that Moseley’s asymptotic solution converges as n → +∞, under the criterion for the existence of asymptotic solutions, that is, nondegeneracy of the critical point κ ∈ Ω of R(x). Thus we prove Theorem 3.27. We take the conformal mapping g : B ≡ {ζ < 1} → Ω such that g(0) = κ ∈ Ω. The Green function G = G(x, y) on Ω is represented by h = g −1 : Ω → B as in G(x, y) = −

󵄨󵄨 h(z) − h(w) 󵄨󵄨 1 󵄨󵄨 󵄨 log󵄨󵄨󵄨 󵄨, 󵄨󵄨 1 − h(w)h(z) 󵄨󵄨󵄨 2π

(3.129)

where z = x1 + 𝚤x2 , w = y1 + 𝚤y2 for x = (x1 , x2 ), y = (y1 , y2 ). We obtain, therefore, G(x, y) = −

1 1 − |h(w)|2 1 log |z − w| + log + O(|z − w|), 2π 2π |h󸀠 (w)|

so that R(x) =

1 2 󵄨 󵄨2 log{󵄨󵄨󵄨g 󸀠 (ζ )󵄨󵄨󵄨 (1 − |ζ |2 ) }, 4π

x = g(ζ ),

w → z,

ζ ∈ B.

Then it follows that 4π

g 󸀠󸀠 (ζ ) 2ζ 𝜕 𝜕 R(x) = 4π R(x) = 󸀠 − , 𝜕ζ g (ζ ) 1 − |ζ |2 𝜕ζ

2

2

g 󸀠󸀠 (ζ ) g 󸀠󸀠 (ζ ) 2ζ 𝜕2 𝜕2 4π 2 R(x) = 4π 2 R(x) = 󸀠 −( 󸀠 ) − , g (ζ ) g (ζ ) 𝜕ζ (1 − |ζ |2 )2 𝜕ζ 4π

𝜕2

𝜕ζ 𝜕ζ

R(x) = −

2 , (1 − |ζ |2 )2

23 The family of solutions in Theorem 3.27 is unique. The construction of multiple blow-up solutions is done by a different method [28]. Then the nondegeneracy is relaxed by the C 1 stability [108].

160 | 3 Complex structure and therefore, κ = g(0) ∈ Ω is a nondegenerate critical point of R(x) if and only if g 󸀠󸀠 (0) = 0 and σ ≡ |g 󸀠󸀠󸀠 (0)/g 󸀠 (0)| ≠ 2.24 Let Ω ⊂ R2 be a simply-connected domain and u = u(x) a solution to −Δu = λeu

in Ω.

Then, by using Liouville’s integral, there is a pair of holomorphic functions {ψ1 , ψ2 } satisfying ω(ψ1 , ψ2 ) = 1 w ≡ e−u/2

in Ω, λ = |ψ1 |2 + |ψ2 |2 , 8

where ω(ψ1 , ψ2 ) = ψ1 ψ󸀠2 − ψ󸀠1 ψ2 denotes the Wronskian of two holomorphic functions ψ1 (z) and ψ2 (z). In fact, {ψ1 (z), ψ2 (z)} forms a system of fundamental solutions to a linear equation of second order with holomorphic coefficients, and therefore, they are single-valued because Ω is simply-connected. The singular limit u0 (x) = 8πG(x, κ) takes the form 󵄨 󵄨2 w0 (x) ≡ e−u0 (x)/2 = 󵄨󵄨󵄨h(x)󵄨󵄨󵄨 by (3.129) and since h(κ) = 0, where h = g −1 : Ω → B, which is pulled-back to B. Hence W = g ∗ w ≡ w ∘ g satisfies W ≡ |Ψ1 |2 +

λ |Ψ |2 8 2

with ω(Ψ1 , Ψ2 ) = Ψ1 Ψ󸀠2 − Ψ󸀠1 Ψ2 = g 󸀠 , W|𝜕B = 1, and W(ζ ) → W0 (ζ ) ≡ |ζ |2 ,

(3.130)

where Ψ1 (x) and Ψ2 (x) are holomorphic functions on B. From the proof of Theorem 3.5, the asymptotics (3.130) splits into Ψ1 (ζ ) → ζ and having {Ψ2 } bounded, uniformly on B, if a one-point blow-up actually occurs. Taking this property into account, we assume Ψ1 = ζ /G and Ψ2 = M/G. Noting ω(ζ /G, M/G) = ω(ζ , M)/G2 , we construct a family {G, M} of pairs of holomorphic functions satisfying ω(ζ , M) = g 󸀠 G2

in B,

‖G − 1‖L∞ (B) → 0,

λ |M|2 8 = O(1).

|G|2 = 1 +

‖M‖L∞ (B)

on 𝜕B, (3.131)

This family {G, M} of holomorphic functions corresponds to {(λ, u)}, the family of onepoint blow-up solutions to (3.111). 24 See [292] for applications to fluid mechanics.

3.4 Singular perturbation

| 161

To see this, we induce the equation for w = e−u/2 to satisfy, that is, wΔw − |∇w|2 =

λ , 2

w>0

in Ω,

w|𝜕Ω = 1.

This relation is pulled back to W = g ∗ w as in WΔW − |∇W|2 =

λ 󵄨󵄨 󸀠 󵄨󵄨2 󵄨g 󵄨 , 2󵄨 󵄨

W >0

in B,

W|𝜕B = 1.

(3.132)

The Liouville integral now takes the form W = |ζ /G|2 +

λ |M/G|2 8

(3.133)

where {G, M} is a pair of holomorphic functions satisfying (3.131).25 The bilinear form 1 {V, W} = (VΔW + WΔV) − ∇V ⋅ ∇W 2 may be useful to confirm (3.132). In fact, equation (3.132) is written as {W} =

λ 󵄨󵄨 󸀠 󵄨󵄨2 󵄨g 󵄨 , 2󵄨 󵄨

W >0

in B,

W|𝜕B = 1,

where {W} stands for {W, W}. Lemma 3.28. For holomorphic functions p(ζ ) and q(ζ ) in ζ ∈ B, it holds that 󵄨 󵄨2 {|p|2 , |q|2 } = 2󵄨󵄨󵄨ω(p, q)󵄨󵄨󵄨 , and, in particular, {|p|2 } = 0. Proof. Writing ζ = ξ1 + 𝚤ξ2 , ζ = ξ1 − 𝚤ξ2 for ξ = (ξ1 , ξ2 ) ∈ B, we obtain {V, W} = 2(VWζ ζ + Vζ ζ W − Vζ Wζ − Vζ Wζ ), and hence 1 󵄨 󵄨 󵄨2 󵄨 󵄨2 󵄨 {|p|2 , |q|2 } = |p|2 󵄨󵄨󵄨g 󸀠 󵄨󵄨󵄨 + 󵄨󵄨󵄨p󸀠 󵄨󵄨󵄨 |q|2 − pp󸀠 q󸀠 q − pp󸀠 q󸀠 q = 󵄨󵄨󵄨pq󸀠 − p󸀠 q󵄨󵄨󵄨. 2 For the function W in (3.133), therefore, it holds that λ󵄨 λ 󵄨2 {|ζ /G|2 , |M/G|2 } = 󵄨󵄨󵄨ω(ζ /G, M/g)󵄨󵄨󵄨 4 2 λ 󵄨󵄨 λ 󵄨󵄨 󸀠 󵄨󵄨2 2 󵄨󵄨2 = 󵄨󵄨ω(ζ , M)/G 󵄨󵄨 = 󵄨󵄨g 󵄨󵄨 , 2 2

{W} = {W, W} =

and hence (3.134) is true. The following lemma is elementary. 25 See [150] for a real-analytic approach to (3.132).

(3.134)

162 | 3 Complex structure Lemma 3.29. Given a holomorphic function K = K(ζ ) there exists a holomorphic function M = M(ζ ) satisfying ω(ζ , M) = (ζ

d − 1)M = K dζ

in B

if and only if K 󸀠 (0) = 0. This M is given as M = α(K) + aζ , where kn n ζ n=1̸ n − 1

α(K)(ζ ) = ∑

(3.135)

n for K(ζ ) = ∑∞ n=0 kn ζ and a ∈ C is arbitrary.

We use the Hardy–Lebesgue space [373] HL = {H(ζ ) : holomorphic in D | ‖H‖2HL < +∞}, 2π

1 󵄨󵄨 󵄨2 ∫ 󵄨󵄨H(re𝚤θ )󵄨󵄨󵄨 dθ. 0≤r ∫ 󵄨α(g )󵄨󵄨 | dζ ̃ |2 . 2π 󵄨 𝜕B

There exists λ = λ (‖H‖X ) such that the mapping a 󳨃→ ϕ(a, H, λ) is a contraction on BR (0) = {|a| ≤ R} ⊂ C, provided that 0 < λ < λ∗ . Hence equation (3.140) is uniquely solvable with respect to a ∈ 𝒞 for 0 < λ ≪ 1, provided that the left-hand side, a linear equation in a, is invertible. This invertibility is equivalent to |I0 (0)/c0 | ≠ 1, or |g 󸀠󸀠󸀠 (0)/g 󸀠 (0)| ≠ 2. More precisely, if a0 ∈ C denotes the unique solution to the linear algebraic equation ∗

∗

a0 c0 + a0 I0 (0) = −

1 󵄨 󵄨2 d ζ ̃ ∫ 󵄨󵄨󵄨α(g 󸀠 )󵄨󵄨󵄨 (ζ ) , 2π𝚤 ζ ̃2 𝜕B

the solution a ∈ C to (3.140) takes the form a(H, λ) = a0 + λa1 (H, λ),

(3.141)

where a1 = a1 (H, λ) : ℳ → C is a bounded operator, locally Lipschitz continuous on every bounded set in ℳ = {(H, λ) | H ∈ X, 0 < λ < λ (‖H‖X )}. ∗

Substituting the relation into the right-hand side of (3.139), we obtain a fixed point equation H = 𝒩 (H, λ) ≡ H0 + λ𝒩1 (H, λ),

(3.142)

for H0 (ζ ) =

|a0 |2 1 + ((a0 c0 + a0 I0 (ζ ))(ζ ) 16 8 1 1 1 1 + − ) dζ ̃ ∈ X ∫ |α(g 󸀠 )(ζ ̃ )|2 ( 2π𝚤 8 ζ ̃ − ζ 2ζ ̃ 𝜕B

and 𝒩1 (H, λ) =

1 λ󵄨 󵄨2 1 Re a1 (H, λ) + 󵄨󵄨󵄨a1 (H, λ)󵄨󵄨󵄨 + (a1 (H, λ)c0 8 16 8 1 1 1 + a1 (H, λ)I0 (ζ ))ζ + − ) dζ ̃ . ∫ Φ(H, a0 + λa1 (H, λ), λ)(ζ ̃ )( 2πι ζ ̃ − ζ 2ζ ̃ 𝜕B

3.4 Singular perturbation

| 165

The operator 𝒩1 : ℳ → X is bounded and locally Lipschitz continuous on every bounded set. To see the unique solvability of (3.142), let B ≡ {H ∈ X | ‖H − H0 ‖X ≤ 1}. Then the mapping H 󳨃→ 𝒩 (H, λ) is a contraction on B if 0 < λ ≪ 1. Hence we complete the proof of Theorem 3.17. 3.4.4 Area functional Inequalities ̃ 󵄩 󵄩󵄩 u/2 −1 󵄩󵄩e 󵄩󵄩󵄩∞ = O(λ ),

󵄩󵄩 u/2 u/2 󵄩 k 󵄩󵄩e − e 󵄩󵄩󵄩∞ = O(λ )

(3.143)

are valid between the genuine solution u and the kth asymptotic solution u,̃ where k = 1, 2, . . . Then we obtain a precise asymptotics of Σ → 8π in accordance with the profile in conformal geometry of Ω [340]. Recall that g(0) = κ ∈ Ω is a critical point of R(x) if and only if g 󸀠󸀠 (0) = 0, and let ∞

g(z) = κ + a1 z + ∑ ak z k .

(3.144)

∫ λeu dx = 8π + Cλ + o(λ)

(3.145)

k=3

Proposition 3.31. It holds that ∗

Ω

with

C π

= −|a1 |2 + ∑∞ k=3

2

k |a |2 . k−2 k

Proof. Let ũ be the kth order asymptotic solution. Then (3.143) implies ‖e O(λk−1 ), and hence ‖e

u−ũ 2

‖∞ = O(1) by k ≥ 1. Then it follows that

u−ũ 2

− 1‖∞ =

󵄩󵄩 u−ũ 󵄩 − 1󵄩󵄩󵄩∞ = O(λk−1 ), 󵄩󵄩e and hence (3.145) is reduced to ̃ Σ̃ ≡ ∫ λeu dx = 8π + Cλ + o(λ) Ω

̃ u−ũ − 1‖∞ . for k = 3, the third asymptotic solution ũ satisfying |Σ̃ − Σ| ≤ Σ‖e 󸀠 2 U Let U = ũ ∘ g for this u.̃ Since −ΔU = λ|g | e in B, we obtain ̃ 󵄨 󵄨2 Σ0 = ∫ λeu dx = ∫ λeU 󵄨󵄨󵄨g 󸀠 󵄨󵄨󵄨 dx = − ∫ ΔU dx B

Ω

𝜕U ds, =−∫ 𝜕r 𝜕B

B

r = |x|.

(3.146)

166 | 3 Complex structure In § 3.4.3 the asymptotic solution U is given by e−U(ζ )/2 =

|ζ |2 + (λ/8)|A(ζ , λ)|2 , |G(ζ , λ)|2

ζ ∈B

with G(ζ , λ) = 1 + λG1 (ζ ) + ⋅ ⋅ ⋅ + λk−1 Gk−1 (ζ ), k = 3, and ζ

g 󸀠 (ζ ) ̂ dζ . A(ζ , λ) = ζ ∫ G(ζ ̂ , λ)2 ζ ̂2 We have −

1 𝜕U −U/2 λ 𝜕 󵄨󵄨 󵄨2 󵄨 󵄨2 e = (2r + 󵄨󵄨A(ζ , λ)󵄨󵄨󵄨 )/󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 2 𝜕r 8 𝜕r λ󵄨 󵄨2 𝜕 󵄨 󵄨2 󵄨 󵄨4 − (r 2 + 󵄨󵄨󵄨A(ζ , λ)󵄨󵄨󵄨 ) 󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 /󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 , 8 𝜕r

| and then, G(ζ , 0) = 1 and U|𝜕B = O(λk ) imply − 21 𝜕U = 2. It thus holds that 𝜕r r=1,λ=0 𝜕U ds = 8π + O(λ). Σ̃ = − ∫ 𝜕r 𝜕B

Next we have −

1 𝜕 𝜕U −U/2 1 𝜕U 𝜕U −U/2 𝜕 λ 𝜕 󵄨󵄨 󵄨2 󵄨 󵄨2 e + e = ((2r + 󵄨A(ζ , λ)󵄨󵄨󵄨 )/󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 ) 2 𝜕λ 𝜕r 4 𝜕r 𝜕λ 𝜕λ 8 𝜕r 󵄨 λ󵄨 𝜕 󵄨2 𝜕 󵄨 󵄨2 󵄨 󵄨4 − ((r 2 + 󵄨󵄨󵄨A(ζ , λ)󵄨󵄨󵄨 ) 󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 /󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 ) 𝜕λ 8 𝜕r = I − II,

with 1 𝜕 󵄨󵄨 󵄨2 λ 𝜕 𝜕 󵄨󵄨 󵄨2 󵄨 󵄨2 󵄨A(ζ , λ)󵄨󵄨󵄨 + 󵄨A(ζ , λ)󵄨󵄨󵄨 )/󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 8 𝜕r 󵄨 8 𝜕λ 𝜕r 󵄨 λ 𝜕 󵄨󵄨 󵄨2 𝜕 󵄨 󵄨2 󵄨 󵄨4 − (2r + 󵄨A(ζ , λ)󵄨󵄨󵄨 ) 󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 /󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 8 𝜕r 󵄨 𝜕λ

I=(

and 1󵄨 󵄨2 λ 𝜕 󵄨󵄨 󵄨2 𝜕 󵄨 󵄨2 󵄨 󵄨4 II = ( 󵄨󵄨󵄨A(ζ , λ)󵄨󵄨󵄨 + 󵄨󵄨A(ζ , λ)󵄨󵄨󵄨 ) 󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 /󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 8 8 𝜕λ 𝜕r λ 󵄨󵄨 󵄨󵄨2 𝜕 𝜕 󵄨 󵄨4 2 + (r + 󵄨󵄨A(ζ , λ)󵄨󵄨 ) |G(ζ , λ|2 /󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 8 𝜕λ 𝜕r λ󵄨 󵄨2 𝜕 󵄨 󵄨2 𝜕 󵄨󵄨 󵄨2 󵄨 󵄨6 − 2(r 2 + 󵄨󵄨󵄨A(ζ , λ)󵄨󵄨󵄨 ) 󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 󵄨G(ζ , λ)󵄨󵄨󵄨 /󵄨󵄨󵄨G(ζ , λ)󵄨󵄨󵄨 , 8 𝜕r 𝜕λ 󵄨

| 167

3.4 Singular perturbation

which implies I|λ=0 =

1 𝜕 󵄨󵄨 󵄨2 󵄨A (ζ )󵄨󵄨 − 4r Re G1 (ζ ), 8 𝜕r 󵄨 0 󵄨

II|λ=0 = 2r 2

𝜕 Re G1 (ζ ), 𝜕r

for A0 (ζ ) = A(ζ , 0). Hence −

1 𝜕 󵄨󵄨 𝜕 1 𝜕 𝜕U 󵄨󵄨󵄨󵄨 󵄨2 = 󵄨 󵄨A (ζ )󵄨󵄨 − 4 Re G1 (ζ ) − 2 G1 (ζ )||ζ |=1 . 2 𝜕λ 𝜕r 󵄨󵄨󵄨λ=0,r=1 8 𝜕r 󵄨 0 󵄨 𝜕r

Here we recall the relations in § 3.4.3, namely 󵄨 󵄨2 16 Re G1 (ζ ) = |c0 |2 + 2 Re(−g 󸀠 (0)c0 ζ + c0 I0 (ζ )) + 󵄨󵄨󵄨−g 󸀠 (0) + ζI0 (ζ )󵄨󵄨󵄨 and A0 (ζ ) = −g 󸀠 (0) + ζI0 (ζ ) + c0 ζ , where c0 ∈ C is a constant and ζ

dζ ̂ I0 (ζ ) = ∫(g 󸀠 (ζ ̂ ) − g 󸀠 (0)) . ζ ̂2 0

Since g is normalized by g 󸀠 (0) > 0, it holds that 2 Re G1 (ζ ) = Furthermore, G1 (ζ ) is holomorphic in B, and hence −∫ 𝜕B

1 |A (ζ )|2 8 0

𝜕 Re G1 ds = − ∫ Δ(Re G1 ) dx = 0. 𝜕r B

Equality (3.146) thus holds with C =−∫− 𝜕B

Setting A0 (ζ ) =

𝜕 𝜕U 󵄨󵄨󵄨󵄨 1 𝜕 󵄨󵄨 󵄨2 ds = ∫ 󵄨A (ζ )󵄨󵄨 − 󵄨 𝜕λ 𝜕r 󵄨󵄨󵄨λ=0,r=1 4 𝜕r 󵄨 0 󵄨 𝜕B

n ∑∞ n=0 bn ζ ,

1 󵄨󵄨 󵄨2 󵄨A (ζ )󵄨󵄨 ds. 2󵄨 0 󵄨

we have ∞

2π

∞

󵄨 󵄨2 ∫ 󵄨󵄨󵄨A0 (ζ )󵄨󵄨󵄨 ds = ∑ ∫ bn bm e𝚤(n−m)θ dθ = 2π ∑ |bn |2 . n=0

n,m=0 0

𝜕B

Similarly, it holds that 2π

∞ 𝜕󵄨 󵄨2 ∫ 󵄨󵄨󵄨A0 (ζ )󵄨󵄨󵄨 ds = ∑ (n + m)bn bm ∫ e𝚤(n−m)θ dθ 𝜕r n+m≥1 0

𝜕B

∞

= 4π ∑ n|bn |2 , n=0

and hence ∞ C = −|b0 |2 + ∑ (n − 1)|bn |2 . π n=2

on |ζ | = 1.

168 | 3 Complex structure n In terms of g(ζ ) = ∑∞ n=0 an ζ with a2 = 0, we have ∞

A0 (ζ ) = −g 󸀠 (0) + C0 ζ + ζI0 (ζ ) = −g 󸀠 (0) + c0 ζ + ∑ an+1 n=2

n+1 n ζ , n−1

which implies b0 = −g 󸀠 (0) = −a1 and bn = an+1 (n + 1)/(n + 1) for n ≥ 2. Then (3.145) follows. If Ω is the exact disc, then we have ak = 0 for k ≥ 3, and hence C/π = −|a1 |2 < 0. The following fact is a consequence of Bieberbach’s area theorem,26 where K(ζ ) denotes the curvature of 𝜕Ω at g(ζ ) ∈ 𝜕Ω for ζ ∈ 𝜕B. If Ω is a disc, again, it holds that K|g 󸀠 | ≡ 1. Proposition 3.32. If K|g 󸀠 | < 2 everywhere on 𝜕B, then C < 0 follows. Proof. Take ξ ∈ (0, 1), and put gξ (ζ ) = (g(ζ ) − g(0))/ξg 󸀠 (0),

∞

hξ (ζ ) = ζgξ󸀠 (ζ ) = ∑ dn ζ n . n=0

Since g(ζ ) is univalent in B, so is gξ (ζ ). We have dn =

n a 1 (n) h (0) = ⋅ n , n! ξ ξ a1

and, in particular, d0 = d2 = 0 and d1 = 1/ξ . Note that C < 0 follows from 2 1 1 ∞ n2 󵄨󵄨󵄨󵄨 an 󵄨󵄨󵄨󵄨 |dn+2 |2 = 2 ∑ 󵄨󵄨 󵄨󵄨 ≤ 1. ξ n=3 n − 2 󵄨󵄨 a1 󵄨󵄨 n=1 n ∞

∑

(3.147)

n The function ωξ (ζ ) = ζ1 + ∑∞ n=1 cn ζ for cn = −dn+2 /n is holomorphic in 0 < |ζ | < 1. If it is univalent, inequality (3.147) follows from the area theorem as ∞

1 |dn+2 |2 ≤ 1. n=1 n ∞

∑ n|cn |2 = ∑

n=1

The image Γr of cr = {|z| = r}, 0 < r < 1, by ωξ is a closed curve. Note that the univalence of ωξ follows if Γr is a Jordan curve and the winding number of ζ ∈ cr 󳨃→ ωξ (ζ ) ∈ Γr is −1 for each r close to 1. To see this property, let 𝒞 be the Riemann sphere 𝒞 ∪ {∞} and τ : 𝒞 → 𝒞 \ {n} be the canonical bijection, n being the north pole. The pole ζ = 0 of ωξ (ζ ) has first order, and hence ωξ (ζ ) extends conformally as τ ∘ ωξ : B → 𝒞 . Then we take ι : 𝒞 → 𝒞 , a rotation of the Riemann sphere, such that the image of ι ∘ τ ∘ ωξ : Br = {|z| < r} → 𝒞 does not contain ∞. Therefore, the mapping τ−1 ∘ ι ∘ τ ∘ ωξ 26 See [245].

3.4 Singular perturbation

| 169

is holomorphic in Br with the image Ωr surrounded by a Jordan curve γr . This time, the winding number of ζ ∈ cr 󳨃→ τ−1 ∘ ι ∘ τ ∘ ωξ (ζ ) ∈ γr is +1 and τ−1 ∘ ι ∘ τ ∘ ωξ is univalent in Br by Darboux’s theorem [245]. Hence the same is true for ωξ (ζ ) in 0 < |ζ | < 1, because r is taken arbitrarily close to 1. Here we note a relation derived from d0 = d2 = 1 and d1 = 1/ξ , that is, ω󸀠ξ (ζ ) = −

1 1 1 1 1 1 hξ (ζ ) + ( − 1) 2 = − 2 gξ󸀠 (ζ ) + ( − 1) 2 . 3 ξ ξ ζ ζ ζ ζ

In fact, we have (ωξ (ζ ) − 1/ζ )󸀠 = ζ −3 (hξ (ζ ) − d1 ζ ), and therefore, 1 1 𝜕 𝜕 ω (re𝚤θ ) = 𝚤ζω󸀠ξ (ζ ) = − 2 ⋅ 𝚤ζHξ󸀠 (ζ ) = − 2 Hξ (re𝚤θ ) 𝜕θ ξ ζ ζ 𝜕θ for ζ = re𝚤θ , 0 ≤ θ < 2π, and Hξ (ζ ) = gξ (ζ ) + (1 − 1/ξ )ζ . Hence we obtain Sr,ξ (θ) = 2−2𝚤θ Tr,ξ (θ)

(3.148)

for Sr,ξ (θ) ∈ 𝜕B and Tr,ξ (θ) ∈ 𝜕B defined by Sr,ξ (θ) =

𝜕 ω (re𝚤θ ) 𝜕θ ξ , 𝜕 | 𝜕θ ωξ (re𝚤θ )|

Tr,ξ (θ) =

𝜕 H (re𝚤θ ) 𝜕θ ξ . 𝜕 | 𝜕θ Hξ (re𝚤θ )|

The holomorphic function gξ (ζ ), ξ > 0, is univalent, so that the winding number of ζ = re𝚤θ ∈ cr 󳨃→ T̃ r,ξ (θ) ∈ 𝜕B is +1, where T̃ r,ξ (θ) =

𝜕 g (re𝚤θ ) 𝜕θ ξ . 𝜕 | 𝜕θ gξ (re𝚤θ )|

The winding number of ζ = re𝚤θ ∈ cr 󳨃→ Tr,ξ (θ) ∈ 𝜕B, therefore, is also +1 whenever ξ is

close to 1, and consequently, that of ζ = re𝚤θ ∈ cr 󳨃→ Sr,ξ (θ) ∈ 𝜕B is equal to −1 by (3.148). We have thus confirmed that (3.147) arises if ωξ (ζ ) is one-to-one on cr = {|z| = r} when ξ and r are close to 1. 𝜕 (Arg Sr,ξ (θ)) < 0 for 0 ≤ θ < 2π, which A simple criterion for this property is 𝜕θ means 𝜕 (Arg Tr,ξ (θ)) < 2, 𝜕θ

0 ≤ θ < 2π.

Inequality (3.149) for ξ and r close to 1 follows from 𝜕 (Arg T(θ)) < 2, 𝜕θ where T(θ) = T1,1 (θ) = g 󸀠 (e𝚤θ )/|g 󸀠 (e𝚤θ )| ∈ 𝜕B.

0 ≤ θ < 2π,

(3.149)

170 | 3 Complex structure Since the unit tangent vector e1 on 𝜕Ω at g(e𝚤θ ) is T(θ), it holds that cos t(θ) ), sin t(θ)

e1 = (

t(θ) = Arg T(θ)

and hence the inner unit normal vector becomes − sin t(θ) ). cos t(θ)

e2 = (

θ

The length parameter, on the other hand, is represented by ℓ = ∫0 |g 󸀠 (e𝚤ω )| dω. Therefore, it holds that Ke2 =

− sin t(θ) 󸀠 dθ d e =( ) t (θ) . cos t(θ) dℓ 1 dt

We thus obtain t 󸀠 (θ) = K|g 󸀠 |, and hence K|g 󸀠 | < 2 everywhere on 𝜕B implies C < 0. The method of scaling guarantees equality (3.145) for general one-point blow-up solution on the simply-connected domain Ω, including the case when x0 ∈ Ω is a degenerate critical point of R(x) [72].

4 Rearrangement Rearrangement reduces a problem to that with more symmetries, and hence fits isoperimetric phenomena. The first section, § 4.1, of this chapter describes the classical theorem of Faber–Krahn, the isoperimetric inequality for the first eigenvalue of the Laplacian under the Dirichlet boundary condition. Meanwhile, a fine feature of the nodal domain based on a theorem of Hartman–Wintner is given. The second section, § 4.2, describes the equimeasurable transformation and its applications, that is, comparison theorem of Talenti for linear elliptic equations, control of the singular set of their solutions, and that of supersolution to the parabolic equation. In the third section, § 4.3, potential theory for spherically harmonic functions is developed as in the Harnack principle, via Nehari’s isoperimetric inequality. Then, it is shown that the theory of Bandle is regarded as a spherically decreasing rearrangement on the round sphere. This observation leads to a theorem for the two-dimensional Gel’fand equation with the aid of spectral analysis, that is, a global bifurcation diagram of the solution. Actually, in the previous chapter we have exposed the asymptotic analysis of this equation, but little was mentioned about the global bifurcation analysis. This analysis is naturally reflected to a question of real analysis, that is, existence of the minimizer of the extremal Trudinger–Moser inequality. The final section, § 4.4, is devoted to the study of the normalized Ricci flow. A pure analytic proof is proposed for the classical result of Hamilton, based on Benilan’s inequality and concentration of probability measures in accordance with the variational functional associated with the Trudinger–Moser inequality.

4.1 Elliptic eigenvalue problems 4.1.1 Vibrating membrane Given a thin elastic vibrating membrane, its displacement u ∈ R is a function of the space variable x ∈ Ω and the time variable t > 0, where Ω ⊂ R2 is a bounded domain with a smooth boundary 𝜕Ω. According to the Lagrange mechanics, this u = u(x, t) is a stationary state of J = (potential energy) − (kinetic energy), and hence d J[u + ηv]|η=0 = 0, dη

∀v ∈ C0∞ (Ω × (0, T)).

If the potential energy is proportional to the variance of the area of the membrane and physical constants are normalized to one, it follows that J[v] =

1 ∬ √1 + |∇v|2 − 1 − vt2 dx dt 2

Ω×(0,T) https://doi.org/10.1515/9783110556285-004

172 | 4 Rearrangement and hence d −1/2 J[u + ηv]|η=0 = ∬ (1 + |∇v|2 ) ∇u ⋅ ∇v − ut υt dx dt dη Ω×(0,T)

= ∬ (−∇ ⋅ Ω×(0,T)

∇u + utt )v dx dt √1 + |∇u|2

if u = u(x, t) is regular. Since the test function v = v(x, t) is arbitrary, we obtain utt = ∇ ⋅

∇u

√1 + |∇u|2

in Ω × (0, T),

(4.1)

which is combined with the initial-boundary condition, say, (u, ut )|t=0 = (u0 (x), u1 (x)),

u|𝜕Ω = 0.

(4.2)

Such an equation derived from the variational problem is called the Euler–Lagrange equation. Direct and indirect methods are used to solve the former and the latter, respectively. Here we adopt the linear approximation in (4.1), assuming |∇u| = o(|u|): utt = Δu in Ω × (0, T).

(4.3)

It is called the wave equation, which is combined with (4.2). The principle of separation of variables is to assume u(x, t) = f (t)ϕ(x) in (4.3). It turns out that f 󸀠󸀠 Δϕ = ≡ −λ, f ϕ and therefore, λ is independent of (x, t). We obtain f (t) = A cos √λt + B sin √λt/√λ, where A, B are constants and sin √λt/√λ = t if λ = 0. The boundary condition in (4.2), on the other hand, becomes ϕ, which results in − Δϕ = λϕ

in Ω,

ϕ|𝜕Ω = 0.

(4.4)

It is an eigenvalue problem, and in particular, (4.4) admits the trivial solution ϕ = 0 for any λ ∈ R. What we do is seek for a λ which admits a nontrivial solution ϕ ≢ 0 to (4.4). Such λ and corresponding ϕ ≢ 0 are called an eigenvalue and an eigenfunction, respectively. To normalize the freedom of having multiple constants, the eigenfunction is taken as to satisfy ‖ϕ‖2 = 1. Given a solution (λ, ϕ(x)) to (4.4), thus, the function u(x, t) = (A cos √λt + B sin √λt/√λ)ϕ(x) solves (4.3) and (4.2).

(4.5)

4.1 Elliptic eigenvalue problems | 173

Here the principle of superposition arises to adjust the initial condition in (4.2). The first observation is that any linear combination of the functions of the form (4.5) satisfies (4.3) and the boundary condition in (4.2). Such eigenfunctions form a countable orthonormal basis of L2 (Ω). Therefore, problem (4.3) with (4.2) will be solved by a countable sum of the functions of the form (4.5). Since u(x, t) vibrates with the period 1/√λ in (4.5), the eigenvalues constitute the tone of the object Ω. Then it holds that 0 < λ1 < λ2 ≤ ⋅ ⋅ ⋅ → ∞, where λ1 < λ2 indicates that λ1 is simple, that is, eigenfunctions corresponding to λ1 form a one-dimensional vector space. To reformulate the problem in an abstract manner, let H be a Hilbert space over R with the inner product (⋅, ⋅) and the norm | ⋅ |, respectively, and b = b(u, v) : H × H → R be symmetric, bilinear, and satisfying b(u, υ) ≥ δ|u|2 ,

󵄨󵄨 󵄨 󵄨󵄨b(u, v)󵄨󵄨󵄨 ≤ M|u| ⋅ |v|,

u, v ∈ V,

where δ, M > 0 are constants. Note that this b provides a complete inner product on H. Let V be the other Hilbert space, continuously embedded in H, with the inner product and the norm denoted by ((⋅, ⋅)) and ‖ ⋅ ‖, respectively, and a = a(u, v) : V × V → R be a symmetric bilinear form satisfying a(u, v) ≥ δ‖u‖2 − C|u|2 ,

󵄨󵄨 󵄨 󵄨󵄨a(u, v)󵄨󵄨󵄨 ≤ M‖u‖ ⋅ ‖v‖,

u, v ∈ V,

where C > 0 is a constant. Then the abstract eigenvalue problem is to find (λ, ϕ) ∈ R×V such that a(ϕ, ψ) = λb(ϕ, ψ),

∀ψ ∈ V.

(4.6)

Any λ ∈ R satisfies (4.6) if ϕ = 0, regarded as the trivial solution, and eigenvalues and eigenfunctions are defined similarly. Here is an abstract form of the Hilbert–Schmidt theory.1 Theorem 4.1. The following facts hold if the embedding V ⊂ H is compact: 1. Problem (4.6) admits countably many eigenvalues with finite multiplicities accumulating to +∞. 2. Eigenfunctions {ϕk }∞ k=1 can be a complete orthonormal system of the Hilbert space H equipped with the inner product b = b(u, v). For each λ, the set of solutions {ϕ} to (4.6) form a subspace of V and its dimension is called the multiplicity of λ. Each eigenvalue λk is counted according to the multiplicity. Hence we write −∞ < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ → +∞. The eigenfunction ϕk , corresponding to λk , is selected uniquely, satisfying b(ϕk , ϕk ) = 1. 1 See [25, 47], and [341].

174 | 4 Rearrangement By Theorem 4.1, any v ∈ H admits k

v = s– lim ∑ b(v, ϕj )ϕj k→∞

in H,

j=1

where s– lim indicates the limit in the strong topology. Then it follows that ∞

b(v, v) = ∑ ck2 , k=1

ck = b(v, v),

2 and, furthermore, v ∈ V if and only if a(v, v) = ∑∞ k=1 λk ck < +∞. Introducing

R[v] =

∞ 2 a(v, v) ∑k=1 λk ck = ∞ 2 , b(v, v) ∑k=1 ck

v ∈ V \ {0},

we obtain the principle of Rayleigh described in § 1.3.1. Proposition 4.2. It holds that λ1 = inf{R[v] | v ∈ V \ {0}}, and the minimum is attained by the first eigenfunction ϕ1 . We have also the identities of Courant–Poincaré λk = max{ min R | Vk ⊂ V, dim V/Vk = k − 1} Vk \{0}

= min{max R|Lk ⊂ V, dim Lk = k}, Lk

which imply λk = inf{R[v] | v ∈ V ∩ Hk−1 \ {0}},

Hk = inf{v ∈ H | b(v, ϕj ) = 0, 1 ≤ j ≤ k − 1}. As an example, let Ω ⊂ Rn be a bounded domain with a C 2 boundary 𝜕Ω. For a portion Γ0 ⊂ 𝜕Ω with positive (n − 1)-dimensional Hausdorff measure, we put H = L2 (Ω) and V = {v ∈ H 1 (Ω) | v|Γ0 = 0}, where ⋅|Γ0 is taken in the sense of trace. Thus, ⋅|Γ0 : H 1 (Ω) → H 1/2 (Γ0 ) is a bounded linear operator and v|Γ0 is the restriction of v to Γ0 if v ∈ C 0 (Ω). Given continuous functions c = c(x) and p = p(x) > 0 on Ω, we define the bilinear forms by a(u, v) = ∫ ∇u ⋅ ∇v + c(x)uv dx,

b(u, v) = ∫ p(x)uv dx.

Ω

Ω

In this case problem (4.6) is nothing but (−Δ + c(x))ϕ = λpϕ in Ω,

ϕ|Γ0 ,

𝜕ϕ 󵄨󵄨󵄨󵄨 󵄨 = 0, 𝜕ν 󵄨󵄨󵄨Γ1

where Γ1 = 𝜕Ω \ Γ0 and ν denotes the outer unit normal vector. Sometimes the third equality is referred to as the natural boundary condition because it is not included in the underlying space V.

4.1 Elliptic eigenvalue problems | 175

4.1.2 Hartman–Wintner’s theorem Here we study local behavior of a real-valued C 2 function u = u(x) satisfying |Δu| ≤ C(|∇u| + u)

in Ω ⊂ R2 ,

(4.7)

where C > 0 is a constant. Without loss of generality, we suppose 0 ∈ Ω an write z = x1 + 𝚤x2 for x = (x1 , x2 ) ∈ Ω. Theorem 4.3 ([155]). If u(x) = o(|x|n ),

|x| ↓ 0

(4.8)

for some integer n ≥ 0, the limit limz→0 uz /z n = a ∈ C exists. Hence it holds that uz = az n + o(|x|n ), and therefore, u(x) = Re(

a n+1 z ) + o(|x|n+1 ). n+1

Proof. Since u is C 2 , the assertion is obvious for n = 0, 1. Let n > 1. Then this theorem is reduced to showing that uz = o(|z|k−1 ),

|z| → 0

(4.9)

implies the existence of lim uz /z k = ak

(4.10)

z→0

for k = 1, 2, . . . , n. In fact, first, (4.9) with k = 1 follows from u ∈ C 2 and n ≥ 2 in (4.8). Furthermore, (4.10) implies u(x) = Re(

a k+1 z ) + o(|z|k+1 ), k+1

and hence ak = 0 by (4.9) if k < n. This property yields (4.9) with k replaced by k + 1, and eventually implies the existence of the limit (4.10) for k = n as desired. To prove this property, we suppose Ω = {|x| < R0 }. If ω ⊂⊂ Ω is a subdomain with a smooth boundary 𝜕ω, Green’s formula assures ∫ guz z̄ + uz gz̄ dz dz̄ = ∫ guz dz,

ω

̄ g ∈ C 2 (ω).

𝜕ω

Recall z = x1 + 𝚤x2 and z̄ = x1 − 𝚤x2 for x = (x1 , x2 ) ∈ Ω, and hence 𝜕 1 𝜕 𝜕 = ( −𝚤 ), 𝜕z 2 𝜕x1 𝜕x2

𝜕 1 𝜕 𝜕 = ( +𝚤 ), 𝜕z̄ 2 𝜕x1 𝜕x2

dz ∧ dz̄ = −2𝚤dx1 ∧ dx2 .

176 | 4 Rearrangement We take ζ ∈ C and R > 0 satisfying 0 < |ζ | < R < R0 and put g(z) = z −k (z − ζ )−1 , where k ≥ 0 is a given integer. For ε > 0 sufficiently small, the subdomain ω ⊂⊂ Ω is taken as 𝜕ω = {|z| = R} ∪ {|z| = ε} ∪ {|z − ζ | = ε}. Then this g is holomorphic in ω, and hence gz̄ = 0 there. Thus we obtain ∫ z −k (z − ζ )−1 uz dz = ∫ z −k (z − ζ )−1 uz z̄ dz dz.̄

(4.11)

ω

𝜕ω

The left-hand side of (4.11) is equal to + ∫ )z −k (z − ζ )−1 uz dz.

(− ∫ −

∫

|z|=ε

|z−ζ |=ε

|z|=R

As ε ↓ 0, the first term disappears by (4.9). The second term converges to the mean value, −2π𝚤ζ −k uz (ζ ). The right-hand side of (4.11), finally, converges to ∫|z|t}

u Jf dx, |Df |

and then Corollary 4.24 implies ∞

∞

u u χEt dH n−1 ds = ∫ ∫ dH n−1 ds. ∫ u dx = ∫ ∫ |Df | |Df |

{f >t}

t f −1 (s)

−∞ f −1 (s)

Let n = 2, s be arc-length parameter to the curve {f = t}, and introduce the arclength parameter σ for the orbits perpendicular to these curves {f = t} in the direction of t increasing. Then it holds that |∇f | = |𝜕f /𝜕σ| = dt/dσ, and hence dxdy = dsdσ = |∇f |−1 dsdt. The above formula in Corollary 4.25 is an extension with measure-theoretic justification. The proof of Propositions 4.21 and 4.22, without sufficient regularity of functions, relies on the extension of geometric isoperimetric inequality and coarea formula. Then we use perimeter, m 𝜕ϕk dx | ϕk ∈ C0∞ , max ϕ2k ≤ 1}, ∑ Rm 𝜕x k k=1 k=1 m

P(E) = sup{∫ ∑ E

defined for a Borel set E ⊂ Rm . Then, De Giorgi’s isoperimetric inequality ensures 󵄨 󵄨 1/m P{x | 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 > t} ≥ mCm μ(t)1−1/m

(4.34)

where μ(t) is the distribution function in (4.30), while Fleming–Rishel’s coarea formula is given by ∞

󵄨 󵄨 ∫ |Du| = ∫ P{x | 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 > ξ } dξ . |u|>t

t

(4.35)

188 | 4 Rearrangement These formulas are valid for u of bounded variation, denoted by u ∈ BV(Ω), which means that u ∈ L1 (Ω) and ∫Ω |Du| < +∞, where ∫ |Du| = sup{ ∫ u∇ ⋅ gdx | g = (g1 , . . . , gm ) ∈ C01 (Ω, Rm ),

Ω

Ω

󵄨 󵄨󵄨 󵄨󵄨g(x)󵄨󵄨󵄨 ≤ 1, x ∈ Ω}. A measurable E ⊂ Rn is called a Caccioppoli set if P(E) < +∞. 4.2.2 Talenti’s comparison theorem Let Ω ⊂ Rm be an open set, a = (aij (x)) a real symmetric matrix with aij = aij (x) ∈ L∞ (Ω), 1 ≤ i, j ≤ n, and 0 ≤ c = c(x) ∈ L∞ (Ω). Let, furthermore, m

m

i,j=1

j=1

∑ aij (x)ξi ξj ≥ ∑ ξj2 ,

x ∈ Ω,

ξ = (ξ1 , . . . , ξm ) ∈ Rm ,

(4.36)

and take the elliptic boundary value problem m

𝜕u 𝜕 (aij (x) ) + c(x)u = f 𝜕x 𝜕x i j i,j=1

−∑

in Ω,

u|𝜕Ω = 0.

(4.37)

This u ∈ H01 (Ω) is a weak solution, which means m

∫ ∑ aij uxj ϕxi + c(x)uϕ − fϕ dx = 0,

Ω i,j=1

∀ϕ ∈ C0∞ (Ω).

2m

Proposition 4.26. Let f ∈ L m+2 (Ω) if m > 2, or let f ∈ Lp (Ω), p > 1, and Ω be bounded if m = 2. Then, there is a unique weak solution u to (4.37). Proof. We take the bilinear form m

a(u, v) = ∫ ∑ aij (x)uxi vxj + c(x)uv dx, Ω i,j=1

u, v ∈ H01 (Ω).

By assumption, it provides a complete inner product to V = H01 (Ω). From the continuous embedding V ⊂ X = L2 (Ω), the inclusion X 󸀠 ⊂ V 󸀠 follows, where Y 󸀠 denotes the dual space of the Banach space Y. Then we identify X 󸀠 with X via the representation theorem of Riesz, to obtain the triplet V ⊂ X ⊂ V 󸀠 . The Sobolev embedding, Theo2m rem 1.10, implies L m+2 (Ω) ⊂ V 󸀠 . The representation theorem of Riesz, applied to V, now assures a unique u ∈ V satisfying a(u, v) = ⟨f , v⟩V 󸀠 ,V , which means that u is a weak solution.

∀v ∈ V,

4.2 Equimeasurable transformations | 189

Theorem 4.27 ([347]). Given f as in Proposition 4.26, let u = u(x) be the weak solution to (4.37). Let, furthermore, v = v(x) be the weak solution to − Δv = f ∗

v|𝜕Ω∗ = 0.

in Ω∗ ,

(4.38)

Then it follows that v ≥ u∗ in Ω∗ and m

∫ |∇v|2 dx ≥ ∫ ∑ aij (x)uxi uxj dx. Ω i,j=1

Ω∗

(4.39)

Any smoothness is not assumed for aij , c, f or 𝜕Ω, but the proof is not easy even when they are smooth. In the latter case, however, u = u(x) becomes smooth, and classical Sard’s lemma is applicable. Below we follow this case. For the general case, we use De Giorgi’s isoperimetric inequality and Fleming–Rishel’s coarea formula.11 Proof of Theorem 4.27 (regular case). Assume that u = u(x) is smooth. First, we show that this theorem is reduced to the case u > 0 in Ω. To this end, let ũ be the solution to (4.37) with f replaced by |f |. Then the maximum principle implies ũ ≥ |u| in Ω, and hence ũ ∗ ≥ u∗ in Ω∗ . We have also |f |∗ = f ∗ and hence supposing f ≥ 0 does not lose any generality. If f ≡ 0, the assertion is obvious because u ≡ 0. Otherwise, we have u > 0 in Ω by the strong maximum principle. Having u > 0 in Ω and also u = 0 on 𝜕Ω, we get Ωt ≡ {u > t} ⊂⊂ Ω for any t ∈ [0, umax ). Then Sard’s lemma ensures that each component of 𝜕Ωt is a smooth (m − 1)-dimensional compact manifold contained in {u = t} for almost every t ∈ (0, umax ). For such t, −∇u/|∇u| coincides with the outer normal vector on 𝜕Ωt . Green’s formula, therefore, is applicable and (4.36) implies − ∫ ∑ {u>t} i,j

ux 𝜕u 𝜕 (aij (x) ) dx = ∫ ∑ ai,j (x)uxj i dH m−1 𝜕xi 𝜕xi |∇u| i,j {u=t}

≥ ∫ |∇u| dH m−1 . {u=t}

Then we obtain ∫ |∇u| dH m−1 ≤ ∫ f (x) dx {u=t}

a. e. t ∈ (0, umax )

(4.40)

{u>t}

by (4.37) and c ≥ 0. Coarea formula (4.33) for μ(t) = ∫{u>t} dx implies − μ󸀠 (t) = ∫ {u=t}

11 See also [348] for quasilinear case.

dH n−1 |∇u|

a. e. ∈ (0, umax )

(4.41)

190 | 4 Rearrangement since Ωt ⊂ Ω, while the isoperimetric inequality for Ωt = {u > t} means 1/m H m−1 ({u = t}) ≥ mCm μ(t)1−1/m .

(4.42)

Combining inequalities (4.40) and (4.41) with the Schwarz inequality, we obtain 2

H m−1 ({u = t}) ≤ −μ󸀠 (t) ∫ f (x) dx. {u>t}

Then, Hardy–Littlewood’s inequality, Proposition 4.21, implies ∞

⋆ (s)f ⋆ (s) ds ∫ f (x) dx = ∫ χ{u>t} (x)f (x) dx ≤ ∫ X{u>t} 0

Ω

{u>t}

μ(t)

= ∫ f ⋆ (s) ds.

(4.43)

0

Inequalities (4.42)–(4.43) are summarized as 1≤

1

μ(t)

(−μ󸀠 (t))μ(t)−2+2/m ∫ f ⋆ (s) ds = Φ󸀠 (t) 2/m

m 2 Cm

0

for a. e. t ∈ (0, umax ), where Φ(t) =

2/m Cm

r

|Ω|

1

∫ r

−2+2/m

dr ∫ f ⋆ (s) ds 0

μ(t)

is a nondecreasing function. Hence it holds that t

∫ Φ󸀠 (s) ds ≤ Φ(t) − Φ(0),

0 ≤ t ≤ umax ,

0

which implies t≤

r

|Ω|

1

2/m m2 Cm

∫ r

−2+2/m

dr ∫ f ⋆ (s) ds,

0 ≤ t ≤ umax

0

μ(t)

by μ(0) = |Ω|. We thus end up with u∗ (s) = inf{t ≥ 0 | μ(t) < s} ≤

1

2/m m2 Cm

|Ω|

∫r s

−2+2/m

r

dr ∫ f ⋆ (s󸀠 ) ds󸀠 , 0

4.2 Equimeasurable transformations | 191

and hence u∗ (|x|) = u⋆ (Cm |x|) ≤

1

2/m m2 Cm

|Ω|

r

Cm |x|

0

∫ r −2+2/m dr ∫ f ⋆ (s) ds

≡ v(|x|). Then we see that this radial function v(|x|) solves (4.38). The proof of (4.39) is similar. 4.2.3 Capacity estimates of the singular set The proof of Theorem 4.27 for the irregular case uses De Giorgi’s isoperimetic inequality (4.34) and Fleming–Rishel’s coarea formula (4.35).12 This argument is applicable to derive a dimension control of the singular set. We use the notion of capacity for this purpose. First, we put K p = {f ∈ Lp (Rn , R) | ∇f ∈ Lp (Rn , Rn )}, ∗

1 1 1 = − p∗ p n

for 1 ≤ p < n, to define the p-capacity of a subset Σ ⊂ Rn by Capp (Σ) = inf{∫ |∇f |p dx | f ≥ 0, f ∈ K p , A ⊂ int{f (x) ≥ 1}}. Rn

This Capp (⋅) is an outer measure on Rn , and we have Capp (Σ) ≤ CH n−p (Σ), where H s denotes the s-dimensional Hausdorff measure. We have Capp (Σ) = 0 if H n−p (A) < +∞, 1 < p < n, and conversely, H s (Σ) = 0 for s > n − p if Capp (Σ) = 0 [112]. Second, given a harmonic function u = u(x) in Ω \ Σ, we say that Σ is removable if ̃ in Ω such that u|̃ Ω\Σ = u. Then, Carleson’ theorem there is a harmonic function ũ = u(x) says that Σ is removable for any harmonic function in Ω \ Σ belonging to L∞ loc (Ω) if and only if Cap2 (Σ) = 0.13 In the other criterion of Serrin, Σ is removable if u ∈ Lq (Ω \ Σ) s [305]. There are several generalizations and Caps (Σ) = 0, where 2 < s ≤ n and q > s−2 [306, 157], but if Σ ⊂⊂ Ω is closed, u is harmonic in Ω \ Σ, and Σ = {|u| = +∞} then it n

follows that Cap2 (Σ) = 0 and u ∈ Lwn−2 (Ω). We thus have different profiles of u according ∞ p p to u ∈ L∞ loc (Ω) or u ∈ ̸ Lloc (Ω). Here, Lw (Ω), 1 < p < ∞, denotes the weak L space on Ω defined by Lpw (Ω) = {v ∈ L1loc (Ω) | ‖v‖p,w < +∞}, ‖v‖p,w = sup{|K|−1+1/p ∫ |v| dx | K ⊂ Ω is a compact set}. K

12 See [140, 116] for the proof. 13 See [66].

192 | 4 Rearrangement We note that u = u(x) below also is Hölder continuous in Ω \ Σ by Nash–Moser’s theorem [139]. Theorem 4.28 ([301]). Let Ω ⊂ Rn , n ≥ 3, be a bounded open set, Σ ⊂ Ω be compact, and L be a differential operator defined by (4.37), m

𝜕 𝜕u (aij (x) ) + c(x)u 𝜕x 𝜕x i j i,j=1

Lu = − ∑

1 with aij , c ∈ L∞ loc (Ω \ Σ), c = c(x) ≤ 0, and (4.36). Let, furthermore, u = u(x) ∈ Hloc (Ω \ Σ) satisfy

Lu = 0 in Ω \ Σ,

(4.44)

and assume the existence of s0 ≥ 0 such that Ωs0 ⊂⊂ Ω and Lipschitz Γ0 = 𝜕Ωs0 , where Ωs = {x ∈ Ω \ Σ | |u(x)| > s} ∪ Σ. Assume, finally, that Ωs is open for any s ≥ s0 . Then, it n

holds that Cap2 (Σ) = 0 and u ∈ Lwn−2 (Ω).

1 Proof. Putting Ω0 = Ωs0 , we have u ∈ Hloc (Ω0 \ Σ) and

Lu = 0,

|u| > s0

in Ω0 \ Σ,

|u| = 0

on 𝜕Ω0 .

(4.45)

For s > s0 , it holds that Σ ⊂⊂ Ωs and therefore, the function 1 φs = (sgn u) ⋅ max{s − |u|, 0} ∈ Hloc (Ω0 \ Σ)

satisfies φs |𝜕Ω0 = (sgn u) ⋅ (s − s0 ), φs = 0 in Ωs \ Σ, and ∇φs = {

−∇u, Ω0 \ Ωs , 0, Ωs \ Σ.

Testing this φs on (4.45), we obtain ∑ ∫ aij Dj uDi u dx = (s − s0 )K + ∫ c|u|(s − |u|) dx, i,j Ω \Ω 0 s

(4.46)

Ω0 \Ωs

𝜕u where K = −⟨ 𝜕ν , sgn u⟩H −1/2 (Γ0 ),H 1/2 (Γ0 ) and L

𝜕 𝜕νL

= ∑i,j νi aij Dj . Then, c ≤ 0 implies

∫ |∇u|2 dx ≤ (s − s0 )K = o(s2 ).

(4.47)

Ω0 \Ωs

Here, we take s1 > s0 and χ = χ(x) ∈ C0∞ (Rn ) such that 0 ≤ χ ≤ 1, supported in Ω0 , and χ = 1 on Ωs1 . Then, fs ≡ s1 min{|u|, s} ⋅ χ ∈ K 2 , s ≫ 1, satisfies Σ ⊂ int{x ∈ Ω0 | fs (x) = 1},

∇fs = {

1 ∇(|u|χ), s

∇χ,

|u| ≤ s,

|u| > s,

4.2 Equimeasurable transformations | 193

and therefore, it holds that Cap2 (Σ) ≤ ∫ |∇fs |2 dx = Rn

1 󵄨2 󵄨 ∫ 󵄨󵄨󵄨∇(|u|χ)󵄨󵄨󵄨 dx = o(1), s2

s ↑ +∞

Ω0 \Ωs

by (4.47) and χ = 1 on Ωs1 . n

To show u ∈ Lwn−2 (Ω), we note first that the mapping s 󳨃→ ∫ c|u|(s − |u|) dx Ω0 \Ωs

is nondecreasing, and hence (4.46) implies d ∫ aij Dj uDi u dx ≤ K ds

a. e. s > s0 .

(4.48)

a. e. s ∈ (s0 , s󸀠 ).

(4.49)

Ω0 \Ωs

If s󸀠 > s0 and s ∈ (s0 , s󸀠 ), therefore, we obtain −

d ds

∫ aij Di uDi u dx ≤ K Ωs \Ωs󸀠

Putting μ(s) = |Ωs |, we show −

d ds

1/2

∫ |∇u| dx ≤ (−μ󸀠 (s)) Ωs \Ωs󸀠

⋅ {−

d ds

1/2

∫ aij Dj uDi u dx}

a. e. s ∈ (s0 , s󸀠 ).

(4.50)

Ωs \Ωs󸀠

In fact, since s ∈ (s0 , s󸀠 ) 󳨃→ ∫Ω \Ω |∇u|dx is nonincreasing, it holds that s

1 [ ∫ − h Ωs \Ωs󸀠

∫ Ωs+h \Ωs󸀠

]|∇u|dx =

s󸀠

1 h

∫

|∇u| dx

Ωs \Ωs+h 1/2

≤{

μ(s) − μ(s + h) 1 } { h h 1/2

≤{

μ(s) − μ(s + h) 1 } { h h 1/2

= {−μ󸀠 (s)} {−

d ds

∫

|∇u|2 dx}

Ωs \Ωs+h

∫

1/2

aij Dj uDi u dx}

Ωs \Ωs+h

∫ aij Dj uDi u dx} Ωs \Ωs󸀠

1/2

1/2

+ o(1),

h ↓ 0.

194 | 4 Rearrangement Then, (4.50) follows. We obtain, on the other hand, ncn1/n μ(s)1−1/n ≤ P(Ωs ) = −

d ds

∫ |∇u| dx

a. e. s ∈ (s0 , s󸀠 )

(4.51)

Ωs \Ωs󸀠

by DeGiorgi’s isoperimetric inequality (4.34) and Fleming–Rishel’s coarea formula (4.35). Inequalities (4.49)–(4.51) thus imply n2 cn2/n ≤ −Kμ(s)−2(1−1/n) μ󸀠 (s), or equivalently, c ≡ n2 cn2/n K −1 ≤ n − where ϕ(μ) = n−2 μ 󸀠 (s0 , s ), and hence

n−2 n

d ϕ(μ(s)) a. e. s ∈ (s0 , s󸀠 ), ds

. This inequality implies ϕ(μ(s)) ≥ ϕ(μ(s0 )) + c(s − s0 ) for s ∈

μ(s) ≤ {

n } (n − 2)(c(s − s0 ) + ϕ(μ(s0 )))

n/(n−2)

,

∀s > s0 . n

Therefore, μ(s)sn/(n−2) = O(1) holds as s ↑ +∞, which implies u ∈ Lwn−2 (Ω).14

1 1 If c = 0, then (4.46) implies u ∈ ̸ Hloc (Ω) in spite of the assumption u ∈ Hloc (Ω \ Σ). Note that the crucial assumption which leads to this property is Σs0 ⊂⊂ Ω. It should be compared to the fact that Cap2 (Σ) = 0 implies H 1 (Ω \ Σ) = H 1 (Ω).15

4.2.4 Parabolic case To describe a parabolic version, let Ω ⊂ Rn be a bounded open set, T > 0, and u = u(x, t) : Ω × [0, T] → (−∞, +∞] be a continuous function such that D = ⋃t∈[0,T] D(t) × {t} ⊂ Ω × [0, T] for D(t) = {x ∈ Ω | u(x, t) = +∞}, and ut − Δu ≥ 0

in (Ω × (0, T)) \ D

(4.52)

in the sense of distributions.16 In this setting, we take a compactification of (−∞, +∞) at +∞, to regard (−∞, +∞] as a topological space. Hence from the continuity of u : Ω × [0, T] → (−∞, +∞] and (Ω × [0, T]) \ D = ⋃ (Ω \ D(t)) × {t} t∈[0,T]

= {(x, t) ∈ Ω × [0, T] | u(x, t) < +∞}, 14 See [359, 375]. 15 See [157]. 16 Theorem 4.29 below is a refinement of [298] where n-dimensional Lebesgue measure is treated.

4.2 Equimeasurable transformations | 195

the set Q = Ω × (0, T) \ D is open. The most important assumption here is D ⊂ Ω × (0, T) again, as in the elliptic case. We assume also the uniform Lipschitz continuity of u = u(x, t) near 𝜕Ω, which can be weakened, for example, to the existence of the trace, 𝜕u | in a weak sense. 𝜕ν 𝜕Ω Theorem 4.29 ([342]). Under above assumption, if n ≥ 2 then T

∫ Cap2 (D(t)) dt ≤ 0

Ln (Ω) , 2

(4.53)

where Ln denotes the n-dimensional Lebesgue measure. The following facts should be noted. First, t 󳨃→ Cap2 (Ω \ D(t)) is lower-semicontinuous by the continuity of u and hence t 󳨃→ Cap2 (D(t)) is upper semicontinuous. Second, (4.53) implies Cap2 (D(t)) < +∞ for a. e. t ∈ (0, T) and hence Capr (D(t)) = 0 for such t, where r < 2. This property implies H s (D(t)) = 0 for s > n − 2, and hence dimH (D(t)) ≤ n − 2 for such t, where dimH (D(t)) denotes the Hausdorff dimension of D(t). Finally, if u is independent of t, then we can take T ↑ +∞ in (4.53). Thus, if there is a continuous function u : Ω → (−∞, +∞] which is superharmonic in Ω \ Σ with Σ = {u = +∞} ⊂ Ω, then Cap2 (Σ) = 0. Thus equality in (4.44) is replaced by Lu ≥ 0, u > −∞, and the continuity of u. Lemma 4.30. Let Ω ⊂ Rn be an open set, u : Ω → (−∞, +∞] be a continuous function 1,p (Ω \ Σ) for 1 ≤ p < N. Then it holds satisfying Σ = {x ∈ Ω | u(x) = +∞} ⊂ Ω, and u ∈ Wloc that Capp (Σ) ≤ lim inf s→+∞

1 (s − s0 )p

|∇u|p dx,

∫ {x∈Ω|s0 s0 , we have Ωs ⊂ Ω and Σ ⊂ Ωs for Ωs = {x ∈ Ω | u(x) > s}. Then we define fs = fs (x) ∈ K p by fs (x) = {

1 (min{u(x), s} s−s0

0,

− s0 ), x ∈ Ωs0 ,

x ∈ Rn \ Ωs0 .

It holds that Σ ⊂ int{x ∈ Rn | fs (x) = 1} and ∇fs = {

1 ∇u, s−s0

0,

Ωs0 \ Ωs ,

Ωs ∪ (Rn \ Ωs0 ).

(4.54)

196 | 4 Rearrangement Thus we have Capp (Σ) ≤ ∫ |∇fs |p dx = Rn

1 (s − s0 )p

∫ |∇u|p dx,

(4.55)

Ωs0 \Ωs

and hence (4.54). Lemma 4.31. Given −∞ < a < b < +∞, ϕ ∈ W 1,1 (a, b), and −∞ < α < β < +∞, it holds that ∫ ϕ󸀠 (t) dt ≤ β − α,

(4.56)

I

provided that I ⊂ {t ∈ [a, b] | α ≤ ϕ(t) < β}. Proof. The left-hand side of (4.56) is equal to the Lebesgue–Stieltjes integral ∫J dϕ for J = ϕ(I), and hence (4.56) follows from J ⊂ [α, β). Proof of Theorem 4.29. By assumption, there is an open set ω ⊂⊂ Ω such that 𝜕ω is Lipschitz continuous, u = u(⋅, t) is Lipschitz continuous near 𝜕ω uniformly in t ∈ [0, T], and D = ⋃ D(t) × {t} ⊂ ω × [0, T].

(4.57)

t∈[0,T]

We write Ω for this ω, without confusion. In particular, u : Ω × [0, T] → (−∞, +∞] is continuous. We may assume also that s0 = sup𝜕Ω×[0,T] u is 0, by taking u − s0 for u. Under these agreements, it follows that ut − Δu ≥ 0 in (Ω × (0, T)) \ D in the sense of distributions, and also u ≤ 0 on 𝜕Ω × [0, T]. Now we take the cut-off of u denoted by uL = min{u, L} : Ω × [0, T] → R for L ≥ 2. From the assumption, it is a continuous function satisfying uLt − ΔuL ≥ 0 in Ω × (0, T) in the sense of distributions and uL ≤ 0 on 𝜕Ω × [0, T]. Then we take the regularization in the space variable, uε,L = ρε ∗ uL + 1, where ρε (x) = ε−N ρ(x/ε) for ρ ∈ C0∞ (Rn ), 0 ≤ ρ ≤ 1, ∫Rn ρ = 1, and supp ρ ⊂ B(0, 1). Let Ωε = {x ∈ Ω | dist x, 𝜕Ω > ε}. Then, above uε,L : Ωε × [0, T] → R is continuous, smooth in Ωε × (0, T), and satisfies ε,L uε,L ≥ 0, t − Δu

uε,L ≥ uL

in Ωε × (0, T),

uε,L |𝜕Ωε ≤ s0,ε

(4.58)

with s0,ε = O(1) as ε ↓ 0. Again, there is an open set ω with Lipschitz boundary 𝜕ω satisfying (4.57), u = u(⋅, t) is Lipschitz continuous near 𝜕ω uniformly in t ∈ [0, T],

4.2 Equimeasurable transformations | 197

and ω ⊂ Ωε for 0 < ε ≪ 1. In particular, we can assume (4.58) with Ωε replaced by ω. Writing Ω for this ω again, we obtain ε,L uε,L ≥ 0, t − Δu

uε,L ≥ uL

in Ω × (0, T),

uε,L |𝜕Ω ≤ s0,ε .

(4.59)

Since 𝜕Ω is Lipschitz continuous in this case, it holds that ε,L ε,L ⋅ (s − uε,L )+ dx ∫ uε,L t (s − u )+ dx ≥ ∫ Δu

Ω

Ω

=

∫ {uε,L (⋅,t)≤s}

≥

∫ {uε,L (⋅,t)≤s}

𝜕uε,L 󵄨󵄨 ε,L 󵄨󵄨2 ⋅ (s − uε,L )+ dS 󵄨󵄨∇u 󵄨󵄨 dx + ∫ 𝜕ν 𝜕Ω

󵄨󵄨 𝜕uε,L 󵄨󵄨 󵄨󵄨 ε,L 󵄨󵄨2 󵄨󵄨 󵄨 󵄨󵄨∇u 󵄨󵄨 dx − (s − s0,ε ) ∫ 󵄨󵄨󵄨 󵄨 dS 󵄨󵄨 𝜕ν 󵄨󵄨󵄨

(4.60)

𝜕Ω

for s > s0,ε . Here, we obtain T

ε,L ∫ dt ∫ uε,L t (s − u )+ dx = ∫ dx 0

Ω

Ω

∫ {uε,L (⋅,t)≤s}

ε,L uε,L t (s − u )+ dt

(4.61)

with ∫ {uε,L (⋅,t)≤s}

ε,L uε,L t (s − u )+ dt =

∫

−

{uε,L (⋅,t)≤s}

1 𝜕 2 (s − uε,L )+ dt 2 𝜕t

1 ≤ (s + Mε,L )2 , 2

Mε,L = − inf uε,L , Ω×[0,T]

(4.62)

by Lemma 4.31. Relations (4.60)–(4.62) imply T

∫ dt 0

∫ {s0,ε L − 1}) ≤

1 (s − s0,ε )2

∫ {s0,ε t} qeh dx, μ(t) = ∫{q>t} eh dx, and apply the coarea formula as − K 󸀠 (t) = ∫ {q=t}

qeh eh ds = t ∫ ds = −tμ󸀠 (t) |∇q| |∇q|

a. e. t.

(4.69)

{q=t}

Then we use Green’ formula as ∫ (−Δ log q) dx = ∫ {q>t}

{q=t}

1 |∇q| ds = ∫ |∇q| ds a. e. t > 1, q t {q=t}

regarding Sard’s lemma, and then it follows that 1 ∫ |∇q| ds ≤ ∫ qeh dx = K(t) t {q=t}

a. e. t > 1

{q>t}

from (4.68). Schwarz’ inequality now implies 2

−K 󸀠 (t)K(t) ≥

1 eh ds ≥ ( ∫ eh/2 ds) ∫ |∇q| ds ⋅ t ∫ t |∇q| {q=t}

{q=t}

≥ 4π ∫ eh dx = 4πμ(t) {q>t}

{q=t}

a. e. t > 1

200 | 4 Rearrangement by (4.66), and hence it follows that d 1 1 (μ(t)t − K(t) + K(t)2 ) = μ(t) + K(t)K 󸀠 (t) dt 8π 4π ≤ 0 a. e. t > 1.

(4.70)

We obtain, on the other hand, K(t + 0) = K(t) ≤ K(t − 0), while j(t) ≡ K(t) − μ(t)t = ∫ (q − t)eh dx {q>t}

is continuous, j(t + 0) = j(t) = j(t − 0). Thus (4.70) implies t=∞

[j(t) +

K(1)2 1 K(t)2 ] = j(1) − ≤ 0. 8π 8π t=1

(4.71)

Using j(1) = ∫ (q − 1)eh dx ≥ ∫(q − 1)eh dx = m(ω) − ∫ eh dx ω

{q>1}

ω

and K(1)2 ≤ m(ω)2 , we deduce m(ω) −

2

1 ℓ(𝜕ω)2 1 m(ω)2 ≤ ∫ eh dx ≤ ( ∫ p1/2 ds) = . 8π 4π 4π ω

𝜕ω

This inequality means (4.65).

4.3.2 Mean value theorems Similarly to (4.71), we obtain j(t) ≤

K(t)2 8π

(4.72)

for any t > 1. This inequality implies the mean value theorem in the following form. Its converse is also true and is proven at the end of this paragraph. Theorem 4.34 ([329]). If Ω ⊂ R2 is an open set, p = p(x) > 0 is a C 2 function satisfying (4.64), and B = B(x0 , r) ⊂⊂ Ω, then it holds that log p(x0 ) ≤

1 1 ∫ log p ds − 2 log(1 − ∫ p dx) . |𝜕B| 8π + 𝜕B

B

(4.73)

4.3 Isoperimetric inequality on surfaces | 201

An analogous result follows similarly as in the following theorem. Its proof is given in the next paragraph. Theorem 4.35. Let Ω ⊂ R2 be an open set, and p = p(x) > 0, x ∈ Ω, be a C 2 function. Then Δ log p ≤ p in Ω if and only if log p(x0 ) ≥

1 1 ∫ log p ds − 2 log(1 + ∫ p dx) |𝜕B| 8π B

𝜕B

for any B = B(x0 , r) ⊂⊂ Ω. Proof of Theorem 4.34. Using (4.72) for ω = B, we obtain μ(t) ≥

K(t)2 1 1 ( − ) t K(t) 8π

(4.74)

for any 1 < t < t0 = maxB q. Furthermore, J(t) =

μ(t) μ(t) 1 μ(t) j(t) − = − − K(t) 8π t tK(t) 8π

is right-continuous, J(t + 0) = J(t), and it holds also that j(t) 1 1 1 { − }+ (μ(t) − μ(t − 0)) t K(t) K(t − 0) 8π 1 1 1 j(t) { − }+ (μ(t) − μ(t − 0)) = t j(t) + μ(t)t j(t) + μ(t − 0)t 8π

J(t − 0) − J(t) =

= −(μ(t) − μ(t − 0)) ⋅ {

1 j(t) − }. K(t)K(t − 0) 8π

Using K(t − 0) − K(t) = ∫{q=t} qeh dx = t(μ(t − 0) − μ(t)) ≥ 0 and j(t) ≥ 0, we obtain J(t − 0) − J(t) ≤ −(μ(t − 0) − μ(t)) ⋅ (

1 j(t) − )≤0 8π K(t)2

by (4.72). We have, on the other hand, 1 K 󸀠 (t) 1 − ) − μ(t) K(t) 8π K(t)2 󸀠 tμ(t) K (t) ≥ μ󸀠 (t) ⋅ − μ(t) ⋅ = 0 a. e. t > 1 2 K(t) K(t)2

J 󸀠 (t) = μ󸀠 (t)(

by (4.72) and (4.69). These relations imply lim J(t) = lim t↑t0

t↑t0

μ󸀠 (t) 1 1 1 = ≥ J(t) = μ(t)( − ), K 󸀠 (t) t0 K(t) 8π

1 ≤ t ≤ t0

202 | 4 Rearrangement by (4.69), where we recall that t0 = maxB q. The right-hand side is estimated below by 2

K(t)2 1 1 ( − ) t K(t) 8π + from (4.74). Then, putting t = 1, we obtain 2

2

1 K(1) m ≥ (1 − ) ≥ (1 − ) , t0 8π + 8π + This inequality means (1 −

m 2 ) 8π +

m = ∫ p dx. B

≥ t0 = maxB pe−h ≥ p(x0 )e−h(x0 ) , or

log p(x0 ) ≤ h(x0 ) − 2 log(1 −

1 ∫ p dx) . 8π + B

Here, using the mean value theorem for the harmonic function, we obtain 1 1 ∫ h ds = ∫ log p ds, |𝜕B| |𝜕B|

h(x0 ) =

𝜕B

𝜕B

and the proof of (4.73) is complete. Corollary 4.36 ([21]). Let B = B(0, R) ⊂ R2 and suppose that p = p(x) is continuous on B, C 2 in B, and satisfies −Δ log p ≤ p in B and ∫B p ≤ 4π. Then it holds that p(0) 1 ≤ ∫ p1/2 ds, 1 + r 2 p(0)/8 |𝜕Br |

0 < r < R.

(4.75)

𝜕Br

Proof. Putting u = log p and m(r) = ∫B eu dx < 8π, we apply Theorem 4.34. It holds r that u(0) ≤

1 m(r) ), ∫ u ds − 2 log(1 − |𝜕Br | 8π

r ∈ (0, R).

𝜕Br

Writing p = eu , we obtain p(0) ≤ (1 −

m(r) 1 ) exp( ∫ u ds) 8π |𝜕Br | −2

𝜕Br

m(r) 1 m(r) 1 ) (1 − ) m󸀠 (r) ∫ p ds = 8π |𝜕Br | 2πr 8π −2

≤ (1 −

−2

𝜕Br

by Jensen’s inequality, and hence it follows that R

R

0

0

1 m󸀠 (r) p(0)R = 2 ∫ p(0)r dr ≤ ∫ dr = 8m(8π − m)−1 , π (1 − m(r)/(8π))2 2

(4.76)

4.3 Isoperimetric inequality on surfaces | 203

where m = m(R) = ∫B p dx. Bol’s inequality, on the other hand, guarantees 1 ℓ2 ≤ m(8π − m) 2 for ℓ = ∫𝜕B p1/2 ds, and therefore, m ≤ m− in the case when m ≤ 4π, where M = m− is the smaller solution to M 2 − 8πM + 2ℓ2 = 0, that is, m− = 4π(1 − √1 − j2 ),

j = ℓ/(2√2π).

Then, we obtain p(0)R2 ≤ 8m− (8π − m− )−1 and hence (4.75) for r = R. Similarly to the harmonic case, the mean value theorem (4.73) implies a Harnack inequality in the following form. Theorem 4.37. If B = B(0, R) ⊂ R2 and v = v(x) ∈ C 2 (B) ∩ C(B) satisfies 0 ≤ −Δv ≤

λev , ∫Ω ev

v≥0

in B

then it holds that v(0) ≤

R + |x| λ v(x) − 2 log(1 − ) , R − |x| 8π +

x ∈ B.

(4.77)

Proof. We obtain −Δ log p ≤ p in B for p = λev / ∫Ω ev , and therefore (4.73) is applicable. It holds that log p(0) ≤

1 λ ) , ∫ log p ds − 2 log(1 − |𝜕B| 8π + 𝜕B

or, equivalently, v(0) ≤

1 λ ) . ∫ v ds − 2 log(1 − |𝜕B| 8π + 𝜕B

Since v(x) ≥ 0 is superharmonic, on the other hand, it holds that 2π

v(re𝚤θ ) ≥

1 R2 − r 2 v(Re𝚤φ ) dφ ∫ 2 2π R − 2Rr cos(θ − φ) + r 2 0

R−r 1 ≥ ∫ v ds, R + r |𝜕B|

0 ≤ r < R.

𝜕B

This formula implies (4.77). We obtain a Harnack principle by Theorem 4.37.

(4.78)

204 | 4 Rearrangement Theorem 4.38 ([329]). If Ω ⊂ R2 is an open set and vk = vk (x), k = 1, 2, . . ., are C 2 functions satisfying 0 ≤ −Δvk ≤

λk evk , ∫Ω evk

vk ≥ 0

in Ω

for λk > 0, then passing to a subsequence, we obtain the following alternatives, where 𝒮 = {x0 ∈ Ω | there exists xk → x0 such that vk (xk ) → +∞}: 1. {vk } is locally uniformly bounded in Ω. 2. vk → +∞ locally uniformly in Ω. 3. 𝒮 ≠ 0 and #𝒮 ≤ lim infk [λk /(8π)]. Inequality (4.76) similarly implies the following fact. Theorem 4.39. A family {pk } of piecewise C 2 functions in Ω ⊂ R2 satisfying −Δ log p ≤ p in Ω has the property that #𝒮I ≤ lim infk [Σk /8π], where pk = ∫Ω pk dx and 𝒮I {pk } = {x0 ∈ Ω | ∃xk → x0 , lim pk (xk ) = +∞}. k

We conclude this section with the following theorem.17 Theorem 4.40. Inequality (4.73) for any x0 ∈ Ω and 0 < r ≪ 1 implies (4.64), where Ω ⊂ R2 is an open set. Proof. Assume on the contrary that − Δ log p > p in B

(4.79)

for B = B(x0 , R) with B ⊂ Ω, in spite of (4.73) for 0 < r ≪ R. Let x0 = 0 without loss of generality. We use the radially symmetric function 1 v(x) = |𝜕B(0, r)|

2π

1 ∫ log p ds = ∫ log p(re𝚤θ ) dθ, 2π

r = |x| < R,

0

𝜕B(0,r)

which satisfies v ∈ C(B) ∩ C 2 (B \ {0}) and Δv ∈ C(B) and 2π

1 Δv(x) = ∫ Δ log p(re𝚤θ ) dθ. 2π 0

Put also w(x) = log p(0) + 2 log(1 −

1 ∫ p dx) , 8π + B(0,r)

17 The argument below is due to [21].

0 < r = |x| ≪ 1

4.3 Isoperimetric inequality on surfaces | 205

and 2π r

Σ = Σ(r) ≡ ∫ p dx = ∫ ∫ rp(re𝚤θ ) dθ. B(0,r)

0 0

We obtain 2π

Σr = r ∫ p(re𝚤θ ) dθ, 0

2π

2π

Σrr = ∫ p(re ) dθ + r ∫ pr (re𝚤θ ) dθ, 𝚤θ

0

0

and hence Σr |r=0 = 0,

1 Σ | = Σrr |r=0 = 2πp(0). r r r=0

Thus w is C 2 near the origin. We show Δ(w − v) > 0

r = |x| ≪ 1

(4.80)

to get a contradiction. In fact, (4.73) implies v ≥ w, in spite of v(0) = w(0) = log p(0), which is impossible by (4.80). To confirm (4.80), we note 1 Φ ≡ (Δv − Δw) 2

2π

1 1 = (8π − Σ)−1 (Σrr + Σr ) + (8π − Σ)−2 Σ2r + ∫ Δ log p(re𝚤θ ) dθ r 4π 0

to confirm 1 Φ|r=0 = (Δ log p(0) + p(0)) < 0, 2 using (4.79).

4.3.3 Reverse inequality Theorem 4.35 is proven by similar arguments as in the previous section. We show this theorem in the following form [332], using p = λeu , where λ is a positive constant.

206 | 4 Rearrangement Theorem 4.41. If Ω ⊂ R2 is an open set and u ∈ C 2 (Ω), it holds that Δu ≤ λeu

in Ω

(4.81)

if and only if u(x0 ) ≥

1 |𝜕Br (x0 )|

∫ u ds − 2 log(1 +

1 ∫ λeu dx) 8π

(4.82)

Br (x0 )

𝜕Br (x0 )

for any x0 ∈ Ω and 0 < r ≪ 1. Proof. Assume (4.81), let x0 ∈ Ω and 0 < r ≪ 1 be such that B ⊂ Ω, where B = Br (x0 ) ≡ B(x0 , r), and take a harmonic function in B satisfying −Δh = 0

in B,

h|𝜕B = log λ + u.

It holds that Δ log q ≤ eh q

in B,

q|𝜕B = 1

for q = λeu−h . The right continuous functions K(t) ≡ ∫ λeu dx = ∫ eh q dx ≥ ∫ Δ log q dx, q≤t

q≤t

q≤t

μ(t) ≡ ∫ eh dx q≤t

enjoy the property K 󸀠 (t) = ∫

q=t

eh qeh ds = t ∫ ds = tμ󸀠 (t) a. e. t, |∇q| |∇q|

(4.83)

q=t

and therefore, it holds that ∫ Δ log q dx = ∫

q≤t

q=t

|∇q| 1 ds = ∫ |∇q| ds q t q=t

a. e. t < 1

because of 𝜕{q ≤ t} = {q = t} for t < 1. Schwarz inequality and Nehari’s isometric inequality now guarantee K(t)K 󸀠 (t) ≥

2

1 eh ds ≥ ( ∫ eh/2 ds) ∫ |∇q| ds ⋅ t ∫ t |∇q| q=t

q=t

h

≥ 4π ∫ e dx = 4πμ(t) q=t

q=t

a. e. t < 1,

4.3 Isoperimetric inequality on surfaces | 207

which implies 1 1 d (μ(t)t − K(t) − K(t)2 ) = μ(t) − K(t)K 󸀠 (t) dt 8π 4π ≤ 0 a. e. t < 1.

(4.84)

We have K(t − 0) = K(t),

K(t − 0) ≤ K(t),

j(t + 0) = j(t) = j(t − 0)

for j(t) ≡ tμ(t) − K(t) = ∫ (t − q)eh dx.

(4.85)

q≤t

Then it follows that [μ(t)t − K(t) −

t=t

1 1 K(t)2 ] = μ(t)t − K(t) − K(t)2 ≤ 0, 8π 8π t=−∞

and hence j(t) ≤

1 K(t)2 , 8π

t ≤ 1,

(4.86)

which implies μ(t) ≤

1 1 K(t)2 ( + ), t K(t) 8π

0 < t ≤ 1.

(4.87)

Let J(t) =

μ(t) μ(t) + . K(t) 8π

It holds that J(t + 0) = J(t) and K(t − 0) − K(t) = t(μ(t − 0) − μ(t)) since j(t − 0) = j(t). Thus we obtain J(t − 0) − J(t)

1 1 1 1 + ) + μ(t)( − ) K(t − 0) 8π K(t − 0) K(t) tμ(t) 1 1 = (μ(t − 0) − μ(t))( + − ) K(t − 0) 8π K(t − 0)K(t) 1 j(t) = (μ(t − 0) − μ(t))( − ). 8π K(t − 0)K(t) = (μ(t − 0) − μ(t))(

(4.88)

208 | 4 Rearrangement Since j(t − 0) = j(t) and K(t − 0) ≤ K(t), inequality (4.86) implies j(t) ≤

1 K(t − 0)K(t), 8π

t ≤ 1.

Then it holds that J(t − 0) − J(t) ≤ 0,

t≤1

by μ(t − 0) ≤ μ(t) and (4.88). From (4.87) and μ󸀠 (t) ≥ 0, on the other hand, we have 1 K 󸀠 (t) 1 + ) − μ(t) K(t) 8π K(t) − 2 tμ󸀠 (t)μ(t) μ(t)K 󸀠 (t) ≥ − = 0 a. e. 0 < t < 1. K(t)2 K(t)2

J 󸀠 (t) = μ󸀠 (t)(

(4.89)

Letting t0 = minB q > 0, we have j(t0 ) = 0 by (4.85), which means t0 μ(t0 ) = K(t0 ). If K(t0 ) ≠ 0, it holds that μ(t0 ) 1 = ≤ J(t0 ) ≤ J(1). t0 K(t0 ) In the other case when K(t0 ) = 0, we obtain μ(t0 ) = 0, and therefore, μ󸀠 (t) 1 = lim 󸀠 = lim J(t) ≤ J(1) t0 t↓t0 K (t) t↓t0 by (4.83) and (4.89). We thus end up with 1 1 1 ≤ J(1) = μ(1)( + ) t0 K(1) 8π ≤ K(1)2 (

2

2

≤ (1 +

2

1 1 1 + ) = (1 + K(1)) K(1) 8π 8π

Σ ), 8π

Σ = ∫ λeu dx

(4.90)

B

by (4.87), and hence q(x0 ) = λeu(x0 )−h(x0 ) ≥ t0 ≥ (1 +

Σ ) . 8π −2

Inequality (4.82) then follows from the mean value theorem for the harmonic function h(x0 ) =

1 |𝜕Br (x0 )|

∫ u ds + log λ. 𝜕Br (x0 )

4.3 Isoperimetric inequality on surfaces | 209

Derivation of (4.81) from (4.82) is similar to the proof of Theorem 4.40. Here we use w(x) = u(0) − 2 log(1 +

1 ∫ λeu dx), 8π

r = |x|.

Br (0)

Let B ⊂ Ω for B = Br (x0 ), u0 = u|𝜕B , and 𝒫 be the Poisson operator: v = 𝒫 u0 ⇐⇒ Δv = 0

in B,

v|𝜕B = u0 .

Inequality (4.90), assured by (4.81), then implies u(x) ≥ 𝒫 u0 (x) − 2 log(1 +

1 ∫ λeu dx), 8π

x ∈ B.

Br (0)

It holds also that 1 ∫ λeu dx) 8π +

u(x) ≤ 𝒫 u0 (x) − 2 log(1 −

Br (0)

if −Δu ≤ λeu

in Ω

is the case, which extends the Harnack principle in the form of Theorem 4.38. To state the result, let B = BR (0) ⊂ R2 , assume ∓Δu ≤ λ± eu + C±

in B,

and put Σ± = ∫ λ± eu , B

where λ± and C± are positive constants. The function f (x) =

1 2 (R − |x|2 ) 4

solves −Δf = 1,

0≤f ≤

R2 4

in B,

f |𝜕B = 0,

which results in 2

−Δ(u − C+ f ) ≤ λ+ eu−C+ f eC+ f ≤ λ+ eR C+ /4 eu−C+ f .

210 | 4 Rearrangement We thus obtain u(x) − C+ f (x) ≤ 𝒫 u0 (x) − 2 log(1 −

1 σ ) , 8π + +

x∈B

for 2

σ+ = ∫ λ+ eR C+ /4 eu−C+ f dx ≤ Σ+ ,

u0 = u|𝜕B ,

B

which means u(x) ≤

C+ 2 Σ (R − |x|2 ) + 𝒫 u0 (x) − 2 log(1 − + ) , 4 8π +

x ∈ B.

Similarly, the inequality Δ(u + C− f ) ≤ λ− eu+C− f e−C− f ≤ λ− eu+C− f implies u(x) + C− f (x) ≥ 𝒫 u0 (x) − 2 log(1 +

σ− ), 8π

x∈B

for 2

σ− = ∫ λ− eu+C− f dx ≤ eR C− /4 Σ− , B

and therefore, 2

u(x) ≥ −

C− 2 eR C− /4 Σ− (R − |x|2 ) + 𝒫 u0 (x) − 2 log(1 + ), 4 8π

x ∈ B.

Using R + |x| R − |x| 𝒫 u (0) ≤ 𝒫 u0 (x) ≤ 𝒫 u (x) R + |x| 0 R − |x| 0 assured by u ≥ 0, we thus obtain u(x) ≤

C+ 2 Σ (R − |x|2 ) − 2 log(1 − + ) 4 8π +

2

C R2 eR C− /4 Σ− R + |x| {u(0) + − + 2 log(1 + )} R − |x| 4 8π

and 2

C eR C− /4 Σ− u(x) ≥ − − (R2 − |x|2 ) − 2 log(1 + ) 4 8π +

C R2 Σ R − |x| {u(0) − − + 2 log(1 − + ) }. R + |x| 4 8π +

4.3 Isoperimetric inequality on surfaces | 211

The second inequality implies u(0) ≤

C C R2 R + |x| 2 u(x) + − (R + |x|) + + R − |x| 4 4 − 2 log(1 −

2

Σ+ eR C− /4 Σ− R + |x| ) + ⋅ 2 log(1 + ), 8π + R − |x| 8π

which generalizes Theorem 4.38 as follows. Theorem 4.42. Let Ω ⊂ R2 be an open set and uk , k = 1, 2, . . . , a family of C 2 functions satisfying ∓Δuk ≤ λk± euk + C± ,

uk ≥ 0

in Ω,

where λk± and C± are positive constants. Suppose, furthermore, Σk− ≡ ∫ λk− euk dx = O(1). Ω

Let 𝒮I be the interior blow-up set of {uk }: 𝒮I = {x0 ∈ Ω | ∃xk → x0 such that uk (xk ) → +∞}.

Then, any x0 ∈ 𝒮I is such that either infBr/2 (x0 ) uk → +∞ or lim inf ∫ λk+ euk dx ≥ 8π k→+∞

Br (x0 )

for 0 < r ≪ 1. In particular, unless infB uk → +∞ for some ball B ≠ 0 with B ⊂ Ω, it holds that # 𝒮I ≤ lim inf[Σk+ /(8π)], k→∞

Σk+ = ∫ λk+ euk dx. Ω

If Σk = O(1) and λk = O(1), furthermore, the family λk+ euk dx, k = 1, 2, . . . , of measures on Ω accumulates to some μ(dx) ∈ ℳ(Ω) ∗-weakly, which takes the form of μ(dx) = μac (dx) + μb (dx) + ∑ c(x0 )δx0 (dx) x0 ∈𝒮I

where μac (dx) denotes the absolutely continuous part of μ(dx), μb (dx) is a measure supported on 𝜕Ω, # 𝒮I < ∞, and c(x0 ) ≥ 8π for each x0 ∈ 𝒮I .

212 | 4 Rearrangement 4.3.4 Bandle’s rearrangement Here we describe the fundamental idea for the proof of Theorem 3.3. Step 1. As in § 3.2.3, we introduce the canonical surface h∗ = (u∗ , λ∗ ) on Ω∗ ≡ {|x| < ∗ 1} ⊂ R2 for a given Σ ∈ (0, 8π). Hence u∗ ∈ C 2 (Ω∗ ) ∩ C 0 (Ω ) and λ∗ satisfy − Δu∗ = λ∗ eu

∗

in Ω∗ ,

u∗ |𝜕Ω∗ = 0,

∫ λ∗ eu dx = Σ. ∗

(4.91)

Ω∗

By Theorem 3.2, we obtain the following fact. Proposition 4.43. The solution h∗ ∗ ∫Ω∗ λ∗ eu dx ∈ (0, 8π).

= (u∗ , λ∗ ) to (4.91) is parametrized by Σ =

Therefore, each Σ ∈ (0, 8π) admits a unique zero h∗ = (u∗ , λ∗ ) of Δu∗ + λ∗ eu ∗ Ψ (⋅, Σ) = ( ): ∫Ω∗ λ∗ eu dx − Σ ∗

∗

C α (Ω∗ ) C02+α (Ω∗ ) 󳨀→ × . × R R

The explicit form of this h∗ is known, which implies p∗ (x) ≡ λ∗ eu (x) = ∗

8μ , (|x|2 + μ)2

μ = μ(Σ) ∈ (0, +∞).

(4.92)

This p∗ = p∗ (|x|) is strictly decreasing in r = |x|, and it holds that limΣ↓0 μ(Σ) = +∞, limΣ↑8π μ(Σ) = 0, and − Δ log p∗ = p∗

in Ω∗ .

(4.93)

These relations guarantee the equality in (4.65) for p = p∗ and Ω = Ω∗ from the proof of Theorem 4.33 for this case. Step 2 (spherically decreasing rearrangement). Let Ω ⊂ R2 be a simply-connected domain, p = p(x) > 0 be a C 2 function satisfying (4.64) and m(Ω) = ∫Ω pdx < 8π. Then we take the canonical surface h∗ = (u∗ , λ∗ ) such that Σ = ∫ p∗ dx = m(Ω),

p∗ = λ ∗ e u . ∗

Ω∗

Given a measurable function v = v(x) of x ∈ Ω and 0 < t < t0 = ‖v‖∞ , let Ωt = {v > t} and define an open concentric ball Ω∗t of Ω∗ by ∫ p∗ dx = ∫ p dx = a(t) ∈ (0, 8π). Ω∗t

Ωt

(4.94)

4.3 Isoperimetric inequality on surfaces | 213

Then, Bandle’s spherically decreasing rearrangement v∗ of v is a nonnegative function in Ω∗ satisfying v∗ (x) = sup{t | x ∈ Ω∗t }. It is an equimeasurable rearrangement, and in particular, it holds that 2

∫ v2 p dx = ∫ t 2 d(−a(t)) = ∫ v∗ p∗ dx. Ω

Ω∗

It is obvious that v∗ is radially symmetric. Proposition 4.44. If v(x) is C 2 on Ω, v ≥ 0 in Ω, and v|𝜕Ω = 0, then it holds that 󵄨 󵄨2 ∫ |∇v|2 dx ≥ ∫ 󵄨󵄨󵄨∇v∗ 󵄨󵄨󵄨 dx.

Ω

(4.95)

Ω∗

Similarly to Theorem 4.15, these facts imply ν1 (p, Ω) ≥ ν1 (p∗ , Ω∗ )

(4.96)

for ν1 (p, Ω) = inf{∫ |∇v|2 dx | v ∈ H01 (Ω), ∫ v2 p dx = 1} Ω

Ω

2

ν1 (p , Ω ) = inf{ ∫ |∇v| dx | v ∈ ∗

∗

Ω∗

H01 (Ω∗ ),

2

∫ v∗ p∗ dx = 1}, Ω∗

provided that m(Ω) ≡ ∫Ω p dx = Σ = ∫Ω∗ p∗ dx ∈ (0, 8π). Hence Theorem 3.3 is reduced to the radially symmetric case. Proof of Proposition 4.44. The function a(t) in (4.94) is right-continuous and strictly decreasing in t > 0. Since v ≥ 0 in Ω, the coarea formula assures −a󸀠 (t) = ∫ {υ=t}

p ds, |∇v|

−

d ∫ |∇v|2 dx = ∫ |∇v| ds dt Ωt

a. e. t ∈ (0, t0 ).

{v=t}

By Sard’s lemma, Schwarz inequality, and Bol’s inequality, now we obtain 2

−

p d ds) ∫ |∇v|2 dx ≥ ( ∫ p1/2 ds) /( ∫ dt |∇v| Ωt

{v=t}

=

2

{v=t}

ℓ({v = t}) (8π − a(t))a(t) ≥ −a󸀠 (t) −a󸀠 (t)

a. e. t ∈ (0, t0 ).

(4.97)

Here, the function j(t) = − ∫Ω |∇v|2 dx is continuous and strictly decreasing in t t and satisfying j(t) − j(t − 0) = ∫ |∇v|2 dx = 0. {v=t}

214 | 4 Rearrangement Thus this j(t) is absolutely continuous, and hence it holds that t0

∫ |∇v|2 dx = ∫ dt(− Ω

0 t0

≥∫ 0

d ∫ |∇v|2 dx) dt Ωt

(8π − a(t))a(t) dt. −a󸀠 (t)

(4.98)

The function v = v∗ (x), on the other hand, is strictly decreasing in r = |x|, and equalities hold at each step in (4.97) for Ω = Ω∗ and v = v∗ . We thus obtain equality in (4.98) for this case, and then (4.95) follows from t0

d 󵄨 󵄨2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨∇v∗ 󵄨󵄨󵄨 dx = ∫ dt(− ∫ 󵄨󵄨󵄨∇v∗ 󵄨󵄨󵄨 dx) dt

Ω∗

0

t0

=∫ 0

Ω∗

(8π − a(t))a(t) dt. −a󸀠 (t)

Step 3. We show ν1 (p∗ , Ω∗ ) > 1 in (4.96), or equivalently, μ1 (p∗ , Ω∗ ) > 0, under the assumption of Σ = ∫Ω∗ p∗ dx < 4π, where μ1 (p∗ , Ω∗ ) denotes the first eigenvalue of −ΔD (Ω∗ ) − p∗ . As is described in § 3.2.4, this fact is proven by studying − Δϕ = Λp∗ (r)ϕ in Ω∗ ,

ϕ|𝜕Ω∗ = 0

(4.99)

for p∗ (r) = 8μ/(r 2 + μ)2 , r = |x|. In fact, from the separation of variables ϕ = Φ(r)e𝚤mθ and the transformation of variables ξ = (μ − r 2 )/(μ + r 2 ), problem (4.99) is reduced to the two-point boundary value problem for the associated Legendre equation [(1 − ξ 2 )Ψξ ]ξ + [2/Λ − m2 /(1 − ξ 2 )]Φ = 0,

ξμ < ξ < 1

(4.100)

with Φ(1) = 1 and Φ(ξμ ) = 0 for ξμ = (μ − 1)/(μ + 1). Let Φ be the solution to (4.100) for Λ = 1, m = 0, and Φ(1) = 1. Then the relation ν1 (p∗ , Ω∗ ) > 1 is equivalent to Φ(ξ ) > 0 for ξμ < ξ < 1. Since such Φ is given by P0 (ξ ) = ξ , it means ξμ > 0, or μ > 1. Finally, p∗ (r) = 8μ/(r 2 + μ)2 implies that μ > 1 if and only if Σ = ∫Ω∗ p∗ dx < 4π. As is seen in § 3.2.4, this fact is proven alternatively by the bending of the set of radially symmetric solutions to (3.36) in the λ–u plane. Furthermore, inequality (4.96) is an isoperimetric inequality of the Laplace–Beltrami operator on a surface with Gaussian curvature bounded above by 1/2. Let S be the two-dimensional round sphere with area 8π, and its canonical metric and area elements be dσ and dV, respectively. Let, furthermore, ι : S \ {n} → R2 be the stereographic projection from the north pole n ∈ S onto the whole space R2 , which is

4.3 Isoperimetric inequality on surfaces | 215

tangent to the south pole s ∈ S. Let ω∗ ⊂ S be a ball (or bowl, more precisely) with center s ∈ S and −ΔS (ω∗ ) be the Laplace–Beltrami operator in ω∗ under the Dirichlet boundary condition on 𝜕ω∗ . Then the injection ι : ω∗ → R2 transforms −ΔS (ω∗ ) to ̃ −Δ/p̃ in Ω̃ = ι(ω∗ ) for some p̃ = p(|x|) > 0, under the Dirichlet boundary condition ̃ ̃ on 𝜕Ω. Since the Gaussian curvature of 𝒮 is 1/2, the radially symmetric function p(|x|) satisfies −Δ log p̃ = p̃ in Ω,̃ besides Σ = ∫ p̃ dx = ∫ dV.

(4.101)

ω∗

Ω̃

If ι∗ denotes the pull-back of ι, the relations 󵄨 󵄨2 ∫ 󵄨󵄨󵄨d(ι∗ v)󵄨󵄨󵄨 dV = ∫ |∇v|2 dx,

ω∗

Ω̃

2

∫ (ι∗ v) dV = ∫ v∗2 p̃ dx ω∗

(4.102)

Ω̃

hold, and hence ν(p,̃ Ω)̃ coincides with the first eigenvalue of −ΔS (ω∗ ) under the Dirichlet boundary condition on 𝜕ω∗ . Here, the operation d in (4.102) denotes the differentiation with respect to the canonical metric on S as is described in § 3.2.5. This observation reveals the reason why the associated Legendre equation arises in the study of (4.99). ̃ Let R be the radius of Ω,̃ and take y = x/R. Then p∗ (y) = R2 p(x) > 0 is radially ∗ ∗ ∗ 2 symmetric and satisfies −Δ log p = p in Ω ≡ {|y| < 1} ⊂ R . Furthermore, we have ∫Ω̃ |∇v|2 dx ∫Ω̃ v2 p̃ dx

=

∫Ω∗ |∇v1 |2 dy ∫Ω∗ v12 p∗ dy

,

̃ v ∈ H01 (Ω),

v1 (y) = v(x).

Hence the eigenvalue problem for −ΔS (ω∗ ) is reduced to that of −Δ/p∗ on the unit ball, − Δϕ = Λp∗ ϕ in Ω∗ ,

ϕ|𝜕Ω∗ = 0,

(4.103)

and it holds also that ∫ p∗ dx = ∫ p̃ dx = Σ ∈ (0, 8π). Ω∗

(4.104)

Ω̃

For the constant λ∗ = p∗ |𝜕Ω∗ > 0, the function p∗ is realized as p∗ = λ∗ eu , where u∗ = u∗ (|y|) is a solution to ∗

−Δu∗ = λ∗ eu

∗

in Ω∗ ,

󵄨 u∗ 󵄨󵄨󵄨𝜕Ω∗ = 0.

8μ

Hence we obtain p∗ (y) = (|y|2 +μ)2 with μ > 0 determined by (4.104), and (4.103) coincides with (4.99). We have thus confirmed that the eigenvalue problem for −ΔS (ω∗ ) is equivalent to (4.99). From this problem, a standard separation of variables for ΔS (ω∗ ), 𝜕2 𝜕2 𝜕2 or three dimensional Laplacian Δ = 𝜕x 2 + 𝜕x 2 + 𝜕x 2 , induces the associated Legen1

2

3

dre equation, which is also true in (4.99) if the inverse stereographic transformation ξ = (μ − r 2 )/(μ + r 2 ) is applied.

216 | 4 Rearrangement From these considerations, we realize that above spherically decreasing rearrangement is nothing but the Schwarz symmetrization on the round sphere. Given a simply-connected domain Ω ⊂ R2 with a sufficiently smooth boundary, and given 0 < p = p(x) ∈ C 2 (Ω) ∩ C 0 (Ω) satisfying −Δ log p ≤ p

in Ω,

Σ = ∫ p dx < 8π, Ω

we take all ball ω∗ ⊂ S satisfying ∫ω∗ dV = Σ. Then, for a measurable function 0 ≤ v = v(x) in x ∈ Ω, we define v∗ : ω∗ → (−∞, +∞] by v∗ (x) = sup{t | x ∈ ωt }, where ωt ⊂ S denotes the open concentric ball of ω∗ such that ∫ω dV = ∫{υ>t} p dx. t

Proposition 4.45. We have the following properties: 1. If ψ : R → R is a continuous monotone function, it holds that ∫(ψ ∘ v)p dx = ∫ (ψ ∘ v∗ ) dV. ω∗

Ω

2.

If v = v(x) is C 2 , v ≥ 0 in Ω, and v|𝜕Ω = 0, we have 󵄨 󵄨2 ∫ |∇v|2 dx ≥ ∫ 󵄨󵄨󵄨dv∗ 󵄨󵄨󵄨 dV. ω∗

Ω

4.3.5 Global analysis We are studying here the global bifurcation of the nonlinear eigenvalue problem (P) Find (u, λ) ∈ C 2 (Ω) ∩ C 0 (Ω) × (0, +∞) such that − Δu = λeu

in Ω,

u|𝜕Ω = 0,

(4.105)

where Ω ⊂ R2 is a bounded domain with smooth boundary 𝜕Ω. Singular limits of the classical solution were classified for this problem, while a one-point blow-up branch 𝒞 ∗ of nonminimal solutions in the λ–u plane was constructed for a simply-connected domain so far. The one-point blow-up of the solution is characterized by Σ = ∫Ω λeu dx → 8π as λ ↓ 0, and in this case the singular limit takes the form u0 (x) = 8πG(x, κ) with κ ∈ Ω satisfying ∇R(κ) = 0,

(4.106)

1 where G(x, y) and R(x) = [G(x, y) + 2π log |x − y|]y=x denote the Green and Robin function, respectively. The first question of the global analysis is to clarify the connectivity of this singular limit to the trivial solution u = 0 in the λ–u plane. They are actually connected if Ω = B due to bending exactly once. This phenomenon is extended to a general situation.

4.3 Isoperimetric inequality on surfaces | 217

Theorem 4.46 ([330]). Let Ω be simply-connected. Suppose that there exists a family of classical solutions {(λ, u)} of (P) satisfying Σ = ∫Ω λeu dx ↑ 8π with λ ↓ 0. Then the singular limit u0 (x) = 8πG(x, κ) is connected with the trivial solution u = 0 in the λ–u plane through a branch 𝒞 bending just once. As for the Weston–Moseley branch 𝒞 ∗ = {(λ, u∗ )} of nonminimal solutions, we have a quantitative criterion for Σ to tend to 8π below. Given a simply-connected domain Ω ⊂ R2 and a point κ ∈ Ω satisfying (4.106), we can take a univalent holomorphic function g : B = {|x| < 1} → Ω with g(0) = κ. It follows from (4.106) that g 󸀠󸀠 (0) = 0. Under the assumption σ ≡ |q󸀠󸀠󸀠 (0)/g 󸀠 (0)| ≠ 2, then the Weston–Moseley branch 𝒞 ∗ k arises. Letting g(z) = κ + a1 z + ∑∞ k=3 ak z , we have (3.145) with ∞ k2 C = −|a1 |2 + ∑ |ak |2 , π k − 2 k=3

and, therefore, if C < 0 then 𝒞 ∗ is connected with 𝒞 , the branch of minimal solutions. We note that C < 0 implies σ = 6|a3 /a1 | < 2. Hence the connectivity of the one-point blow-up singular limit and the trivial solution is assured if Ω is close to a disc. In fact, if Ω is the exact disc, then we have ak = 0 for k ≥ 3, and hence C/π = −|a1 |2 < 0. By Proposition 3.32, if K|g 󸀠 | < 2 everywhere on 𝜕B, then C < 0 follows, where K(ζ ) denotes the curvature of 𝜕Ω at g(ζ ) ∈ 𝜕Ω for ζ ∈ 𝜕B. If Ω is a disc, again, it holds that K|g 󸀠 | ≡ 1. 4.3.6 A priori estimates and bending The geometric structure of (P) is useful in the proof of Theorem 4.46. Here we employ the method of rearrangement to reduce a theorem for the radial case Ω = B. Henceforth, simply-connectedness of Ω is assumed.18 To parametrize the solution {h = (u, λ)} by Σ = ∫ λeu dx,

(4.107)

Ω

we use the operator C a (Ω) C02+a (Ω) Δu + λeu 󳨀→ ) : × × , ∫Ω λeu dx − Σ R R

Ψ = Ψ(⋅, Σ) ≡ (

h=(

u ). λ

Recall that C α (Ω), 0 < α < 1, denotes the Schauder space and C02+α (Ω) = {v ∈ C 2+α (Ω) | v|𝜕Ω = 0}. Each zero of Ψ(⋅, Σ) represents a solution h = (u, λ) of (P) satisfying (4.107). This formulation has a geometric meaning. The solution h = (u, λ) is associated with a conformal mapping g from Ω ⊂ R2 into a two-dimensional round sphere of area 18 The result is also true for multiply-connected domains [29]. For the proof, Bol’s inequality is extended in this case, under the cost that p = p(x) is constant on 𝜕Ω. Then Lemma 4.48 below is proven directly [29].

218 | 4 Rearrangement 8π, and Σ indicates the area of g(Ω) as an immersion. We parametrize the immersed surfaces by their areas, using the following lemmas. Lemma 4.47. The set {h = (u, h) | Ψ(h, Σ) = 0, Σ ∈ [0, 8π −δ]} is compact for each δ > 0. Lemma 4.48. If Ψ(h, Σ) = 0 holds with Σ ∈ [0, 8π), the linearized operator C02+α (Ω) C α (Ω) dh Ψ(⋅, Σ) : 󳨀→ × × R R is an isomorphism. Lemma 4.49. Under the assumption of the Lemma 4.48, it holds that μ2 (p, Ω) > 0 for p = λeu .

Here and henceforth, μj (p, Ω) for j = 1, 2, . . . denotes the jth eigenvalue of the differential operator −Δ − p in Ω under the Dirichlet boundary condition. That operator will be denoted by −ΔD (Ω) − p. Thus, Lemma 4.49 indicates that the second eigenvalue of the linearized operator for (P) with respect to u is positive whenever Σ < 8π. We confirm that above lemmas imply Theorem 4.30. First, consider the set of zero points of Ψ in the Σ–h plane. Every zero (h, Σ) of Ψ generates a branch whenever 0 < Σ < 8π by the implicit function theorem. This branch continues up to Σ = 0 by Lemma 4.47, but only the trivial solution h = (0, 0) is admitted to (P) and (4.107) with Σ = 0. There arises also local uniqueness of zero of Ψ around h = (0, 0). Thus we obtain a unique nonbending and nonbifurcating branch 𝒞 formed by the zeros of Ψ in the Σ–h plane, starting from (Σ, h) = (0, 0), to approach the hyperplane Σ = 8π, on which the singular limit h∗ = (8πG(⋅, κ), 0) lies. In the case when Σ ↑ 8π along the singular limit, therefore, the branch 𝒞 connects the trivial solution and the one-point blow-up singular limit, 𝒞 = {(Σ, h(Σ)) | 0 < Σ < 8π}

0 lim h(Σ) = ( ) , 0 Σ↓0

8πG(⋅, κ) ). 0

lim h(Σ) = (

Σ↑8π

Then, we are left with representing {h(Σ) | 0 < Σ < 8π} in the λ–u plane. Lemma 4.49 and the implicit function theorem guarantee that u in (u, λ) ∈ 𝒞 is parametrized by λ, unless μ1 (p, Ω) = 0 occurs for p = λeu . At this degenerate point, however, Theorem 2.14 is applicable. From the convexity of f (u) = eu , the family {(λ(Σ), u(Σ))} exhibits a bending at this degenerate point, changing the solution from the minimum to a nonminimum. Then, only one possibility of such a degeneracy is permitted by the uniqueness of the minimal solution, which completes the proof of Theorem 4.46. The following proposition is a consequence of Theorem 4.34 applied to p = λeu for h = (u, λ). Then it implies Lemma 4.47 from the elliptic estimate for u and the upper bound λ of λ in § 3.2.1.19 19 Theorem 3.5, however, provides a direct proof. In the form of Boltzmann–Poisson equation (3.37), it assures the local compactness of the solution set in any component in [0, +∞) \ 8πN. This property leads to a topological study of the solution set [201, 81, 82, 220].

4.3 Isoperimetric inequality on surfaces | 219

Proposition 4.50. If h = (u, λ) solves (P) with Σ < 8π, it follows that ‖u‖∞ ≤ −2 log(1 −

Σ ). 8π

(4.108)

Lemma 4.49, on the other hand, is a consequence of the following lemma, derived from Proposition 6.1. Proposition 4.51. If 0 < p = p(x) ∈ C 2 (Ω) ∩ C 0 (Ω) satisfies (4.64) with Σ = ∫Ω pdx < 8π, it holds that μ2 (p, Ω) > 0. Proof. First, ν1 (p, Ω) > 1 is equivalent to μ1 (p, Ω) > 0 under the notation of § 4.3.4. By Corollary 4.13, the second eigenfunction ψ2 of −ΔD (Ω) − p has two nodal domains denoted by Ω± = {±ψ2 > 0}. Hence both Ω± are open connected sets, and their boundaries consist of a number of piecewise C 2 Jordan curves by Proposition 4.9. Here we confirm μ2 (p, Ω) = μ1 (p, Ω± ), where μ1 (p, Ω± ) ≡ inf{ ∫ |∇v|2 dx | v ∈ H01 (Ω± ), ∫ v2 p dx = 1}. Ω±

Ω±

In fact, for μ = μ2 (p, Ω), the function ϕ± = ±ψ2 |Ω± ∈ C 2 (Ω± ) ∩ C 0 (Ω± ) ∩ H01 (Ω± ) satisfies −Δϕ± = μpϕ± ,

ϕ± > 0

in Ω± ,

ϕ± |𝜕Ω± = 0,

which implies μ ≡ μ2 (p, Ω) = μ1 (p, Ω± ) by Corollary 4.12. We have, on the other hand, Σ = ∫ p dx + ∫ p dx < 8π, Ω+

Ω−

and hence either ∫Ω pdx < 4π or ∫Ω pdx < 4π, which implies μ2 (p, Ω) > 0 by Proposi+ − tion 6.1. 4.3.7 Connectivity via isoperimetric inequality Proof of Lemma 4.48. First, we note that the lemma is obvious when Σ = 0. In fact, in this case, Ψ(h, 0) = 0 implies h = (0, 0), and hence20 Δu + λeu dh Ψ(h, 0) = ( ∫Ω λeu ⋅ dx

eu Δ )=( ∫Ω eu dx 0

1 ): |Ω|

C02+α (Ω) C α (Ω) 󳨀→ × . × R R

20 The other and simpler linearized analysis is done via the variational functional associated with the Trudinger–Moser inequality. See § 4.3.8.

220 | 4 Rearrangement In the case of Σ > 0, we have λ > 0. Hence Ψ(h, Σ) = 0 is equivalent to Φ(h, Σ) = 0, where Δu + λeu ): Φ = Φ(⋅, Σ) ≡ ( ∫Ω eu dx − Σλ

C α (Ω) C02+α (Ω) 󳨀→ × , × R R

u h = ( ). λ

Hence the isomorphy of dh Ψ(h, Σ) is reduced to that of dh Φ(h, Σ) if Σ > 0. Then the linearized operator Δu + λeu dh Φ(h, Σ) = ( ∫Ω eu ⋅ dx

C α (Ω) C02+α (Ω) 󳨀→ × × Σ): λ2 R R

eu

has a natural self-adjoint extension in L2 (Ω) × R denoted by −T with the domain D(−T) = (H01 ∩H 2 )(Ω)×R. By virtue of the elliptic regularity, the isomorphy of dh Φ(h, Σ) is equivalent to that of T. This operator T is associated with the bilinear form 𝒜(⋅, ⋅) on H01 (Ω) × R: u

u

u

2

𝒜(ξ , η) = ∫ ∇v ⋅ ∇ω − (λe vw + e κw + e vρ) dx − Σκρ/λ , Ω

where ξ = (v, κ), η = (w, ρ) ∈ H01 (Ω) × R, that is, 𝒜(ξ , η) = ⟨Tξ , η⟩,

ξ ∈ D(T),

H01 (Ω) η∈ × R

defined for ⟨ξ , η⟩ = ∫ vw dx + κρ, Ω

v ξ = ( ), κ

w η = ( ). ρ

Since h = (u, λ) is a zero of Φ(⋅, Σ), it holds that Σ = ∫Ω λeu dx, which implies ρ κ A(ξ , η) = ∫ ∇v ⋅ ∇w − λeu (υ + )(ω + ) dx. λ λ Ω

For Hc1 (Ω) = {v ∈ H 1 (Ω) | v = constant on 𝜕Ω}, we have the isomorphy v ξ =( )∈ κ

H01 (Ω) κ × 󳨃→ v + ∈ Hc1 (Ω). λ R

4.3 Isoperimetric inequality on surfaces | 221

Hence we take the bilinear form v, w ∈ Hc1 (Ω),

ℬ(v, w) = ∫ ∇v ⋅ ∇w − vwp dx, Ω u

for p = λe . Let  p be the self-adjoint operator in L2 (Ω) associated with this ℬ(⋅, ⋅) on Hc1 (Ω). Then, the above T = −dh Φ(h, Σ) is an isomorphism if and only if  p is invertible. ̂ Let σ( p ) = {μ̂ j (p, Ω)}∞ j=1 , −∞ < μ̂ 1 (p, Ω) ≤ μ̂ 2 (p, Ω) ≤ ⋅ ⋅ ⋅, be the eigenvalues of Ap . First, the Rayleigh principle implies μ̂ 1 (p, Ω) = inf{ℬ(v, v) | v ∈ Hc1 (Ω), ‖v‖2 = 1} ≤ − 1/2

because ζ = 1/|Ω| tion.21

∈

Hc1 (Ω)

1 ∫ p dx < 0, |Ω| Ω

and ‖ζ ‖2 = 1. Second, we apply the following proposi-

Proposition 4.52. If 0 < p = p(x) ∈ C 2 (Ω) ∩ C 0 (Ω) satisfies − Δ log p ≤ p

in Ω,

Σ ≡ ∫ p dx < 8π,

(4.109)

Ω

then it holds that K > 1, where K ≡ inf{∫ |∇v|2 dx | v ∈ Hc1 (Ω), ∫ v2 p dx = 1, ∫ vp dx = 0}. Ω

Ω

(4.110)

Ω

̂ Ω) > 0 by the minimax principle in § 4.1.1, Inequality (4.110) implies μ(p, μ̂ 2 (p, Ω) =

sup

X1 ⊂Hc1 (Ω),codim X1 =1

inf{∫ |∇v|2 − v2 p dx | v ∈ X1 , ‖v‖2 = 1}, Ω

and hence Lemma 4.48 follows. Proof of Proposition 4.52. First, K in (4.110) is the second eigenvalue of the following problem: (EP) Find ϕ ∈ Hc1 \{0} and K ∈ R such that ∫Ω ∇ϕ⋅∇v dx = K ∫Ω ϕvp dx for any v ∈ Hc1 (Ω). In fact, the first eigenvalue of (EP) is 0 with constant eigenfunction, and therefore, the second eigenvalue of (EP) is given by K in (4.110). In particular, the minimizer ϕ ∈ Hc1 (Ω) of K in (4.110) satisfies − Δϕ = Kpϕ

in Ω,

ϕ|𝜕Ω = constant,

∫ 𝜕Ω

where ν denotes the outer unit normal vector on 𝜕Ω. 21 The proof is valid even in the case Σ = 8π.

𝜕ϕ ds = 0, 𝜕ν

(4.111)

222 | 4 Rearrangement Let {Ωi }i∈I be the nodal domains of ϕ, the set of connected components of {ϕ ≠ 0}. Each 𝜕Ωi consists of a number of piecewise C 2 Jordan curves by Proposition 4.9, and it holds that ∫ 𝜕Ωi

𝜕ϕ ϕ ds = 0, 𝜕ν

i∈I

(4.112)

by (4.111). In fact, each 𝜕Ωi is a portion of nodal lines {ϕ = 0} or the boundary 𝜕Ω, which implies 𝜕Ωi = γ0 ∪ γ1 with γ0 ⊂ 𝜕Ω and γ1 ⊂ {ϕ = 0} satisfying γ0 ∩ γ1 = 0. Since ∫ 𝜕Ωi

𝜕ϕ 𝜕ϕ ϕ ds = ∫ ϕ ds, 𝜕ν 𝜕ν γ0

we have only to take the case γ0 ≠ 0 to prove (4.112). If γ0 ∩ γ1 ≠ 0, then γ0 ⊂ {ϕ = 0} 𝜕ϕ because ϕ is constant on 𝜕Ω, which implies ∫γ 𝜕ν ϕ ds = 0 and hence (4.112). The 0 other case of γ0 ∩ γ1 = 0 induces γ0 = 𝜕Ω by γ0 ≠ 0 because Ω is simply-connected and hence 𝜕Ω is of one component.22 Then it follows that ∫

γ0

𝜕ϕ 𝜕ϕ ϕ ds = ∫ ds ⋅ ϕ|𝜕Ω = 0, 𝜕ν 𝜕ν 𝜕Ω

and hence (4.112). As is noted, from the proof of Theorem 4.11, property (4.112) implies #I ≤ 2 for the second eigenfunction ϕ, while it cannot be of definite sign due to ∫Ω ϕp dx = 0 in (4.110). Therefore, each Ω± = {±ϕ > 0} is a nodal domain of ϕ, and hence is an open connected set with the boundary composed of a number of piecewise smooth Jordan curves. Let κ = ϕ|𝜕Ω ∈ R. If κ = 0, this ϕ ∈ Hc1 (Ω) satisfies −Δϕ = Kpϕ,

±ϕ > 0

in Ω± ,

ϕ|𝜕Ω± = 0.

Since ∫Ω p dx < 8π, one either has ∫Ω p dx < 4π or ∫Ω p dx < 4π, which implies K > 1 + − by Theorem 3.3. If κ ≠ 0, on the other hand, any nodal line of ϕ does not touch 𝜕Ω, and hence either Ω+ or Ω− is simply-connected. Without loss of generality, suppose that Ω− is simply-connected, that is, 𝜕Ω ⊂ 𝜕Ω+ , to put Σ± = ∫ p dx. Ω±

Since Σ+ + Σ− = Σ < 8π, we have either Σ− < 4π or Σ+ < 4π ≤ Σ− . In the first case of Σ− < 4π, we obtain K > 1 again by Theorem 3.3, because ϕ− ≡ ϕ|Ω− ∈ H01 (Ω− ) holds by the above topological assumption on Ω± . The proof is thus reduced to the case Σ+ < 4π ≤ Σ− with a simply-connected Ω− . 22 This property is used later in accordance with the process of rearrangement.

4.3 Isoperimetric inequality on surfaces | 223

Putting Γ = 𝜕Ω, γ = 𝜕Ω− , and ϕ+ = ϕ|Ω+ , we obtain −Δϕ+ = Kpϕ+ ,

ϕ+ > 0

in Ω,

ϕ+ |γ = 0,

ϕ+ |Γ = constant,

∫ Γ

𝜕ϕ+ ds = 0. 𝜕ν

This property means that K is its first eigenvalue. Hence we obtain K = inf{ ∫ |∇v|2 dx | v ∈ H̃ c1 (Ω+ ), ∫ v2 p dx = 1} Ω+

(4.113)

Ω+

for H̃ c1 (Ω+ ) = {v ∈ H 1 (Ω) | v|γ = 0, v|Γ = constant}, and the minimum K in (4.113) is attained by a constant times ϕ+ . We take the zero extension of this minimizer ψ to Ω− , and show the desired conclusion K > 1 in (4.113) by reducing it to the radially symmetric case. Let τ = ψ|Γ , Ω1 = {ψ ≤ τ}, and Ω2 = {ψ > τ}. It holds that Ω = Ω1 ∪ Ω2 . The open set Ω2 may be empty, but otherwise we take the spherically decreasing rearrangement ψ∗2 of ψ2 = ψ|Ω2 in § 4.3.4. Henceforth, S denotes a round sphere with area 8π, and dσ and dV stand for its canonical metric and volume element, respectively. Then we take a disc B ⊂ S satisfying ∫ dV = ∫ p dx. B

(4.114)

Ω2

The function ψ∗2 on B is defined by ψ∗2 (x) = sup{t | x ∈ ωt }, with ωt standing for the open concentric disc of B such that ∫ω dV = ∫{ψ >t} p dx. It holds that t

2

∫ ψ2 p dx = ∫(ψ∗2 ) dV,

Ω2

B

2

󵄨 󵄨2 ∫ |∇ψ2 |2 dx ≥ ∫ 󵄨󵄨󵄨dψ∗2 󵄨󵄨󵄨 dV, B

Ω2

ψ∗2 |𝜕B = τ.

We apply the following procedure to ψ1 = ψ|Ω1 , which may be called an annular increasing rearrangement. First, we take open concentric discs B0 ⊂ B1 ⊂ S such that ∫ dV = ∫ p dx = Σ− ,

B0

Ω−

∫ dV = ∫ p dx, B1

(4.115)

Ω1

noting Ω− ⊂ Ω1 . Second, the function ψ1∗ on the annulus A ≡ B1 \ B0 is defined by ψ1∗ = inf{t | x ∈ At }, where At is the closed concentric annulus of A satisfying that At ∪ B0 ⊂ S is a closed ball and ∫A ∪B dV = ∫{ψ≤t} p dx. Since this process is an t

0

equimeasurable rearrangement, it holds that

∫ ψ2 p dx = ∫(ψ1∗ )2 dV. Ω+ ∩Ω1

A

224 | 4 Rearrangement A decrease of the Dirichlet integral, on the other hand, is achieved as in Proposition 4.44, ∫ |∇ψ|2 dx ≥ ∫ |dψ1∗ |2 dV, A

Ω+ ∩Ω1

because 0 ≤ ψ ≤ τ in Ω1 , ψ|γ = 0, and ψ|𝜕Ω1 \γ = τ imply 𝜕{x ∈ Ω1 ∪ Ω− | ψ(x) ≤ t} = {x ∈ Ω1 ∪ Ω− | ψ(x) = t}

a. e. t ∈ (0, τ).

The relations ψ1∗ |γ∗ = 0 and ψ1∗ |Γ∗ = τ, finally, are obvious, where γ ∗ = 𝜕B0 and Γ∗ = 𝜕B1 . Regarding Σ+ < 4π ≤ Σ− and Σ+ + Σ− < 8π, we arrange the disc B and the annulus A so as to be concentric with the center at the south pole s on S. Thus B0 and B1 are concentric discs with the center north and south poles, respectively, and B = {ψ1∗ > τ} and A = {0 < ψ1∗ < τ} are concentric to B1 . Furthermore, ω+ = A ∪ B is a disjoint sum contained in the south hemisphere of S, and γ ∗ and Γ∗ are the outscribing and inscribing boundaries of A, respectively. We thus obtain K ≥ K ∗ ≡ inf{ ∫ |dv|2 dV | v ∈ H̃ c1 (ω+ )}

(4.116)

ω+

where H̃ c1 (ω+ ) = {v ∈ H 1 (ω+ ) | v|γ∗ = 0, v = τ ∈ R on Γ∗ ∪ 𝜕B}. This proposition is thus reduced to K ∗ > 1. Through the stereographic projection ι : S → R2 ∪ {∞} used in § 4.3.4, this K ∗ is realized as the first eigenvalue of −Δϕ = Kp∗ ϕ in Ω∗ = A∗ ∪ B∗ , ϕ=τ∈R

on Γ2 ∪ Γ3 ,

∫ Γ2 ∪Γ3

ϕ|Γ1 = 0, 𝜕ϕ ds = 0, 𝜕ν

(4.117)

where A∗ = ι(A), B∗ = ι(B), Γ1 = ι(γ ∗ ), Γ2 = ι(Γ∗ ), and Γ3 = ι(𝜕B). The above function p∗ is derived from the projection ι of the Laplace–Beltrami operator −ΔS on S. The annulus A∗ and the disc B∗ are in the flat plane, concentric, and disjoint. Through the scaling transformation introduced in the previous section, the outer radius of A∗ may be equal to 1. Then, it holds that p∗ (x) =

8μ , (|x|2 + μ)2

μ > 0,

∫ p∗ dx = ∫ dV < 4π. Ω∗

(4.118)

ω+

The first eigenfunction ϕ(x) of (4.117) is radially symmetric and positive. Therefore, in terms of ξ = (μ − r 2 )/(μ + r 2 ) this Ψ(ξ ) = ϕ(x) satisfies, with some a and b in a < b < 1, [(1 − ξ 2 )Φξ ]ξ + (2/K ∗ )Φ = 0,

Φ(ξ ) > 0,

ξμ < ξ < a,

b 0. μ+1

(4.119)

Inequality K ∗ > 1, therefore, follows if ̂ ) > 0, Φ(ξ

ξμ < ξ < a,

b 0, Φ(ξ

0 < ξ < a,

b −∞,

v∈E

E = {v ∈ H 1 (Ω) | ∫ v = 0}.

(4.129)

Ω

23 The linearized operator of the solution to this equation is associated with the sesquilinear form derived from the second derivative of the functional. This form is equivalent to ℬ on Hc1 (Ω) described in § 4.3.7. 24 This minimizer is the minimal solution, vλ ∈ H01 (Ω).

4.3 Isoperimetric inequality on surfaces | 227

Both λ = 8π and λ = 4π are best in (4.126) and (4.129), respectively, for inf Jλ (v) > −∞

(4.130)

v∈E

to hold. In fact, the Euler–Lagrange equation of (4.130), − Δv = λ(

1 ev − ), v |Ω| ∫Ω e

𝜕v 󵄨󵄨󵄨󵄨 󵄨 = 0, 𝜕ν 󵄨󵄨󵄨𝜕Ω

∫ v = 0,

(4.131)

Ω

can admit a sequence of solutions (λk , vk ), k = 1, 2, . . . , such that λk → 4π and ‖vk ‖∞ → +∞, taking a blow-up point on 𝜕Ω [304]. If 𝜕Ω has corners with the minimal angle θ, furthermore, the constant 4π in (4.129) is reduced to 4θ. A sharp form of (4.127) is log(

1 1 ‖∇v‖22 + 1, ∫ ev ) ≤ |Ω| 16π Ω

v ∈ H01 (Ω)

(4.132)

with the constant K = 1 on the right-hand side being the best possible [234]. Trudinger–Moser inequality has a real analytic background of the embedding theorems of Sobolev and Morrey, W01,p (Ω)

np

󳨅→ {

L n−p (Ω), C

1− pn

1 ≤ p < n,

(Ω), p > n.

(4.133)

Recall that Ω ⊂ Rn is an open set and W01,p (Ω) is the closure of C0∞ (Ω) in W 1,p (Ω). In the critical case p = n in (4.133), the space W01,p (Ω) is embedded in an Orlicz space where Moser’s inequality provides the best exponent [231]: v ∈ H01 (Ω),

2

‖∇v‖2 ≤ 1 󳨐⇒ ∫ e4πv ≤ C.

(4.134)

Ω

Given v ∈ H01 (Ω) \ {0}, let w = v/‖∇v‖2 ∈ H01 (Ω). Since ‖∇w‖2 = 1, 2

∫ e4πw ≤ C, Ω

and then v = w‖∇v‖2 ≤ 4πw2 +

1 ‖∇v‖22 16π

implies

∫ ev ≤ eC ⋅ exp( Ω

1 ‖∇v‖22 ), 16π

a rough form of (4.132) or (4.127): log ∫ ev ≤ Ω

1 ‖∇v‖22 + K, 16π

v ∈ H01 (Ω).

(4.135)

228 | 4 Rearrangement If Ω is a compact Riemannian surface without boundary, there is a form of (4.135) with v ∈ H01 (Ω) replaced by v ∈ E, where E = {v ∈ H 1 (Ω) | ∫Ω v = 0}: log ∫ ev ≤ Ω

1 ‖∇v‖22 + K, 16π

v ∈ E.

(4.136)

Note that the Euler–Lagrange equation of (4.130), − Δv = λ(

1 ev − ) ∫Ω ev |Ω|

in Ω,

∫ v = 0,

(4.137)

Ω

does not have a boundary blow-up point, differently from (4.131). The inequality analogous to (4.134), v ∈ E,

2

‖∇v‖2 ≤ 1 󳨐⇒ ∫ e4πv ≤ C

(4.138)

Ω

is also known, and (4.138) and (4.136) are called Fontana’s inequality [119]. A sharp form of (4.136) is known if Ω is a round sphere. For Ω = S2 with radius 1, Onofri’s inequality [266, 164] holds as follows: log(

1 1 1 ‖∇v‖22 + ∫ ev ) ≤ ∫ v, 4π 16π 4π S2

v ∈ H 1 (S2 ),

S2

that is, K = 0 in (4.136). As we noticed, uniqueness of (3.37) is valid even for λ = 8π. If there is actually a solution on this hyperplane, this solution is connected with 𝒞 . Then we see that the minimum I8π (Ω) in (4.127) is achieved if and only if 𝒞 does not blow-up as λ ↑ 8π. This 1 case does not arise if Ω is a disc. There is only a singular limit 4 log |x| at λ = 8π, without any classical solutions. However, there can be multiple singular limits at λ = 8π except for a possible unique classical solution [226]. Assume the simply-connectedness of Ω. Let κ ∈ Ω be a critical point of the Robin function R = R(x), and g : B = {|z| < 1} → Ω be a conformal mapping such that g(0) = κ, which is equivalent to g 󸀠󸀠 (0) = 0: ∞

g(z) = κ + a1 z + ∑ ak z k . k=3

Then the nondegeneracy of this κ is equivalent to σ = |g 󸀠󸀠󸀠 (0)/g 󸀠 (0)| ≠ 2, or |a3 /a1 | ≠ 1/3. In this case uλ , 0 < 8π − λ ≪ 1 coincides with the Weston–Moseley solution, and hence (3.145) in § 3.4.4 arises.25 In the form of Boltzmann–Poisson equation (3.37), it reads as λk = 8π + π(D(x0 ) + o(1))σk 25 Equality (4.139) is valid even if κ is degenerate [72].

(4.139)

4.3 Isoperimetric inequality on surfaces | 229

with k2 |ak |2 − |a1 |2 , k − 2 k=3 ∞

D(x0 ) = ∑

σk =

λk . ∫Ω evk dx

Then we can classify simply-connected domains which attain I8π , using the above value D(x0 ) at the maximum point x0 of R(x). Henceforth, K ⊂ Ω denotes the set of critical points of R = R(x). Each x0 ∈ K in D(x0 ) < 0 admits |a3 /a1 | < 1/3, and hence a branch of classical solutions for λ < 8π, converging to a singular solution at λ = 8π. Thus 𝒞 = {(λ, uλ ) | 0 < λ < 8π} blows-up as λ ↑ 8π, lim ‖uλ ‖∞ = +∞,

(4.140)

λ↑8π

and therefore, I8π (Ω) in (4.127) is not attained. Assume the existence of x0󸀠 ∈ K \ {x0 } such that D(x0󸀠 ) ≤ 0 in (4.139), and perturb Ω, or g, in such a way that the perturbed set K takes two distinct elements with negative D. Then there are two distinct Weston– Moseley’s solutions for 0 < 8π − λ ≪ 1, contradicting the uniqueness of the solution there. In other words, if there is x0 ∈ K such that D(x0 ) < 0, then it is a unique critical point of the Robin function, and hence this function attains its maximum. If there is x0 ∈ K such that D(x0 ) > 0, on the contrary, equality (4.140) does hold and I8π (Ω) is attained. Since uniqueness of the solution is still valid at λ = 8π, this minimum is attained by this solution, and furthermore, 𝒞 extends to 0 < λ − 8π ≪ 1 because of the nondegeneracy of the linearized operator. These properties are refined including the case D(x0 ) = 0. Theorem 4.53 ([72]). Let Ω ⊂ R2 be a simply-connected domain with a smooth boundary 𝜕Ω, and K ⊂ Ω be the set of critical points of the Robin function R = R(x). Then the value I8π = I8π (Ω) is attained if and only if there is a maximum point x0 ∈ K of R(x) such that D(x0 ) > 0. If this is the case, D(x0 ) > 0 holds for any x0 ∈ K. If there is x0 ∈ K with D(x0 ) ≤ 0, on the contrary, this x0 is the unique maximum point of R(x), and (4.140) holds. Proof. For the first part, it suffices to show that the existence of x0 ∈ K such that D(x0 ) = 0 implies (4.140). In this case there is a sequence of univalent functions on B = {|z| < 1}, denoted by {gj }, satisfying the following conditions, where Kj ⊂ Ω stands for the set of critical points of the Robin function in Ωj = gj (B), denoted by Rj = Rj (x): 1. gj → g uniformly on B. 2. There is a sequence xj → x0 such that xj ∈ Kj and Dj (xj ) < 0. j

Letting the set of minimizers of Jλ for 0 < λ < 8π be 𝒞 j = {(λ, uλ ) | 0 < λ < 8π}, j

therefore, we obtain limλ↑8π ‖uλ ‖∞ = +∞ for each j. Hence each c ≫ 1 admits 0
0 implies that of the solution to (3.37) at λ = 8π, lim sup ‖uλ ‖∞ < +∞. λ↑8π

(4.144)

Then, (4.144) assures that I8π (Ω) is attained. Hence if this is the case, it holds that D(x0 ) > 0 for any x0 ∈ K by (4.143). To show the last part, we assume D(x0 ) ≤ 0 for x0 ∈ K. Then, we have x0 ∈ K0 , and u0 (x) = 8πG(x, x0 ) is the singular limit of the solution by (4.144). Hence this x0 ∈ K is unique. By Theorem 4.53, the value I8π (Ω) is attained if #𝒮0 ≥ 2. The other necessary and π sufficient condition, I8π (Ω) > 1 + 4π supΩ R + log |Ω| , is also confirmed by Theorem 4.53 [72].26

4.4 Normalized Ricci flow 4.4.1 Geometric motivation Normalized Ricci flow in two space dimensions describes the time evolution of the metric g = g(t) on the compact Riemann surface Ω without boundary, given by 𝜕g = (r − R) g, 𝜕t 26 This inequality was first presented as a sufficient condition by [62].

(4.145)

4.4 Normalized Ricci flow

| 231

where R = R(t) is the scalar (Gauss) curvature of (Ω, g(t)), r = r(t) is the volume mean r=

∫Ω R(t) dμt ∫Ω dμt

,

and μ = μt is its area element. The profile of the solution, global-in-time existence, and convergence as t ↑ +∞ to the metric with constant mean curvature [151], has opened the resolution of the Poincaré conjecture. Following the argument, first, we deduce 𝜕R = Δt R + R(R − r) 𝜕t

(4.146)

from (4.145), where Δt denotes the Laplace–Beltrami operator associated with g = gt . Second, we concentrate on the geometrically important case R(0) > 0 on Ω [86]. It follows from (4.146) that R(t) > 0 on Ω in this case, as far as the solution exists. We use, third, Gauss–Bonnet’s formula in the form of ∫ R(t) dμt = 4 πχ(Ω),

(4.147)

Ω

where χ(Ω) = 2 − 2 k(Ω) denotes the Euler characteristics of Ω, with k(Ω) standing for the genus of Ω. Since R(t) > 0 implies k(Ω) = 0 by (4.147), therefore, the uniformization theorem allows assuming Ω = S2 ,

g(t) = ew(⋅,t) g0 ,

(4.148)

where S2 denotes the two-dimensional sphere, g0 is its standard metric, and w = w(⋅, t) is a smooth function. Let R0 be the scalar curvature associated with g0 and Δ = Δg0 . Then, (4.148) implies R(t) = e−w (−Δw + R0 ),

(4.149)

∫ R(t) dμt = 8π

(4.150)

while

S2

by (4.147). We thus obtain r=

8π 8π = ∫S2 dμt ∫S2 ew dx

(4.151)

for dx = dμg0 . Note also that (4.150) implies 󵄨󵄨 2 󵄨󵄨 󵄨󵄨S 󵄨󵄨R0 = 8 π because R0 is a constant.

(4.152)

232 | 4 Rearrangement Substitute (4.149) into (4.145), use (4.151)–(4.152), and obtain 𝜕ew ew 1 = Δw + 8π( − 2 ) w 𝜕t |S | ∫2e S

in S2 × (0, T),

w|t=0 = w0 (x).

(4.153)

Here and henceforth, dx is omitted. The above geometric result of [151] means, in the context of (4.153), that the solution w = w(⋅, t) exists globally-in-time and satisfies w(⋅, t) → w∞ ,

t ↑ +∞

(4.154)

in C ∞ topology, where w∞ is a stationary solution: − Δw∞ = 8π(

ew∞ 1 − 2 ). w ∫S2 e ∞ |S |

(4.155)

From the view point of the theory of dynamical systems, this result is composed of three factors, that is, global-in-time existence of the solution, compactness of the orbit, and the ω-limit set being a singleton. The proof, on the other hand, depends heavily on the geometric structure, as all the quantities are derived from the metric, which results in the Harnack inequality for scalar curvatures, monotonicity in time of the geometric entropy, classification of the Ricci solitons, deformation of the Ricci flow via the transformation groups, and so forth. An analytic method was applied to simplify this proof [30]. First, the a priori estimate |∇S2 w| ≤ C

(4.156)

is derived via the moving plane on the sphere, called the moving sphere method. Second, inequality (4.156) induces global-in-time existence of the solution and the compactness of the orbit via the Harnack inequality. Finally, the normalized stationary solution, − Δw = 8π(

ew 1 − ) ∫S2 ew |S2 |

in S2 ,

∫ w = 0,

(4.157)

S2

is unique, which assures that its ω-limit set is a singleton.27 In fact, the functional 1 Jλ (w) = ‖∇w‖22 − λ{log ∫ ew−w }, 2 Ω

w=

1 ∫w |Ω| Ω

2

for λ = 8π and Ω = S takes the role of the Lyapunov function in (4.153):

d ew 1 Jλ (w) = ∫ ∇w ⋅ ∇wt − λ ∫( − )wt = − ∫ ew wt2 ≤ 0, w dt |Ω| ∫ e Ω

Ω

Ω

Ω

27 Geometrically, this uniqueness means that the metric on the round sphere with constant Gaussian curvature is standard. An analytic proof to show directly the uniqueness of the solution to (4.157) is given in [73, 84, 210].

4.4 Normalized Ricci flow

| 233

and therefore, the ω-limit set of the compact, global-in-time orbit is contained in the set of stationary solutions. Equation (4.153), on the other hand, implies d ∫ ew = 0, dt S2

and therefore, if w = w∞ belongs to the ω-limit set to (4.153), it satisfies (4.155) and ∫ ew∞ = ∫ ew0 . S2

(4.158)

S2

Then uniqueness of the solution to (4.157) implies that the solution w∞ to (4.155) is a constant defined by w∞ = log(

1 ∫ ew0 ). |S2 | S2

In particular, the ω-limit set to (4.155) is a singleton, and (4.154) holds in C ∞ topology.

4.4.2 Analytic approach Henceforth, we investigate a general form of (4.153), 𝜕ew 1 ew − = Δw + λ( ), 𝜕t ∫ ew |Ω|

w|t=0 = w0 (x),

Ω

(4.159)

where Ω is a compact Riemannian surface without boundary and λ > 0 is a constant. This problem is no longer about a normalized Ricci flow unless Ω = S2 and λ = 8π. Similar conclusion, however, is derived for any Ω and 0 < λ ≤ 8π, which provides also an analytic proof of the above result concerning (4.145). First, we have d ∫ ew = 0 dt Ω

in (4.159), similarly to (4.150), and hence r = 𝜕ew ≤ Δw + rew , 𝜕t

λ ∫Ω ew

is a constant. Second,

w|t=0 = w0 (x),

and therefore, the comparison theorem implies ew ≤ e‖w0 ‖∞ ⋅ ert .

234 | 4 Rearrangement An estimate of w = w(⋅, t) from below, however, is not easy to get due to the lack of a geometric structure of (4.159). Here, it may be worth noting the following part of a geometric argument [151]. We rewrite (4.159) as wt = r − R,

(4.160)

using R = e−w (−Δw +

λ ). |Ω|

Then ∫ wt dμt = ∫ wt ew = Ω

Ω

d ∫ ew = 0, dt Ω

and so the curvature potential f = f (⋅, t) is determined by Δt f = R − r, where Δt is the Laplace–Beltrami operator associated with the metric g(t) = ew(⋅,t) g0 . Then we obtain28 Δt (

𝜕f ) = Δt (Δt f + rf ), 𝜕t

to normalize this f as 𝜕f = Δt f + rf . 𝜕t If (4.159) is a normalized Ricci flow, the Bochner–Weitzenböck formula induces 𝜕H = Δt H − 2|M|2 + rH, 𝜕t

(4.161)

where 󵄨 󵄨2 H = R − r + 󵄨󵄨󵄨∇t f 󵄨󵄨󵄨 ,

1 Mij = ∇it ∇jt f − Δt f ⋅ gij 2

(4.162)

with ∇it , i = 1, 2, being covariant derivatives associated with g(t) = ew(⋅,t) g0 . We obtain H ≤ ‖H0 ‖∞ ert , 28 Chapter 5 of [87].

R ≤ r + ‖H0 ‖∞ ert

(4.163)

4.4 Normalized Ricci flow

| 235

by (4.161)–(4.162), and then wt ≥ −‖H0 ‖∞ ert ,

w ≥ w0 − r −1 ‖H0 ‖∞ ert

follows from (4.160) and (4.163). This estimate of w = w(⋅, t) from below ensures T = +∞. Coming back to (4.159), we put u = rew , t = r −1 τ: uτ = Δ log u + u −

1 ∫ u, |Ω|

u|τ=0 = u0 (x) > 0,

Ω

∫ u0 = λ.

(4.164)

Ω

An analogous logarithmic diffusion equation is uτ = Δ log u in Ω × (0, T),

u|t=0 = u0 (x) > 0

(4.165)

derived from the Ricci flow. This (4.165) is known to represent several physical models. If Ω = R2 and u0 ∈ L1 (R2 ), the total mass of u ∈ C([0, T), L1 (Ω)) decreases as ∫ u(⋅, τ) = ∫ u0 − 4πτ, R2

R2

and T = ∫R2 u0 /4π is the maximal existence time of the solution.29 We have, actually, u ≡ 0 at t = T, and singularities arise on the right-hand side of (4.165). We thus have quenching in this case. Equation (4.164) takes the form uτ = Δ(log u − v),

−Δv = u −

1 ∫ u, |Ω| Ω

∫ v = 0, Ω

which is comparable to the Smoluchowski–Poisson equation described in § 7.5.4, uτ = ∇ ⋅ u∇(log u − v),

−Δv = u −

1 ∫ u, |Ω| Ω

∫ v = 0.

(4.166)

Ω

Henceforth, we employ t for τ in (4.164), to study ut = Δ log u + u −

1 ∫ u, |Ω|

u|t=0 = u0 (x) > 0.

(4.167)

Ω

We write, furthermore, (−Δ)−1 u = v for − Δv = u −

1 ∫ u, |Ω| Ω

29 Chapter 8 of [358].

∫ v = 0. Ω

(4.168)

236 | 4 Rearrangement From equation (4.166) for τ = t, the free energy 1 2

ℱ (u) = ∫ u(log u − 1) − ⟨(−Δ) u, u⟩ Ω

−1

takes the role of Lyapunov function in (4.167), where ⟨⋅, ⋅⟩ is the paring between the field and particles identified with the L2 -inner product. More precisely, since the operator (−Δ)−1 is self-adjoint in L2 (Ω), it follows that d 󵄨 󵄨2 −1 ℱ (u) = ⟨ut , log u − (Δ) u⟩ = − ∫󵄨󵄨󵄨∇(log u − v)󵄨󵄨󵄨 ≤ 0. dt

(4.169)

Ω

From the total mass conservation, furthermore, ∫ u(⋅, t) = ∫ u0 ≡ λ, Ω

(4.170)

Ω

equation (4.167) is reduced to ut = Δ log u + u −

λ , |Ω|

u|t=0 = u0 (x) > 0.

(4.171)

Then, u(t) = ‖u0 ‖∞ et is a supersolution to (4.171), and it holds that 0 < u(x, t) ≤ ‖u0 ‖∞ et

in Ω × [0, T).

4.4.3 Benilan’s inequality To estimate u(⋅, t) > 0 from below in (4.171), we put w = log u and reduce (4.171) to a similar form of (4.159), that is, λ 𝜕ew = Δw + ew − , 𝜕t |Ω|

∫ ew = λ.

(4.172)

Ω

Equation (4.171) is regarded as a limit case of the porus media equation, which admits the Benilan–Crandall inequality30 ut (x, t) ≤ g(t), u(x, t) with prescribed g(t). This property is also valid for (4.171). 30 See Chapter 1 of [358], together with the analogous inequality of Aronson–Benilan.

4.4 Normalized Ricci flow

| 237

Lemma 4.54. The solution u = u(⋅, t) to (4.171) satisfies ut (x, t) et ≤ t u(x, t) e −1

in Ω × (0, T).

(4.173)

Proof. Equation (4.171) implies utt = Δ(

ut ) + ut , u

and hence 2

utt u − u2t u u u 1 = Δ( t ) + t − ( t ) . 2 u u u u u We thus obtain pt = e−w Δp + p − p2 for p = ut /u. If C > 0 is a constant, the function q(x, t) = 1 +

in Ω × (0, T)

(4.174)

1 et+C − 1

satisfies (4.174). Take 0 < δ ≪ 1 and define C = Cδ by Cδ = log(1 + Then we obtain

1 ) > 0. |‖p(⋅, δ)‖∞ − 1|

q(x, 0) = 1 + and therefore,

1 ≥ p(x, δ), −1

eCδ

p(x, t + δ) ≤ q(x, t) = 1 +

1

et+Cδ

−1

≤1+

et

1 −1

in Ω × (0, T − δ) by the comparison theorem. Then follows that ut (x, t) et = p(x, t) ≤ t u(x, t) e −1

as δ ↓ 0. Inequality (4.173) implies

and therefore,

u − u ⋅ et /(et − 1) 𝜕 u ( t )= t ≤ 0, 𝜕t e − 1 et − 1

(4.175)

lim u(x, t) = u(x, T) ∈ [0, ∞)

(4.176)

t↑T

exists for each t ∈ [0, T), in case T = Tmax < +∞.

238 | 4 Rearrangement 4.4.4 Estimate from below We turn to w = log u, satisfying (4.172). Inequality (4.173) then means wt (x, t) ≤

et

et −1

in Ω × (0, T).

(4.177)

Put w=

1 ∫w |Ω| Ω

and define the field functional 1 Jλ (w) = ‖∇w‖22 − λ{log(∫ ew ) − w} 2 Ω

1󵄩 󵄩2 = 󵄩󵄩󵄩∇(w − w)󵄩󵄩󵄩2 − λ log ∫ ew−w . 2

(4.178)

Ω

For the solution w = w(⋅, t) to (4.172), we obtain ew 1 d Jλ (w) = ∫ ∇w ⋅ ∇wt − λ( − )w dx w dt |Ω| t ∫ e Ω

Ω

= −∫e

w

Ω

wt2 ,

(4.179)

in accordance with (4.169). The Trudinger–Moser inequality on compact Riemannian surface without boundary arises as Fontana’s inequality [119] in the form of J8π (w) ≥ −C,

w ∈ H 1 (Ω),

∫ w = 0.

(4.180)

Ω

Here is a key lemma for the estimate of w from below. Lemma 4.55. Let w = w(⋅, t) be the solution to (4.172) for 0 < λ ≤ 8π. Then it holds that lim inf w(t) > −∞. t↑T

(4.181)

To show this lemma, we use the following fact proven in the next paragraph.31 31 The case for a bounded domain in R2 under the Dirichlet condition is treated in [259, 263]. See also Lemma 6.1 of [63].

4.4 Normalized Ricci flow

| 239

Lemma 4.56. Let Ω be a compact Riemannian manifold without boundary, embedded in RN isometrically. Let {uk } be a family of positive measurable functions on Ω satisfying ‖uk ‖1 = 8π,

ℱ (uk ) ≤ C,

lim ⟨(−Δ)−1 uk , uk ⟩ = +∞,

k→∞

lim ∫ xuk = 8πx∞ ∈ RN .

k→∞

Ω

Then, it holds that x∞ ∈ Ω and uk (x)dx ⇀ 8πδx∞ (dx)

in ℳ(Ω).

Proof of Lemma 4.55. In (4.178) we have Jλ (w) = (8π − λ) log(∫ ew−w ) + J8π (w) Ω

= (8π − λ)(log λ − w) + J8π (w), and therefore, this lemma is obvious for 0 < λ < 8π by (4.179)–(4.180). Assuming λ = 8π with T < +∞, we confirm the pointwise convergence (4.176) and the monotonicity (4.175). Since ∫ ew = 8π, Ω

it holds that 1 J8π (w) = ‖∇w‖22 − 8π log(∫ ew−w ) 2 Ω

1 = ‖∇w‖22 + 8πw − 8π log(8π), 2 and hence 1 − C ≤ ‖∇w‖22 + 8πw ≤ C 2

(4.182)

by (4.179)–(4.180). Trudinger–Moser–Fontana’s inequality (4.180), on the other hand, assures 1 1 1 󵄨 󵄨2 ∫ |∇w|2 ⋅ = ∫󵄨󵄨󵄨∇(w/2)󵄨󵄨󵄨 2 4 2 Ω

Ω

≥ 8π log(∫ ew/2 − Ω

1 (w/2)− |Ω|1 ∫Ω (w/2) e )−C |Ω|

= 8π log(∫ ew/2 ) − Ω

8π ∫(w/2) − C, |Ω| Ω

240 | 4 Rearrangement and hence 1 8π ⋅ 2 ∫ w − C. ∫ |∇w|2 ≥ 4 ⋅ 8π log(∫ ew/2 ) − 2 |Ω| Ω

Ω

(4.183)

Ω

Then it follows that w ≥ 4 log(∫ ew/2 ) − C

(4.184)

Ω

from (4.182)–(4.183). Recall that T < +∞. From inequality (4.184) it follows that lim w = −∞ 󳨐⇒ lim inf ∫ ew/2 = lim inf ∫ u(x, t)1/2 dx = 0, t↑T

t↑T

t↑T

Ω

Ω

which implies ∫ u(x, T)1/2 dx = 0 Ω

by the monotone convergence theorem. We thus obtain u(x, T) = 0

a. e. x ∈ Ω,

and therefore, 8π = lim ∫ u(x, t) dx = ∫ u(x, T) dx = 0 t↑T

Ω

Ω

again by the monotone convergence theorem. This equality is impossible, and hence (4.181) holds in this case. Let, finally, λ = 8π and T = +∞. Since ew w ≥ −e for w ∈ R, it holds that H(t) ≡ ∫ ew w ≥ −e|Ω|, Ω

and furthermore, (4.172) and (4.180) imply d 𝜕ew dH = ∫ ew (wt + wwt ) = ⋅w ∫ ew + ∫ dt dt 𝜕t Ω

Ω

8π = ∫[Δw + (e − )]w |Ω|

Ω

w

Ω

= −‖∇w‖22 + ∫ ew w dx − 8 πw Ω

w

≤ ∫ e w dx + 8πw + C = H + 8πw + C. Ω

4.4 Normalized Ricci flow

| 241

Assume, on the contrary to (4.181), that lim inf w(t) = −∞. t↑+∞

Then inequality (4.177) assures the existence of tk ↑ +∞ and δ > 0 such that tk+1 > tk +δ and 8πw(t) + C ≤ −k,

tk − δ < t < tk

for k = 1, 2, . . . Then it holds that d −t (e H) ≤ −ke−t , dt

tk − δ < t < tk .

t

Take t ∈ (tk − δ, tk − δ/2), apply ∫t k ⋅ dt to this inequality, and obtain e−tk H(tk ) ≤ e−t H(t) + k(e−tk − e−t ), which means H(t) ≥ et−tk H(tk ) + k(1 − et−tk )

≥ −e−δ+1 |Ω| + k(1 − e−δ/2 ),

tk − δ < t < tk − δ/2.

Hence lim

∫(ew w)(⋅, t) = +∞.

inf

k→∞ t∈(tk −δ,tk −δ/2)

(4.185)

Ω

We have, on the other hand, ∞

tk −δ/2

∞

∑ ∫ dt ∫ ew wt2 ≤ ∫ dt ∫ ew wt2 < +∞

k=1 t −δ k

0

Ω

(4.186)

Ω

by (4.178) and (4.180), together with 󵄩󵄩 w 󵄩󵄩2 w w 2 w 2 󵄩󵄩e wt 󵄩󵄩1 ≤ ∫ e ⋅ ∫ e wt = 8π ∫ e wt . Ω

Ω

Ω

Inequalities (4.186)–(4.187) then imply lim

k→∞

tk −δ/2

󵄩 󵄩2 ∫ 󵄩󵄩󵄩ew wt (⋅, t)󵄩󵄩󵄩1 dt = 0,

tk −δ

(4.187)

242 | 4 Rearrangement and therefore, equality (4.185) assures the existence of tk󸀠 ∈ (tk − δ, tk − δ/2) such that 󵄩󵄩 󵄩󵄩 𝜕ew 󵄩 󵄩 lim 󵄩󵄩󵄩 (⋅, tk󸀠 )󵄩󵄩󵄩 = 0. 󵄩󵄩1 k→∞󵄩 𝜕t 󵄩

lim ∫ ew w(⋅, tk󸀠 ) = +∞,

k→∞

Ω

(4.188)

Writing u = ew and uk = u(⋅, tk󸀠 ), we obtain lim ∫ uk log uk = +∞

k→∞

(4.189)

Ω

by the first equality of (4.188), and also (4.190)

ℱ (uk ) ≤ ℱ (u0 )

by (4.169). Then (4.189)–(4.190) imply lim ⟨(−Δ)−1 uk , uk ⟩ = +∞.

k→∞

It is, on the other hand, obvious that ‖uk ‖1 = 8π by (4.170) and since λ = 8π. Finally, passing to a subsequence, we obtain x∞ ∈ RN such that lim ∫ xuk = x∞ ∈ RN .

k→∞

Ω

Lemma 4.56, then, assures x∞ ∈ Ω, and, passing to a subsequence, we obtain ew(⋅,tk ) = uk ⇀ 8 πδx∞ 󸀠

in ℳ(Ω).

(4.191)

We now apply the second equality of (4.188) and (4.191) to (4.172). From the elliptic L1 -estimate [56], it follows that w(⋅, tk󸀠 ) → 8πG(⋅, x∞ )

in W 1,q (Ω),

1 ≤ q < 2,

where G = G(x, x󸀠 ) denotes the Green function to (4.168). Then (4.192) implies lim ∫ ew(⋅,tk ) = +∞, 󸀠

k→∞

Ω

by Fatou’s lemma,32 which contradicts ∫Ω ew = 8π. 32 [53]; see § 5.3.3.

(4.192)

4.4 Normalized Ricci flow

| 243

4.4.5 Concentration of probability measures Lemma 4.56 is regarded as concentration of absolutely continuous probability measures. Put P(Ω) = {ρ ∈ L1 (Ω) | ρ ≥ 0, ‖ρ‖1 = 1} and ℐ (ρ) =

1 1 ∬ G(x, x󸀠 )ρ ⊗ ρ − ∫ ρ log ρ, 2 8π Ω×Ω

ρ ∈ P(Ω).

Ω

Given 0 ≤ u ∈ L1 (Ω) in ‖u‖1 = 8π, we have ρ = u/8π ∈ P(Ω) and ℐ (ρ) = −

1 1 {∫ u(log u − 1) − ∬ G(x, x 󸀠 )u ⊗ u} 2 64π 2 Ω

Ω×Ω

1 1 − {1 − log(8π)} = − ℱ (u) + constant. 64π 2 64π 2 Since the dual form of Fontana’ inequality reads33 inf{ℱ (u) | u ≥ 0, ‖u‖1 = 8π} > −∞, we obtain sup{ℐ (ρ) | ρ ∈ P(Ω)} < +∞. Let, furthermore, 𝒦(ρ) =

1 ∬ G(x, x󸀠 )ρ ⊗ ρ, 2 Ω×Ω

ℰ (ρ) = − ∫ ρ(log ρ − 1). Ω

It holds that ℐ = 𝒦 + ℰ /(8π).

Lemma 4.56 is reduced to the following lemma by (4.193). 33 It is the Toland duality; see [335].

(4.193)

244 | 4 Rearrangement Lemma 4.57. If {ρk } ⊂ P(Ω) satisfies lim 𝒦(ρk ) = +∞,

k→∞

lim ℐ (ρk ) = I∞ > −∞,

k→∞

lim ∫ xρk = x∞

k→∞

Ω

for x∞ ∈ RN , it holds that x∞ ∈ Ω and ρk (x)dx ⇀ δx∞ (dx)

in ℳ(Ω).

For the proof, we use the following lemma comparable to the improved Trudinger– Moser inequality to be stated in § 5.6.5, where ℐβ = 𝒦 + ℰ /β.

Lemma 4.58. Any d > 0 and m > 0 admit C > 0 and β > 8π such that if Ai ⊂ Ω, i = 1, 2, are measurable sets satisfying dist(A1 , A2 ) ≥ d,

∫ ρ ≥ m,

i = 1, 2

(4.194)

Ai

for ρ ∈ P(Ω), it holds that ℐβ (ρ) ≤ C. Proof. Let A3 = Ω \ (A1 ⋃ A2 ), and put χi = χAi ρ, 1 ≤ i ≤ 3. It holds that 𝒦(ρ) =

1 ∬ G(x, x󸀠 )ρ ⊗ ρ = E11 + E22 + E33 + 2E12 + 2E23 + 2E31 , 2

where G = G(x, x󸀠 ) is the Green function of −Δ and Eij =

1 ∬ G(x, x 󸀠 )ρi ⊗ ρj . 2 Ω×Ω

Since G(x, x 󸀠 ) is smooth in x ≠ x󸀠 , it holds that E12 ≤ C = C(d, m) by (4.194). Thus we obtain 2

2

2

𝒦(ρ) ≤ a + b + c + 2bc + 2ca + C

(4.195)

for E11 = a2 , E22 = b2 , E33 = c2 , because the Schwarz inequality implies Eij2 ≤ Eii Ejj . Inequality (4.193), on the other hand, implies Si ≤

−8πEii + C , Mi

1≤i≤3

for Si = − ∫ ρ log ρ, Ai

Mi = ∫ ρ, Ai

4.4 Normalized Ricci flow

| 245

and therefore, ℰ (ρ) = − ∫ ρ log ρ ≤ −8π( Ω

b2 c2 a2 + + )+C M1 M2 M3

(4.196)

with M1 , M2 ≥ m,

M1 + M2 + M3 = 1.

(4.197)

Inequalities (4.195), (4.196), and (4.197), then imply34 the existence of δ = δ(m) > 0 such that ℰ (ρ) ≤ −(8π + δ)𝒦(ρ) + C,

and the proof is complete. Proof of Lemma 4.57. We define the concentration function of ρk ∈ P(Ω) by Qk (r) = sup y∈Ω

∫

ρk ,

0 < r ≪ 1.

Ω∩B(y,r)

Take xk ∈ Ω as in ρk = Qk (r/2),

∫ Ω∩B(xk ,r/2)

to note 1 − Qk (r) ≤ 1 −

ρk =

∫ Ω∩B(xk ,r)

∫

ρk .

Ω\B(xk ,r)

Then it holds that min{Qk (r/2), 1 − Qk (r)} ≤ min{

∫

Ω∩B(xk ,r/2)

ρk ,

∫

ρk }

Ω\B(xk ,r)

and Lemma 4.58 is applicable with d = r/2. Given m > 0, we have β > 8π such that m ≤ min{Qk (r/2), 1 − Qk (r)} 󳨐⇒ ℐβ (ρk ) ≤ C. We have, on the other hand, ℐβ (ρk ) =

β 8π {( − 1)𝒦(ρk ) + ℐ (ρk )} → +∞ β 8π

34 See the proof of Lemma 6.2 of [63].

(4.198)

246 | 4 Rearrangement from the assumption, and therefore, lim min{Qk (r/2), 1 − Qk (r)} = 0

k→∞

by (4.198). Here we apply Vitali’s covering theorem to assure the existence of c0 > 0 such that Qk (r) ≥ c0 r 2 ,

k = 1, 2, . . . ,

0 < r ≪ 1,

to obtain lim {1 − Qk (r)} = 0,

0 < r ≪ 1.

k→∞

(4.199)

Given 0 < r ≪ 1, this (4.199) implies 1 − Qk (r/2) ≤ r for k ≫ 1. Letting xk = ∫Ω xρk , therefore, we obtain 󵄨󵄨 󵄨󵄨 󵄨 󵄨 |xk − xk | = 󵄨󵄨󵄨∫(x − xk )ρk 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

≤

|x − xk |ρk +

∫ Ω∩B(xk ,r)

∫

|x − xk |ρk

Ω\B(xk ,r)

≤ r + diam Ω ⋅

ρk

∫ Ω\B(xk ,r/2)

= r + diam Ω ⋅ (1 − Qk (r/2)) ≤ (1 + diam Ω)r.

It holds that limk→∞ |xk − xk | = 0, and hence x∞ ∈ Ω. Given ζ = ζ (x) ∈ C(Ω), similarly, we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨ζ (xk ) − ∫ ζρk 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

≤

∫ Ω∩B(xk ,r)

󵄨󵄨 󵄨 󵄨󵄨ζ (xk ) − ζ (x)󵄨󵄨󵄨ρk +

∫ Ω\B(xk ,r)

󵄨󵄨 󵄨 󵄨󵄨ζ (xk ) − ζ (x)󵄨󵄨󵄨ρk

󵄩 󵄩 ≤ 󵄩󵄩󵄩ζ − ζ (xk )󵄩󵄩󵄩L∞ (B(x ,r)) + 2 ⋅ diam Ω ⋅ ‖ζ ‖∞ k 󵄩 󵄩 ≤ 󵄩󵄩󵄩ζ − ζ (xk )󵄩󵄩󵄩L∞ (B(x

k ,r))

∫ Ω\B(xk ,r/2)

+ 2 ⋅ diam Ω ⋅ ‖ζ ‖∞ r.

Since ζ ∈ C(Ω) is uniformly continuous, we obtain 󵄨󵄨 󵄨󵄨 󵄨 󵄨 lim 󵄨󵄨󵄨ζ (xk ) − ∫ ζρk 󵄨󵄨󵄨 = 0 󵄨󵄨 k→∞󵄨󵄨 Ω

with r ↓ 0, and hence ρk (x)dx ⇀ δx∞ (dx).

ρk

4.4 Normalized Ricci flow

| 247

4.4.6 Compactness of the orbit The dynamics of the analytic normalized Ricci flow is described by the following theorem. Theorem 4.59. The solution w = w(⋅, t) to (4.159) exists globally-in-time, provided that 0 < λ ≤ 8π. Its orbit is precompact in C ∞ topology, and the ω-limit set ω(w0 ) = {w∞ ∈ C 2 (Ω) | ∃tk ↑ +∞ such that lim ‖w(⋅, tk ) − w∞ ‖C2 = 0} k→∞

is not empty, compact, connected, and contained in the set of stationary solutions to − Δw∞ = λ(

1 ew∞ − ), w ∞ |Ω| ∫Ω e

∫ ew∞ = ∫ ew0 . Ω

(4.200)

Ω

Proof. Assume 0 < λ ≤ 8π. It suffices to show that the solution u = ew to (4.171) satisfies the a priori estimate 0 < u(x, t),

u(x, t)−1 ≤ C

in Ω × [0, T),

from the parabolic regularity. First, we confirm 󵄩󵄩 󵄩 󵄩󵄩∇w(⋅, t)󵄩󵄩󵄩2 ≤ C,

0 ≤ t < T.

(4.201)

In fact, (4.201) is obvious for 0 < λ < 8π by (4.179)–(4.180) and λ λ 1 )‖∇w‖22 + J (w). Jλ (w) = (1 − 2 8π 8π 8π In the case of λ = 8π, we use 1 J8π (w) = ‖∇w‖22 − 8π log(∫ ew−w ) 2 Ω

1 = ‖∇w‖22 + 8πw − 8π log(8π). 2 Then, equality (4.179) and Lemma 4.55 imply (4.201). Second, we note that 󵄨󵄨 󵄨 󵄨󵄨w(t)󵄨󵄨󵄨 ≤ C holds by exp(

1 1 λ ≡α ∫ w) ≤ ∫ ew = |Ω| |Ω| |Ω| Ω

Ω

(4.202)

248 | 4 Rearrangement derived from Jensen’s inequality and Lemma 4.55. Inequalities (4.201)–(4.202) then assure ‖ log u‖H 1 (Ω) = ‖w‖H 1 (Ω) ≤ C by Poincaré–Wirtinger’s inequality, and therefore, 󵄩󵄩 p log u 󵄩󵄩 󵄩󵄩 −p log u 󵄩󵄩 󵄩󵄩1 ≤ Cp , 󵄩󵄩e 󵄩󵄩1 , 󵄩󵄩e

p≥1

by Fontana’s inequality, which results in 󵄩󵄩 󵄩 󵄩 −1 󵄩 󵄩󵄩u(⋅, t)󵄩󵄩󵄩p , 󵄩󵄩󵄩u (⋅, t)󵄩󵄩󵄩p ≤ Cp ,

p ≥ 1.

(4.203)

To estimate u from above, we use (4.171) to deduce 1 d p+1 p p ‖u‖p+1 p+1 = − ∫ ∇u ⋅ ∇ log u dx + ‖u‖p+1 − α‖u‖p p + 1 dt Ω

p = − ∫ u−1 ∇up ⋅ ∇u dx + ‖u‖p+1 p+1 − α‖u‖p Ω

4 󵄩 p 󵄩2 p = − 󵄩󵄩󵄩∇u 2 󵄩󵄩󵄩2 + ‖u‖p+1 p+1 − α‖u‖p , p recalling α = λ/|Ω|. We thus obtain 4 󵄩󵄩 p2 󵄩󵄩2 1 d ‖u‖p+1 ∇u 󵄩󵄩2 + α‖u‖pp = ‖u‖p+1 p+1 + 󵄩 p+1 p + 1 dt p󵄩 p

2 for p ≥ 1. Write the right-hand side of (4.204) as ‖u‖p+1 p+1 = ‖u ‖

rdo–Nirenberg inequality in two space dimensions, ‖w‖qq ≤ Cq0 ‖w‖q−1 ‖w‖1 , H1

2 p+1 p

2 p+1 p

(4.204) , and apply Gaglia-

1 ≤ q ≤ q0 < ∞

(4.205)

for q = 2 p+1 ∈ [2, 4]. From Poincaré–Wirtinger’s inequality, (4.202), and since 1 < p 1+

2 p

≤ 3, it follows that

󵄩󵄩 p2 󵄩󵄩 󵄩󵄩 p2 󵄩󵄩 1+ p2 󵄩󵄩 p2 󵄩󵄩 ‖u‖p+1 󵄩∇u 󵄩󵄩2 + 󵄩󵄩u 󵄩󵄩1 ) 󵄩󵄩u 󵄩󵄩1 p+1 ≤ C(󵄩 2

2

󵄩 p 󵄩1+ 󵄩 p 󵄩 󵄩 p 󵄩2+ ≤ C(󵄩󵄩󵄩∇u 2 󵄩󵄩󵄩2 p 󵄩󵄩󵄩u 2 󵄩󵄩󵄩1 + 󵄩󵄩󵄩u 2 󵄩󵄩󵄩1 p ).

(4.206)

Here, we take p > 2 and apply Young’s inequality to the right-hand side of (4.206) 2p for q = p+2 > 1, q1 + q1󸀠 = 1. It follows that ‖u‖p+1 p+1 ≤

2 1 󵄩󵄩 p2 󵄩󵄩1+ p2 q 1 −1 󵄩󵄩 p2 󵄩󵄩 q󸀠 󵄩 p 󵄩2+ (a󵄩󵄩∇u 󵄩󵄩2 ) + 󸀠 (a C16 󵄩󵄩u 󵄩󵄩1 ) + C 󵄩󵄩󵄩u 2 󵄩󵄩󵄩1 p q q

4.4 Normalized Ricci flow

| 249

with a > 0 satisfying aq 2 = . q p Since q → 2,

q󸀠 =

2p → 2, p−2

as p → ∞ and it holds that q󸀠 > 2 + ‖u‖p+1 p+1 ≤

2 p

aq ∼ p−1 ,

aq ∼ p−1 󸀠

for p > 2, we obtain

2p 2 󵄩󵄩 p2 󵄩󵄩2 󵄩 p󵄩 󵄩󵄩∇u 󵄩󵄩2 + C ⋅ p(󵄩󵄩󵄩u 2 󵄩󵄩󵄩1p−2 + 1). p

(4.207)

It thus holds that 2p p d 󵄩󵄩 p2 󵄩󵄩2 2 󵄩 󵄩󵄩u 2 󵄩󵄩󵄩 p−2 + 1) ‖u‖p+1 + 2 ∇u ≤ C ⋅ p ( 󵄩 󵄩 󵄩 󵄩2 󵄩 󵄩1 p+1 dt

(4.208)

for p ≥ 3 by (4.204) and (4.207). Adjusting a > 0, on the other hand, we get 2p

p 󵄩󵄩 p2 󵄩󵄩2 󵄩󵄩 2 󵄩󵄩 p−2 ‖u‖p+1 󵄩∇u 󵄩󵄩2 + C(󵄩󵄩u 󵄩󵄩1 + 1) p+1 ≤ 󵄩

(4.209)

similarly to (4.207). Adding (4.208)–(4.209) then ensures 2p p d p+1 2 󵄩 󵄩u 2 󵄩󵄩󵄩 p−2 + 1). ‖u‖p+1 󵄩 󵄩 p+1 + ‖u‖p+1 ≤ C ⋅ p (󵄩 1 dt

(4.210)

Use 2p

2p

p−2 (p+2) (p+1)(p−2) 󵄩󵄩 p2 󵄩󵄩 p−2 󵄩󵄩u 󵄩󵄩1 = (∫ u 2p ⋅ u 2p )

Ω

on the right-hand side of (4.210), and apply Hölder’s inequality for q = q = 󸀠

3p p+4

∈ (1, 3) satisfying

1 q

+

1 q󸀠

= 1. It follows that

1 2p 󸀠 p−2

4

2p

3 q 3 󵄩󵄩 p2 󵄩󵄩 p−2 q󸀠 (p+2) (p+1) ) (∫ u 2p ) 󵄩󵄩u 󵄩󵄩1 ≤ (∫ u 4

Ω

Ω

with (∫ u Ω

q󸀠 (p+2) 2p

1 2p q󸀠 p−2

)

≤C

3 2p 4 p−2

> 1 and

250 | 4 Rearrangement from (4.203), to reach 4

3 3 d p+1 (p+1) 2 4 ) + 1], ‖u‖p+1 p+1 + ‖u‖p+1 ≤ Cp [(∫ u dt

p ≥ 3.

(4.211)

Ω

The power of u is homogeneous in (4.211), similarly to the linear equation, and then the estimate 󵄩󵄩 󵄩 󵄩󵄩u(⋅, t)󵄩󵄩󵄩∞ ≤ C is derived from Moser’s iteration scheme [9]. First, inequality (4.211) implies sup [‖u‖p+1 p+1 + 1]

0≤t 0}. Proof. From the above conditions, we have 0 ≤ f (u) ≤ C(up + uq ),

u≥0

with C > 0 and p, q ∈ (1, n∗ ). Then Sobolev’s inequality implies 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 󵄨󵄨∫ f (v+ )v+ dx󵄨󵄨󵄨 ≤ ρ(‖∇v‖2 ) ‖∇v‖2 , 󵄨󵄨 󵄨󵄨 Ω

v∈X

(5.6)

258 | 5 Supplementary topics with a continuous function ρ(R) > 0 satisfying limR↓0 ρ(R) = 0. Hence there exists δ ∈ (0, 1) such that δ‖∇v‖22 ≤ I(v) for v ∈ X with R = ‖∇v‖22 ≪ 1, which implies the conclusion. Proposition 5.5. There is δ > 0 such that δ‖∇v‖22 ≤ J(v) for v ∈ X with R = ‖∇v‖2 ≪ 1. Proof. The inequality 0 ≤ F(u) ≤ C(up+1 + uq+1 ), u ≥ 0, holds by (5.6). The rest of the proof is similar. Now we use (f4). Proposition 5.6. For each v ∈ 𝒩+ ≡ {v ∈ X+ \{0} | I(v) = 0}, the function j(t) = J(tv) is strictly increasing and decreasing on [0, 1] and [1, ∞), respectively. Proof. Given v ∈ 𝒩+ , we have f (v) f (tv) 2 d J(tv) = t ∫{ − }v dx, dt v tv

t > 0,

Ω

which implies the conclusion. Corollary 5.7. It holds that d ≡ inf J = inf J > 0. 𝒩

𝒩+

(5.7)

Proof. By Propositions 5.4 and 5.6, we have d ≥ inf𝜕BR J for 𝜕BR = {v ∈ X | ‖∇v‖2 = R} with R > 0 sufficiently small. Hence (5.7) follows from Proposition 5.5. Proposition 5.8. Scalar multiplication realizes Q : X+ \{0} → N+ . Proof. For each v ∈ X+ \{0}, the continuous function t ∈ (0, ∞) 󳨃→ b(t) = ∫Ω is strictly increasing by (f4). It holds also that

f (tv) 2 v dx tv

lim b(t) = f 󸀠 (0) ∫ v2 dx = 0, t↓0

Ω

lim b(t) ≥ lim inf ∫ t→ +∞

t↑+∞

{v≥δ}

for 0 < δ ≪ 1 due to limu↑+∞ that

f (u) u

∫ Ω

f (tv) 2 v dx = +∞ tv

= +∞. Hence there is a unique t = t(v) ∈ (0, ∞) such f (tv) 2 v dx = ∫ |∇v|2 dx, tv

(5.8)

Ω

which means tv ∈ 𝒩+ . Proposition 5.9. The value t(v) ∈ (0, ∞) defined in the proof of Proposition 5.8 satisfies t(v) > 1 and t(v) < 1 for v ∈ 𝒩 i and v ∈ 𝒩 e , respectively.

5.1 Nehari principle |

259

Proof. The result follows from (5.8) and (f4). Proof of Theorem 5.3. In the definition of j, the path γ may be supposed to lie in X+ , by replacing γ(t) by γ(t)+ if necessary. Then it meets 𝒩+ and hence 𝒥 (γ) = max J(γ(t)) ≥ d, 0≤t≤1

which implies j ≥ d. Each ε > 0, on the contrary, admits v ∈ 𝒩+ such that J(v) < d + ε. Let γ1 and γ2 be the rays from the origin, containing v and e, respectively: γ1 = {tv | t ≥ 0},

γ2 = {te | t ≥ 0}.

If R ≫ 1, there is a curve γ3 ⊂ 𝜕BR in the plane generated by 0, e, and v which bridges γ1 and γ2 to create a path γ∗ ∈ Γ. Since R ≫ 1, one has J|γ3 ≤ 0, which implies j ≤ 𝒥 (γ∗ ) = J(v) < d + ε, and hence j ≤ d. In the proof of the above theorem, we have constructed a minimizer path γ ∈ Γ in the mountain pass critical value j of (1.58), under the cost of (f4). 5.1.3 Minimizing sequences To establish the proof of Theorem 5.1, we introduce the other functional 1 H(v) = ∫ f (v+ )v+ − F(v+ ) dx, 2

v ∈ X.

Ω

Lemma 5.10. It holds that H(v) > d for v ∈ 𝒩 e ∩ X+ . Proof. Since (f4) implies uf 󸀠 (u) > f (u) for u > 0, we have d 1 H(tv) = ∫ v{tvf 󸀠 (tv) − f (tv)} dx > 0, dt 2 Ω

v ∈ X+ \{0},

t > 0.

Hence the conclusion follows from Proposition 5.9 and d = inf𝒩+ J = inf𝒩+ H. Proof of Theorem 5.1. This theorem is proven by using the Lagrangian multiplier. First, we take a minimizing sequence {vk } ⊂ 𝒩+ for d in (5.4). Second, we show the boundedness of this {vk } in X. In fact, each ε > 0 admits 1 F(u) ≤ ( − ε)f (u)u + Cε , 2

u≥0

(5.9)

v ∈ 𝒩+ .

(5.10)

by (f 1), and hence J(v) ≥ ε ∫ f (v)v dx − Cε |Ω|, Ω

260 | 5 Supplementary topics Then it follows that ‖∇vk ‖22 = ∫ f (vk )vk dx = O(1), Ω

which implies also ∫Ω F(vk ) dx = O(1) by (5.9). Passing to a subsequence, we have the weak convergence of vk ⇀ u in X, the strong convergence in Lp for any 1 ≤ p < N∗ , and convergence a. e. in Ω. The dominated convergence theorem and Fatou’s lemma then imply J(vk ) = H(vk ) → H(u) = d,

∫ F(vk ) dx → ∫ F(u) dx, Ω

Ω

and I(u) ≤ lim inf I(vk ) = 0, respectively. Then there are the alternatives: u = 0, u ∈ 𝒩+ , or u ∈ 𝒩 e ∩ X+ . If u = 0, then d = H(u) = 0, which contradicts (5.7). The case u ∈ 𝒩 e ∩ X+ is not permitted by Lemma 5.10. Thus we obtain u ∈ 𝒩+ , and hence H(u) = J(u) = d. Then we obtain ‖vk ‖22 = J(vk ) + ∫ F(vk ) dx → d + ∫ F(u) dx = ‖u‖22 , Ω

Ω

and hence the strong convergence vk → u in X. Having u ∈ 𝒩+ which attains d in (5.7), we examine its constrained qualification in Lagrangian multiplier principle, that is, (5.11)

R(I 󸀠 (u)) = R. In fact, we have I 󸀠 (u)[v] = ∫ 2∇u ⋅ ∇v − f 󸀠 (u)uv − f (u)v dx,

v ∈ X,

Ω

and hence I 󸀠 (u)[v] = ∫{ Ω

f (u) − f 󸀠 (u)}u2 dx < 0 u

by (f 4), and then (5.11) follows. Therefore, there exists λ ∈ R such that ∫ ∇u ⋅ ∇v − f (u)v + λ{2∇u ⋅ ∇v − f 󸀠 (u)uv − f (u)v} dx = 0 Ω

for any v ∈ X. Taking v = u ∈ 𝒩+ again, we have 0 = ∫ |∇u|2 − f (u)u dx = −λI 󸀠 (u)[u], Ω

and hence λ = 0. Thus, the minimizer u is a classical solution to (5.1) by (5.12).

(5.12)

5.1 Nehari principle

| 261

5.1.4 Iterative sequences Here we show Theorem 5.2. Recall Q in Proposition 5.8. Lemma 5.11. The operator Q : X+ \0 → 𝒩+ is continuous in X. Proof. It suffices to show the continuity of v ∈ X+ \{0} 󳨃→ t(v) ∈ (0, +∞), where t = t(v) indicates the unique root of (5.8). Given {vk } ⊂ X+ \{0} converging to v0 ∈ X+ \{0} in X, Propositions 5.9 and 5.4 imply α ≡ inf{‖∇v‖2 | v ∈ 𝒩+ } > 0. Hence it holds that lim tk > 0,

k→∞

tk = t(vk ).

(5.13)

To show lim sup tk < +∞,

(5.14)

k→∞

assume on the contrary that tk → ∞, passing to a subsequence. Since v0 ∈ X+ \{0}, there are ε > 0 and ω ⊂ Ω such that |ω| > 0 and v0 (x) ≥ √2ε in ω. Since vk → v0 in X = H01 (Ω), one gets vk → v0 a. e. in ω, up to a subsequence, and then there is A ⊂ ω such that |A| > 0 and vk2 ≥ ε in A. Then we obtain ‖∇vk ‖22 = ∫ Ω

f (tk vk ) 2 f (t v ) f (s) v dx ≥ ∫ k k vk2 dx ≥ ε|A| inf → +∞ tk vk k tk vk s≥√εtk s A

as k → ∞, a contradiction. Inequalities (5.13) and (5.14) imply limk→∞ tk = t(v0 ) by the unique solvability of (5.8). Lemma 5.12 ([247, 248]). The operator T : X+ \{0} → 𝒩+ in (5.5) satisfies J(Tv) ≤ J(v),

v ∈ 𝒩+

(5.15)

with the equality if and only if v is a solution to (5.1). Proof. Given v ∈ 𝒩+ , let w = (−Δ)−1 f (v). We have to show J(tw) ≤ J(v),

t > 0,

(5.16)

to prove (5.15). From (f1) and (f3), it follows that F(v) = ϕ(v2 ) for a C 1 function ϕ : (v) [0, +∞) → [0, +∞). Since ϕ󸀠 (v2 ) = f 2v , this ϕ is convex by (f4), which implies 1 f (v) Φ(η) − Φ(v) ≥ (η2 − v2 , ), 2 v

η ∈ X,

where Φ(η) = ∫Ω F(η+ ) dx and (⋅, ⋅) is the L2 -inner product.

(5.17)

262 | 5 Supplementary topics Let (⋅, ⋅)σ be the L2 -inner product with respect to the measure dσ = w = (−Δ)−1 f (v), we have ‖∇w‖22 = (w, v)σ ,

f (v) dx. v

Since

(v, v)σ = (f (v), v) = (∇v, ∇w).

Since v ∈ 𝒩+ , it holds also that ‖∇v‖22 = (f (v), v) = (∇v, ∇w) ≤ ‖∇v‖2 ‖∇w‖2 , and hence ‖∇v‖2 ≤ ‖∇w‖2 . Writing ‖ ⋅ ‖σ = (⋅, ⋅)1/2 σ , therefore, we have ‖v‖2σ ≤ ‖∇v‖2 ‖∇w‖2 ≤ ‖∇w‖22 = (w, v)σ ≤ ‖w‖σ ‖v‖σ , and hence ‖v‖σ ≤ ‖w‖σ .

(5.18)

Now we recall (5.17) to conclude 1 J(tw) − J(v) = (t 2 ‖∇w‖22 − ‖∇v‖22 ) − (Φ(tw) − Φ(v)) 2 1 1 f (v) ≤ (t 2 ‖∇w‖22 − ‖∇v‖22 ) − (t 2 w2 − v2 , ) 2 2 v =

t2 ((w, v)σ − ‖w‖2σ ) ≤ 0 2

by (5.18), which means (5.17). In the above arguments, the equality in (5.17) holds if and only if w is a constant multiple of v. Hence v is a solution to (5.3) with a constant λ. Since v ∈ 𝒩+ , however, we have λ = 1, and therefore, v is a solution to (5.2). Proof of Theorem 5.2. First, the mapping T : X+ \{0} → 𝒩+ is compact by Rellich– Kondrachov’s theorem and Lemma 5.11. Take v0 ∈ 𝒩+ , and put vk = T k v0 , for k = 1, 2, . . . By Lemma 5.12 and Corollary 5.7, the sequence {J(vk )} has a limit d̃ ≥ d. Hence {vk } ⊂ X is bounded by (5.10). Given {vk󸀠 }, we have its subsequence denoted by the same symbol, converging weakly to some v ∈ X and satisfying also J(vk󸀠 ) = H(vk󸀠 ) → H(v) = d̃ ≥ d, which excludes the case v = 0. We have {vk2 } ⊂ {vk } satisfying vk󸀠 = Tvk2 . Similarly, we obtain {vk3 } ⊂ {vk2 } which converges weakly to an element w ∈ X+ \{0}. Then {vk4 = Tvk3 } ⊂ {vk󸀠 } ⊂ 𝒩+ converges strongly to v, which implies v ∈ 𝒩+ . Furthermore, {J(vk )} converges, and hence it holds that J(Tv) = lim J(Tvk3 ) = lim J(vk4 ) = J(v). k

Thus v is a solution to (5.2) by Lemma 5.12.

k

5.2 Moving plane revisited | 263

5.2 Moving plane revisited 5.2.1 A priori estimate of the solution In § 2.1, an a priori estimate is used to establish the existence of the solution to − Δu = f (u),

u>0

in Ω,

u|𝜕Ω = 0.

(5.19)

The method of the moving plane is applicable for this purpose. Let Σγ be the maximal cap of the domain Ω generated by the unit vector γ ∈ Rn . If Ω is strictly convex, the set ⋃γ Σγ is an Ω-neighborhood of 𝜕Ω. By the proof of Theorem 2.32, u decreases toward the boundary 𝜕Ω nontangentially. This Ω-neighborhood of 𝜕Ω is independent of the solution and the nonlinear term. Thus we have readily obtained the following fact. This Ix is a cone, having a distance more than ε/2 and less than ε to the boundary 𝜕Ω. Lemma 5.13. If Ω ⊂ Rn is a strictly convex domain with a C 2 boundary 𝜕Ω, there is a pair δ, ε > 0 determined by Ω such that any x ∈ Ω satisfying dist(x, 𝜕Ω) < ε/2 admits a measurable set Ix , satisfying |Ix | ≥ δ and Ix ⊂ {x ∈ Ω | dist(x, 𝜕Ω) > ε/2}, such that if f = f (u) is C 1 and u ∈ C 2 (Ω) in (5.19), then it follows that u(y) > u(x) for any y ∈ Ix . Theorem 5.14 ([99]). Let Ω ⊂ Rn be a strictly convex domain with a C 2 boundary 𝜕Ω, and λ1 be the first eigenvalue of −Δ with Dirichlet boundary condition. Assume that f is C 1 , n ≥ 3, and lim inf t→∞

f (t) > λ1 , t

lim

t→∞

f (t)

t

n+2 n−2

= 0,

lim sup t→∞

tf (t) − θF(t) ≤ 0, t 2 f (t)2/n

(5.20)

u

in (5.19), for θ ∈ [0, 2n/(n − 2)) and F(u) = ∫0 f (u) du. Then there is C > 0 such that any solution u = u(x) ∈ C 2 (Ω) to (5.19) admits ‖u‖∞ ≤ C. Proof. We divide the proof into four steps. Step 1. To apply Kaplan’s method, let ϕ = ϕ1 (x) > 0 be the first eigenfunction corresponding to λ1 > 0: −Δϕ1 = λ1 ϕ1 ,

ϕ1 > 0

in Ω,

ϕ1 |𝜕Ω = 0,

∫ ϕ1 dx = 1. Ω

Then (5.19) and (5.20) imply ∫Ω f (u)ϕ1 dx = λ1 ∫Ω uϕ1 dx and λt ≤ f (t) + C, t ≥ 0, for λ > λ1 , and hence λ ∫ uϕ1 dx ≤ ∫ f (u)ϕ1 dx + C = λ1 ∫ uϕ1 dx + C. Ω

Ω

Ω

We thus obtain ∫Ω uϕ1 dx ≤ C and ∫Ω f (u)ϕ1 dx ≤ C, and therefore, any compact set K ⊂ Ω admits C(K) > 0 such that ‖u‖L1 (K) ,

󵄩󵄩 󵄩 󵄩󵄩f (u)󵄩󵄩󵄩L1 (K) ≤ C(K).

(5.21)

264 | 5 Supplementary topics Step 2. Recall Lemma 5.13, and let K ≡ {x ∈ Ω | dist(x, 𝜕Ω) > ε/2} ⊂ Ω and dist(x, 𝜕Ω) < ε/2 for x ∈ Ω. Then it holds that C(K) ≥ ∫ u(y) dy ≥ u(x) ⋅ δ Ix

̂ by Ix ⊂ K. Thus we obtain a neighborhood ω̂ of 𝜕Ω such that ‖u‖L∞ (ω) ≤ C for ω = ω∩Ω, which implies 󵄩󵄩 󵄩 󵄩󵄩f (u)󵄩󵄩󵄩L1 (Ω) ≤ C

‖u‖L1 (Ω) ,

(5.22)

by (5.21). Furthermore, (5.19) and the boundary Schauder estimate described in § 1.3.4 imply ‖∇u‖L∞ (ω) ≤ C

(5.23)

for ω made smaller. Step 3. Inequality (5.23) implies ∫ F(u) dx ≤

n−2 ∫ uf (u) dx + C 2n

(5.24)

Ω

Ω

by the Pohozaev identity (1.6). Sobolev’s inequality, (5.22), and Hölder’s inequality, on the other hand, imply 2n 1 ∫ uf (u) dx = ∫ |∇u| dx ≥ (∫ u n−2 dx) C

2

Ω

Ω

n−2 n

Ω

2n 1 ≥ (∫ u n−2 dx) C

n−2 n

Ω

2

n 󵄨 󵄨 (∫󵄨󵄨󵄨f (u)󵄨󵄨󵄨 dx)

1 󵄨 󵄨2 ≥ ∫ u2 󵄨󵄨󵄨f (u)󵄨󵄨󵄨 n dx. C

Ω

(5.25)

Ω

By (5.20), furthermore, each 0 < ε ≪ 1 admits 󵄨 󵄨2 tf (t) ≤ θF(t) + εt 2 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 n + Cε ,

t ≥ 0,

and hence 󵄨 󵄨2 ∫ uf (u) dx ≤ θ ∫ F(u) dx + ε ∫ u2 󵄨󵄨󵄨f (u)󵄨󵄨󵄨 n dx + C.

Ω

Ω

(5.26)

Ω

Since 0 ≤ θ < 2n/(n − 2), we obtain ∫Ω F(u) dx ≤ C by (5.24)–(5.26), and then, the Pohozaev identity (1.6) implies ∫ f (u)u dx = ∫ |∇u|2 dx ≤ C. Ω

Ω

(5.27)

5.2 Moving plane revisited | 265

Step 4. Equation (5.19) implies ∫ f (u)up dx = p ∫ |∇u|2 up−1 dx = Ω

Ω

4p 󵄨 p+1 󵄨2 ∫󵄨󵄨󵄨∇u 2 󵄨󵄨󵄨 dx 2 (p + 1)

(5.28)

Ω

for 1 < p < ∞. By Sobolev’s inequality, a constant multiple of the right-hand side of (5.28) is estimated below by q

(∫ u dx)

n−2 n

,

q = n(p + 1)/(n − 2).

Ω

By (5.20), on the other hand, we have f (t)t p ≤ εt p+σ + Cε ,

t ≥ 0,

for σ = (n + 2)/(n − 2) and 0 < ε ≪ 1, and therefore, the left-hand side of (5.28) is estimated above by Cε ∫ u

n−2 nq

u

4 n−2

q

dx + Cε ≤ Cε(∫ u dx)

Ω

Ω

n−2 n

(∫ u

2n n−2

2 n

dx) + Cε

Ω

using Hölder’s inequality. Since (5.27) implies ‖u‖ 2n ≤ C, again by Sobolev’s inequaln−2 ity, it follows that ‖u‖q ≤ C for any 1 < q < ∞. Hence any Lp -norm of the right-hand side of (5.19) for 1 < p < ∞ is estimated above by a constant, which implies ‖u‖∞ ≤ C by the elliptic estimate and Morrey’s inequality. Another method different from that of the moving plane to derive a priori estimates of the solution is the blow-up analysis, where the convexity of the domain is not necessary under the cost of a sharp profile of the nonlinearity. This argument assures also the optimality of the exponent (n + 2)/(n − 2) of the growth order of the nonlinearity [136]. A typical example is − Δu = up ,

u>0

in Ω,

u|𝜕Ω = 0

(5.29)

n+2 for 1 < p < n−2 , where Ω ⊂ Rn is a bounded domain with a smooth boundary 𝜕Ω. To show the existence of C = C(Ω) such that ‖u‖∞ ≤ C for any classical solution u = u(x) to (5.29), we note the invariance of −Δu = up under the scaling transformation 2 uμ (x) = μ p−1 u(μx) for μ > 0. Thus, suppose, on the contrary, the existence of a family of solutions to (5.29), denoted by {uk }, such that Mk = ‖uk ‖∞ ↑ +∞. Letting uk (xk ) = Mk with xk ∈ Ω, we may assume xk → x0 ∈ Ω, passing to a subsequence. Then ũ k (x)̃ = uk (μk x̃ + xk )/Mk with μk = Mk−(p−1) satisfies

−Δũ k = ũ pk ,

0 < ũ k ≤ ũ k (0) = 1

in Ω̃ k = μ−1 k (Ω − {xk }),

ũ k |𝜕Ω̃ k = 0.

266 | 5 Supplementary topics Since μk ↓ 0, if x0 ∈ Ω and B(x0 , R) ⊂ Ω, one has R/μk ↑ +∞, and then the elliptic regularity implies the local uniform convergence ũ k → ũ in Rn , passing to a subsequence, with ũ = u(̃ x)̃ satisfying − Δũ = ũ p ,

̃ 0 ≤ ũ ≤ u(0) =1

in Rn .

(5.30)

n+2 Equation (5.30), however, does not admit a solution if 1 < p < n−2 , which is called the Liouville property [135], and we obtain a contradiction. Letting x0 ∈ 𝜕Ω, we assume that 𝜕Ω is represented by xn = φ(x󸀠 ) with a smooth φ = φ(x󸀠 ) for x󸀠 = (x1 , . . . , xn−1 ), satisfying Ω ∩ B(x0 , R) ⊂ {xn > φ(x 󸀠 )}, 0 < R ≪ 1, and ∇φ(x0󸀠 ) = 0. Then ũ k = ũ k (x)̃ is defined for x̃ ∈ B(0, R/μk ) ∩ {x̃n > φ̃ k (x̃ 󸀠 )} ≡ Dk , where

1 (−xkn + φ(xk󸀠 + μk x̃ 󸀠 )) μk 1 ̃ x̃k󸀠 )) + x̃ 󸀠 ⋅ ∇φ(xk󸀠 ) + o(1) = (−xkn + φ( μk

φ̃ k (x̃ 󸀠 ) =

locally uniformly in x̃ 󸀠 . Also −Δũ k = ũ pk ,

0 < ũ k ≤ ũ k (0) = 1

in Dk ,

ũ k |Γk = 0

for Γk = B(0, R/μk ) ∩ {x̃k = φ̃ k (x̃ 󸀠 )}. Since −xkn + φ(xk󸀠 ) < 0 and ∇φ(xk󸀠 ) = o(1), one gets 1 (−xkn +φ(xk󸀠 )) → c ∈ [−∞, 0], passing to a subsequence. If c = −∞, we obtain ũ k → ũ μk locally uniformly in Rn , again up to a subsequence, to reach (5.30), a contradiction. In the other case of −∞ < c ≤ 0, we obtain − Δũ = ũ p ,

̃ 0 ≤ ũ ≤ u(0) =1

in {x̃n > c},

u|̃ x̃n =c = 0

(5.31)

n+2 similarly. Obviously, c ≠ 0, while (5.31) does not hold even for c < 0 if 1 < p < n−2 [136]. All the possibilities are thus excluded, and we obtain the result. Classification of entire solutions is called the Liouville property. It implies a priori estimates of the solution under the property of scaling invariance of the equation [337]. For the sign-changing case of (5.29),

−Δu = |u|p−1 u in Ω,

u|𝜕Ω = 0

n+2 with 1 < p < n−2 , the property of uniform boundedness of the solution is equivalent to that of boundedness of its Morse indices [18].

5.2.2 Positivity of the solution We recall that the proof of Lemma 2.33 is divided into the cases of f (0) ≥ 0 and f (0) < 0. The former is easier, using Δu + f (u) − f (0) ≤ 0 in Ω. In fact, since f is C 1 , we find an L∞ -function c(x) satisfying −Δu ≥ c(x)u in Ω, and then Hopf’s lemma and the strong maximum principle guarantee the following result.

5.2 Moving plane revisited | 267

Proposition 5.15. If Ω ⊂ Rn is a bounded domain with a C 1 boundary 𝜕Ω, f = f (u) is a C 1 function satisfying f (0) ≥ 0, and u ∈ C 2 (Ω) ∩ C 1 (Ω) solves − Δu = f (u),

u ≥ 0,

u ≢ 0 in Ω,

u|𝜕Ω = 0,

(5.32)

then it follows that u>0

in Ω,

𝜕u 󵄨󵄨󵄨󵄨 󵄨 < 0. 𝜕ν 󵄨󵄨󵄨𝜕Ω

(5.33)

Here we concentrate on the case b(x) ≡ 0 in (2.60). By Proposition 5.15, the requirement of u > 0 in Ω in Theorem 2.32 is relaxed to u ≥ 0 and u ≢ 0, provided that f (0) ≥ 0. In fact, if n ≥ 2, this relaxation is justified even for f (0) < 0. From the proof, the following theorem is valid if Ω is close to a ball. Theorem 5.16 ([68]). If Ω = B ≡ {|x| < 1} ⊂ Rn , n ≥ 2, and f = f (u) is C 1 , a solution u ∈ C 2 (Ω) to (5.32) satisfies the first property of (5.33), and therefore, it holds that u = u(r) and ur (r) < 0 for 0 < r = |x| ≤ 1 by Theorem 2.20. Proof. It suffices to investigate the case f (0) < 0. To begin with, we observe that the proof of Lemma 2.33 is valid even if u ≥ 0 and u ≢ 0 in Ω. Assume the converse, and let y maximize |y| among y󸀠 ∈ B such that u(y󸀠 ) = 0. From the above observation, we may assume y = (δ, 0, . . . , 0) with 0 ≤ δ < 1. Then it holds that u > 0 in A = {x ∈ B | δ < |x| < 1}. For λ = (1 + δ)/2, the first equation of (2.61) does not hold from the proof of Lemma 2.33 and the assumption u(y) = 0. Since λ > 1/2, we obtain ζ ≡ inf{s ∈ [0, 1) | (2.61), ∀λ ∈ (s, 1]} > 1/2. By the continuity of u and u1 =

𝜕u , 𝜕x1 ζ

inequalities u1 ≤ 0 and u(x) ≤ u(x λ ) are valid

for λ = ζ . Since n ≥ 2, if u(x) ≡ u(x ) in Σ(ζ ), the reflection with respect to Tζ of the portion of 𝜕B cut by Σ(ζ ), denoted by C, is inside B, to reach 𝜕B transversally. Since u = 0 on C, however, this property contradicts the refinement of Lemma 2.33 stated above. All the assumptions in Lemma 2.34 are thus verified, and we reach (2.61) for λ = ζ . This property contradicts the definition of ζ so that u does not attain any zero in B. Proposition 5.17. If n ≥ 2, 𝜕Ω is C 2 , f = f (u) is C 1 , and u ∈ C 2 (Ω), it holds that u ≡ 0 or u > 0 in ω ≡ ⋃γ Σγ in (5.32) even if f (0) < 0. Proof. This proposition follows from the proof of the above theorem. Let f (0) < 0. If Ω is close to a ball, the first conclusion of (5.33) is valid in Proposition 5.15. The second relation 𝜕u/𝜕ν|𝜕Ω < 0, however, is not always true. For a C 1

268 | 5 Supplementary topics nonlinearity f = f (u), there is, thus, a solution u = u(|x|) ∈ C 2 (B) to (5.19) for Ω = B satisfying ur (1) = 0. To illustrate this property, we note that u(x) = cos x + 1 satisfies −uxx = u − 1,

−3π < x < 3π,

u|x=±3π = 0,

ux |x=±3π = 0.

This relation is also a counterexample to Theorem 5.16 for n = 1. In Proposition 5.15 it is possible to relax the regularity of f = f (u) to Lipschitz continuity. There is a counterexample, however, for Hölder continuous f = f (u), that is, v(x) = {

(1 − |x|2 )p , |x| ≤ 1, 0, |x| > 1.

If p > 2, this v = v(x) is a C 2 function in Rn , satisfying −Δv = f (v)

in B = B1 (0),

v|𝜕B = 0

for f (u) = −2p(p − 2)u1−2/p + 2p(n + 2p − 2)u1−1/p ,

u > 0.

This f (u) is Hölder continuous with the exponent close to 1 as p ↑ ∞, while

𝜕v | 𝜕ν 𝜕B

= 0.

5.2.3 Uniqueness of the solution Fredholm alternative assures the equivalence of uniqueness and existence of the solution to linear equations with symmetry. In nonlinear problems, however, this equivalence does not arise. Multiple existence or nonexistence of the solution arises in accordance with the shape of domain, including its dimension and the nonlinearity. This problem of global bifurcation is related to the geometrical properties of the solution. The latter properties have also left a number of open questions, even if limited to those of the first eigenfunction of the linear eigenvalue problem for −ΔD , the differential operator −Δ with the Dirichlet boundary condition. Let Ω ⊂ Rn be a bounded domain with a smooth boundary 𝜕Ω and 1 < p < ∞. As is described in Chapter 1, Emden–Fowler equation − Δu = up ,

u>0

in Ω,

u|𝜕Ω = 0,

admits a solution u ∈ C 2 (Ω) for 1 < p < 2∗ − 1 for 2∗ ≡ n∗ + 1 = {

2n , n−2

n ≥ 3, +∞, n = 2,

(5.34)

5.2 Moving plane revisited | 269

while if n ≥ 3, p ≥ 2∗ − 1, and Ω is star-shaped there is no solution satisfying u ∈ C 2 (Ω) ∩ C 1 (Ω). If p = 2∗ − 1, however, existence of the solution is assured by topological argument in accordance with the shape of Ω. As discussed in § 2.2.1, if Ω = B ≡ {|x| < 1} ⊂ Rn , the solution u ∈ C 2 (Ω) to (5.34) is unique. Several uniqueness results are valid if the domain is two-dimensional, convex, and symmetric with respect to both axes. We say that Ω ⊂ R2 is convex with respect to both axes, if any segment connecting two points in Ω and parallel to the coordinate axes lies in Ω. By Theorem 2.32, any solution u = u(x) to (5.34) in the following theorem is symmetric with respect to both axes. Degeneracy of the linearized operator is then a central issue, where topological properties of the nodal domain of the eigenfunction take a role. The technical assumption that 𝜕Ω is C 3 is used to work in a Schauder space for this spectral analysis, where the first part of Proposition 5.15 concerning the case f (0) ≥ 0 plays a role. Theorem 5.18. Let Ω ⊂ R2 be a domain with a C 3 boundary 𝜕Ω, symmetric and convex with respect to both axes of R2 . Given f ∈ C 1 (R) with f (0) ≥ 0, let 𝒮 = {(λ, u)} for 0 < λ < +∞ and u ∈ C 2 (Ω) be the total set of solutions to − Δu = λf (u),

u > 0 in Ω,

u|𝜕Ω = 0.

(5.35)

Then, each component of 𝒮 is homeomorphic to R. Lemma 5.19 ([163]). Under the assumption of Theorem 5.18, let (λ, u) ∈ 𝒮 . Then there is no nontrivial v = v(x) ∈ C 2 (Ω), symmetric with respect to both axes, such that v(0) = 0 and − Δv = λf 󸀠 (u)v

in Ω,

v = 0,

v < 0,

v>0

on 𝜕Ω, exclusively.

(5.36)

Proof. Assume the existence of such v = v(x) ∈ C 2 (Ω), and recall that each nonempty connected component of {x ∈ Ω | v(x) ≠ 0} is called a nodal domain of v. Theorem 4.3 is applicable since v(0) = 0. From the boundary condition and the symmetry of v = v(x), there is a nodal domain D of v such that either D ⊂ Ω1 or D ⊂ Ω2 , where Ω1 = Ω∩{x1 > 0} and Ω2 = Ω∩{x2 > 0}. Without loss of generality, we assume D ⊂ Ω1 ∩{v > 0}. Hence the differential operator −Δ − λf 󸀠 (u) in D under the Dirichlet boundary condition, denoted by AD , has the first eigenvalue equal to zero. Since f (0) ≥ 0, Lemma 2.33 casts a sharp form, and hence u1 = 𝜕u/𝜕x1 satisfies −Δu1 = λf 󸀠 (u)u1 ,

u1 < 0

in Ω1 = Ω ∩ {x1 > 0},

u1 |𝜕Ω∩{x1 >0} < 0.

This property, particularly the last inequality, implies that the first eigenvalue of −Δ − λf 󸀠 (u) in Ω1 under the Dirichlet boundary condition is positive. Then the Rayleigh principle guarantees inf{∫ |∇ϕ|2 − c(x)ϕ2 dx|ϕ ∈ H01 (Ω1 ), ‖ϕ‖2 = 1} > 0 Ω1

270 | 5 Supplementary topics for c(x) = λf 󸀠 (u). Taking the zeroth order expansion of the first eigenfunction of AD , however, we see that this infimum is nonpositive, a contradiction. Henceforth, we index the function space of symmetric functions with respect to both axes by Σ and the function space of functions being zero on the boundary by 0. Under the assumption of Lemma 5.19, any classical solution to (5.35) is symmetric with respect to both axes, and the linearized operator 2+θ θ L = −Δ − λf 󸀠 (u) : C0,Σ (Ω) 󳨀→ C0,Σ (Ω),

0 < θ < 1,

is Fredholm with the exponent zero. The first condition in Lemma 5.20 indicates a simple degeneracy, while the second is called the transversality condition. They are used in the abstract theorem of bifurcation from simple eigenvalues.1 Lemma 5.20. Under the assumption of Lemma 5.19, if L is nondegenerate it holds that dim 𝒩 (L) = 1 and f (u) ∈ ̸ ℛ(L). Proof. The first statement follows from Lemma 5.19. In fact, if the dimension of 𝒩 (L) is larger than 2 then a linear combination of two independent elements in this space creates a nontrivial solution v ≢ 0 satisfying (5.36), a contradiction. 2+θ Turning to the second part, we take v0 ≡ x ⋅ ∇u ∈ C0,Σ (Ω), which satisfies (−Δ − λf 󸀠 (u))v0 = 2f (u)

in Ω,

v0 |𝜕Ω < 0.

2+θ Assume on the contrary that f (u) ∈ ℛ(L), and take v1 ∈ C0,Σ (Ω) satisfying

(−Δ − λf 󸀠 (u))v1 = f (u) in Ω,

v1 |𝜕Ω = 0.

Then v = v0 −2v1 satisfies (−Δ−λf 󸀠 (u))v = 0 in Ω and v|𝜕Ω < 0. Since L is degenerate, on 2+θ the other hand, there exists a nontrivial v2 ∈ C0,Σ (Ω) satisfying (−Δ − f 󸀠 (u))v2 = 0 in Ω, and v2 |𝜕Ω = 0. Then a linear combination of v and v2 creates v ≢ 0 satisfying (5.36), a contradiction. Proof of Theorem 5.18. From the symmetry of the solution, it follows that 𝒮 ⊂ (0, +∞)× 2+θ C0,Σ (Ω). Then we apply the first part of Lemma 2.13 and obtain the result. If f (u) is asymptotically linear, bifurcation from infinity is applicable, which guarantees the connectedness of 𝒮 , which is parametrized by u(0), the value of the origin of the solution [163]. Transforming (5.34) to the nonlinear eigenvalue problem − Δu = λup ,

u>0

in Ω,

u|𝜕Ω = 0,

(5.37)

however, is not efficient to deduce uniqueness of the solution to (5.34) because of the − 1 trivial correspondence of its solution u0 and that u to (5.37), u = λ p−1 u0 .2 1 See Theorem 6.13. 2 Regarding p > 1 as a parameter is useful [209], but then the transversality condition does not follow.

5.2 Moving plane revisited | 271

5.2.4 Lipschitz continuity of the nonlinearity Here we try to relax the assumption f ∈ C 1 and u ∈ C 2 (Ω) in Theorem 2.32. We have observed in § 5.2.2 that if f (0) ≥ 0, the requirement on u is relaxed to u ∈ C 2 (Ω)∩C 0 (Ω), and Lemma 2.33 is improved to Lemma 5.15. The other proof of Lemma 2.33 due to Berestycki–Nirenberg [37] is valid when f (0) < 0. It is based on a careful use of the Poincaré inequality, where u ∈ C 2 (Ω) is relaxed to u ∈ C 2 (Ω) ∩ C 0 (Ω), but the other conditions f ∈ C 1 and u > 0 in Ω are necessary [37]. We still assume u ∈ C 2 (Ω) and concentrate on the condition f ∈ C 1 . This condition is used twice in the proof of Theorem 2.20, that is, in Lemmas 2.33 and 2.34. It is easy to see that this condition can be relaxed to the local Lipschitz continuity of f on [0, ∞). In the proof of Lemma 2.33, however, it is essentially used in the case of f (0) ≥ 0. Actually, there is a counterexample to the conclusion of Theorem 2.20 for Hölder continuous f satisfying f (0) ≥ 0, as is shown at the end of § 5.2.2, while if f (0) < 0 the conclusion of Lemma 2.33 follows only from f ∈ C 0 with 0 ≤ u ∈ C 2 (Ω) and u ≢ 0. If f (0) < 0, furthermore, the alternative proof [37] is valid for u ∈ C 2 (Ω) ∩ C 0 (Ω) under the cost of local Lipschitz continuity of f and the positivity of u in Ω. In contrast to this situation of Lemma 2.33, local Lipschitz continuity of f is used to derive an equation for the difference of u(x) and uλ (x) to satisfy. The desired property, however, is just u(x) > uλ (x) in Σ󸀠 (λ), which is obvious if f is monotone nondecreasing. Thus we obtain the following fact.3 Proposition 5.21. The first part of Theorem 2.32 is valid under the assumption that 𝜕Ω is C 2 , f = f1 +f2 with Lipschitz continuous f1 and monotone nondecreasing f2 , and u ∈ C 2 (Ω). To describe the method of Berestycki–Nirenberg in short, we recall the argument in § 2.2.3 again. The conclusion of Lemma 2.34 used in the proof of Lemma 2.34 is just needed in a left neighborhood of the hyperplane Tλ . The assertion of Lemma 2.33 is in the same sprit. For these proofs, the strong maximum principle in a narrow domain for the operator −Δ + c(x) is applicable, regardless of the sign of c(x) ∈ L∞ (Ω). Under this process, the requirement on the solution is relaxed, and at the same time, that for the monotonicity of f is reversed. Thus we obtain the following fact. Proposition 5.22 ([179]). The same conclusion of the first part of Theorem 2.32 holds in (5.19), provided that 𝜕Ω is C 1 , f = f1 + f2 , f1 is locally Lipschitz continuous on [0, ∞), f2 ∈ C(R) is locally Lipschitz in (0, ∞) and monotone nonincreasing near 0, and u ∈ C 2 (Ω) ∩ C 0 (Ω). 2,n The method of Berestycki–Nirenberg relaxes the condition on u(x) to u ∈ Wloc (Ω)∩ C (Ω) under the cost of the local Lipschitz continuity of f on [0, ∞). The other advantage is on the smoothness of 𝜕Ω, which is replaced by the convexity of Ω in the 0

3 Proposition 5.21 is noticed in the original paper [137].

272 | 5 Supplementary topics γ-direction. Under these improvements, the symmetry of a positive solution in a rectangular domain is proved as in § 2.2.3.4 A motivation of the above study is the degenerate parabolic equation vt − Δvm = g(v)

in Rn × [0, T),

m > 1.

(5.38)

Regarding the property of finite propagation of (5.38), we take the stationary solution w with compact support. Putting u = wm and f (u) = g(w), we thus reach − Δu = f (u),

u>0

in Ω,

u=

𝜕u =0 𝜕ν

on 𝜕Ω,

(5.39)

where Ω ⊂ Rn is a bounded domain. There is a classification of radially symmetric solutions, assuming Ω to be a ball. Obstruction to conclude radially symmetry of the free boundary of solution, however, is the lack of local Lipschitz continuity of f (u) = u1/m for u ≥ 0 at u = 0. The following theorem proven by the method of Proposition 5.22, actually, is not applicable. Theorem 5.23 ([179]). Assume that 𝜕Ω is C 2 , f = f1 + f2 , f1 is locally Lipschitz continuous on [0, ∞), f2 ∈ C(R) is locally Lipschitz in (0, ∞) and monotone nondecreasing, and u ∈ C 2 (Ω). Then, Ω in (5.39) is a ball, and the solution u = u(x) satisfies the conclusion of Theorem 2.20.

5.2.5 Poincaré metric and symmetry Besides the radial symmetry of the solution u = u(r), r = |x|, the decreasing ur < 0, 0 < r ≤ 1, is established in Theorem 2.20. We call such a property the radial decreasingness in short. If we find such decreasing lines in a general domain, the point which attains the maximum of the solution is prescribed. Assume that Ω ⊂ R2 is a simply-connected domain, and take a conformal mapping g : D = {|ζ | < 1} → Ω. This g is determined up to the Möbius transformation ϕz (ζ ) =

ζ +z , 1 + zζ

z ∈ D.

Then we obtain a family of functions Uz = u ∘ g ∘ ϕz , z ∈ D on D, which leads, for 𝜕 example, to w = 𝜕s U𝚤s |s=0 . We thus arrive at w = ξ ⋅ ∇U0 for ξ = (−2x1 x2 , 1 + x12 − x22 ). If this function is nonnegative in a part of D, then u does not increase the value in the corresponding part of Ω along the vector field g ∘ ξ . For the general vector field ξ , we have the following result, where the solution is said to be mild if the second eigenvalue of the linearized operator is positive. 4 This fact is implicitly assumed in the study of sinh-Gordon equation by [318].

5.2 Moving plane revisited | 273

Proposition 5.24 ([240]). Given K ∈ C 1 (D), let u = u(x) ∈ C 2 (D) be a mild solution to −Δu = K(x)f (u) ≥ 0

in D,

u|𝜕D = 0,

symmetric with respect to the x1 axis, and let h = ξ1 + 𝚤ξ2 be a holomorphic function satisfying x⋅ξ ≥0

on 𝜕D ∩ {x2 > 0},

ξ1 |D∩{x2 =0} = 0,

and ∇ ⋅ (Kξ ) ≤ 0

in D ∩ {x2 > 0}.

(5.40)

Then u is nonincreasing along ξ in D+ . To treat the general case, let K = K(r) satisfy (5.40) for ξ = (−2x1 x2 , 1 + x12 − x22 ). First, we note that this condition means the nonincreasingness of the function r 󳨃→ (1 − r 2 )2 K(r). This observation leads to the idea of using the Poincaré metric ds2 = (1 − r 2 )−2 dx2 in D to develop the moving plane method concerning ds. Here we recall the fundamental properties of this metric. First, the geodesic of Dg is an arc in D and it intersects 𝜕D orthogonally. Let Tλ be a geodesic, symmetric with respect to the x1 axis. Here λ indicates the x1 -coordinate of the intersection of Tλ and the x1 axis. If Tλ ⊂ {x1 > 0}, therefore, it holds that 0 < λ < 1. Second, the symmetric point xλ of x ∈ Tλ in Dg is determined so that the geodesic connecting x and xλ is perpendicular to Tλ and has the same distance to x from the crossing point of these two geodesics. This position coincides with the usual mirror reflection of x with respect to the arc Tλ . Let Σ(λ) and Σ󸀠 (λ) be the smaller and larger sections of D cut by Tλ , which are symmetric with respect to Tλ in the sense of ds. In such a situation, 󵄨 󵄨 x ∈ Σ󸀠 (λ) 󳨐⇒ |x| < 󵄨󵄨󵄨x λ 󵄨󵄨󵄨,

(5.41)

and the Laplace–Beltrami operator in Dg is given by Δg = (1 − r 2 )2 Δ. Given f ∈ C([0, 1] × [0, ∞)) in fu ∈ C([0, 1] × [0, ∞)), let u ∈ C 2 (D) ∩ C 0 (D) be the solution to − Δu = f (|x|, u) in D,

u|𝜕D = 0.

(5.42)

The standard method guarantees radial decreasingness of the solution under the assumption of the nonincreasingness of r 󳨃→ f (r, u) [137]. To confirm the validity of the Poincaré metric, let uλ (x) = u(xλ ) and define Λ ≡ {λ ∈ (0, 1) | u > uλ in Σ󸀠 (λ),

𝜕u 󵄨󵄨󵄨󵄨 󵄨 < 0} , 𝜕ν 󵄨󵄨󵄨Tλ

where ν denotes the outer normal unit vector on 𝜕Σ󸀠 (λ). Following § 2.2.3, we confirm three facts:

274 | 5 Supplementary topics 1. 2. 3.

0 < λ0 − λ ≪ 1 󳨐⇒ λ ∈ Λ. λ∗ ∈ Λ, 0 < λ∗ − λ ≪ 1 󳨐⇒ λ ∈ Λ. λj ∈ Λ, λj ↓ λ∗ ∈ (0, 1) 󳨐⇒ λ∗ ∈ Λ.

Here we apply the method of Berestycki–Nirenberg as in the proof of Theorem 5.23. Then we see that the above claims are reduced to the analogous assertions in Lemmas 2.33 and 2.34, and thus, eventually the maximum principle and Hopf’s lemma. What we have to do first is, therefore, to derive an equation satisfied by uλ . In fact, equality (5.42) implies 2

−Δg u = (1 − |x|2 ) f (|x|, u)

in D.

Since the map x 󳨃→ xλ is isometric with respect to ds, 󵄨 󵄨2 2 󵄨 󵄨 −Δg uλ = (1 − 󵄨󵄨󵄨xλ 󵄨󵄨󵄨 ) f (󵄨󵄨󵄨x λ 󵄨󵄨󵄨, uλ ) in D. If 2

r 󳨃→ (1 − r 2 ) f (r, u)

is nonincreasing, ∀u > 0,

(5.43)

then (5.41) assures −Δg (u − uλ ) ≥ (1 − |x|2 )2 (f (|x|, u(x)) − f (|x|, uλ (x))) in Σ󸀠 (λ) and hence − Δv ≥ c(x)v

in Σ󸀠 (λ)

(5.44)

for v = u − uλ , where 1

c(x) = ∫ fu (|x|, tu(x) + (1 − t)uλ (x)) dt.

(5.45)

0

A different situation from the standard case, here, is that Σ(λ) and Σ󸀠 (λ) divide D. From this property, a boundary point of Σ(λ) on 𝜕D is mapped on 𝜕D by the reflection with respect to Tλ , and hence v=0

on 𝜕Σ󸀠 (λ).

(5.46)

Since we cannot exclude v ≡ 0 in Σ󸀠 (λ∗ ) from u > 0 inside D, which causes a problem unless “nonincreasing” is changed to “decreasing” in (5.43), we just get either λ∗ ∈ Λ or u ≡ uλ in Σ󸀠 (λ) in the above third step. In the two-dimensional case, this property does not induce any trouble, but when considering higher dimensions, here we assume the decreasingness of r 󳨃→ (1 − r 2 )2 f (r, u) to proceed to the proof of the first and second steps. The essential property needed in the first step is v > 0 near 𝜕Σ(λ󸀠 ) and uν < 0 on Tλ , where uν denotes the normal component of u. Then D = Σ󸀠 (λ) ∪ Σ(λ) causes trouble again, to ensure the conclusion of Lemma 2.33. The method of Castro–Shivaji is not

5.2 Moving plane revisited | 275

effective also, because Tλ intersects 𝜕D orthogonally under the Poincaré metric. Such a difficulty, however, arises similarly when using standard moving planes in the whole space, or on rectangles, which is overcome by the method of Berestycki–Nirenberg, based on the positivity of the solution and the strong maximum principle in a narrow domain.5 We can discuss the first step as follows. Given a solution u, first, we recall that c(x) defined by (5.45) is uniformly bounded in λ. Hence there is r0 satisfying 0 < 1−r0 ≪ 1 such that the strong maximum principle is valid in the annulus A = D \ Br0 for −Δ − c(x) under the Dirichlet boundary condition. Then we take r1 ∈ (r0 , 1) satisfying max{u(x) | r1 ≤ |x| ≤ 1} < min{u(x) | |x| ≤ r0 }. Then, for r1 < λ < 1 and x ∈ Σ󸀠 (λ), it holds that |xλ | > r1 , and therefore, v(x) > 0 on Br0 . Regarding (5.44) and (5.46), now we apply the strong maximum principle II, Theorem 2.31, and then the Hopf lemma III, Lemma 2.29, on A, to deduce (r1 , 1) ⊂ Λ. For the second step, we use the strong maximum principle in A ∩ Σ󸀠 (λ∗ ) for A defined above. First, Theorem 2.31 implies v > 0 in A ∩ Σ󸀠 (λ∗ ), and then v > 0 in Σ󸀠 (λ). Second, Lemma 2.29 implies 𝜕v/𝜕ν < 0, and hence 𝜕u/𝜕ν < 0 on 𝜕A ∩ Tλ∗ . Since these estimates extend near the boundary 𝜕Σ(λ), finally, the standard argument guarantees λ ∈ Λ. Using the Poincaré metric in n ≥ 3, we obtain the following result. Theorem 5.25 ([239]). Let B = B(0, 1) ⊂ Rn , n ≥ 2, be the unit ball and f = f (r, s) ∈ C([0, 1] × [0, ∞)) be Lipschitz continuous in s, and u ∈ C 2 (B) ∩ C 0 (B) satisfy − Δu = f (|x|, u),

u>0

in B,

u|𝜕B = 0.

(5.47)

Assume, furthermore, that r 󳨃→ (1 − r 2 )

n+2 2

f (r, (1 − r 2 )

− n−2 2

s)

is monotone decreasing, ∀s > 0.

(5.48)

Then it holds that u = u(r),

((1 − r 2 )

n−2 2

u)r < 0,

0 < r = |x| < 1.

(5.49)

The second inequality in (5.49) is improved to ur < 0 for 0 < r ≤ 1 if f (r, s) ≥ 0. The solution u(x) may take a singularity, u(x) → +∞ as x → 0. In the arguments of Gidas–Ni–Nirenberg, the nonincreasingness of r 󳨃→ f (r, s) is needed. In general, to obtain such an estimate, we need some assumptions on r 󳨃→ f (r, s) for the following example. Radial symmetry of the solution does not arise in (5.47) for general f = f (r, s). For example, if w = w(x) is an asymmetric eigenfunction of −Δ under the Dirichlet boundary condition in B with the eigenvalue μ, then u(x) = 1 − |x|2 + εw(x) with 0 < ε ≪ 1 is positive in B and satisfies (5.47) for f (r, u) = λu + 4 − μ(1 − r 2 ). 5 This property is assured by modifying the proof of the Poincaré inequality in Lemma 7.3.

276 | 5 Supplementary topics

5.3 Method of scaling 5.3.1 Boltzmann–Poisson equation revisited The blow-up mechanism to the Gel’fand equation in two-space dimensions, (3.36), is described by Theorems 3.5 and 3.6. The latter theorem is concerned with the mass quantization and recursive hierarchy, which takes the following form for the Boltzmann–Poisson equation (3.37): − Δu =

λeu ∫Ω eu dx

in Ω,

u|𝜕Ω = 0.

(5.50)

Theorem 5.26 ([235]). Let Ω ⊂ R2 be a bounded domain with a smooth boundary 𝜕Ω, and {(λk , vk )} be a solution sequence to (5.50) satisfying λk → λ0 ∈ (0, +∞). Then, λ0 = 8πℓ with ℓ ∈ N, and, passing to a subsequence, we obtain vk → v0 locally uniformly in Ω \ 𝒮 with v0 = v0 (x) satisfying ℓ

v0 (x) = 8π ∑ G(x, xj∗ ), j=1

1 ∇R(xj∗ ) + ∑ ∇x G(xi∗ , xj∗ ) = 0, 2 i=j̸

1 ≤ j ≤ ℓ,

(5.51)

for 𝒮 = {x1∗ , . . . , xℓ∗ }, standing for the blow-up set of {vk }. Recall, first, that ‖vk ‖W 1.q = O(1) follows from the L1 -estimate [320, 56], where 1 ≤ n since n = 2. Second, this estimate implies a uniform boundary estimate q < 2 = n−1 indicated by ‖vk ‖L∞ (ω) = O(1),

(5.52)

using the reflection argument combined with the Kelvin transformation [137, 99], where ω = ω̂ ∩ Ω and ω̂ is an open set satisfying 𝜕Ω ⊂ ω.̂ Third, the original proof described in § 3.3 uses the complex structure. In fact, we obtain e−v/2 = c|φ1 |2 +

σc−1 |φ2 |2 , 8

σ=

λ ∫Ω ev dx

for (5.50), using s(z) of (3.21) with u replaced by v, where {φ1 (z), φ2 (z)} is the fundamental system of solutions to (3.22) defined by (3.23) with z ∗ corresponding to a critical point of v, denoted by x∗ ∈ Ω. Thus, there is a family of holomorphic functions {sk (z)} defined by (3.23) for u = vk , and this family is uniformly bounded on Ω by (5.52). Passing to a subsequence, therefore, we obtain sk → s0 locally uniformly in Ω.

| 277

5.3 Method of scaling

Introducing the fundamental system of solutions {φ1k (z), φ2k (z)} to (3.21) for s = sk (z), we take x∗ = xk∗ as a maximum point of vk . Passing to a subsequence, the convergence sk → s mentioned above guarantees that of φ1k → φ10 and φ2k → φ20 locally uniformly as analytic functions in Ω, because {xk∗ } is in Ω \ ω.̂ Then, it holds that ck = exp(−‖vk ‖∞ /2) → 0 in an analogous relation to (3.27), e−vk /2 = ck |φ1k |2 +

σk ck−1 |φ2k |2 . 8

(5.53)

Since {vk } is bounded in W 1,q (Ω) for 1 ≤ q < 2, any blow-up point of {vk } must be a zero of the analytic function φ10 , and therefore, each blow-up point is isolated. We obtain finiteness of the blow-up points in this way, while classification of the singular limit, (5.51), is derived by residue analysis, more precisely, singularity vanishing of 2 s0 (z) = v0zz − 21 v0z . In this proof, we obtain σk ck−1 ≈ 1, i. e., ‖vk ‖∞ ≈ −2 log σk as k → ∞. From the proof j of Theorem 5.33 described below, on the other hand, each xj∗ has a sequence xk → xj∗ , j

j

where xk is a local maximum point of vk . Thus, we can reformulate x ∗ = xk in (5.53), j vk (xk )

and consequently, the rates of blow-up → +∞, 1 ≤ j ≤ ℓ, are proportional to each other. The other consequence is a local uniform estimate j

uk (xk ) − uk (x) 2

󵄨 󵄨 j󵄨 j 󵄨2 = log{1 + o(󵄨󵄨󵄨x − xk 󵄨󵄨󵄨) + γk ak 󵄨󵄨󵄨x − xk 󵄨󵄨󵄨 (1 + o(1))} j

j

valid when |x − xk | ≪ 1, where for γk = ck−1 σk and ak = 8−1 evk (xk )/2 .6 An alternative proof of Theorem 5.26 uses Theorems 5.29–5.30 in the following subsection and the Pohozaev identity, instead of the complex structure [219]. This argument is valid to the nonhomogeneous coefficient case. The second equality of (5.51) means that (x1∗ , . . . , xℓ∗ ) ∈ Ω × ⋅ ⋅ ⋅ × Ω is a critical point of H = H(x1 , . . . , xℓ ) =

1 ∑ R(xi ) + ∑ G(xi , xj ). 2 i i 0.8 This property is a basis of blow-up analysis, as well as classification of the entire solutions, and that of the linearized equation.9 Theorem 5.27 ([78]). If − Δv = ev

in R2 ,

∫ ev dx < +∞,

(5.54)

R2

it holds that v(x) = log{

8μ2 }, (1 + μ2 |x − x0 |2 )2

∃x0 ∈ R2 ,

∃μ > 0

(5.55)

and hence ∫R2 ev dx = 8π. The linearized problem to (5.54) for v = v(x) satisfying v ≤ v(0) = 0 is given by − Δw =

w

{1 +

|x|2 2 } 8

in R2 .

(5.56)

Then its bounded solutions are created by differentiating v = vμ,x0 in (5.55) with respect to the parameters μ and x0 . Theorem 5.28 ([28]). One has 2

v(x) = ∑ i=1

ai xi 8 − |x|2 + b ⋅ , 8 + |x|2 8 + |x|2

∃ai , b ∈ R

if v = v(x) is a bounded solution to (5.56). 5.3.2 Blow-up analysis Scaling invariance of the model causes the lack of compactness of the family of (approximate) solutions, and this mechanism is clarified by the blow-up analysis at the end of § 5.2.1. Its ingredients are summarized as follows: 1. Scaling invariance of the problem; 2. Classification of the entire solutions; 3. Control at infinity of the rescaled solution; 4. Hierarchical argument. 8 See Chapters 9 and 12 of [335]. 9 See also §12.7 of [335] and [141, 299], for the proof of Theorems 5.27 and 5.28, respectively.

5.3 Method of scaling

| 279

The following theorem, free from boundary condition, is useful in such a study because the effect of the boundary condition is lost in the scaling argument. It deals with the nonhomogeneous coefficient case with the lack of the complex structure. Theorem 3.5 is associated with this theorem for vk − log σk . Actually, we obtain v > 0 in (5.50) with u replaced by v. Theorem 5.29 ([53]). Let Ω ⊂ R2 be a bounded domain and vk = vk (x), k = 1, 2, . . ., be a sequence of solutions to − Δvk = Vk (x)evk ,

0 ≤ Vk (x) ≤ C1

in Ω,

∫ evk ≤ C2 ,

(5.57)

Ω

where C1 , C2 are constants. Then, passing to a subsequence, there are the following alternatives: 1. {vk } is locally uniformly bounded in Ω. 2. vk → −∞ locally uniformly in Ω. 3. There is a finite set 𝒮 = {xj∗ } ⊂ Ω and mj ≥ 4π such that vk → −∞ locally uniformly in Ω \ 𝒮 and Vk (x)evk dx ⇀ ∑ mj δx∗ (dx) j

j

in ℳ(Ω).

(5.58)

Furthermore, 𝒮 is the blow-up set of {vk } in Ω. The following theorem describes mass quantization without boundary condition, with possible collision of delta functions.10 The boundedness of Palais–Smale sequence relative to the Trudinger–Moser inequality does not always follow. This theorem is then applied to compensate the difficulty when creating nontrivial solutions to the mean field equation [327, 104]. Theorem 5.30 ([202]). In the third case of the above theorem, mj = 8πnj for some nj ∈ N, provided that Vk → V in Cloc (Ω). A rough estimate derived from the prescaled analysis is sufficient for the proof of Theorem 5.29. It is proven by a basic inequality concerning linear equation. Lemma 5.31. If Ω ⊂ R2 is a bounded domain, f ∈ L1 (Ω), and −Δv = f (x) in Ω with v|𝜕Ω = 0, it holds that ∫ exp( Ω

4π − δ 󵄨󵄨 4π 2 󵄨 (diam Ω)2 , 󵄨󵄨v(x)󵄨󵄨󵄨) dx ≤ ‖f ‖1 δ

0 < δ < 4π.

This lemma implies ε-regularity in the form of the following lemma, and then a standard argument guarantees Theorem 5.29.11 10 The proofs of Theorems 5.29 and 5.30 are described in [350, 335]. 11 See §12.6 of [335] for details.

280 | 5 Supplementary topics Lemma 5.32. Let Ω ⊂ R2 be a bounded domain, K ⊂ Ω a compact set, c1 , c2 > 0, and ε0 ∈ (0, 4π). Then, there is C > 0 such that −Δv = V(x)ev ,

0 ≤ V(x) ≤ c1

in Ω,

‖v+ ‖1 ≤ c2 ,

∫ V(x)ev dx ≤ ε0 Ω

implies ‖v+ ‖L∞ (K) ≤ C. Once Theorem 5.29 is proven, Theorem 5.30 is localized as follows. Let B = B(0, R) ⊂ R2 and Br = B(0, r). Theorem 5.33. If −Δvk = Vk (x)evk , Vk (x) ≥ 0 in B, Vk → V in C(B), maxB vk → +∞, maxB\B vk → −∞ for any 0 < r < R, ∫B Vk (x)evk dx → α, and ∫B evk dx ≤ C0 , then it r holds that α ∈ 8π 𝒩 . In spite of the case α = 8πℓ, ℓ ≥ 2 in [76], there is the following fact with C > 0 standing for a generic constant. The original proof [201] is completed by using the moving plane, based on the fact that the profile of vk = vk (x) in the outer region x ∈ B \ B(0, δk ), δk = e−uk (xk )/2 for uk (xk ) = ‖uk ‖∞ , is almost similar to that of the Kelvin transformation of vk = vk (x) on B(0, δk ), under the assumption. An alternative proof is based on the mass identity, where the collapsed mass 8π is equal to the total mass of the entire solution, to control the outer region [211]. Theorem 5.34 ([201]). It holds that α = 8π in the previous theorem, provided that max𝜕B vk − min𝜕B vk ≤ C and ‖∇Vk ‖∞ ≤ C, and furthermore, 󵄨󵄨 󵄨󵄨 󵄨󵄨vk (x) − log 󵄨󵄨 (1 +

evk (0)

󵄨󵄨 󵄨󵄨 󵄨 ≤ C, Vk (0) vk (0) 2 2 󵄨󵄨 e |x| ) 󵄨 8

∀k, x ∈ B.

(5.59)

Then we obtain a generalization of Theorem 5.26 concerning − Δv =

λV(x)ev ∫Ω V(x)ev dx

in Ω,

v|𝜕Ω = 0

(5.60)

for λ > 0 and 0 < V = V(x) ∈ C 1 (Ω) ∩ C(Ω), where Ω ⊂ R2 denotes a general bounded domain with a smooth boundary 𝜕Ω.12 Theorem 5.35 ([219]). Given 0 < V = V(x) ∈ C 1 (Ω), let (λk , vk ) be a solution to (5.60) for k = 1, 2, . . . such that λk → λ0 and ‖vk ‖∞ → +∞. Then it holds that λ0 = 8πℓ, ℓ ∈ N, and, passing to a subsequence, one has V(x)evk dx ⇀ 8π ∑x0 ∈𝒮 δx0 (dx), with the blow-up set 𝒮 = {x1∗ , . . . , xℓ∗ } subject to 1 1 ∇R(xj∗ ) + ∑ ∇x G(xi∗ , xj∗ ) + ∇ log V(xj∗ ) = 0, 2 8π i=j̸ 12 Related results are described in [241, 260, 299, 334].

1 ≤ j ≤ ℓ.

5.3 Method of scaling

| 281

Theorem 5.33 is proven by the blow-up analysis.13 We take xk ∈ B satisfying vk (xk ) = ‖vk ‖∞ with xk → 0, to put ṽk (x) = vk (δk x +xk )+2 log δk and δk = e−vk (xk )/2 → 0. It holds that −Δṽk = Vk (δk x + xk )evk ,

ṽk ≤ ṽk (0) = 0

̃

in B(0,

R ), 2δk

evk dx ≤ C0 , ̃

∫ B(0, 2δR ) k

and Theorem 5.29 is applicable to this {ṽk }. Thus, {ṽk } is locally uniformly bounded in R2 , and, passing to a subsequence, we have ṽk → ṽ locally uniformly in R2 with −Δṽ = V(0)ev , ̃

in R2 ,

̃ ṽ ≤ v(0) =0

∫ ev dx ≤ C0 ̃

R2

by the elliptic regularity. Then we get V(0) > 0, and assume 0 < a ≤ Vk (x) ≤ b < +∞,

x∈B

without loss of generality, with a, b independent of k. We obtain, furthermore, ṽ = ̃ v(|x|) by the method of the moving plane [78], which results in ̃ v(x) = log

1

(1 +

V(0) |x|2 )2 8

,

∫ V(0)ev = 8π ̃

R2

by Theorem 5.27.

5.3.3 Brezis–Merle’s inequality and related topics Proof of Lemma 5.31. We take B = B(x0 , R) containing Ω for R = sion of f (x) outside Ω, and v(x) =

1 2

diam Ω, zero exten-

1 2R 󵄨 󵄨 ⋅ 󵄨󵄨f (x 󸀠 )󵄨󵄨󵄨 dx 󸀠 . ∫ log 2π |x − x󸀠 | 󵄨 B

It holds that −Δv = |f | in R2 and v ≥ 0 in B by 2R ≥ |x − x 󸀠 | for x, x 󸀠 ∈ B, and, therefore, |v| ≤ v from the maximum principle. We thus obtain ∫ exp( Ω

4π − δ 󵄨󵄨 4π − δ 󵄨 v(x)) dx. 󵄨v(x)󵄨󵄨󵄨) dx ≤ ∫ exp( ‖f ‖1 󵄨 ‖f ‖1

(5.61)

Ω

13 Recent developments based on the blow-up analysis include the asymptotic nondegeneracy of the solution in accordance with the Morse indices [141, 142, 145, 264, 143, 300].

282 | 5 Supplementary topics Since Jensen’s inequality implies exp(

4π − δ 2R |f (x 󸀠 )| 󸀠 4π − δ v(x)) = exp(∫ ⋅ log ⋅ dx ) ‖f ‖1 2π |x − x 󸀠 | ‖f ‖1 B

≤ ∫( B

δ 2− 2π

2R ) |x − x󸀠 |

|f (x 󸀠 )| 󸀠 dx , ‖f ‖1

the right-hand side of (5.61) is estimated above by δ 2− 2π

|f (x󸀠 )| 󸀠 2R dx ⋅ ∫( ) ∫ ‖f ‖1 |x − x 󸀠 | B

dx.

B

Here, B = B(x0 , R) ⊂ B(x󸀠 , 2R) for any x󸀠 ∈ B, and it holds that |f (x󸀠 )| 󸀠 4π − δ 󵄨󵄨 󵄨 dx ⋅ ∫ exp( 󵄨󵄨v(x)󵄨󵄨󵄨) dx ≤ ∫ ‖f ‖1 ‖f ‖1 B

Ω

=

δ 2− 2π

2R ) ∫ ( |x − x 󸀠 |

dx

B(x󸀠 ,2R)

4π 2 4π 2 (2R)2 = (diam Ω)2 . δ δ

Corollary 5.36. Let Ω ⊂ R2 be a bounded domain with a smooth boundary 𝜕Ω, and u = u(x) a classical solution to −Δu = V(x)eu in Ω with u|𝜕Ω = 0. Take 1 < p ≤ ∞ and 0 < γ < 4π/p󸀠 . Then each M > 0 admits C(M) > 0 such that 󵄩󵄩 u 󵄩󵄩 󵄩󵄩e 󵄩󵄩1 ≤ γ,

‖V‖p ≤ M 󳨐⇒ ‖u‖∞ ≤ C.

Proof. We take δ > 0 satisfying 4π − δ > γ(p󸀠 + δ). For f = V(x)eu , one has 4π − δ 4π − δ ≥ > p󸀠 + δ, ‖f ‖1 γ and hence 󸀠

∫ e(p +δ)|u| dx ≤ ∫ exp( Ω

Ω

4π − δ 󵄨󵄨 4π 2 󵄨 (diam Ω)2 , 󵄨󵄨u(x)󵄨󵄨󵄨) dx ≤ ‖f ‖1 δ

which results in ‖eu ‖p󸀠 +δ ≤ C, ‖V‖p ≤ M, and hence ‖Veu ‖q = ‖ − Δu‖q ≤ C for some q > 1. Then elliptic regularity and Morrey’s inequality assure the result. In spite of Theorem 5.30, the upper bound 4π/p󸀠 of γ is the best possible. To see this, we take fk (x) = {

4 2 k , p󸀠

0,

|x| < k1 ,

otherwise,

k = 1, 2, . . . ,

5.3 Method of scaling

| 283

and define uk = uk (x) by −Δuk = fk in B = B(0, 1) ⊂ R2 with uk |𝜕B = 0. Then 1 2 2 { p󸀠 (−k r + 1 + 2 log k), uk (x) = { 2 − log r, { p󸀠

0 < r = |x| < k1 , 1 k

< r < 1,

and hence 4

{ 󸀠 k 2/p e(−1+k Vk (x) = { p { 0,

2 2

r )/p󸀠

,

0 < r = |x| < k1 , 1 k

< r < 1,

for Vk = e−uk fk . Although the values ‖Vk ‖pp = (

p

p

2 2 󸀠 󸀠 4 4 ) k 2 ∫ e(−1+k r )/p dx = ( 󸀠 ) πp󸀠 (1 − e−1/p ), p󸀠 p

|x|< k1

4π 󵄩󵄩 u 󵄩 󵄩󵄩Vk e k 󵄩󵄩󵄩1 = ‖fk ‖1 = 󸀠 p are independent of k, it holds that ‖uk ‖∞ = uk (0) =

1 (1 p󸀠

+ 2 log k) ↑ +∞ as k ↑ +∞.

The following theorem is a refinement of Theorems 3.5–3.6.

Theorem 5.37 ([372]). Let Ω ⊂ R2 be a bounded domain with a smooth boundary 𝜕Ω, and 0 ≤ f (u) = eu + o(eu ) as u ↑ +∞. Let, furthermore, {(λk , uk )}, be a family of solutions to − Δu = λf (u)

in Ω,

u|𝜕Ω = 0

(5.62)

for (λ, u) = (λk , uk ), such that λk > 0, ‖uk ‖ → +∞, and Σk = ∫Ω λk euk dx = O(1). Then it holds that λk → 0, and the conclusion of Theorem 3.6 holds. Proof. By the proof of Theorem 3.5, there is a neighborhood of 𝜕Ω, denoted by ω, such that ‖uk ‖L∞ (Ω∩ω) ≤ C. Writing Vk = e−uk f (uk ) and vk = uk + log λk , we obtain −Δvk = Vk (x)evk in Ω by (5.62). Then 0 ≤ Vk (x) ≤ C and ∫Ω evk dx = Σk = O(1), and hence Theorem 5.29 is applicable, so that λk → 0 follows from the boundary estimate of {uk } above. It holds also that ℓ

Vk (x)evk = λk f (uk ) ⇀ ∑ ai δκi i=1

in ℳ(Ω)

(5.63)

with ai > 0, κi ∈ Ω, and 1 ≤ i ≤ ℓ. Let f (u) = eu + p(u). Since λk ↓ 0, we obtain ‖λk euk p(uk )‖1 → 0 due to p(u) = o(eu ) as u ↑ +∞, and therefore, the right-hand side of (5.63) may be replaced by λk euk : ℓ

λk euk ⇀ ∑ ai δκi i=1

in ℳ(Ω).

284 | 5 Supplementary topics Let, furthermore, H = u2z in (5.62), using the complex variable z = x1 + 𝚤x2 for x = (x1 , x2 ). It holds that 1 1 Hz = − λuz f (u) = − λ(eu + P(u))z 2 2

u

in Ω,

P(u) = ∫ p(u) du. 0

Here we put T = − πz1 2 = ( π2 log |z|)zz and W = (λeu + P(u))χΩ ∈ L1 (R2 ), to have α = W ∗ T in the sense of distributions. Since ( π2 log |z|)zz = δ, αz = W ∗ Tz = Wz ∗ δ = Wz in R2 . We thus end up with H̃ z = 0 in Ω,

α H̃ = H + . 2

Then we can argue similarly as in the proof of Theorem 3.6, to reach (3.61)–(3.62).

5.3.4 Vanishing residual At the end of § 5.3.2 we detected the principal collapse formed at the origin for the proof of Theorem 5.33. Collecting the other collapses, we show the vanishing of the residual part. This is done by the sup + inf inequality proven by Alexandroff’s inequality originally. Alexandroff’s inequality is also an isoperimetric inequality on a surface described by its Gaussian curvature, regarded as a refinement of Bol’s inequality [25]. Thus, we can show the following lemma.14 Lemma 5.38 ([310]). Given B = B(0, 1) ⊂ R2 and a, b > 0, we have C0 > 0 and α0 > 4π such that −Δv = V(x)ev ,

a ≤ V(x) ≤ b

in B,

∫ V(x)ev dx ≤ α0 B

implies v(0) ≤ C0 . This lemma is a refinement of Lemma 5.32 under the cost of V(x) ≥ a. If V = V(x) is restricted to a compact family in C(Ω), which is sufficient for later arguments, then we can apply the blow-up analysis for the proof. In this case, the above α0 can be arbitrary as long as α0 < 8π and, furthermore, the case a = 0 is permitted. Using the above lemma and the scaling invariance of the equation, next we show the following lemma [310].15 14 See §12.5 of [335] for the proof using Alexandroff’s inequality. 15 See §2.8 of [336] for the proof.

5.3 Method of scaling

| 285

Lemma 5.39 (Sup+Inf inequality). If Ω ⊂ R2 is a bounded domain, K ⊂ Ω is a compact set, and a, b > 0 are constants, then there are c1 = c1 (a, b) ≥ 1 and c2 = c2 (a, b, dist(K, 𝜕Ω)) > 0 such that − Δv = V(x)ev ,

a ≤ V(x) ≤ b

in Ω 󳨐⇒ sup v + c1 inf v ≤ c2 . Ω

K

(5.64)

In another version of (5.64) proven by the blow-up analysis [57], the condition c1 = 1 is achieved under the cost of ‖∇V‖∞ ≤ C. In any case, this sup + inf inequality induces the key estimate, again by the scaling.16 Recall BR = B(0, R). Lemma 5.40 ([202]). Given a, b > 0 and C1 > 0, we have γ > 0 and C2 > 0 independent of 0 < R0 ≤ R/4 such that −Δv = V(x)ev ,

a ≤ V(x) ≤ b

in BR \ BR0

v(x) + 2 log |x| ≤ C1

in BR ,

implies ev(x) ≤ C2 e−γv(0) ⋅ |x|−2(γ+1) for 2R0 ≤ |x| ≤ R/2. To complete the proof of Theorem 5.33, first, we refine the blow-up argument for the principal collapse. Thus, let ‖vk ‖∞ = vk (xk0 ) → +∞, xk0 → 0, and δk0 = 0

e−vk (xk )/2 → 0. By a diagonal argument, then we obtain rk0 → 0 satisfying rk0 /δk0 → +∞,

∫

Vk (x)evk → 8π.

B(xk0 ,2rk0 )

If 󵄨 󵄨 sup{vk (x) + 2 log󵄨󵄨󵄨x − xk0 󵄨󵄨󵄨 | x ∈ B \ B(xk0 , rk0 )} < +∞,

(5.65)

Lemma 5.40 guarantees 0 󵄨 󵄨−2(γ+1) evk (x) ≤ Ce−γvk (xk ) 󵄨󵄨󵄨x − xk0 󵄨󵄨󵄨 ,

x ∈ BR/2 \ B(xk0 , rk0 ).

Therefore, we obtain ∫

vk

Vk (x)e ≤ b ⋅ C ⋅

BR/2 \B(xk0 ,rk0 )

2γ (δk0 )

+∞

⋅ 2π ⋅ ∫ r −2(γ+1) r dr rk0

=

πbC 0 0 2γ (δk /rk ) → 0, γ

which implies ∫B Vk (x)evk → 8π because vk → −∞ locally uniformly in B \ {0}, and hence α = 8π. 16 See §12.8 of [335] for the proof.

286 | 5 Supplementary topics If (5.65) does not hold, there is {xk1 } ⊂ B such that 󵄨 󵄨 󵄨 󵄨 {vk (x) + 2 log󵄨󵄨󵄨x − xk0 󵄨󵄨󵄨} = vk (xk1 ) + 2 log󵄨󵄨󵄨xk1 − xk0 󵄨󵄨󵄨 → +∞,

sup

x∈B\B(xk0 ,rk0 )

which implies vk (xk1 ) → +∞, xk1 → 0, and σk1 = dk /δk1 → +∞ for dk = |xk1 − xk0 | and 1

δk1 = e−vk (xk )/2 . Given |x| ≤ σk1 /2, we have

󵄨󵄨 1 󵄨 1 1 0󵄨 0󵄨 1 󵄨󵄨δk x + xk − xk 󵄨󵄨󵄨 ≥ 󵄨󵄨󵄨xk − xk 󵄨󵄨󵄨 − δk |x| ≥

1 󵄨󵄨 1 0󵄨 󵄨x − xk 󵄨󵄨󵄨 2󵄨 k

and therefore, ṽk1 (x) ≡ vk (δk1 x + xk1 ) + 2 log δk1 󵄨 󵄨 󵄨 󵄨 ≤ vk (xk1 ) + 2 log󵄨󵄨󵄨xk1 − xk0 󵄨󵄨󵄨 − 2 log󵄨󵄨󵄨δk1 x + xk1 − xk0 󵄨󵄨󵄨 + 2 log δk1 1󵄨 󵄨 󵄨 󵄨 ≤ vk (xk1 ) + 2 log δk1 + 2 log󵄨󵄨󵄨xk1 − xk0 󵄨󵄨󵄨 − 2 log 󵄨󵄨󵄨xk1 − xk0 󵄨󵄨󵄨 2 = 2 log 2.

We thus obtain ̃1

−Δṽk1 = Vk (δk1 x + xk1 )evk ,

ṽk1 ≤ 2 log 2

in Bσ1 /2 , k

ṽk1 (0) = 0,

1,α 2 (R ) with ṽ1 = ṽ1 (x) satisfying and, passing to a subsequence, one has ṽk1 → ṽ1 in Cloc

ṽ1 (x) = log

a2 , (1 + μ2 a2 |x − x|2 )2

ṽ1 (x) ≤ 2 log 2,

x ∈ R2 ,

ṽ1 (0) = 0

for some μ, a > 0 and x ∈ R2 . This convergence allows us to redefine xk1 and δk1 , denoted 1

by the same symbols, satisfying vk (xk1 ) = ‖vk ‖L∞ (B(x1 ,2r1 )) → +∞, δk1 = e−vk (xk )/2 → 0, k

k

and rk1 /δk1 → +∞ for rk1 = dk /4. Similarly to the above case, it follows that ∫

Vk (x)evk → 8π

B(xk1 ,2rk1 )

with B(xk1 , 2rk1 ) ∩ B(xk0 , 2rk0 ) = 0. We shall show α = 16π, if 1

󵄨 j󵄨 j j sup{vk (x) + 2 log min󵄨󵄨󵄨x − xk 󵄨󵄨󵄨 | x ∈ B \ ⋃ B(xk , rk )} < +∞. j=0,1

j=0

(5.66)

It suffices to prove ∫ B(xk0 ,2dk )

Vk (x)evk → 16π

(5.67)

5.3 Method of scaling

| 287

because ∫B\B(x0 ,2d ) Vk (x)evk → 0 follows from (5.66) similarly. For this purpose, we k

k

take ṽk (x) = vk (dk x + xk0 ) + 2 log dk , to obtain

−Δṽk = Vk (dk x + xk0 )evk

in dk−1 (B − {xk0 }).

̃

j

Put, furthermore, x̃k = j

xkj −xk0 j , δ̃ k dk

j

rk̃ r = kj → +∞, ̃δj δk k

j

= e−vk̃ (x̃k )/2 =

δkj , dk

j

and rk̃ =

rkj dk

for j = 0, 1. Then

B(x̃k0 , 2rk0̃ ) ∩ B(x̃k1 , 2rk1̃ ) = 0, 1

󵄨 j󵄨 j j sup{ṽk (x) + 2 log min󵄨󵄨󵄨x − x̃k 󵄨󵄨󵄨 | x ∈ BR/dk \ ⋃ B(x̃k , rk̃ )} < +∞, j=0,1

∫

Ṽ k (x)e

vk̃

j=0

→ 8π,

j = 0, 1,

(5.68)

B(x̃kj ,2rkj̃ )

for Ṽ k (x) = Vk (dk x + xk0 ).

We obtain x̃k0 = 0 and |x̃k1 − x̃k0 | = 1, and therefore, x̃k1 → x̃ 1 with |x̃ 1 | = 1, passing to a subsequence. The third relation of (5.68) and Theorem 5.29 now imply ṽk → −∞ locally uniformly in R2 \ {0, x̃ 1 }. Therefore, if rk1̃ → r 1̃ > 0, passing to a subsequence, then ∫

̃ Ṽ k (x)evk → 8π

(5.69)

B(x̃k1 ,1/2)

and hence (5.67). If rk1̃ → 0, we apply the scaling around x̃k1 . It holds that (5.69) by the third relation of (5.68). If (5.66) does not hold, we continue the process and obtain xk2 → 0 and rk2 → 0 j j satisfying vk (xk2 ) = ‖vk ‖L∞ (B(x2 ,2r2 )) → +∞, rk2 /δk2 → 0, B(xki , 2rki ) ∩ B(xk , 2rk ) = 0 for k k 0 ≤ i < j ≤ 2, and Vk (x)evk → 8π,

∫

2

δk2 = e−vk (xk )/2 .

B(xk2 ,2rk2 )

To show α = 24π in the case of 2

󵄨 j󵄨 j j sup{vk (x) + 2 log min 󵄨󵄨󵄨x − xk 󵄨󵄨󵄨 | x ∈ B \ ⋃ B(xk , rk )} < +∞, 0≤j≤2

j=0

j

we classify the rate di,j = |xki −xk | of concentration to the origin for 0 ≤ i < j ≤ 2. First, we j

j

show the residual vanishing inside the ball containing B(xk , 2rk ) with a proportional rate. These balls are contained in a larger ball, where the residual vanishing occurs similarly. We end this procedure in finitely many steps, and obtain the conclusion.17 17 See [337] for a complete set of the proof of the above lemmas and later developments.

288 | 5 Supplementary topics 5.3.5 Higher-dimensional case Putting w = v + log λ − log ∫Ω ev in (5.50), we obtain − Δw = ew

in Ω,

w = constant on γ = 𝜕Ω,

∫ ew = λ.

(5.70)

Ω

Conversely, if w = w(x) solves (5.70), then v = w − wγ is a solution to (5.50) for wγ = w|γ ∈ R. By Theorem 5.26, we can show the quantized blow-up mechanism to (5.70). Theorem 5.41. If Ω ⊂ R2 is a bounded domain with a smooth boundary 𝜕Ω and {(λk , wk )} is a solution sequence to (5.70) satisfying λk → λ0 , then, passing to a subsequence, the following alternatives hold: 1. ‖wk ‖∞ = O(1). 2. supΩ wk → −∞. 3. λ0 = 8πℓ for some ℓ ∈ N. In the third case there exist xj∗ ∈ Ω, j = 1, . . . , ℓ, satisfying the second relation of (5.51) j

j

j

and xk → xj∗ , such that x = xk is a local maximum point of wk = wk (x), wk (xk ) → +∞, k

w → −∞ locally uniformly in Ω \

{x1∗ , . . . , xℓ∗ },

and

k

ew dx ⇀ ∑ 8πδx∗ (dx) j

j

in ℳ(Ω).

Thus, 𝒮 = {x1∗ , . . . , xℓ∗ } is the blow-up set of {wk }. Proof. First, Theorems 5.29–5.30 and their proofs guarantee the following alternatives, passing to a subsequence: (a) {wk } is locally uniformly bounded in Ω. (b) wk → −∞ locally uniformly in Ω. (c) There is a finite set 𝒮 ⊂ Ω such that wk → −∞ locally uniformly in Ω \ 𝒮 , any x0 ∈ 𝒮 admits xk → x0 with xk a local maximum point of wk = wk (x) satisfying k

wk (xk ) → +∞, and ew dx → ∑x0 ∈𝒮 m(x0 )δx0 (dx) in ℳ(Ω) for some m(x0 ) ∈ 8π 𝒩 .

We obtain, on the other hand, vk = wk − wγk ≥ 0 in Ω by the maximum principle, and also ‖vk ‖L∞ (ω) ≤ C

(5.71)

from the proof of Theorem 3.5, where ω is an Ω-neighborhood of γ = 𝜕Ω. Since wk ≥ wγk ,

we have either wγk → −∞ or wγk = O(1), passing to a subsequence.

Actually, wγk = O(1) occurs in the case of (a), and then {vk } is uniformly bounded on

Ω by (5.71). Thus, the first case of the theorem arises. In the other case of wΓk → −∞, if

5.3 Method of scaling

| 289

{vk } is uniformly bounded, then the second case of the theorem follows. If not, we obtain ‖vk ‖∞ → +∞, passing to a subsequence, and therefore, one gets the second case k

of Theorem 3.5, that is, Theorem 5.26, which results in −Δvk dx = −Δwk dx = ew dx ⇀ ∑j 8πδx∗ (dx) in ℳ(Ω). j

Equation (5.70) is regarded as a free boundary problem associated with plasma confinement, where {w > 0} indicates the plasma region [121, 335]. Higher-dimensional mass quantization is observed in an analogous problem − Δw = w+q

in Ω,

w = constant on γ,

∫ w+q = λ,

(5.72)

Ω

m

where Ω ⊂ R , m ≥ 3, is a bounded domain with a smooth boundary 𝜕Ω = γ, and m . Furthermore, we can formulate it as the equilibrium self-gravitating fluid q = m−2 equation described by the field component similarly to Theorem 5.41. The quantized mass m∗ > 0 is defined by m∗ = ∫B U q dx for U = U(x), satisfying − ΔU = U q ,

U>0

in B,

U|𝜕B = 0

(5.73)

with B = B(0, R). This U is radially symmetric, and hence (5.73) is reduced to Urr +

n−1 Ur + U q = 0, r

Ur (0) = 0,

0 = U(R) < U(r),

0 ≤ r < R,

which has a unique solution for each R > 0 [250]. The scaling invariance of (5.73) 2 described by Uμ (x) = μ q−1 U(μx), μ > 0, on the other hand, assures m∗ independent of R > 0. A local theory arises as in Theorems 5.29–5.30 because of ε-regularity, selfsimilarity, classification of the entire solution, and sup + inf inequality. The first difference is a lack of geometrical structure, which is used for sup + inf inequality in two space dimensions. This lack, however, is replaced by the blow-up analysis. This argument implies also boundary ε regularity. Lemma 5.42. Given q = C0 such that −Δw = w+q

m , m−2

in Ω ∩ B,

B = B(0, 1) ⊂ Rm , and B/2 = B(0, 1/2), we have ε0 > 0 and w = c ∈ (−∞, 0] on 𝜕Ω ∩ B,

∫ w+q dx < ε0

Ω∩B

implies ‖w‖L∞ (B/2) ≤ C0 . The second difference is a profile of the entire solution − Δw = w+q ,

w ≤ w(0) = 1

in Rm ,

∫ w+q < +∞,

(5.74)

Rm

provided with compact support, which, however, makes the argument simpler.18 18 See [344] for a generalization.

290 | 5 Supplementary topics Theorem 5.43 ([362]). Let Ω ⊂ Rm , m ≥ 3, be a bounded domain and let each w = wk , k = 1, 2, . . ., satisfy −Δw = w+q

∫ w+q ≤ C

in Ω,

Ω

m and C > 0. Then, passing to a subsequence, we have the following alternafor q = m−2 tives: 1. {wk } is locally uniformly bounded in Ω. 2. wk → −∞ locally uniformly in Ω. j j 3. There exist ℓ ∈ N and xj∗ for j = 1, . . . , ℓ, and xk → xj∗ such that x = xk is a local j

maximum point of wk = wk (x), wk (xk ) → +∞, wk → −∞ locally uniformly in Ω \ {x1∗ , . . . , xℓ∗ }, and wk (x)q+ dx ⇀ ∑j m∗ nj δx∗ (dx) in ℳ(Ω), where nj ∈ N. j

Turning to the global theory as in Theorem 5.41 for two space dimensions, let G = G(x, x󸀠 ) be the Green’s function of −Δ on Ω with the Dirichlet boundary condition and R(x) = [G(x, x󸀠 ) − Γ(x − x󸀠 )]x󸀠 =x , with Γ(x) =

1 ωm (m − 2)|x|m−2

standing for the fundamental solution to −Δ and ωm denoting the (m − 1)-dimensional volume of the boundary of the unit ball in Rm . We begin with the exclusion of the boundary blow-up. Lemma 5.44. Let {(λk , wk )} be a family of solutions to (5.72) for λ = λk > 0, w = wk (x), satisfying λk → λ0 , ‖wk+ ‖∞ → +∞. Then, passing to a subsequence, it holds that #𝒮 < +∞, ck = wk |𝜕Ω → −∞, and q wk+ dx ⇀ ∑ m(x0 )δx0 (dx) in ℳ(Ω) x0 ∈𝒮

wk → −∞

locally uniformly in Ω \ 𝒮 ,

(5.75)

where 𝒮 = {x0 ∈ Ω | there exists xk → x0 such that wk (xk ) → +∞} is the blow-up set. Furthermore, 𝒮 ⊂ Ω. Proof. By Lemma 5.42 and Theorem 5.43, one has a rough estimate which guarantees #𝒮 < +∞ and (5.75), up to a subsequence. If ck ≥ −C, the maximum principle assures wk ≥ −C in Ω, which contradicts the second relation of (5.75). Assume, finally, x0 ∈ 𝜕Ω ∩ 𝒮 and take 0 < R ≪ 1 such that B(x0 , 2R) ∩ 𝒮 = {x0 }. Then it holds that wk − ck → ∑ m(x0 )G(⋅, x0 ) locally uniformly in Ω ∩ B(x0 , 2R) x0 ∈𝒮

including their derivatives of any order.

(5.76)

5.3 Method of scaling

| 291

Here we apply the Kazdan–Warner identity ∫ Δu∇u dx = ∫

ω

𝜕ω

𝜕u 1 ∇u − |∇u|2 ν dS 𝜕ν 2

for ω = Ω ∩ B(x0 , R). Letting f (w) = w+q and F(w) =

1 wq+1 q+1 +

(5.77)

in (5.72), first, we have

− ∫ Δw∇w dx = ∫ f (w)∇w dx = ∫ ∇F(w) dx = ∫ ν ⋅ F(w) dS. ω

ω

ω

𝜕ω

Then (5.77) and ck → −∞ imply ∫− 𝜕ω

𝜕wk 1 ∇wk + |∇wk |2 ν dS = 0, 𝜕ν 2

k ≫ 1.

(5.78)

Since (5.76) implies ∫Ω∩𝜕ω |∇wk |2 dS = O(1), equality (5.78) guarantees ∫ 𝜕Ω∩𝜕ω

1 |∇wk |2 ν dS = O(1) 2

because wk = ck on 𝜕Ω. Then it follows that ∫𝜕Ω∩𝜕ω |∇wk |2 dS = O(1), which implies 2

∫ ( 𝜕Ω∩𝜕ω

𝜕G (⋅, x0 )) dS < +∞. 𝜕ν

(5.79)

Inequality (5.79), however, is impossible due to 2

(

𝜕 G(x, x0 )) ≈ |x − x0 |−2m+2 , 𝜕ν

x → x0 .

Theorem 5.45. If Ω ⊂ Rm , m ≥ 3, is a bounded domain with a smooth boundary 𝜕Ω and m {(λk , wk )} is a solution sequence to (5.72) with q = m−2 satisfying λk → λ0 , then, passing to a subsequence, one has the following alternatives: 1. ‖wk ‖∞ = O(1). 2. supΩ wk → −∞. 3. λ0 = m∗ ℓ for some ℓ ∈ N. j

In the third case there are xj∗ ∈ Ω and xk → xj∗ , j = 1, . . . , ℓ, such that 𝒮 = {x1∗ , . . . , xℓ∗ } ⊂ Ω

is the blow-up set of {wk } on Ω satisfying

1 ∇R(xj∗ ) + ∑ ∇x G(xi∗ , xj∗ ) = 0, 2 i=j̸ j

1 ≤ j ≤ ℓ, j

(5.80)

x = xk is a local maximum point of wk = wk (x), wk (xk ) → +∞, wk → −∞ locally uniformly in Ω \ 𝒮 , and wk (x)q+ dx ⇀ ∑j m∗ δx∗ (dx) in ℳ(Ω). j

292 | 5 Supplementary topics Proof. It holds that lim ck = −∞,

k→∞

ck = wk |𝜕Ω

(5.81)

by Lemma 5.44. Henceforth, we drop the index k and apply the method of symmetrization for the dual variable [335]. Using u = w+q ,

q=

m , m−2

(5.82)

we transform (5.72) into w − wγ = ∫ G(⋅, x󸀠 )u(x 󸀠 ) dx󸀠 , Ω

∫ u = λ,

(5.83)

Ω

which implies ∇w(x) = ∫Ω ∇x G(x, x󸀠 )u(x󸀠 ) dx󸀠 . Given ψ ∈ C 1 (Ω)n , we have ∫(ψ ⋅ ∇w)u dx = ∬ ψ(x) ⋅ [∇x G(x, x󸀠 )]u(x)u(x󸀠 ) dx dx󸀠 . Ω

(5.84)

Ω×Ω

By (5.81), the left-hand side of (5.84) is equal to ∫(ψ ⋅ ∇w)u dx = Ω

1 1 ∫ ψ ⋅ ∇w+q+1 dx = − ∫ w+q+1 ∇ ⋅ ψ dx q+1 q+1 Ω

(5.85)

Ω

for k sufficiently large. Let 0 ≤ φ = φx0 ,R ≤ 1 be a smooth cut-off function with support in B(x0 , R) and equal to 1 on B(x0 , R/2). We put 𝒮 = {x1∗ , . . . , xℓ∗ } ⊂ Ω, recalling Lemma 5.44. Let x0 ∈ 𝒮 and take 0 < R ≪ 1 so that B(x0 , 2R) ⊂ Ω and B(x0 , 2R) ∩ 𝒮 = {x0 }. We put ψ(x) = (x − a)φ(x) for a ∈ Rm and φ = φx0 ,R . By (5.75), (5.85), and since ∇ ⋅ ψ = mφ + (x − a) ⋅ ∇φ, it holds that ∫(ψ ⋅ ∇w)u dx = − Ω

m ∫ w+q+1 φ dx + o(1), q+1

k → ∞,

Ω

which implies m ∫ w+q+1 φ + ∬ ψ(x) ⋅ [∇x G(x, x 󸀠 )]u(x)u(x 󸀠 ) dx dx󸀠 = o(1) q+1 Ω

(5.86)

Ω×Ω

by (5.84). Let φ̂ = φx0 ,2R . The second term on the left-hand side of (5.86) is equal to ∬ ψ(x) ⋅ [∇x G(x, x 󸀠 )]u(x)u(x 󸀠 ) dx dx󸀠 Ω×Ω 󸀠 ̂ = ∬ ψ(x) ⋅ [∇x G(x, x 󸀠 )]u(x)φ(x)u(x ) dx dx󸀠 Ω×Ω

5.3 Method of scaling

| 293

󸀠 ̂ 󸀠 ) dx dx󸀠 ̂ )φ(x = ∬ ψ(x) ⋅ [∇x G(x, x󸀠 )]u(x)φ(x)u(x Ω×Ω 󸀠 ̂ ̂ 󸀠 )) dx dx 󸀠 . + ∬ ψ(x) ⋅ [∇x G(x, x󸀠 )]u(x)φ(x)u(x )(1 − φ(x

(5.87)

Ω×Ω

The second term of the right-hand side of (5.87), furthermore, is equal to 󸀠 ̂ ̂ 󸀠 )) dx dx 󸀠 )(1 − φ(x ∫ ∫ ψ(x) ⋅ [∇x G(x, x 󸀠 )]u(x)φ(x)u(x Ω×Ω

= m(x0 )(x0 − a) ⋅

∑

x0󸀠 ∈𝒮\{x0 }

m(x0󸀠 )∇x G(x0 , x0󸀠 ) + o(1),

while the method of symmetrization19 is applicable to the first term of the right-hand side of (5.87). In fact, since K(x, x󸀠 ) = G(x, x󸀠 ) − Γ(x − x 󸀠 ) is smooth in (Ω × Ω) ∪ (Ω × Ω), this term is equal to 󸀠 ̂ ̂ 󸀠 ) dx dx󸀠 )φ(x ∬ ψ(x) ⋅ [∇x G(x, x󸀠 )]u(x)φ(x)u(x Ω×Ω

=

1 ∬ ρ0ψ (x, x󸀠 )u0 (x)u0 (x󸀠 ) dx dx󸀠 2 Ω×Ω

+ ∬ ψ(x) ⋅ [∇x K(x, x󸀠 )]u0 (x)u0 (x󸀠 ) dx dx󸀠

(5.88)

Ω×Ω

for u0 = uφ̂ and ρ0ψ (x, x󸀠 ) = (ψ(x) − ψ(x󸀠 )) ⋅ ∇Γ(x − x 󸀠 ). For the second term on the right-hand side of (5.88), we can write ∬ ψ(x) ⋅ [∇x K(x, x󸀠 )]u0 (x)u0 (x󸀠 ) dx dx󸀠 Ω×Ω

= m(x0 )2 (x0 − a) ⋅ ∇x K(x0 , x0 ) + o(1),

k → ∞,

(5.89)

while ρ0ψ (x, x󸀠 ) = −(n − 2)Γ(x − x󸀠 ) in B(x0 , R/2) × B(x0 , R/2) for the first term. Putting ũ 0 = uφ̃ and φ̃ = φx0 ,R/2 , we obtain ρ0ψ (x, x󸀠 )u0 (x)u0 (x󸀠 ) = −(n − 2)Γ(x − x󸀠 )ũ 0 (x)ũ 0 (x󸀠 )

̃ ̃ 󸀠 )u0 (x)u0 (x 󸀠 ) + ρ0ψ (x, x󸀠 )(1 − φ(x)) φ(x ̃ 󸀠 ))u0 (x)u0 (x󸀠 ). + ρ0ψ (x, x 󸀠 )(1 − φ(x

(5.90)

The second and third terms on the right-hand side of (5.90) are symmetric with respect to x and x󸀠 , which are treated similarly. Concerning the third term, we use |ρ0ψ (x, x󸀠 )| ≤ 19 See Section 13.1 of [335].

294 | 5 Supplementary topics CΓ(x − x󸀠 ), ̃ 󸀠 ))u0 (x)u0 (x󸀠 ) dx dx 󸀠 0 ≤ ∬ Γ(x − x󸀠 )(1 − φ(x Ω×Ω

̃ 0 ⟩, = ⟨Γ ∗ u0 , (1 − φ)u and 󵄩󵄩 ̃ 0 󵄩󵄩󵄩󵄩∞ = o(1), 󵄩󵄩(1 − φ)u

󵄩󵄩 0󵄩 󵄩󵄩Γ ∗ u 󵄩󵄩󵄩1 ≤ C‖u‖1 = O(1),

k → ∞,

which implies ̃ 󸀠 ))u0 (x)u0 (x󸀠 ) dx dx󸀠 = o(1). ∬ Γ(x − x󸀠 )(1 − φ(x

(5.91)

Ω×Ω

By (5.90)–(5.91), we obtain 1 ∬ ρ0ψ (x, x󸀠 )u0 (x)u0 (x 󸀠 ) dx dx󸀠 2 Ω×Ω

=−

n−2 ∬ Γ(x − x󸀠 )ũ 0 (x)ũ 0 (x󸀠 ) dx dx󸀠 + o(1). 2

(5.92)

Ω×Ω

From (5.87)–(5.89) and (5.92), equality (5.86) is reduced to n n−2 ∫ w+q+1 φ − ∬ Γ(x − x󸀠 )ũ 0 (x)ũ 0 (x 󸀠 ) dx dx󸀠 q+1 2 Ω

Ω×Ω

+ m(x0 )(x0 − a) ⋅

∑

x0󸀠 ∈𝒮\{x0 }

m(x0󸀠 )∇x G(x0 , x0󸀠 )

+ m(x0 )2 (x0 − a) ⋅ ∇x K(x0 , x0 ) = o(1),

k → ∞.

(5.93)

For simplicity, we take the zero extension of u = u(x) in (5.82) where it is not defined, to put ℱ0 (u) =

1 1 ∫ uγ dx − ⟨Γ ∗ u, u⟩, γ 2

γ =1+

Rn

1 2 =2− . q n

From n n−2 n−2 γ ∫ w+q+1 φ dx = ∫ uγ φ dx = ∫(ũ 0 ) + o(1) q+1 γ γ Ω

Ω

Ω

in (5.93), it follows that ̃ + m(x0 )(x0 − a) o(1) = (n − 2)ℱ0 (φu) ⋅[

∑

x0󸀠 ∈𝒮\{x0 }

m(x0󸀠 )∇x G(x0 , x0󸀠 ) + m(x0 )∇x K(x0 , x0 )].

(5.94)

5.3 Method of scaling

| 295

Since a ∈ Rn is arbitrary, the second term on the right-hand side of (5.94) vanishes. Hence each x0 ∈ 𝒮 satisfies the relation m(x0 ) ∇R(x0 ) + ∑ m(x0󸀠 )∇x G(x0 , x0󸀠 ) = 0 2 x󸀠 ∈𝒮\{x }

(5.95)

̃ = o(1). ℱ0 (φu)

(5.96)

0

0

and also

Having (5.95), we show the simplicity of the collapse, that is, m(x0 ) = m∗ . The 2

scaling wμ (x) = μ q−1 w(μx) to (5.72) implies that of uμ (x) = μn u(μx + x0 ) to (5.82) for μ > 0. Then it holds that n−2

ℱ0 (uμ ) = μ

ℱ0 (u).

To execute the blow-up analysis below, we write the index k explicitly again. First, from the proof of Theorem 5.43, each x0 ∈ 𝒮 admits a local maximizer x = xk0 → x0 of uk = uk (x) in x ∈ B(x0 , 2R), for k sufficiently large. Under the scaling 0 ũ k (x) = μnk uk (μk x + xk0 ), μk = uk (xk0 )−1/n → 0, and w̃ k (x) = μn−2 k wk (μk x + xk ), there is a subsequence denoted by the same symbol such that locally uniformly in Rn

w̃ k → w̃

(5.97)

̃ with w̃ = w(x) satisfying n

− Δw̃ = w̃ +n−2 ,

̃ w̃ ≤ w(0) =1

in Rn ,

n

∫ w̃ +n−2 dx < +∞.

(5.98)

Rn

From the classification of entire solutions, equation (5.98) implies ̃ w̃ = w(|x|),

supp w̃ ⊂ B,

n

∫ w̃ +n−2 dx = m∗

(5.99)

Rn

for B = B(0, L) with L ≫ 1. Henceforth, we put ̃ k )(μk x + xk0 ). û k (x) = μnk (φu

(5.100)

First, by (5.96) it holds that n−2

̃ k ) → 0. ℱ0 (û k ) = μk ℱ0 (φu

(5.101)

Second, we apply Theorem 5.43 to ŵ k (x) = w̃ k (x), x ∈ μ−1 k (B(x0 , R/4) − {x0 }). Similarly to (5.97), one has ŵ k (x) → w̃

locally uniformly in Rn ,

(5.102)

296 | 5 Supplementary topics passing to a subsequence. From (5.82) and (5.102), it follows that n

û k → ũ ≡ w̃ +n−2

locally uniformly in Rn ,

and hence ∇w̃ = ∇Γ ∗ ũ by Liouville’s theorem for harmonic functions. This relation implies x ̃ ∗ u,̃ ⋅ ∇w+q+1 = x ⋅ u∇Γ q+1 and hence 1 n ∫ w̃ +q+1 dx + ∬ (x − x 󸀠 ) ⋅ ∇Γ(x − x 󸀠 )ũ ⊗ ũ dx dx󸀠 = 0, q+1 2 Rn

Rn ×Rn

which means 1 1 ∫ ũ γ dx − ⟨Γ ∗ u,̃ u⟩̃ = 0 γ 2

ℱ0 (u)̃ =

(5.103)

Rn

since q =

n n−2

and x ⋅ ∇Γ(x) = −(n − 2)Γ(x). We have, on the other hand, ‖û k ‖Lp (Rn ) ≤ C,

p = 1, ∞,

(5.104)

and hence ‖Γ ∗ û k ‖W 2,p (Rn ) ≤ C,

1 < p < ∞.

(5.105)

By (5.104)–(5.105), a subsequence denoted by the same symbol satisfies ̃ lim ⟨Γ ∗ û k , û k ⟩ = ⟨Γ ∗ u,̃ u⟩.

k→∞

(5.106)

Relations (5.101), (5.103), and (5.106) imply γ

lim ∫ û k dx = ∫ ũ γ dx,

k→∞

Rn

Rn

and therefore, it holds that û k → ũ in Lγ (Rn ).

(5.107)

We complete the proof of the simplicity of the collapse at x0 . By the proof of Theorem 5.43, condition m(x0 ) > m∗ implies the collision of collapses. Hence we have a local maximizer xk1 ≠ xk0 of uk = uk (x) and rk0 , rk1 → 0 such that xk1 → x0 and lim

k→∞

∫ B(xk0 ,rk0 )

uk dx = lim

k→∞

∫ B(xk1 ,rk1 )

uk dx = m∗ ,

5.3 Method of scaling

| 297

B(xk0 , 2rk0 ) ∩ B(xk1 , 2rk1 ) = 0. This fact follows from the scaling (5.100). Namely, we have L󸀠 > L, xk󸀠 ∈ R2 , and rk󸀠 > 0 such that lim

k→∞

∫ û k dx = lim

k→∞

B(0,L󸀠 )

û k dx = m∗ ,

∫ B(xk󸀠 ,rk󸀠 )

B(0, 2L󸀠 ) ∩ B(xk󸀠 , 2rk󸀠 ) = 0.

(5.108)

Furthermore, ũ k → 0 locally uniformly in B(0, 2L󸀠 )c , and therefore, it holds that 󵄨 󵄨 lim 󵄨󵄨󵄨xk󸀠 󵄨󵄨󵄨 = +∞.

(5.109)

k→∞

Here we take the second scaling ũ 󸀠k (x) = (μ󸀠k )n û k (μ󸀠k x + xk󸀠 ) for μ󸀠k = û k (xk󸀠 )

−1/n

≥ 1.

We show lim μ̃ 󸀠k = +∞,

(5.110)

k→∞

passing to a subsequence. In fact, if (5.110) does not hold then μ󸀠k = ũ k (xk󸀠 )

−1/n

≈ 1.

(5.111)

󸀠 󸀠 Then we apply Theorem 5.43 to ũ 󸀠󸀠 k = ũ k (⋅ + xk ). Since 0 ≤ û k ≤ ũ k ≤ C, a subsequence satisfies n

󸀠󸀠 󸀠󸀠 n−2 ũ 󸀠󸀠 k → ũ = (w̃ )+

locally uniformly in R2

with w̃ 󸀠󸀠 = w̃ 󸀠󸀠 (x) satisfying n

−Δw̃ 󸀠󸀠 = (w̃ 󸀠󸀠 )+n−2

in Rn ,

0 < w̃ 󸀠󸀠 (0) = 1 ≤ max w̃ 󸀠󸀠 < +∞, n R

n n−2

∫ (w̃ )+ dx < +∞. 󸀠󸀠

Rn

This property implies ∫Rn u󸀠󸀠 dx = m∗ , and therefore, rk󸀠 ≈ 1

(5.112)

in (5.108) by (5.111). We thus obtain lim k

∫ B(0,2L)c

γ û k dx ≥ lim k

∫ B(xk󸀠 ,2rk󸀠 )

γ

(û 󸀠k ) dx > 0

(5.113)

298 | 5 Supplementary topics by (5.108), (5.109), and (5.112), which contradicts (5.107), recalling B = B(0, L) in (5.99). Hence relation (5.110), which implies uk (xk0 ) = +∞. k→∞ uk (x 1 ) k lim

From the same argument with changing the roles of xk0 and xk1 , however, it follows that uk (xk0 ) = 0, k→∞ uk (x 1 ) k lim

a contradiction.

5.4 Differential inequality −Δu ≤ f (u) 5.4.1 Blow-up profile We study the family of blow-up functions {u(⋅, t)}, 0 ≤ t < T, satisfying the differential inequality − Δu ≤ f (u) in B = BR ≡ {|x| < R} ⊂ Rn

(5.114)

for n ≥ 2. The nonlinearity f : [0, +∞) → [0, +∞) is supposed to satisfy f ∈ C2 ,

f 󸀠 (u) > 0,

0 < u < +∞,

f (+∞) = +∞.

(5.115)

Then the limiting function u = u(⋅, T) subject to (5.114) is estimated from below in accordance with the elliptic equation with two parameters σ > 0 and a ∈ I ≡ {f (u) | 0 < u < +∞} = (f (0), +∞), − Δv = σ 2 f (v + f −1 (a))

in B ≡ B1 ,

v|𝜕B = 0.

(5.116)

It holds that v = v(|x|) by Theorem 2.20. Let {u(⋅, t)}, 0 ≤ t < T, be a family of C 1 functions on BR satisfying (5.114). Suppose that t ∈ [0, T) 󳨃→ ∫ f (u(x, t)) dx,

0 < r ≪ 1,

Br

is continuous and 0 ∈ BR is a blow-up point of this {u(⋅, t)}: ∃xk ∈ BR ,

tk ↑ T

Then we obtain the following result.

such that u(xk , tk ) → +∞.

(5.117)

5.4 Differential inequality −Δu ≤ f (u)

| 299

Theorem 5.46. Suppose the existence of a classical solution to (5.116) for a = A(σ) ∈ I and 0 < σ ≪ 1. Then, it holds that u∗ (σ, T) ≡ lim sup sup u(x, t) ≥ f −1 (A(σ)). t↑T

(5.118)

|x|=σ

We perform a comparison of (5.114) and (5.116) in the following cases. Example 1. f (u) = eu . Equation (5.116) means −Δv = λev

in B,

v|𝜕B = 0

for λ = σ 2 a. There exists the least upper bound λ < +∞ of λ for the existence of classical solutions by the argument in § 2.2.1. It holds actually that = 2, n = 2, { { λ { > 2(n − 2), 2 < n < 10, { { = 2(n − 2), n ≥ 10, and λ = λ(n) admits a solution if and only if 2 ≤ n < 10. Thus we can take A(σ) = λσ −2 ,

λ{

= λ(n), < λ(n),

2 ≤ n < 10, n ≥ 10.

Inequality (5.118) means u∗ (r, T) = lim sup sup ≥ 2 log t↑T

|x|=r

1 + log λ, r

and therefore, it holds that u∗ (r, T) = lim sup sup ≥ 2 log t↑T

|x|=r

1 + log λ(n), r

0 < r ≪ 1.

(5.119)

This result is applicable for u = u(⋅, t) satisfying ut − Δu = eu

in B × (0, T),

u|t=0 = u0 (x) ≥ 0,

u|𝜕B = 0

(5.120)

and limt↑T ‖u(⋅, t)‖∞ = +∞. Under the assumption − Δu0 ≤ eu0 ,

(5.121)

we have ut ≥ 0, and hence (5.114) holds for f (u) = eu . From the monotonicity of u(x, t) in t for each x, the blow-up profile is described by u(x, T) = lim u(x, t) ∈ (0, +∞]. t↑T

300 | 5 Supplementary topics If u0 = u0 (|x|), therefore, it holds that20 u(x, T) ≥ 2 log

1 + log λ(n), |x|

0 < |x| ≪ 1.

In this case we have [34] u(x, T) ≤ 2 log

1 1 + log log + C, |x| |x|

0 < |x| ≪ 1,

and there is a blow-up pattern 1 1 + log log + log 8 + o(1), |x| |x|

u(x, T) = 2 log

0 < |x| ≪ 1

for a class of initial values [45]. Example 2. f (u) = up , 1 < p < ∞. Equation (5.116) reads − Δw = (w + λ)p , 2

2

w>0

in B,

w|𝜕B = 0

(5.122)

1

for w = σ p−1 v and λ = σ p−1 a p . There is the least upper bound of λ for the existence of classical solutions, similarly, by the proof of Theorem 2.1. Taking the first eigenfunction ϕ1 = ϕ1 (x) as in −Δϕ1 = λ1 ϕ1 ,

ϕ1 > 0

in B,

ϕ1 |𝜕B = 0,

∫ ϕ1 dx = 1, B

we use Jensen’s inequality to deduce p

λ1 ∫ wϕ1 dx = ∫(w + λ)p ϕ1 dx ≥ (∫ wϕ1 dx + λ) . B

B

B

Then it follows that λ ≤ sup{(λ1 J)1/p − J} < +∞. J>0

For the existence of the solution to (5.122) for 0 < λ ≪ 1, on the other hand, we take 1 1 w = μ p−1 z and λ = μ p−1 to derive − Δz = μ(1 + z)p , 20 A related result is in [32, 33].

z>0

in B,

z|𝜕B = 0,

(5.123)

5.4 Differential inequality −Δu ≤ f (u)

|

301

which possesses a solution for 0 < μ ≪ 1 via the bifurcation from the trivial solution z = 0. Hence we obtain 1

u∗ (r, T) = lim sup sup u(x, t) ≥ μ p−1 r t↑T

2 − p−1

|x|=r

,

0a

(5.126)

BR

it holds that ∫ p dx ≤ ∫ dx,

p>t

q>t

∀t ≥ a,

and hence t0 ≡ max p ≤ t0 ≡ q(0). Ω

(5.127)

302 | 5 Supplementary topics Proof. Given t ∈ I ≡ [a, t0 ], put Ωt = {p > t} and Γt = 𝜕Ωt = {p = t}. The distribution function is defined by μ(t) = Ln (Ωt ), where Ln denotes the n-dimensional Lebesgue measure. Since it is right continuous and strictly decreasing, it has a continuous and nonincreasing (and hence absolutely continuous) left inverse t = t(μ) satisfying t(μ) = t and μ(t(μ)) ≥ μ. The coarea formula produces −μ󸀠 (t) = ∫ Γt

ds |∇p|

a. e. t ∈ I,

while Sard’s lemma and Green’s formula imply D(t) ≡ ∫ p dx ≥ − ∫ Γt

Ωt

𝜕 ϕ(p) ds 𝜕ν

= ϕ(t) ∫ |∇p| ds

a. e. t ∈ I,

(5.128)

Γt

where ds and ν denote the area element and outer unit normal vector, respectively. Then we obtain ∫ |∇p| ds ≤ Γt

D(t) , ϕ󸀠 (t)

∫ Γt

ds D󸀠 (t) =− |∇p| t

a. e. t ∈ I

since ϕ󸀠 > 0 and ∞

D(t) = ∫ rd(−μ(r)).

(5.129)

t

Schwarz’s inequality now implies H n−1 (Γt )2 ≤ −

D(t)D󸀠 (t) 1 󸀠 = − 󸀠 (D(t)2 ) tϕ󸀠 (t) 2ϕ (t)

a. e. t ∈ I,

(5.130)

where H n−1 denotes the (n−1)-dimensional Hausdorff measure. This inequality is combined with Ln (Ωt )γn ≤ cn H n−1 (Γt )2

a. e. t ∈ I,

where γn =

2(n − 1) ∈ [1, 2), n

cn = n−2 ω−2/n , n

and ωn = π n/2 /Γ(1 + n/2) stands for the volume of an n-dimensional unit ball.

(5.131)

5.4 Differential inequality −Δu ≤ f (u)

|

303

In fact, since (5.129) implies ∞

Ln (Ωt ) = μ(t) = ∫ t

∞

1 D(t) D(r) d(−D(r)) = − ∫ 2 dr, r t r t

we obtain γn

∞

−

cn D(t) D(r) 󸀠 (D(t)2 ) ≥ tϕ󸀠 (t){ − ∫ 2 dr} 2 t r

a. e. t > a.

t

(5.132)

It holds also that D(a) = ∫ p dx, p>a

D(t) > 0,

a ≤ t < t0 ,

D(t0 ) = 0.

(5.133)

We can follow the same procedure for the continuous strictly decreasing function q = q(ρ), ρ = |x|. We have the equalities, first, in (5.128) for D(t) ≡ ∫ q dx q>t

by (5.125), second, in (5.130) because |∇q| = −q󸀠 (r) is constant on {q = t}, and finally, in (5.131) because {q > t} is a ball. Hence it follows that γn

∞

−

cn D(t) D(r) 󸀠 (D(t)2 ) = tϕ󸀠 (t){ − ∫ 2 dr} 2 t r

a. e. t > a,

t

(5.134)

and furthermore, D(a) = ∫ q dx,

D(t) > 0,

a ≤ t < t0,

D(t 0 ) = 0.

(5.135)

B

To perform a comparison between D(t) and D(t) using (5.132)–(5.133) and (5.134)– (5.135), let e(t) = D(t) − D(t). Since γn ≥ 1, it holds that cn 󸀠 {e(t)(D(t) + D(t))} 2

γn

∞

∞

γn

D(t) D(r) D(t) D(r) ≥ tϕ (t)[{ − ∫ 2 dr} − { − ∫ 2 dr} ] t t r r 󸀠

t

t

∞

= −tϕ󸀠 (t)k(t){

e(t) e(r) − ∫ 2 dr} t r t

a. e. t > a,

304 | 5 Supplementary topics where k = k(t) ≥ 0 is a right-continuous function. Thus we obtain ∞

a. e. t ∈ I ≡ [a, max{t0 , t 0 }],

e (t) ≥ K(t)e(t) + ∫ L(r, t)e(r) dr 󸀠

t

with e(a) ≥ 0 by (5.126), where D󸀠 (t) + D (t) + 󸀠

K(t) = −

2 󸀠 ϕ (t)k(t) cn

D(t) + D(t)

L(r, t) =

,

2 󸀠 ϕ (t)k(t)t cn

(D(t) + D(t))r 2

≥ 0.

The right-continuous function t

E(t) = e(t) exp(− ∫ K(t 󸀠 ) dt 󸀠 ), 0

therefore, satisfies ∞

E (t) ≥ ∫ M(r, t)E(r) dr 󸀠

t

a. e. t ∈ I

r

with E(a) ≥ 0 for M(r, t) = L(r, t) exp(∫t k(t 󸀠 ) dr 󸀠 ), which implies t

t

∫ E (t ) dt + E(a) ≥ ∫ N(r, t)E(r) dr, 󸀠

󸀠

󸀠

a

a

t∈I

r∧t

for N(r, t) = ∫a L(r, t 󸀠 ) dt 󸀠 ≥ 0. Since D = D(t) and D = D(t) are nonincreasing and continuous, respectively, we have t

E(t − 0) = e(t − 0) exp(− ∫ K(t 󸀠 ) dt 󸀠 ) t

0

≤ e(t) exp(− ∫ K(t 󸀠 ) dt 󸀠 ) = E(t),

t ≥ a,

0

and therefore, t

t

E(t) ≥ ∫ E (t ) dt + E(a) ≥ ∫ N(r, t)E(r) dr, 󸀠

a

󸀠

󸀠

a

Thus we obtain E(t) ≥ 0 for t ∈ I, and hence (5.127).

t ∈ I.

5.4 Differential inequality −Δu ≤ f (u)

|

305

In the next lemma, the functions p, q are defined on the same domain B = B1 ≡ {|x| < 1} ⊂ Rn . Lemma 5.48. Let p, q ∈ C 1 (B) ∩ C(B) be functions satisfying − Δϕ(p) ≤ p,

−Δϕ(q) = q

in B.

(5.136)

Assume, furthermore, that q = q(r), r = |x|, and r 󳨃→ q(r) is strictly decreasing. Then it holds that q(1) ≤ max p,

(5.137)

∫ q dx = ∫ p dx.

(5.138)

𝜕B

provided that

B

B

Proof. In the setting of the previous lemma, assume more strongly that b ≡ max p < a. 𝜕B

Then, there is an R󸀠 < R such that q|𝜕B 󸀠 ≤ b, and hence R

∫ p dx ≤ ∫ q dx < ∫ q dx. BR

BR󸀠

p>b

In this setting of (5.136), however, equality (5.138) implies ∫ q dx ≥ ∫ p dx, B

p>b

and hence (5.137).

5.4.3 Mean value theorem in higher dimension If the solution set for (5.116) is not empty, it possesses the minimal element denoted by v, called the minimal solution. This v is, thus, a solution by itself and satisfies v ≥ v in B for any solution v. If (5.116) admits a solution for σ = σ1 ≥ 0 and a = a1 ∈ I, the same happens for σ ∈ [0, σ1 ] and a ∈ I ∩ (−∞, a1 ]. Furthermore, the minimal solution v = vσ,a is continuous in C(B) with respect to (σ, a). It is, furthermore, monotone increasing componentwise in (σ, a) and pointwise in x ∈ B. See § 2.1.2 for the argument to prove these facts.

306 | 5 Supplementary topics Henceforth, vσ,a denotes the minimal solution to (5.116). Recall (5.115), I = {f (u) | 0 < u < +∞}, and ϕ = f −1 . Given r > 0, let p ∈ C 1 (Br ) ∩ C(Br ) satisfy − Δϕ(p) ≤ p in Br

(5.139)

and put p(x) ∈ I for any x ∈ Br . Suppose, furthermore, that (5.116) admits a solution for σ = r and a = A ∈ I, and put m = ∫ r 2 f (vr,A (x) + f −1 (A)) dx,

B = vr,A (0).

B

(5.140)

The following theorem may be regarded as a form of the mean value theorem involving the volume integration. Theorem 5.49. If m≡

1 ∫ p dx ≤ m, r n−2 Br

max p ≤ A, 𝜕Br

it holds that max p ≤ f (B + f −1 (A)). Br

Proof. The continuous function Φ(σ, a) = σ n−2 ∫ σ 2 f (vσ,a (x) + f −1 (a)) dx B

is defined for 0 ≤ σ ≤ r and max𝜕Br p ≤ a ≤ A. It holds that ϕ(0, a) = 0,

Φ(r, A) = r n−2 m,

and therefore, there is (σ∗ , a∗ ) ∈ [0, r] × [max𝜕Br p, A] such that Φ(σ∗ , a∗ ) = r n−2 m. Put v∗ = vσ∗ ,a∗ and q(x) = f (v∗ (σ∗−1 x) + f −1 (a∗ )),

x ∈ B∗ ≡ Bσ∗ .

By Theorem 2.20, it holds that q = q(ρ), ρ = |x|, and ρ 󳨃→ q(ρ) is strictly decreasing. We have, furthermore, −Δϕ(q) = q

in B∗ ,

q|𝜕B∗ = a∗ ≥ max p. 𝜕Br

5.4 Differential inequality −Δu ≤ f (u)

|

307

Putting x̃ = σ∗−1 x, we obtain ∫ q dx = σ∗n−2 ∫ σ∗2 f (v∗ (x)̃ + f −1 (a∗ )) dx̃

B∗

B

= Φ(σ∗ , a∗ ) = r n−2 m = ∫ p dx, Br

and therefore, Lemma 5.47 implies max p ≤ q(0) = f (v∗ (0) + f −1 (a∗ )) ≤ f (B + f −1 (A)). Br

5.4.4 Pattern formation This section is devoted to the proof of Theorem 5.46. The idea lies in the exclusion of the equality for continuously depending functions [167]. Lemma 5.50. Under the assumption of Theorem 5.49, if max p < A

(5.141)

m < m.

(5.142)

𝜕Br

more strongly, it follows that

̃ ̃ = r 2 p(r x)̃ for x̃ ∈ B = B and ϕ(p)̃ = ϕ(r −2 p). ̃ It holds that p̃ ∈ C 1 (B)∩C(B) Proof. Put (x) 1 2 and p(̃ x)̃ ∈ I ̃ ≡ r I and −Δϕ(̃ p)̃ ≤ p̃ in B,

∫ p̃ dx̃ = B

1 ∫ p dx = m. r n−2 Br

The function ̃ q(̃ x)̃ = r 2 f (vr,A (x)̃ + f −1 (A)),

x̃ ∈ B,

̃ and satisfies on the other hand, is radially symmetric, strictly increasing in ρ = |x|, −Δϕ(̃ q)̃ = q̃

in B,

∫ q̃ dx̃ = m. B

By Lemma 5.48, therefore, if m = m, it holds that ̃ = r 2 A ≤ max p̃ = r 2 max p. q(1) 𝜕B

Inequality (5.141), thus implies (5.142).

𝜕Br

308 | 5 Supplementary topics Proof of Theorem 5.46. Let p = f (u) in (5.114). From the assumption, there is vr,A(r) for 0 < r ≪ 1 such that lim A(r) = +∞.

(5.143)

r↓0

Suppose on the contrary to (5.118), and take t0 ∈ [0, T) and rk ↓ 0 such that sup p(x, t) < A(rk ),

|x|=rk

t0 ≤ ∀t < T,

∀k = 1, 2, . . .

(5.144)

Put mt (r) =

1 ∫ p(x, t) dx, r n−2 Br

m(r) = ∫ r 2 f (vr,A(r) (x) + f −1 (A(r))) dx, B

and confirm mt0 (r) < m(r),

0 < r ≪ 1.

(5.145)

In fact, we have lim r↓0

mt0 (r) r2

= ωn p(0, t0 ),

recalling that ωn denotes the volume of an n-dimensional ball, while it holds that m(r) > ωn A(r) → +∞, r2

r↓0

by (5.143). Hence we obtain (5.145). By (5.144), therefore, there is 0 < r ≪ 1 such that sup p(x, t) < A(r), |x|=r

t0 ≤ ∀t < T,

(5.146)

and simultaneously (5.145) holds. Now apply Lemma 5.50. Since t 󳨃→ mt (r) is continuous, the equality mt (r) = m(r) never occurs in t0 ≤ t < T by (5.145) and (5.146). Thus we obtain mt (r) < m(r),

t0 ≤ ∀t < T.

Theorem 5.49 then implies max p(⋅, t) ≤ f (B(r) + f −1 (A(r))) t0 ≤ ∀t < T Br

for B(r) = vr,A(r) (0), which implies that x = 0 is not a blow-up point of {p(⋅, t)} as t ↑ T, a contradiction.

5.5 Trudinger–Moser inequality revisited | 309

5.5 Trudinger–Moser inequality revisited 5.5.1 Threshold of asymptotics This section is devoted to the Trudinger–Moser inequality in the form of 2

sup{∫ e4πv dx | v ∈ H01 (Ω), ‖∇v‖2 = 1} < +∞,

(5.147)

Ω

which implies (4.127), where Ω ⊂ R2 is a bounded domain. By the Schwarz symmetrization in § 4.1.5, inequality (5.147) is reduced to the case Ω = B ≡ B(0, 1) ⊂ R2 , and v = v(r) monotone decreasing in r = |x|. We begin with the associated Euler equation in the general form. Given f ∈ C 1 (R, R), we thus take the eigenvalue problem − Δu = λf (u),

u>0

in B,

u|𝜕B = 0,

(5.148)

for λ > 0. We assume a family {(uj , λj )} of solutions satisfying ‖uj ‖∞ ↑ +∞,

λj ↓ 0.

(5.149)

If the nonlinearity takes the form f (u) = eu + g(u) with g(u) = o(eu ) as u ↑ +∞ and |g(u) − g 󸀠 (u)| ≤ G(u), satisfying G(u) + |G󸀠 (u)| = O(eγu ) for γ < 1/4, it holds that u(x) → 1 4 log |x| in W01,p (B) ∩ L∞ loc (B \ {0}) for 1 ≤ p < ∞ by Corollary 3.8 in § 3.3.1. Inequality (5.147), however, is associated with the nonlinearity of higher growth rate. First, we actually have u = u(r) and ur < 0 for 0 < r = |x| ≤ 1 in (5.148). Second, it holds also that f (+∞) = +∞ by (5.149), and hence f (u) > 0 for u ≫ 1. By the following theorem, we see that strong and weak nonlinearities induce the convergence to zero except for the blow-up point and the entire blow-up, respectively, and therefore, a singular limit arises only for exponentially dominated nonlinearities. Theorem 5.51 ([258]). It holds that lim (log f )󸀠 (u) = +∞ 󳨐⇒ lim uj (x) = 0

locally uniformly in x ∈ B \ {0},

lim (log f )󸀠 (u) = 0 󳨐⇒ lim uj (x) = +∞

locally uniformly in x ∈ B.

j→∞

u↑+∞

j→∞

u↑+∞

Proof. Given c ∈ R and a bounded domain Ω ⊂ Rn with a smooth boundary 𝜕Ω, the Pohozaev identity for −Δu = λf (u) in Ω,

u|𝜕Ω = c

arises as λn ∫ F(u) − F(c) dx + λ Ω

2

2−n 1 𝜕u ∫ f (u)(u − c) dx = ∫ ( ) x ⋅ ν ds, 2 2 𝜕ν Ω

𝜕Ω

310 | 5 Supplementary topics u

where F(u) = ∫0 f (u) du. Applying it for n = 2 and Ω = {|x| < r}, we see that the solution u = u(r) to (5.148) satisfies u2r (r) + 2λF(u(r)) =

2λ ∫ F(u) dx, πr 2

0 < r < 1.

(5.150)

|x| 0 for u ≥ K. In the first case of limu↑+∞ (log f )󸀠 (u) = +∞, it holds that lim Ck = 0,

k↑+∞

Ck ≡ sup u≥k

F(u) . f (u)

Hence for k sufficiently large, we obtain ∫ λF(uj ) dx = ∫ λj F(uj ) dx + ∫ λj F(uj ) dx B

{uj ≥k}

{uj 0,

g (u) ≥ 0, 󸀠

u ≫ 1,

0 < lim inf ϕ󸀠󸀠 (u) ≤ lim sup ϕ󸀠󸀠 (u) < +∞, u↑+∞

u↑+∞

lim (log g) (u) = 0,

u↑+∞

󸀠

0 < lim inf ϕ󸀠 (u)g(u)u−m ≤ lim sup ϕ󸀠 (u)g(u)u−m < +∞ u↑+∞

u↑+∞

(5.152)

for m ∈ R. A typical example is ϕ(u) = uα , 1 < α ≤ 2, and g(u) is a positive polynomial, both for u ≫ 1. By the proof of Theorem 5.51, the asymptotics (5.149) of the family {(uj , λj )} of classical solutions ensures η2j ≡ max r 2 u2jr (r) → 0, 0≤r≤1

uj (r) → 0 locally uniformly in 0 < r ≤ 1.

22 Another argument without Harnack principle is also known [258].

(5.153)

312 | 5 Supplementary topics 2 u The function v0 (y) = 2 log 1+|y| 2 is the unique classical solution to (5.148) for f (u) = e and λ = 2. The following theorem says that the function ϕ(u(x)) exhibits the profile v0 (x) around x = e−rj /2 under a scaling variable.

Theorem 5.52. Assuming (5.152), there exists a subsequence such that λj ϕ󸀠 (uj (e−τj /2 ))f (uj (e−τj /2 ))e−τj = 2 + o(1)

(5.154)

with some τj ↑ +∞. Then it holds that ϕ(uj (e−τj /2 y)) = ϕ(uj (e−τj /2 )) + 2 log

2 + o(1) 1 + |y|2

(5.155)

locally uniformly in y ∈ R2 \ {0}. Step 1 of the proof. For the moment we drop the index j. The function h = ϕ󸀠 g satisfies lim (log h)󸀠 (u) = 0,

u↑+∞

lim sup max v↑+∞

0≤u≤v

h(u) < +∞, h(v)

h(ϕ−1 (ϕ(u) + 2 log 2)) lim sup < +∞. h(u) u↑+∞

(5.156)

In fact, first, since (log h)󸀠 = ϕ󸀠󸀠 /ϕ󸀠󸀠 + g 󸀠 , it holds that lim (log h)󸀠 (u) = 0,

u↑+∞

(log h)󸀠 (u) ≥ 0,

u ≫ 1.

Therefore, h(u) is nondecreasing for u ≫ 1, and hence it holds that lim sup max v↑+∞

0≤u≤v

h(u) < +∞. h(v)

By (5.153), second, there exists M > 0 such that m

h(ϕ−1 (ϕ(u) + 2 log 2)) ϕ−1 (ϕ(u) + 2 log 2) ≤ M( ) . h(u) u Then we obtain lim sup u↑+∞

ϕ−1 (ϕ(u) + 2 log 2) ϕ−1 (v + 2 log 2) = lim < +∞ v↑+∞ u ϕ−1 (v)

because ϕ(v) has quadratic growth, and then the result follows for m ≥ 0. We have, on the other hand, ϕ󸀠 (u) > 0 for u ≫ 1, and hence holds for m < 0.

ϕ−1 (ϕ(u)+2 log 2) u

≥ 1, there. Hence (5.156)

5.5 Trudinger–Moser inequality revisited | 313

Step 2 of the proof. For the moment we drop the index j. Using r = e−t/2 , U(t) = u(r), d , we obtain and ⋅ = dt λ Ü + g(U)eϕ(U)−t = 0, 4

U > 0,

t > 0,

U(0) = 0,

̇ t/2 = 0. lim Ue

t↑+∞

Given τ > 0, let Uτ (t) ≡ U(t + τ) and V(t) ≡ ϕ(Uτ (t)) − ϕ(U(τ)). Note U(t) = Uj (t), Uτ (t) = Uτj (t), and V(t) = Vτj (t). Then it holds that λ V̈ + ϕ󸀠 (Uτ )g(Uτ )eϕ(U(τ))−τ eV−t = ϕ󸀠󸀠 (Uτ )Uτ2̇ , 4 ̇ t/2 = 0, V(0) = 0, lim Ve

t > −τ,

t↑+∞

(5.157)

and hence ∞

∞

K(s) V(s)−s e ds − V(+∞), ∫ (s − t)ϕ (Uτ (s))U̇ τ2 (s) ds = V(t) + ∫ (s − t) 4 󸀠󸀠

t

K(t) ≡ λϕ (Uτ (t))g(Uτ )e 󸀠

ϕ(U(τ))−τ

t

(5.158)

.

Since 1 U̇ τ (t) = − rur |r=e−(t+r)/2 , 2

(5.159)

lim ‖U̇ τ ‖L∞ (−τ,∞) = 0.

(5.160)

one has τ↑+∞

Under this translation, we put ρ = ρ(|y|) ≡ 4et ϕ󸀠󸀠 (Uτ (t))U̇ τ2 (t),

v = v(|y|) ≡ V(t),

k = k(|y|) ≡ K(t)

(5.161)

for |y| = e−t/2 , to reach − Δv = kev − ρ,

> 0, { { v { = 0, { { < 0,

|y| < 1, |y| = 1, 1 < |y| < eτ /2.

(5.162)

|∇u|2 dx.

(5.163)

Since |ϕ󸀠󸀠 | ≤ C, it holds that ‖ρ‖L1 (R1 0.

(5.164)

Two cases are distinguished. Case 1. M ≡ max0≤r≤1 λϕ󸀠 (u(r))f (u(r))r 2 → +∞. We take r1 ∈ (0, 1) satisfying λϕ󸀠 (u(r1 ))f (u(r1 ))r12 = 2 for j ≫ 1. By the second relation of (5.153) and when λ ↓ 0, it holds that r1 → 0. Then τ = 21 log(1/r1 ) → +∞ with λϕ󸀠 (U(τ))g(U(τ))eϕ(U(τ))−τ = 2.

(5.165)

From this relation it follows that U(τ) → +∞, similarly, as well as h(Uτ (t)) 1 K(t) = →1 2 h(U(τ))

locally uniformly in t ∈ (−∞, +∞)

and 2

ρ(|y|) = 4|y|−2 ϕ󸀠󸀠 (u(e−τ/2 |y|))(e−τ/2 |y|) u2r (e−τ/2 |y|). We thus end up with k → 2,

locally uniformly in R2 \{0}

ρ→0

(5.166)

as j → ∞. We have v ≤ 0,

0 ≤ k ≤ 2 max h(v)/h(U(τ)) = O(1),

|y| ≥ 1,

0≤v≤U(τ)

and therefore, the elliptic estimate implies 0 ≤ −vr (1) = O(1),

‖v‖L∞ (|y|>1−δ) = O(1) loc

with δ > 0. Furthermore, it holds that e−τ/2

‖ρ‖L1 (ε 0, u↑+∞ uα 1 ϕ(u) − uϕ󸀠 (u) > 0, u > 0, 2

lim ϕ(u) = lim ϕ󸀠 (u) = +∞,

u↑+∞

ϕ (u) ≥ 0, 󸀠󸀠

u↑+∞

ϕ󸀠 (u) > 0,

lim

(5.181)

for α ∈ (1, 2). Then, there is a family {(u, λ)} of classical solutions to (5.148). The parameter τ → +∞ given in Theorem 5.52, moreover, satisfies 󵄨󵄨 2 󵄨󵄨󵄨󵄨 󵄨 max 󵄨󵄨󵄨ϕ(u(eτ/2 y)) − ϕ(u(eτ/2 )) − 2 log 󵄨 ↛ 0. 1 + |y|2 󵄨󵄨󵄨 |y|0

in Ω,

u|𝜕Ω = 0,

(5.188)

where Ω ⊂ R2 is a bounded domain with a smooth boundary 𝜕Ω and f = f (u) ≥ 0 is 2 a smooth function satisfying 0 ≤ f (u) ≤ A(1 + um )eu , u ≥ 0, with m ≥ 0 and A > 0, where μ > 0 is a constant. Given μ > 0, we thus put Sμ = {v ∈ H01 (Ω) | ∫ F(v+ ) dx = μ},

u

F(u) = ∫ f (u) du. 0

Ω

‖∇v‖22

Theorem 5.55 ([311]). If there is v ∈ Sμ such that = γ < 4π, the minimum inf{‖∇w‖2 | w ∈ Sμ } is attained by a minimizer u such that ‖∇u‖22 ≤ γ. From the Lagrangian multiplier principle, this minimizer u solves (5.188) with λ > 0. Theorem 5.55 is a two-dimensional version of Theorems 1.19–1.20. We recall that 2n Sobolev’s embedding H01 (Ω) ⊂ L n−2 (Ω) for n > 2 is continuous but not compact. The best constant S0 of this inclusion is determined only by n, and is not attained unless Ω = Rn . In the limiting case n = 2, there arises a continuous embedding of H01 (Ω) into an Orlicz space [352], which is refined by [231] as 2

sup{∫ ew dx | ‖∇w‖22 ≤ 4π} < +∞,

(5.189)

Ω

where 4π is the best possible constant. This maximum, however, is attained if Ω = B, in contrast with the Sobolev inequality [67]. Theorem 5.55 is derived by a standard argument based on the following lemma.

5.5 Trudinger–Moser inequality revisited | 321

Lemma 5.56. If vj → v weakly in H01 (Ω) and supj ‖∇vj ‖22 < 4π/α for α > 0, it follows that 2

2

∫Ω eαvj dx → ∫Ω eαv dx.

By the Schwarz symmetrization, this lemma is reduced to the radially symmetric case of Ω = B and vj = vj (|x|). Letting f (u) = g(u)eϕ(u) , the argument in § 1.5 is applicable in the subcritical case indicated by f (0) = 0,

lim sup ϕ(u)u−α < +∞,

α < 2.

u↑+∞

(5.190)

Proposition 5.57. Assuming (5.190) instead of (5.152), there is a family of classical solutions to (5.148) satisfying (5.149). Any such family, on the other hand, has the property E = ‖∇v‖22 → +∞. Proof. Since the mountain pass lemma is applicable to (5.148), there is a family {(λ, u)} of solutions to (5.148) such that E ≡ ‖∇v‖22 ↛ 0 and λ ↓ 0. Here, the elliptic estimate and Trudinger–Moser inequality guarantee that the condition ‖u‖∞ = O(1) implies E → 0, and therefore, we have the existence of a family of classical solutions to (5.148) satisfying (5.149). For the same reason, this family satisfies E → +∞, for otherwise ‖u‖∞ = O(1) holds when λ ↓ 0. In the critical case, the behavior of E is different. To apply Theorem 5.55, let lim sup ϕ(u)u−2 ≤ 1.

(5.191)

u↑+∞

By (5.152), there is n ≥ 0 such that lim supu↑+∞ g(u)u−n < +∞. Then, if there is v ∈ H01 (B) satisfying v ≥ 0, ‖F(v)‖1 = μ, and ‖∇v‖22 = γ < 4π, there is a classical solution to (5.152), denoted by (u, λ), such that ∫ F(u) dx = μ,

∫ |∇u|2 dx ≤ γ.

B

B

Similarly to the proof of Proposition 5.57, on the other hand, any family {(u, λ)} of classical solutions to (5.148) with (5.149) holding satisfies lim inf E ≥ 4π. Lemma 5.58. For a continuous function k = k(u) satisfying limu↑+∞ k(u) = +∞, there 2

exists a family {wj } ⊂ H01 (B) such that wj ≥ 0, ‖∇wj ‖22 < 4π, and ∫B k(wj )ewj dx → +∞. Proof. Writing W(t) = w(r) for r = e−t/2 , we have ∞

∫ Ẇ 2 dt = 0

1 ∫ |∇u|2 dx, 4π B

∞

∫ k(W)eW

2

−t

dt =

0

2 1 ∫ k(w)ew dx. π

B

1

Then we construct a family {W} ⊂ C [0, +∞) satisfying W ≥ 0,

W(0) = 0,

∞

∞

∫ Ẇ 2 dt < 1,

∫ k(W)eW

0

0

2

−t

dt → +∞.

322 | 5 Supplementary topics These requirements are actually fulfilled for W(t) = ε−1/2 ηε (εt), where ηε (s) = min(s, 1− ε) with 0 < ε ≪ 1. In fact, W(0) = 0 is obvious, while we obtain ∞

∞

0

0

∫ Ẇ 2 dt = ε ∫ η󸀠ε 2 (s) ds = (1 − ε)2 < 1 and ∞

∫ k(W)e

W 2 −t

∞

dt = ∫ k(ε−1/2 ηε (s))eε

0

0

≥

inf

u≥e−1/2 (1−ε)

−1

ηε (s)−ε−1 s −1

ε ds

∞

−1

k(u) ∫ e(1−ε−s)ε ε−1 ds = 1−ε

inf

u≥ε−1/2 (1−ε)

k(u) → +∞

as ε ↓ 0. Theorem 5.59. Under the assumptions of Proposition 5.53, assume, furthermore, (5.191) and lim g(u)/u = +∞,

lim inf ϕ(u)u−2 > 0.

u↑+∞

u↑+∞

(5.192)

Then there is a family {(u, λ)} of classical solutions to (5.148) satisfying ‖u‖∞ → +∞,

E ≡ ‖∇u‖22 ↑ 4π.

λ ↓ 0,

u

(5.193)

2

Proof. Writing F(u) = ∫0 g(u)eϕu du = k(u)eu , we obtain limu↑+∞ k(u) = +∞, and hence Lemma 5.58 is applicable. Then Theorem 5.55 assures a family {(u, λ)} of classical solutions to (5.148) satisfying ‖∇u‖22 < 4π,

∫ F(u) dx → +∞. B

The last relation implies ‖u‖∞ → +∞, and therefore, supposing the conditions of Proposition 5.53 implies that λ ↓ 0. Consequently, lim inf E ≥ 4π holds. 5.5.5 Uniformness near the origin 2

Here we assume ϕ(u) = u2 . The requirement in Theorem 5.59 for f (u) = g(u)eu > 0 becomes g 󸀠 (u) ≥ 0,

(log g)󸀠󸀠 (u) ≥ −2,

u ≫ 1,

lim (log g)󸀠 (u) = 0,

u↑+∞

0 < lim inf g(u)u−n ≤ lim sup g(u)u−n < +∞, u↑+∞

u↑+∞

lim sup(log g)󸀠󸀠 (u) < +∞, u↑+∞

n ≥ 0,

lim g(u)/u = +∞,

u↑+∞

1 T3 (u) ≡ log{ug(u)} − u(log g)󸀠 (u) → +∞, 2

u ↑ +∞.

5.5 Trudinger–Moser inequality revisited | 323

Then we have a family {(u, λ)} of classical solutions to (5.148) satisfying (5.193). A typical example is g(u) = un for n > 1. There is, however, a qualitative theorem independent of the existence of such a family. Theorem 5.60 ([257]). If ϕ(u) = u2 and lim sup E < 6π in Theorem 5.52, equality (5.155) includes y = 0. It is, thus, locally uniform in y ∈ R2 , where E = ‖∇v‖22 . Lemma 5.61. The function k = k(|y|) defined by (5.158) and (5.161) satisfies ‖k‖Lp (|y| 0, it holds that 4π ≤ lim inf

∫

j→∞

−τj /2

{Re

|∇uj |2 dx.

0 and M > 0, there exists a unique solution v = v(r) to (6.7) satisfying (6.8). If we take F(z) = βz α for α = 1 + defined by (3.28),

L 2

and log(β2 ) = λM/2(2 + L)2 , the function v = v(r)

v(r) = log

8α2 β2 r 2(α−1) , λ(1 + β2 r 2α )2

(6.9)

satisfies (6.8) and (rv󸀠 )󸀠 +λrev = 0, and hence boundary condition (6.5), or equivalently (6.6), classifies radial solutions to (6.4). Several properties of G = G(ζ ) suffice for this purpose. First, it is positive in (0, a), and satisfies limζ ↓0 G(ζ ) = G(a) = 0, limζ ↓0 G = +∞, and G󸀠 (a) < 0.2 Lemma 6.3. For each 0 < a < 1, the function G = G(ζ ) has one critical point in (0, a). Proof. Putting ζ = aη, we have G(ζ ) = a2 (log aη)2 with g 󸀠 (η) = (log aη)2

(1−η)(1−a2 η) { log1aη (1−a2 η2 )2

h(η) ≡ 1 +

(1 − η)η(1 − a2 η) ≡ a2 g(η) (1 − a2 η2 )2

− h(η)} and

1 1 2 2 + − − . 2 1 − η 1 − a η 1 − aη 1 + aη

Using A2 + B2 ≥ 2AB, we find h󸀠 (η) = ≥

1 a2 2a 2a + − + 2 (1 − η) (1 − a2 η)2 (1 − aη)2 (1 + aη)2

8a2 η 2a 2a − = (1 − η)(1 − a2 η) (1 − a2 η2 )2 (1 − η)(1 − a2 η)(1 − a2 η2 )2 ⋅ {(1 − aη)4 + 4a(a − 1)2 η2 } > 0,

0 < η < 1,

and hence h(η) is monotone increasing in η ∈ (0, 1). Since 1/(log aη) is monotone decreasing in η ∈ (0, 1), we conclude that g 󸀠 (η) changes sign only once in (0, 1). 2 The following proof is due to T. Hanada.

360 | 6 Equations on annulus Lemma 6.3 implies the following theorem, where m∗ denotes the maximum of G(ζ ), ζ ∈ (0, a), and λ∗ = 8m∗ /a2 (log a)2 . Theorem 6.4. There exists λ∗ > 0 such that equation (6.4) has two, one, and none radial solutions for λ ∈ (0, λ∗ ), λ = λ∗ , and λ ∈ (λ∗ , ∞), respectively. We have examined that the radial solution corresponds uniquely to ζ ∈ (0, a), or equivalently, to α ∈ (1, ∞), and this correspondence is continuous. Each λ ∈ (0, λ∗ ) admits two roots ζ and ζ of (6.6): 0 < ζ < ζ < a. The corresponding radial solutions vλ and vλ to (6.4) will turn out to be the nonminimal and minimal, respectively. Lemma 6.5. For each a < r < 1, it holds that lim vλ (r) = +∞, λ↓0

lim vλ (r) = 0. λ↓0

(6.10)

Proof. The minimal solution vλ corresponds to ζ , α, and β satisfying ζ → a, α → 1, and β → 0, and hence it follows that 8 2 2 2 α β = (1 + β2 ) → 1, λ

λ ↓ 0.

By (6.9) and this relation, the second convergence of (6.10) is obvious. Turning to vλ , we regard λ = λ(ζ ) as a smooth function of 0 < ζ ≪ 1, and α, β, similarly. Then using κ ≡ ( 8λ )1/2 αβ = 1 + β2 , we introduce A(ζ ) ≡ B(ζ ) ≡

1 − aζ ζ 1 = = 1 − + O(ζ 2 ), a κaα−1 1 − ζ2

a−ζ β2 ζ = = 1 − + O(ζ 2 ), κ a a(1 − ζ 2 )

(6.11)

to obtain α−1

a vλ (r) = −2 log{A(ζ )( ) r

+ B(ζ )r σ+1 }.

(6.12)

This time we have ζ → 0, α → +∞, β → +∞, and consequently, A → 1, B → 1 as λ ↓ 0. Hence the first convergence in (6.10) follows. In Theorem 3.2, all solutions to (3.36) for Ω = B are parametrized by Σ = ∫B λeu dx, where Σ represents the area of F(Ω) on the Riemann sphere 𝒞 for F(z) = Cz. For this case of Ω = A, the analytic function F(z) = βz α on A is α-fold, and therefore, the area F(Ω) on C is given by Σ/α. Hence there comes an idea of parameterizing radial solutions to (6.4) by αλ ∫A eu dx. This property is actually examined for (λ, vλ ) with 0 < λ ≪ 1. Let σ≡

λ ∫ evλ dx. α A

(6.13)

6.1 2D case

| 361

Lemma 6.6. The mapping ζ 󳨃→ σ is one-to-one if 0 < ζ ≪ 1. It holds that σ = 8π{1 − (a +

1 )ζ + O(ζ 2 )}, a

ζ ↓0

(6.14)

and hence ζ =

8π − σ 1 (a + ) {1 + O(8π − σ)}, 8π a −1

σ ↑ 8π.

(6.15)

Proof. Using f ≡

(a − ζ )(1 − aζ ) 1 λa = = 1 − (a + )ζ + O(ζ 2 ), a 8α2 ζ a(1 − ζ 2 )2

(6.16)

we obtain σ=

α−1

λ a ∫{A(ζ )( ) α r A

−2

+ B(ζ )r α+1 } dx

1

= 16παf (ζ ) ∫{A(ζ )(√a/r)α−1 + aB(ζ )(r/√a)a+1 } r dr. −2

a

Then, the transformation t = (r/√a)2α ensures σ=

t=ζ −1

8πf (ζ ) 1 [− ] B(ζ ) A(ζ ) + aB(ζ )t t=ζ

=

8πa(1 − ζ 2 )f (ζ ) (A(ζ ) + aB(ζ )ζ )(A(ζ )ζ + aB(ζ ))

and then the convergence (6.14) follows from (6.11) and (6.16). We thus have limζ ↓0 −8π(a +

1 ), a

and hence (6.15).

dσ dζ

=

By Lemma 6.6, the quantities μ ≡ ∫A evλ dx and ν ≡ ∫A |∇vλ |2 dx are regarded as functions of σ for 0 < λ ≪ 1. Lemma 6.7. It holds that μ(σ) =

8π 2 (a2 + 1) log a {1 + o(1)}, (8π − σ) log(8π − σ)

σ ↑ 8π.

Proof. Since log ζ a2 (log a)(1 − ζ 2 )2 α a log a 1 = = = {1 + (a + )ζ + O(ζ 2 )} λ λ log a 8 log ζ (a − ζ )ζ (1 − aζ ) 8ζ (log ζ ) a follows from (6.11), we obtain μ(σ) =

α πa log a ⋅σ = {1 + O(ζ 2 )}, λ ζ log ζ

by (6.14). Then the convergence (6.17) holds by (6.15).

ζ ↓ 0,

(6.17)

362 | 6 Equations on annulus Lemma 6.8. It holds that ν(σ) =

2

1 −8π (log ) {1 + o(1)}, log a 8π − σ

σ ↑ 8π.

(6.18)

Proof. Since ν(σ) = − ∫A vλ ⋅ Δvλ dx = λ ∫A evλ vλ dx, equality (6.11) implies 1

α−1

a ν(σ) = −4πλ ∫{A(ζ )( ) r a

α−1

a + B(ζ )r α+1 } {A(ζ )( ) r −2

+ B(ζ )r α+1 }r dr.

Using t = (r/√a)2α as above, we obtain ζ −1

−2πλa2 −2πλa2 −2 ν(σ) = (I1 + I2 + I3 ) ≡ ∫ {A(ζ ) + aB(ζ )t} aζ aζ ζ

⋅{

α−1 1−α log a + log t + log(A(ζ ) + aB(ζ )t)} dt. 2 2α

First, it holds that 1 − ζ2 a−1 (log a) 2 (A(ζ )ζ + aB(ζ ))(A(ζ ) + aB(ζ )ζ ) α−1 = (log a){1 + O(ζ 2 )} 2a

I1 =

by (6.11). Second, putting τ = B(ζ )A(ζ )−1 in I2 , we have τ−

1−α −1 I2 = {A(ζ )B(ζ )a} ∫ (1 + τ)−2 {log τ − log B(ζ )A(ζ )−1 a} dτ, 2α τ+

where τ± = B(ζ )A(ζ )−1 aζ ±1 . Since A(ζ ) → 1, B(ζ ) → 1, and α ↑ +∞, one gets ∞

1 I2 = − ∫ (1 + r)−2 (log τ − log a) dτ + o(1) = O(1), 2a

ζ ↓ 0.

0

Similarly, we obtain ∞

1 I3 = ∫ (1 + τ)−2 log(1 + τ) dτ + o(1) = O(1), a

ζ ↓ 0.

0

Recalling (6.5), we obtain ν(σ) = −

8π(log ζ )2 πλa log a {1 + o(1)} = − {1 + o(1)}, ζ log a

by (6.19). Then (6.15) implies (6.18).

ζ ↓0

(6.19)

6.1 2D case

| 363

We note that pλ (r) ≡ λevλ (r) satisfies lim pλ (r) = { λ↓0

+∞, 0,

r = √a, r ≠ √a.

(6.20)

In fact, since pλ (r) = 8α2 (1 −

α−1

√a ζ −2 )(1 − aζ )(1 − ζ 2 ) {A(ζ )( ) a r

+ aB(ζ )(

α+1 −2

r ) √a

} ,

equality (6.20) follows from ζ → 0, A(ζ ) → 1, B(ζ ) → 1, and α → +∞. Let 𝒞∞ be the total set of radial solutions, and 𝒞 ∗ be the branch composed of nonminimal solutions (λ, vλ ) parametrized by σ in Lemma 6.6. Given (λ, v) ∈ 𝒞∞ , we define σ, μ, and ν by (6.13) and (6.17), with v replaced by vλ . Theorem 6.9. The solution (λ, vλ ) ∈ 𝒞 ∗ is parametrized by σ and satisfies (6.17) and (6.18). In particular, μ and ν are unbounded in 𝒞 ∗ , while the other branch 𝒞∞ \ 𝒞 ∗ is uniformly bounded. Proof. It suffices to prove the last part. The values μ and ν are uniformly bounded for solutions (λ, v) with λ ∈ [ϵ0 , λ∗ ], where ϵ0 > 0. Meanwhile, the uniform boundedness of minimal solutions is obvious. Since 𝒞∞ \ 𝒞 ∗ does not contain any other solutions, the result follows.

6.1.3 Generation of nonradial solutions Here we take the variational approach to guarantee the existence of nonradial solutions. If Ω ⊂ R2 is a bounded domain with a smooth boundary, the argument in Chapter 1 works for the semilinear eigenvalue problem − Δv = λf (v)

in Ω,

v|𝜕Ω = 0,

(6.21)

where λ ∈ R and f = f (v) is a continuous function of u ∈ R satisfying3 0 ≤ f (v) ≤ C exp(|v|q ),

0 < q < 2.

Let Ω = A and Tm be the rotation by 2π/m, Tm (r cos θ, r sin θ) = (r cos(θ + 2π/m), r sin(θ + 2π/m)), and define the closed subspaces V∞ and Vm of H01 (A) by V∞ = {v ∈ H01 (A) | v = v(|x|)},

Vm = {v ∈ H01 (A) | v(Tm x) = v(x) a. e. x ∈ A}.

3 This growth order comes form the Trudinger–Moser inequality.

364 | 6 Equations on annulus Let V = Vm for m = 1, 2, . . . , ∞. Each (v, λ) ∈ V × R admits a solution w to −Δw = λf (v)

in A,

w|𝜕A = 0,

which belongs also to V. Hence the symmetric criticality in § 2.4 is assured by Lemma 2.51. u Let F(u) = ∫0 f (u) du and put Φ(v) ≡ ∫ F(v) dx,

J(v) ≡

A

1 ∫ |∇v|2 dx, 2 A

v ∈ H01 (A).

We define Kc = {v ∈ V | Φ(v) = c and jc = inf{J(v) | Kc }} for c ∈ R. If Kc ≠ 0, a minimizer vc ∈ Kc of jc = J(vc ) exists, which satisfies −Δvc = λc f (vc )

in A,

vc |𝜕A = 0

with a Lagrange multiplier λc ∈ R. Hence the method of § 2.4.3 is applicable. For each m = 1, 2, . . . , ∞, the value jm [μ] = inf{J(v) | v ∈ Km,μ },

Km,μ = {v ∈ Vm | Φ(v) = μ}

is attained by a solution v = vm,μ ∈ Km,μ to (6.21) for Ω = A with some λ = λm,μ ∈ R. If f (v) = ev in (6.21), it holds that F(v) = ev − 1. Given m = 1, 2, . . . , ∞ and μ > 0, we obtain v = vm,μ ∈ Km,μ and λ = λm,μ ∈ R satisfying (6.4). In case λ ≤ 0, one has u ≤ 0 in A by the maximum principle, which contradicts Φ(u) = μ > 0. Therefore, a solution (λ, u) of (3.36) arises with the property that v ∈ Km,μ . Hence it holds that λ = λm,μ > 0. The following lemma implies the generation of nonradial solutions to (6.4).4 Lemma 6.10. Each m = 1, 2, . . . admits μm > 0 such that j∞ [μ] > jm [μ],

μ ≥ μm .

(6.22)

Proof. The uniqueness of the solution (λ, u∞ ) to (6.4) is assured in § 6.1.1, provided that u∞ ∈ K∞,μ and μ ≫ 1. Hence j∞ [μ] is attained by v = u∞ for μ ≫ 1. Moreover, (λ, u∞ ) is parametrized by σ in 0 < 8π − σ ≪ 1, satisfying μ = μ(σ) =

8π 2 (a2 + 1) log a {1 + o(1)} (8π − σ) log(8π − σ) 2

1 −4 1 j∞ [μ] = ν(σ) = (log ) {1 + o(1)}, 2 log a 8π − σ

σ ↑ 8π.

(6.23)

Hence there is a one-to-one correspondence between μ ≫ 1 and 0 < 8π − σ ≪ 1. 4 The proof motivates the quantized blow-up mechanism, Theorem 3.5, and the scaling argument to control the solution.

6.1 2D case

| 365

Now we use radially symmetric solutions in Ω = B to control the value jm [μ] for finite m. Let Am = {x = (r cos θ, r sin θ) | a < r < 1, |θ| < π/m} and take a disk ωm in Am with center x0 and radius ϵ. We define the functions U ∈ H01 (Am ) and wΣ ∈ Vm by U(x) = {

uΣ ((x − x0 )/ϵ), 0,

x ∈ ωm , x ∈ A\ωm ,

m−1 and wΣ (x) = U(x) + U(Tm x) + ⋅ ⋅ ⋅ + U(Tm x), where uΣ , 0 < Σ < 8π, denotes the solution to (3.19) for Ω = B given in Theorem 3.2. It holds that

Φ(wΣ ) = mϵ2 m(Σ) + |A| − mϵ2 π,

1 J(wΣ ) = mD(Σ) 2

with m(Σ) in (3.46) and D(Σ) in (3.47). Similarly to the case of σ, the value μ ≫ 1 determines Σ in 0 < 8π − Σ ≪ 1 by Φ(wΣ ) = μ. Accordingly, σ and Σ are related as mϵ2 ⋅

8π 2 (a2 + 1) log a 8π 2 + |A| − mϵ2 π = {1 + o(1)}. 8π − Σ (8π − σ) log(8π − σ)

Since wΣ ∈ Km,μ , it holds that 1 1 ){1 + o(1)} jm [μ] ≤ mD(Σ) = 8πm(log 2 8π − Σ 1 = 8πm(log ){1 + o(1)}, σ ↑ 8π. 8π − σ

(6.24)

Then we obtain j∞ [μ] > jm [μ] for μ ≫ 1 by (6.23) and (6.24). Theorem 6.11 ([236]). Any positive integer m admits μm such that for any μ > μm , problem (3.36) has a nonradial solution (λ, u) satisfying Φ(u) = μ such that the largest integer ℓ satisfying v ∈ Vℓ is m. Proof. The result follows from the argument in § 2.4.3, once inequality (6.24) is established. 6.1.4 Nonradial bifurcation The family of m-mode solutions in Theorem 6.11 bifurcates from the branch of nonminimal radial solutions in the λ–u plane. This fact is confirmed using associated Legendre equation because O(1)-symmetry of the equation reduces the problem to the bifurcation from simple eigenvalues. Let 𝒞 ∗ = {(λ, uλ )} be the branch of nonminimal radially symmetric solutions to (6.4). First, we examine the degeneracy of the linearized operator −Δ − p(r) under the Dirichlet condition, where p(r) = λeuλ =

8α2 a−1 j , + a−1 j(√a/r)α ]2

r 2 [(r/√a)α

j=

1 − aζ . 1 − a−1 ζ

366 | 6 Equations on annulus By the separation of variables, the linearized equation −Δϕ =

p(r) ϕ in A, Λ

ϕ|𝜕A = 0

is reduced to 1 (rψr )r + [p(r)/λ − k 2 /r 2 ]ψ = 0, r

a < r < 1,

ψ|r=a,1 = 0

for k = 0, 1, 2, . . . Letting t = (r/√a)α and q(t) = 8a−1 j/(t 2 + a−1 j)2 , we obtain 2

1 k (tψt )t + [q(t)/Λ − ( ) /r 2 ]ψ = 0, t α

ζ −1/2 < t < ζ 1/2 ,

ψ|t=ζ ±1/2 = 0.

Through the transformation ξ = (a−1 j − t 2 )/(a−1 j + t 2 ) used in § 3.2.4, there arises the associated Legendre equation 2

[(1 − ξ 2 ) ψξ ]ξ + [ where μ = k/a and ξ± =

2 − μ2 /(1 − ξ 2 )]ψ = 0, Λ

a−1 j−ζ ± . a−1 j+ζ ±

ξ− < ξ < ξ+ ,

ψ|ξ =ξ ± = 0,

Here we note a classical fact.

Proposition 6.12. The fundamental solutions to [(1 − ξ 2 )ψξ ]ξ + [2 − μ2 /(1 − ξ 2 )]ψ = 0,

−1 < ξ < 1

are given as follows: 1. When μ ≠ 0, ±1, μ

μ

ϕ1 (ξ ) = (1 − ξ ) 2 (1 + ξ )− 2 (ξ + μ), ξ 2

μ

1+ξ

2.

When μ = 0, ϕ1 (ξ ) = ξ , ϕ2 (ξ ) = −1 +

3.

When μ = ±1, ϕ1 (ξ ) = (1 − ξ 2 ) 2 , ϕ2 (ξ ) = {log 1−ξ +

tion

1

μ

ϕ2 (ξ ) = (1 − ξ )− 2 (1 + ξ ) 2 (ξ − μ).

log 1−ξ .

1+ξ

2ξ }(1 1−ξ 2

1

− ξ 2) 2 .

Then, the degeneracy of the linearized operator is reduced to the algebraic equaϕ2 ϕ (ξ+ ) = 2 (ξ− ), ϕ1 ϕ1

which implies the symmetry breaking in any mode from the branch 𝒞 ∗ of non-minimal radial solutions [206, 328]. We can also examine the transversality condition. Hence each bifurcated set locally forms a cone, a two-dimensional manifold in the λ–u plane [206]. We apply the following theorem, called bifurcation from simple eigenvalues, for this purpose, using O(1)-symmetry of A [357].

6.2 Higher-dimensional case | 367

Theorem 6.13. Let X, Y be Banach spaces, V a neighborhood of 0 in X, and F : (−1, 1) × V 󳨀→ Y satisfy the following conditions: 1. F(t, 0) = 0 for |t| < 1. 2. The Frechét derivatives Ft , Fx , and Ftx exist and are continuous. 3. 𝒩 (Fx (0, 0)) and Y \ ℛ(Fx (0, 0)) are one-dimensional. 4. Ftx (0, 0)x0 ∈ ̸ ℛ(Fx (0, 0)), where 𝒩 (Fx (0, 0)) = ⟨x0 ⟩. Then, if X = 𝒩 (Fx (0, 0)) ⊕ Z, there are a neighborhood U of (0, 0) in R × X, an interval (−a, a), and continuous ϕ : (−a, a) → R and ψ : (−a, a) → Z such that ϕ(0) = 0, ψ(0) = 0, and F −1 (0) ∩ U = {(ϕ(α), αx0 + αψ(α)) | |α| < a} ∩ {(t, 0) ∈ U}. If Fxx is also continuous, then the functions ψ and ϕ are once continuously differentiable.

6.2 Higher-dimensional case 6.2.1 Radial solutions in higher dimensions Here we take the ODE approach for the study of radially symmetric solutions on an annulus. The fundamental assumptions on the nonlinearity are positivity and superlinearity, f (u) > 0,

u ≥ 0,

lim f (u)/u = +∞,

u↑+∞

(6.25)

which ensures c0 = infu≥0 f (u) > 0. Equation (6.3) is equivalent to urr +

n−1 u + λf (u) = 0, r r

u > 0,

a1 < r < a0 ,

u|r=a1 ,a0 = 0,

(6.26)

where the Liouville transformation is applicable to eliminate the first order term. This section is devoted to the case n > 2. Then this transformation takes the form s = r 2−n , and it holds that uss + λρ(s)f (u) = 0,

u(s) > 0,

s0 < s < s1 ,

u|s=s0 ,s1 = 0,

(6.27)

2

where ρ(s) = (n − 2)−2 s−(2+ n−2 ) and sk = a2−n for k = 0, 1. The function u = u(s) is thus k concave, and hence attains a maximum at the unique point s2 ∈ (s0 , s1 ): max[s0 ,s1 ] u = u(s2 ). Theorem 6.14. It holds that s0 < s0̂ ≤ s2 ≤ s1̂ < s1 with {s0̂ , s1̂ } determined by a0 , a1 , n. < 21 (a1 + a0 ) for a2 , Proof. Theorems 2.36 and 2.35 imply s2 < 21 (s0 + s1 ) and a2 = s2−n 2 respectively, and then we obtain the result.

368 | 6 Equations on annulus Theorem 6.15 ([207]). Let {(λ, u)} be a family of solutions to (6.27) such that λ → λ0 ∈ [0, +∞], ‖u‖∞ → +∞, and let I ⊂⊂ (s0 , s1 ) be an arbitrary compact interval. Then it holds that infI u → +∞, |us (s)| → +∞ for s = s0 , s1 , and λ0 = 0. Proof. First, concavity of u(s) implies v(s) ≤ u(s) for s0 ≤ s ≤ s1 , where v = v(s), s0 ≤ s ≤ s1 , is a graph composed of two segments, connecting (s0 , 0), (s2 , u(s2 )), and (s1 , 0). Given I ⊂⊂ (s0 , s1 ), second, it holds that mI = infs∈I ρ(s) f (u(s)) > 0, and then, u(s)

Strum’s comparison theorem implies λmI ≤ π 2 /|I|2 , with the right-hand side being the first eigenvalue on I. Finally, one gets mI → +∞ from the assumption.

We extend the solution u(r) to (6.26) outside the interval a1 < r < a0 and specify the behavior as r ↑ +∞ and r ↓ 0, fixing λ > 0. For this purpose we extend f = f (u) to u ≤ 0 as f (u) > 0,

u ≤ 0,

c1 = sup f (u) < +∞, u≤0

(6.28)

to consider (6.29) in the following theorem. In its second assertion, the first two cases correspond to singular and classical solutions on the ball, respectively, as u > 0 holds in (0, r0 ). In the third case, u(r) possesses a unique zero point r1 in (0, r0 ) and hence corresponds to a solution on an annulus. Theorem 6.16. A solution u(r) to urr +

n−1 u + λf (u) = 0, r r

(6.29)

satisfying u(r∗ ) > 0 for some r∗ > 0, has the following properties: 1. This u = u(r) exists in r > r∗ , possessing a zero point r0 > r∗ , and it holds that u(r) < 0 for r > r0 . 2. This u = u(r) exists also in 0 < r < r∗ , satisfying one of the following properties: (a) limr↓0 u(r) = +∞. (b) limr↓0 u(r) > 0 exists and limr↓0 ur (r) = 0. (c) There exist L > 0 and M ∈ R such that lim{ur (r) − r↓0

L } = 0, r n−1

lim{u(r) + r↓0

L } = M. (n − 2)r n−2

(6.30)

Proof. Since u(s) is concave, u(r) does not attain a local minimum. Therefore, once it has a zero point it must be negative afterwards. Then u(s) can be continued up to r = 0 or r = +∞ by (6.28). To prove the first assertion, we derive a contradiction if u(r) > 0 occurs in the ̂ In fact, if this is the case, maximal existence interval as r increases, denoted by (r∗ , r). it holds that (r n−1 ur )r ≤ −λc0 r n−1 ,

r∗ < r < r.̂

6.2 Higher-dimensional case | 369

r

Then, applying ∫r ⋅dr, we obtain ∗

ur (r) ≤ −

λc0 λC r + n−12 , n r

r∗ < r < r ̂

with a constant C2 . This inequality implies r ̂ = +∞ due to u > 0, and hence limr↑+∞ ur = −∞, which, however, is impossible again since u(r) > 0 for r∗ < r < +∞. To prove the second assertion, let the maximal existence interval as r decreases be (r,̃ r∗ ). First, we assume that u(r) > 0 in (r,̃ r∗ ). The function r n−1 ur is decreasing in r similarly, and hence ur (r) ≥ −C3 /r n−1 ,

r < r < r∗

(6.31)

with C3 > 0. Furthermore, if ur (r3 ) ≥ 0 occurs for r3 ∈ (r, r∗ ), then it holds that ur (r) > 0,

r ̃ < r < r3 .

(6.32)

Now we exclude this possibility. In fact, inequality (6.32) and u > 0 imply r ̃ = 0 and then r n−1 ur (r) ≥ r4n−1 ur (r4 ) ≡ C4 > 0,

0 < r < r4

holds for r4 < r3 since (r n−1 ur )r < 0. We thus obtain u(r4 ) ≥ [−

r=r

4 C4 −n+2 r ] , n−2 r=r

0 < r < r4 ,

a contradiction as r ↓ 0. Now we have ur < 0 in (r,̃ r∗ ), which implies r ̃ = 0 and the existence of u(+0) = limr↓0 u(r) ∈ (0, +∞]. The case of u(r) > 0 in r ̃ < r < r∗ , therefore, corresponds to the first two alternatives in the theorem. When u(r1 ) = 0 for some r1 ∈ (r,̃ r), on the other hand, we have r ̃ = 0 and u < 0 in (0, r1 ) as confirmed. This property implies (r n−1 ur )r ≥ −λc1 r n−1 ,

0 < r < r1 .

(6.33)

Since u(s) is concave, on the other hand, we obtain ur > 0 in (0, r1 ). It holds also that r n−1 ur (r) > C5 ≡ r1n−1 ur (r1 ),

0 < r < r1 ,

since (r n−1 ur )r < 0, and therefore, L = limr↓0 r n−1 ur (r) > 0 exists. Letting ρ ↓ 0 in r ∫ρ (r n−1 ur )r dr < 0, we obtain ur (r) −

L ≤ 0, r n−1

0 < r < r1 .

(6.34)

370 | 6 Equations on annulus Inequality (6.33), on the other hand, implies λC1 L r ≤ ur (r) − n−1 , n r

−

0 < r < r1 ,

(6.35)

L and then it follows that (6.30) is true. Finally, the function u(r) + (n−2)r n−2 is decreasing by (6.34). It is bounded below by r1

λC L + ∫ 1 r dr n−2 n (n − 2)r1 r

from (6.35). Hence the second convergence in (6.30) follows. 6.2.2 Emden transformation Coming back to the exponential nonlinearity, we seek the radial solution − Δu = λeu ,

u = u(|x|) in A ≡ {a < |x| < 1} ⊂ Rn ,

u|𝜕A = 0.

(6.36)

Relations (6.25) and (6.28) are satisfied, so that the solution u(r) for 0 < r = |x| < 1 is continued up to r ↓ 0 with the property lim{ur (r) − r↓0

L } = 0, r n−1

lim{u(r) + r↓0

L }≡M (n − 2)r n−2

(6.37)

for L > 0. This asymptotics can be used for ur (0) = 0 and u(0) = α to modify the transformation in § 2.2.1. We introduce the function w(t) by u(r) = w(t) − 2t + M, Then, urr +

n−1 ur r

r = Bet ,

1/2

B = {2(n − 2)/λeM } .

(6.38)

+ λeu = 0 with (6.37) is reduced to

ẇ + (n − 2)ẇ + 2(n − 2)(ew − 1) = 0, ̇ − 2 − α(n − 2)e−(n−2)t } = 0 lim {w(t) − 2t + αe−(n−2)t } = lim e−t {w(t)

t↓−∞

t↓−∞

(6.39)

for α = LB−(n−2) /(n − 2) > 0. Then α = 0 corresponds to the case when the domain is the unit ball. To establish u(a) = u(1) = 0 from (6.39), we take several preparatory steps. Proposition 6.17. Each α ≥ 0 admits a unique solution w(t) = w(α, t) to (6.39) in (−∞, −T1 ) for T1 sufficiently large. Proof. Letting v(t) = w(t) − 2t + αe−(n−2)t , we reach {e(n−2)t v󸀠 } + 2(n − 2)ent+v(t)−αE(t) = 0, 󸀠

lim v(t) = lim e−t v󸀠 (t) = 0,

t↓−∞

t↓−∞

E(t) ≡ e−(n−2)t ,

(6.40)

6.2 Higher-dimensional case

| 371

which are reduced to η

t

v(t) = −2(n − 2) ∫ e

−(n−2)η

−∞

dη ∫ enξ −αE(ξ )+v(ξ ) dξ .

(6.41)

−∞

We take the Banach space X ≡ {v ∈ C(−∞, −T1 ) | limt↓−∞ v(t) = 0} provided with the maximum norm where the positive constant T1 is chosen later. Letting the righthand side of (6.41) be Φα (v), we show the existence of a fixed point of Φα in X by the contraction mapping principle. In fact, putting 𝒦 ≡ {v ∈ X | ‖v‖ ≤ 1}, we obtain 󵄩󵄩 󵄩 󵄩󵄩Φα (v1 ) − Φα (v2 )󵄩󵄩󵄩 t

≤ 2(n − 2) ∫ e −∞ T1

−(n−2)η

η

󵄨 󵄨 dη ∫ enξ −αE(ξ ) 󵄨󵄨󵄨ev1 (ξ ) − ev2 (ξ )󵄨󵄨󵄨 dξ −∞ η

≤ 2(n − 2) ∫ e−(n−2)η dη ∫ enξ ⋅ e‖v1 − v2 ‖ dξ −∞

−∞

n−2 e ⋅ e−2T1 ‖v1 − v2 ‖, = n

vi ∈ 𝒦,

i = 1, 2,

and ‖Φα (v1 )‖ ≤ n−2 e ⋅ e−2T1 for v ∈ 𝒦. Hence Φα has a unique fixed point vα = vα (t) ∈ X, n provided that T1 ≫ 1, which implies the result. This w = w(α, t) extends to −∞ < t < +∞ by the argument below. Given such ̇ t)) | −∞ < t < +∞} with w(α, t) = v(α, t) + 2t − αe−(n−2)t , we put 𝒪a = {(w(α, t), w(α, ̇ t) = 𝜕t𝜕 wα (α, t). This orbit is generated by the autonomous system w(α, w ẇ d ( )=( ) w −2(n − 2)(e − 1) − (n − 2)ẇ dt ẇ

(6.42)

with the condition as t ↓ −∞ in (6.39). Its stationary state is given by putting (0, 0) in the right-hand side, and hence the origin O(0, 0). If 2 < n < 10, this O(0, 0) is a spiral attractor at t = +∞, as is described in § 2.2.1. Equation (6.39) implies dẇ n−2 =− {ẇ − g(w)}, dw ẇ

g(w) = −2(ew − 1),

(6.43)

which determines the direction of the orbit 𝒪a in 𝒟1 = {(w, w)̇ | ẇ > 0, ẇ > g(w)}, ̇ | ẇ < 0, ẇ > g(w)}, 𝒟3 = {(w, w)̇ | ẇ < 0, ẇ < g(w)}, and 𝒟4 = {(w, w)̇ | 𝒟2 = {(w, w) ẇ > 0, ẇ < g(w)}. Proposition 6.18. The orbit 𝒪α with α ≥ 0 moves right and down in 𝒟1 , left and down in 𝒟2 , left and up in 𝒟3 , and right and up in 𝒟4 . Without escaping to the infinity, it proceeds

372 | 6 Equations on annulus from 𝒟i to 𝒟i+1 , crossing either the w-axis vertically or the curve ẇ = g(w) horizontally, where i = 1, 2, 3, 4 with 𝒟5 = 𝒟1 . It eventually approaches O(0, 0) spirally and nonspirally if 2 < n < 10 and n ≥ 10, respectively. Proof. The former assertion is obvious. We show that the orbit 𝒪σ never escapes to the ̇ remains infinity in any Di , 1 ≤ i ≤ 4. First, if it escapes to the infinity in 𝒟1 , then w(t) ẇ | → ∞ by (6.43), which is impossible from the movement of 𝒪 bounded. Hence | ddw in 𝒟1 . This property implies the vertical entrance of 𝒪 into 𝒟2 . The other cases are ̇ t))} continues up to t = +∞ similar. Then we see that the orbit 𝒪α = {(w(α, t), w(α, and approaches to O(0, 0) because the system (6.42) has no other singular point than O(0, 0). Then the last results follows from the linearization around O(0, 0) described in § 2.2.1. Propositions 6.17–6.18 imply the following property. Proposition 6.19. There exists a unique orbit of (6.42) asymptotic to the line ẇ = 2 as w → −∞, that is, 𝒪0 . We proceed to several facts. Proposition 6.20. The family of orbits {𝒪α }α≥0 has the property that 𝒪α ∩ 𝒪β = 0 if α ≠ β. Each 𝒪α with α > 0 crosses the line ẇ = 2 just once. Proof. The fact 𝒪α ∩ 𝒪0 = 0 for α > 0 is obvious because wα̇ → +∞ as t ↓ −∞ by (6.39). To prove 𝒪a ∩ 𝒪β = ϕ for α, β > 0, α ≠ β, we show that the asymptotic relation ̇ t) as t ↓ −∞ is different according to α > 0. between w(α, t) and w(α, ̇ t) for simplicity. By (6.40), To this end, we fix α > 0 and put w = w(α, t), ẇ = w(α, ̈ = 0, which implies we have limt↓−∞ v(t) ̈ = −α(n − 2)2 e−(n−2)t + v(t) ̈ < 0, w(t)

t ↓ −∞.

̇ Hence we can regard t = t(w)̇ as a function of ẇ given by w(t) = 2 + α(n − 2)e−(n−2)t + ̇ such that limt↓−∞ ẇ = +∞, due to the implicit function theorem. Accordingly, we v(t) obtain w=

2 2 ẇ − 2 ẇ − 2 log α − log − + o(1), n−2 n−2 n−2 n−2

ẇ ↑ +∞.

Thus the first part has been proved. The second assertion follows from the fact that 𝒪α with α > 0 moves as in Proposition 6.18 and never meets 𝒪0 . Lemma 6.21. For each 0 < ϵ ≪ 1 and T ≫ 1, the solution w(α, ⋅) in Proposition 6.17 is continuous in C 1 [−T, T] with respect to α ∈ [ϵ, +∞). Proof. Given α, β ∈ [ϵ, +∞), let vα , vβ be as in the proof of Proposition 6.17. Below the constants Ci (ϵ), 1 ≤ i ≤ 5, are independent of T.

6.2 Higher-dimensional case

| 373

The mean value theorem implies 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨vα (t) − vβ (t)󵄨󵄨󵄨 = 󵄨󵄨󵄨Φα (vα )(t) − Φβ (vβ )(t)󵄨󵄨󵄨 t

≤ 2(n − 2) ∫ e

−(n−2)η

−∞

η

󵄨 󵄨 dη ∫ enξ 󵄨󵄨󵄨e−αE(ξ )+vα (ξ ) − e−βE(ξ )+vβ (ξ ) 󵄨󵄨󵄨 dξ

t

−∞ η

−∞

−∞

≤ 2(n − 2) ∫ e−(n−2)η dη ∫ enξ −ϵE(ξ )+1 {|α − β|E(ξ ) 󵄨 󵄨 + 󵄨󵄨󵄨vα (ξ ) − vβ (ξ )󵄨󵄨󵄨} dξ ≤ C1 (ϵ)|α − β| + C2 (ϵ)e−2T1 ‖vα − vβ ‖ for t ∈ (−∞, −T1 ], and hence ‖vα − vβ ‖ ≤ C3 (ϵ)|α − β|

(6.44)

by taking T1 ≫ 1 in Proposition 6.17. In the same way, we have ‖v̇α − v̇β ‖ ≤ C4 (ϵ)|α − β|.

(6.45)

From (6.40), (6.44) and (6.45), it follows that 󵄨󵄨 󵄨 󵄨 ̇ ̇ −T1 )󵄨󵄨󵄨󵄨 ≤ C5 (ϵ)|α − β|. −T1 ) − w(β, 󵄨󵄨wα (−T1 ) − wβ (−T1 )󵄨󵄨󵄨 + 󵄨󵄨󵄨w(α, As a consequence, the assertion holds because the solution w = w(α, t), t ≥ −T1 , to (6.39) depends continuously on its value at t = −T1 . Proposition 6.22. w(α, t) is real analytic with respect to α > 0 and t ∈ R. Proof. We can extend the value α > 0 to the complex variable in the right-half plane. Then the proof of Lemma 6.21 is valid and the conclusion follows. If α > β > 0, the orbit 𝒪α lies in the left-hand side of 𝒪β as t increases. These orbits {𝒪α }α≥0 cover the phase plane R2 except for the origin O(0, 0). Proposition 6.23. Each P(η0 , ζ0 ) ∈ R2 \ {(0, 0)} admits a unique α ≥ 0 such that P ∈ 𝒪α . Proof. If P exists on 𝒪0 , the assertion is obvious. Otherwise, the orbit 𝒪 of (6.42) enters necessarily into ẇ > 2 as t ↓ −∞ by Proposition 6.18, and hence we may assume that ζ0 > 2. We also suppose that 𝒪 = {(w(t), z(t))} passes through P at t = t0 ≡ ζ0 /2, regarding that system (6.42) is autonomous, which makes later description simpler. Using U(r) = w(t) − 2t, r = B0 et , and B0 = {2(n − 2)/λ}1/2 , we obtain a solution U(r) to (6.36) satisfying the assumption in Theorem 6.16 for r0 = Bet0 . Hence there exist L, M > 0 such that lim{U 󸀠 (r) − r↓0

L } = 0, r n−1

lim{U(r) + r↓0

L } = M. (n − 2)r n−2

374 | 6 Equations on annulus We define W = W(t) by U(r) = W(t)+M−2t, r = Bet , and B = {2(n−2)/λeM }, to obtain the

orbit 𝒪α = {(W(t), Z(t))} for α = LB /(n − 2). Then it follows that W(t0 + 21 M) = η0 , Z(t0 + 21 M) = ζ0 , and therefore, W(t + 21 M) = w(t), Z(t + 21 M) = z(t). Thus we obtain 𝒪 = 𝒪α . −(n−2)

The above considerations are summarized as follows. Theorem 6.24. Each α ≥ 0 admits a global-in-time solution w = w(α, t) to (6.39). The ̇ t)) | −∞ < t < +∞} is absorbed into the origin O(0, 0) as orbit 𝒪α = {(w(α, t), w(α, t ↑ +∞. This family of orbits {𝒪α }α≥0 forms a foliation, 𝒪α ∩ Oβ = 0,

α ≠ β;

⋃ 𝒪α = R2 \{0}.

α≥0

The orbit 𝒪0 is asymptotic to ẇ = 2 as w → −∞, while other orbits 𝒪α , α > 0 cross with ẇ = 2 just once. The above analysis also brings us a characterization of singular radial solution to − Δu = λeu

in Rn \{0},

lim u(r) = +∞.

(6.46)

r↓0

Theorem 6.25 ([237, 224]). Each λ > 0 admits a unique radial solution u∗ = u∗ (|x|) to (6.46). Then it holds that u∗ (|x|) = 2 log

λ 1 − log . |x| 2(n − 2)

(6.47)

Proof. Given such u(|x|), we define a solution w∗ = w∗ (t) to (6.39) by u(r) = w∗ (t) − 2t, r = |x| = B0 et , and B0 = {2(n − 2)/λ}1/2 . Let 𝒪∗ = {(w∗ (t), ẇ ∗ (t))}. By the proof of Theorem 6.16, it holds that u󸀠 (r) < 0 for 0 < r ≪ 1, which means ẇ ∗ (t) < 2, or 𝒪∗ remains below the line ẇ = 2 as t ↓ −∞. As a consequence, 𝒪∗ is equal to either 𝒪0 or {O(0, 0)}. In the former case, however, one has limt↓−∞ e−t {ẇ ∗ (t) − 2} = 0 and hence limr↓0 u󸀠 (r) = 0. This property contradicts the blow-up profile of u(r) at r = +0, and therefore, 𝒪∗ consists of the single point O(0, 0). Therefore, (6.47) is derived from u(r) = −2t and r = B0 et = {2(n − 2)/λ}1/2 et . 6.2.3 Boundary pair Given A = {a < |x| < 1} ⊂ Rn , a radial solution (λ, u(|x|)) to (6.36) was realized in the foliation {𝒪α }α , prescribing boundary condition. This condition on 𝒪α for α > 0 is represented by the pair of time variables {t ± } satisfying w(t − ) − 2t − + M = w(t + ) − 2t + + M = 0,

t − = log a/B,

t + = log 1/B,

6.2 Higher-dimensional case

| 375

where abbreviation w(t) = w(α, t) is adopted. Writing w± = w(t ± ), we obtain w+ − w− = −2 log a,

t + − t − = − log a.

(6.48)

For any pair {P ± (w± , ẇ ± )} of points in 𝒪α satisfying (6.48), conversely, a radial solution (λ, u(|x|)) to (6.36) is recovered by u(r) = w(t) − 2t + M and r = Bet , using −

+

B = ae−t = e−t ,

M = 2t − − w− = 2t + − w+ ,

L = α(n − 2)an−2 e−(n−2)t = α(n − 2)e−(n−2)t , +

−

λ = 2(n − 2)a−2 ew− = 2(n − 2)ew . +

We call {P ± = Pα± } the boundary pair. Then 𝒦α = {Pα+ | α > 0} characterizes the set of radial solutions denoted by 𝒞a = {(λ, u(|x|)) | solving (6.36)}.

̇ t + (α)) for Pα+ = (w+ , ẇ + ), where By the definition, we have w+ = w(α, t + (α)), ẇ + = w(α, t + (α) is the root of F(α, t) ≡ w(α, t) − w(α, t + log a) + 2 log a = 0,

(6.49)

and furthermore, each Pα+ (w+ , ẇ + ) ∈ 𝒦α produces (λ, u(|x|)) ∈ 𝒞a for λ = 2(n − 2)ew

+

(6.50)

Lemma 6.26. Given a ∈ (0, 1), there is a unique boundary pair Pα± on each 𝒪α , continuous in α > 0. Proof. We fix α > 0, to write F(t) = F(α, t) and w(t) = w(α, t), in short. We have ̇ limt↑+∞ F(t) = 2 log a < 0 and limt↓−∞ F(t) = +∞, due to limt↑0 (w(t), w(t)) = (0, 0) + + and (6.39), respectively. Hence there is t ∈ R such that F(t ) = 0, that is, w(t + ) − w(t + + log a) = −2 log a.

(6.51)

Put t − = t + +log a. By the mean value theorem, (6.51) implies the existence of τ ∈ (t − , t + ) ̇ ̇ − )) and P + (w(t + ), w(t ̇ + )) lie such that w(τ) = dw (τ) = 2. Consequently, P − (w(t − ), w(t dt above and below the line ẇ = 2, respectively. To show the uniqueness of such a pair {P ± }, assume the existence of t1+ < t2+ such that F(t1+ ) = F(t2+ ) = 0, and put Pi± (wα (ti± ), ẇ α (ti± )) for ti− = ti+ + log a, i = 1, 2. Then Pi+ and Pi− are also separated by the line ẇ = 2. We have also t ∗ ∈ (t1+ , t2+ ) such that F 󸀠 (t ∗ ) = 0, or equivalently, ̇ ∗ ) = w(t ̇ ∗ + log a). w(t

(6.52)

̇ ∗ + log a) > 2 > w(t ̇ ∗ ), a Since t ∗ + log a ∈ (t1− , t2− ) and t ∗ ∈ (t1+ , t2+ ), one gets w(t ± contradiction. Once we obtain the uniqueness of Pα , the final assertion follows from Lemma 6.21.

376 | 6 Equations on annulus The points Pα+ and Pα− lie below and above the ẇ = 2, respectively, and henceforth we call Pa+ and Pa− the outer and inner boundary points, respectively. Lemma 6.26 assures the following fact. Proposition 6.27. The set 𝒦a forms a continuous curve in R2 , homeomorphic to R. Using (6.38), we arrive at ̇ t + (α)) = 2 + ur (1), w(α,

̇ t + (α) + log a) = 2 + ur (a) ⋅ a. w(α,

(6.53)

This equality shows again that the boundary pair {Pa± } separates ẇ = 2 on 𝒪α . Each outer boundary point (w+ , ẇ + ) ∈ 𝒦a corresponds to a radially symmetric solution u = u(r) to (6.36) for λ defined by (6.50). Therefore, the decreasingness and increasingness of λ correspond to these movements of the w-coordinate of the outer boundary point Pα+ , and therefore, we have to study the shape of the continuous curve 𝒦a to draw a bifurcation diagram of radially symmetric solutions to (6.36). Proposition 6.28. The outer boundary point Pα+ eventually lies in the part of w < 0, ẇ < 0 and is absorbed into infinity. Proof. As is noticed in § 6.2.1, the component of radially symmetric solutions containing the branch of strictly minimal solutions is unbounded in R+ × C 0 (A). Then Theorem 6.15 implies λ ↓ 0 and wr (a) → +∞ along this branch, hence the conclusion by (6.50) and (6.53). The above profile of 𝒦a is realized as α ↑ +∞ by the following lemma. Lemma 6.29. Each a ∈ (0, 1) admits lim wα+ = lim wα− = −∞, α↓0

α↓0

lim ẇ α+ = lim ẇ α− = 2, α↓0

α↓0

(6.54)

where {Pα± (wα± , ẇ α± )} denotes the boundary pair on 𝒪α associated with A = {a < |x| < 1}. Proof. Concerning the point w(α, τ(α)) where 𝒪a crosses the line ẇ = 2, one has limα↓0 w(α, τ(α)) = −∞. Since w(a, t + (α) + log a) < w(α, τ(α)) by t − (α) = t + (α) + log a < τ(α), therefore, it holds that lim wα− = lim w(α, t + (α) + log a) = −∞, α↓0

α↓0

which implies limα↓0 wα+ = −∞ by (6.48). d Equality (6.39) implies dt {(ẇ − 2)e(n−2)t } = −2(n − 2)ew+(n−2)t , while wα− ≤ w(t) ≤ wα+ in t − (α) ≤ t ≤ t + (α). Hence it holds that −2(n − 2)ewα +(n−2)t ≤ +

− d ̇ t) − 2)e(n−2)t } ≤ −2(n − 2)ewα +(n−2)t {(w(α, dt

6.2 Higher-dimensional case

| 377

for t − (α) ≤ t ≤ t + (α). Integration over (τ(α), t) for t ∈ (τ(α), tα+ ) implies ̇ t) − 2 ≤ −2ewα {1 − e−(n−2)(t−τ(α)) }, −2ewα {1 − e−(n−2)(t−τ(a)) } ≤ w(α, −

+

̇ τ(α)) = 2. Similar inequality holds by the integration over (t, τ(α)) for t ∈ since w(α, − (t (α), t(τ)). Consequently, we obtain + 󵄨 ̇ 󵄨 max+ 󵄨󵄨󵄨w(α, t) − 2󵄨󵄨󵄨 ≤ 2ewa , t−(α)≤t≤t (α)

and then (6.54) follows from wα+ → −∞. Lemma 6.30. For each a ∈ (0, 1), the radially symmetric solution (λα , uα ) corresponding to Pα+ (wα+ , ẇ α+ ) satisfies limα↓0 ‖uα ‖L∞ (A) = 0 and limα↓0 λα = 0. Proof. The second relation follows from (6.50) and Lemma 6.29. We have, on the other hand, 󵄨 󵄨 󵄨 󵄨 ‖uα ‖∞ = max󵄨󵄨󵄨uα (r)󵄨󵄨󵄨 = max󵄨󵄨󵄨w(α, t) − 2t + (2t − (α) − wα− )󵄨󵄨󵄨, a≤r≤1 t∈I(α) for I(α) = [t − (α), t + (α)]. Then the mean value theorem implies 󵄨 ̇ 󵄨 󵄨 󵄨 ‖uα ‖∞ ≤ max󵄨󵄨󵄨w(α, t) − 2󵄨󵄨󵄨 ⋅ max󵄨󵄨󵄨(t − ta− )󵄨󵄨󵄨 t∈I(α)

t∈I(α)

󵄨 󵄨 󵄨 󵄨 ≤ max{󵄨󵄨󵄨ẇ α− − 2󵄨󵄨󵄨, 󵄨󵄨󵄨ẇ α+ − 2󵄨󵄨󵄨} ⋅ (− log a) → 0,

α ↓ 0,

by Lemma 6.29. We have proven the following fact concerning the total set 𝒞a ⊂ R+ ×C 0 (A) of radial solutions to −Δu = λeu in A = {a < |x| < 1} ⊂ Rn with u|𝜕A = 0 for n ≥ 3 and 0 < a < 1. Theorem 6.31 ([237]). The set 𝒞a forms a branch, a one-dimensional manifold homeomorphic to R, starting from (0, 0) and approaching λ = 0 with ‖u‖∞ → +∞. 6.2.4 Bending arbitrary many times This subsection is devoted to the following theorem. Theorem 6.32 ([237]). If 2 < n < 10, then there are ai ↓ 0 in 0 < ai < 1, i = 1, 2, . . . , such that the bending of 𝒞a with respect to the λ axis occurs i times, provided that 0 < a < ai . Thus lima↓0 b(a) = +∞, where b(a) denotes the number of bendings of 𝒞a . The following lemma says that every point on 𝒪α \{(w, w)|ẇ = 2} corresponds to or Pα− for a suitably chosen a ∈ (0, 1), where {Pα± } denotes the boundary pair for A = {a < |x| < 1}. Pα+

378 | 6 Equations on annulus ̇ t)) ∈ 𝒪α with w(α, ̇ t) < 2, there exists a Lemma 6.33. Given α > 0 and (w(α, t), w(α, unique a = a∗ (t) ∈ (0, 1) such that w(α, t) − w(α, t + log a∗ ) = −2 log a∗ , and therefore, ̇ t)) and P − (w(α, t+log a∗ ), w(α, ̇ t+log a∗ )) form outer and inner boundary P + (w(α, t), w(α, points on 𝒪α associated with {a∗ < |x| < 1}, respectively. The function t ∈ (τ, +∞) 󳨃→ a∗ (t) is monotone decreasing, where τ = τ(α) denotes the time when Oα crosses the line ẇ = 2: 𝒪α ∩ {(w, w)̇ | ẇ = 2} = {(w(α, τ), 2)}. ̇ ̇ < 2, we put Proof. We write w(t) for w(α, t). Given (w(t), w(t)) ∈ 𝒪α with w(t) G(a) ≡ w(t) − w(t + log a) + 2 log a,

a ∈ (0, 1).

̇ ∗ ) − 2} holds for some For 0 < 1 − a ≪ 1, one has t + log a > τ, while G(a) = (− log a){w(t ∗ ∗ ∗ ̇ ) < 2, and hence G(a) < 0 in this case. t ∈ (t + log a, t). Since t > τ, we obtain w(t We have, on the other hand, lima↓0 ta = −∞ for ta = t + log a, and hence lim G(a) = lim{w(t) − w(ta ) + 2 log a} a↓0

a↓0

= lim{w(t) − (w(ta ) − 2ta + αe−(n−2)ta ) − 2t + αae−(n−2)ta } = +∞ a↓0

by (6.39). Thus there is a∗ = a∗ (t) such that G(a∗ ) = 0, which means that ̇ ̇ + log a∗ ))} {P + (w(t), w(t)), P − (w(t + log a∗ ), w(t

(6.55)

forms a boundary pair to A∗ = {a∗ < |x| < 1}. To show the uniqueness of such a∗ , assume on the contrary that there exist of ∗ a1 < a∗2 such that G(a∗1 ) = G(a∗2 ) = 0. Then there is ã ∈ (a∗1 , a∗2 ) such that G(̇ a)̃ = 0, that is, ̇ + log a)̃ = 2. w(t

(6.56)

̃ w(t ̃ lies above ̇ + log a)) However, since t + log ã < t + log a∗2 < τ, the point (w(t + log a), the line ẇ = 2, which contradicts (6.56). ̇ ̇ + log a)} does not vanish at By the same reason, the function G(a) = a1 {2 − w(t ∗ ∗ a = a , and hence t ∈ (τ, +∞) 󳨃→ a (t) is differentiable in t by the implicit function theorem. Actually, it follows that ̇ − w(t ̇ + log a∗ )}a∗ /{w(t ̇ + log a∗ ) − 2}, ȧ ∗ (t) = {w(t) the right-hand side of which is negative because {P + , P − } in (6.55) forms a boundary ̇ < 2 < w(t ̇ + log a∗ ). pair on 𝒪α and w(t) The proof of the following lemma is similar to that of Lemma 6.33. ̇ t)) ∈ 𝒪α with w(α, ̇ t) > 2, there exists a Lemma 6.34. Given α > 0 and (w(α, t), w(α, unique a∗ = a∗ (t) ∈ (0, 1) such that w(α, t − log a∗ ) − w(α, t) = −2 log a∗ , and therefore, ̇ t−log a∗ )) and P − (w(α, t), w(α, ̇ t)) form outer and inner boundary P + (w(α, t−log a∗ ), w(α, points on 𝒪α associated with {a∗ < |x| < 1}, respectively. The function t ∈ (−∞, τ) 󳨃→ a∗ (t), furthermore, is monotone increasing.

6.3 Bifurcation from symmetry | 379

Lemma 6.35. Given α > 0, the values a∗ (t) and a∗ (t) defined by Lemmas 6.33 and 6.34, respectively, satisfy limt↓τ a∗ (t) = limt↑τ a∗ (t) = 1, where τ = τ(α) denotes the crossing time of 𝒪α with ẇ = 2. Proof. From the monotonicity of t ∈ (τ, +∞) 󳨃→ a∗ (t), there is a = limt↓τ a∗ (t). If a ≠ 1, it holds that a < 1, and then we take a0 ∈ (a, 1). By Lemma 6.26, there is t0 > τ ̇ t0 )) ∈ 𝒪α is the outer boundary point associated such that the point (w(α, t0 ), w(α, with {a0 < |x| < 1}. This property means a∗ (t0 ) = a0 > a, which contradicts the monotonicity of a∗ (t). The other part is proven similarly. Proof of Theorem 6.32. Since 2 < n < 10, the orbit 𝒪0 crosses the w-axis infinitely many times. Let XN (ξN , 0), N = 1, 2, . . . , be their successive points as t ↑ +∞. It holds that ξ2 < ⋅ ⋅ ⋅ < ξ2j < ⋅ ⋅ ⋅ < 0 < ⋅ ⋅ ⋅ < ξ2j−1 < ⋅ ⋅ ⋅ < ξ3 < ξ1 for j = 1, 2, . . . Let σα,N be the time when the orbit 𝒪α crosses the segment XN XN+2 on the w-axis. From Proposition 6.23 it follows that XN XN+2 = {(w(α, σα,N ), 0) | α > 0}. Let a∗α = a∗ be the value in Lemma 6.33, and put aN = supα>0 a∗α (σα,N ) ∈ (0, 1]. We show that this sequence {aN } has the desired property. In fact, each a ∈ (0, aN ) admits β > 0 in a < a∗β (σβ,N ), and t ∗ > σβ,N in a∗β (t ∗ ) = a by Lemma 6.33. Hence the curve 𝒦a defined in the previous subsection passes the point Q(wβ (t ∗ ), ẇ β (t ∗ )), while both ends of 𝒦a lie in w = −∞. By this property, the curve 𝒦a meets Xj Xj+2 , 1 ≤ j ≤ N, at least twice. Accordingly, 𝒞a bends at least (2N + 1) times by (6.50).

6.3 Bifurcation from symmetry 6.3.1 Morse indices We show the existence of nonradially symmetric solutions, particularly, their generation from radial solutions called symmetry breaking. Theorem 6.36 ([238, 269]). Infinitely many nonradial bifurcations occur to 𝒞a before it is absorbed into the hyperplane λ = 0. To employ a topological argument, we introduce an index, invariant if nothing happens topologically on 𝒞a , the total set of radially symmetric solutions {(λ, u(|x|))} to (6.36). Since this index varies arbitrary many times along 𝒞a , the breaking of symmetry infinitely many times is proven. Here we use the Morse index, the number of negative eigenvalues of the linearized operator. In the general setting, nonlinearity f ∈ C 1 (R, R) and parameter λ > 0 are given, and u = u(r) > 0, r = |x|, is a radially symmetric classical solution to − Δu = λf (u) in A,

u|𝜕A = 0.

(6.57)

The linearized operator is L = Lλ,u ≡ −Δ − λf 󸀠 (u(r)) with null Dirichlet boundary condition, and it accepts the separation of variables with respect to r = |x| and ω ∈ Sn−1 ,

380 | 6 Equations on annulus |ω| = 1. Thus this operator L is identified with the direct sum of the ordinary differential ρ 1 d n−1 d m ⋅) + rm2 − λf 󸀠 (u(r)) operators Lm = Lm λ,u for m = 0, 1, 2, . . . , defined by L = − r n−1 dr (r dr

with ⋅|r=a,1 = 0, realized as a self-adjoint operator in L2 ((a, 1), r n−1 dr). Here, {ρm }∞ m=0 stands for the set of eigenfunctions of the Laplace–Beltrami operator Λ on the unit sphere Sn−1 . If n > 2, it is given by ρm = m(m + n − 2), m = 0, 1, 2, . . . , with the multiplicity κm = (

m+n−2 ) ⋅ (n + 2m − 2) ⋅ (n + m − 2)−1 . m

The Morse index i = iu,λ of the solution (λ, u(r)) is defined by i = #{negative eigenvalues of L}, and iR = #{nonpositive eigenvalues of L0 } is called the augmented radial Morse index. The augmented Morse index i and the radial Morse index iR are defined similarly. The former indicates the number of negative eigenvalues of L, while the latter the number of nonpositive eigenvalues of L0 . The proof of Theorem 6.36 is carried over using the following principle. Let 𝒞 and 𝒞 ∞ be the total set of solutions and radially symmetric solutions to (6.57), respectively. Theorem 6.37 ([207]). Given {(λα , uα )}α∈A ⊂ 𝒞 ∞ and 𝒜 ⊂ R, let La = Lλα ,uα and Lm α = m Lm for m = 0, 1, 2, . . . The eigenvalues of L are denoted by μ (α) < μ (α) < ⋅ ⋅ ⋅. Let m,1 m,2 a λα ,uα m ≥ 1 and suppose the following conditions: 1. μm,2 (α) > 0 for α ∈ 𝒜. 2. μm,1 (α) = 0 on [α, α] ⊂ int 𝒜. It changes sign between α < α and α > α in 𝒜. Then, [α, α] is a bifurcation interval (or bifurcation point if α = α) so that there is a nonempty open connected set 𝒞m ⊂ 𝒞 \{(λα , uα )}α∈𝒜 such that {(λα , uα )}α≤α≤α ∩ 𝒞 m ≠ 0. Theorem 6.38. If f (u) = eu in (6.57), we have i → +∞ and iR = 1 after some point on 𝒞a , until it is absorbed into λ = 0. The first eigenvalues of L and L0 coincide, so that iR = 1 means μ0,1 < 0 < μ0,2 . If iR = 1, therefore, the first condition in Theorem 6.37 holds since μm,2 ≥ μ0,2 for m = 0, 1, 2, . . . The second condition of Theorem 6.37 also holds if m is large, as 𝒞a is absorbed into λ = 0. In fact, μm,1 > 0 for any m on the branch of minimal solutions, which becomes eventually negative for large m, because of i → +∞. We thus have infinitely many bifurcations of the solutions from the branch of radially symmetric solutions 𝒞a by Theorems 6.37–6.38. These bifurcated solutions are nonradial, as no radially symmetric solutions are permitted other than 𝒞a , which completes the proof of Theorem 6.36.

6.3 Bifurcation from symmetry | 381

From the asymptotics of the solution u = u(r) described in Theorem 6.15, the fact i → +∞ is easy to confirm, while iR = 1 may look strange. In fact, we have arbitrarily many bendings, and at each bending point the radial Morse index changes. The truth will be that iR increases and decreases one by one, as the curve 𝒦a in the w–ẇ plane studied in Sections 6.2.2–6.2.4 bends clockwise and counterclockwise with respect to the w-axis, respectively. This property is studied in the following section. We also note that the solution produced by the mountain pass lemma has the Morse index equal to 1 if it is not degenerated [198]. Even for the supercritical case, nonminimum radial solution on an annulus is constructed by the lemma based on the symmetric criticality [338]. This subsection is concluded with the proof of the first part of Theorem 6.38 and Theorem 6.37. Theorem 6.39 ([319, 207]). Let {(λα , uα )}α≫1 be a family of radially symmetric solutions to (6.57) such that λα ↓ 0 and ‖uα ‖∞ → +∞ as α ↑ +∞. Suppose f 󸀠 (u) > 0 for u > 0 and limu↑+∞ f (u)/(uf 󸀠 (u)) = 0 besides (6.25). Then it holds that i → +∞, where i denotes the Morse index. Proof. For the moment, we drop index α. Let p = λf 󸀠 (u), Q(v) = ∫ |∇v|2 − pv2 dx, A

v ∈ X = H01 (A),

and X0 be the maximum subspace of X1 such that Q is negative definite on X1 . This X0 is a finite-dimensional subspace of X and it holds that i = dim X0 . We use the identity valid to w ∈ C 2 (A) and v ∈ H01 (A): Q(wv) = ∫ w2 {|∇v|2 − pv2 } − (wΔw)v2 dx. A κ

m Take v = u(r) and w = ϕj (x), where {ϕj }j=1 denotes the orthonormalized eigenfunctions

of Λ corresponding to the eigenvalue ρm , that is, Λϕj = ρm ϕj in Sn−1 and ∫Sn−1 ϕi ϕj dS = δij . Then it holds that Q(ϕj u) = ∫ ϕ2j {|∇u|2 − pu2 } − (ϕj Δϕj )u2 dx A

= ∫ |∇u|2 − pu2 + ρm r −2 u2 dx ≡ Qρm (u) A

for u = u(r) and Δ = that

1 𝜕 n−1 𝜕 (r 𝜕r ⋅) r n−1 𝜕r

−

1 Λ, r2

and therefore i = dim X0 ≥ ∑m j=0 κj , provided

Qρm (u) < 0.

(6.58)

382 | 6 Equations on annulus r

Since u(r) = ∫a ur dr, we obtain 1

1

a

a

dr u(r) ≤ ∫ n−1 ∫ u2r r n−1 dr = C1 ‖∇u‖22 , r 2

and hence 1

∫ r u(r) dx = ωn ∫ u(r)2 r n−3 dr ≤ C2 ‖∇u‖22 2

−2

a

A

where ωn denotes the area of Sn−1 . Then we obtain Qρ (u) ≡ ∫ |∇u|2 − pu2 + ρr −2 u2 dx A 2

≤ ∫ pu dx ⋅ {(C2 ρ + 1) ⋅ A

∫A |∇u|2 dx ∫A pu2 dx

− 1},

and therefore, the equality lim

α↑+∞

∫A |∇uα |2 dx ∫A pu2α dx

=0

(6.59)

implies (6.58) for any ρ > 0, and hence i → +∞ as α ↑ +∞. To show (6.59), we use ∫A |∇u|2 dx ∫A pu2 dx

=

∫A uf (u) dx

∫A u2 f 󸀠 (u) dx

.

The function m(s) = supu≥s f (u)/(uf 󸀠 (u)) satisfies lims↑+∞ m(s) = 0. Theorem 6.15, on the other hand, implies 󵄨󵄨 −1 󵄨 −1 󵄨󵄨{uα ≤ ϵ }󵄨󵄨󵄨 = ∫ χuα ≤ ϵ dx → 0,

α ↑ +∞

(6.60)

A

for any ϵ > 0 by the dominated convergence theorem, where | ⋅ | denotes the volume. This theorem guarantees also ∫A u2α f 󸀠 (uα ) dx ≥ δ with δ > 0 independent of α ≫ 1, and therefore, ∫A uf (u) dx ∫A

u2 f 󸀠 (u) dx

≤ ≤

∫u≤ϵ−1 uf (u) dx ∫A

u2 f 󸀠 (u) dx

+

∫u≥ϵ−1 uf (u) dx

∫u≥ϵ−1 u2 f 󸀠 (u) dx

1 󵄨󵄨 −1 󵄨 −1 󵄨{u ≤ ϵ }󵄨󵄨󵄨 max−1 {uf (u)} + m(ϵ ) δ󵄨 0≤u≤ϵ

6.3 Bifurcation from symmetry | 383

by the definition of m = m(s). Then we obtain lim sup

α↑+∞

∫A uα f (uα ) dx

∫A u2α f 󸀠 (uα ) dx

≤ m(ϵ−1 )

for u = uα by (6.60). Letting ϵ ↓ 0, we obtain (6.59). Proof of Theorem 6.37. Writing x = (x1 , x2 , . . . , xn ), the function v(x) is said to be O(n−1) invariant if v(Tx󸀠 , xn ) = v(x󸀠 , xn ) for any T ∈ O(n − 1), where x 󸀠 = (x1 , x2 , . . . , xn−1 ). Let Em be the eigenspace of Λ corresponding to ρm , and Ẽ m be its set composed of O(n − 1) invariant functions [314, 313, 316, 315, 126]. Then dim Ẽ m = 1,

m ≥ 1.

(6.61)

Given α ∈ 𝒜, take the radial solution (λα , uα ) to (6.57). Putting Φα (w) = λα (−Δ)−1 {f (uα + w) − f (uα )},

w = u − ua ,

we see that equation (6.57) is reduced to Ψα (w) ≡ w − Φα (w) = 0,

1+γ

w ∈ X ≡ C0 (A)

for 0 < γ < 1. The mapping Φα : X → X is compact, and the invertible compact linear T ≡ Φ󸀠α (0), Tv = λα (−Δ)−1 (f 󸀠 (uα )v) is self-adjoint in L2 (A, f 󸀠 (uα ) dx) so that nμ = dim ker(μI −T) is finite for each eigenvalue μ. If X̃ ⊂ X is an O(n−1) invariant subspace, the mapping Φα and operator T are restricted there, denoted by Φ̃ α : X̃ → X and ̃ By (6.61), we get T̃ ∈ B(X,̃ X). ñ μ ≡ algebraic multiplicity of μ for T̃ = 1

(6.62)

for any eigenvalue μ of T.̃ If [α, α] is not a bifurcation interval, 0 is an isolated zero point of Ψα for α − δ < α < α + δ and 0 < δ ≪ 1. Hence the fixed point index i(Ψ̃ a , 0) is defined and constant in (α − δ, α + δ), where Ψ̃ a = I − Φ̃ a . The eigenvalue μ > 1 of T corresponds to the negative eigenvalue of L = −Δ − λf 󸀠 (u) including the multiplicity. Therefore, it holds that i(Ψ̃ α , 0) = (−1)ℓ for ℓ = #{m | μm,1 < 0} by (6.61), (6.62), and the first hypothesis of the theorem. Thus we get a contradiction of the discrepancy of the fixed point index i(Ψ̃ α , 0) in (α − δ, α) and (α, α + δ) from the second hypothesis of the theorem. If A is replaced by B = {|x| < 1} ⊂ Rn in (6.36), Morse index of the solution coincides with that of radial Morse index by a general theory. Then it increases one by one as the bending occurs to the total set of solutions in the λ–u plane described in Proposition 2.24 [238].

384 | 6 Equations on annulus 6.3.2 Radial Morse indices Finite bending of the total set of radial solutions denoted by 𝒞a to (6.36), to be proven below, is closely related to the phenomenon of nonradial bifurcation. First, this 𝒞a has an analytic parametrization. Second, each section 𝒞aλ = {u | (u, λ) ∈ 𝒞a } admits only two elements for 0 < λ ≪ 1. To this end, we recall the homeomorphism between 𝒞a and 𝒦a , which is characterized by the root t = ta+ of F(α, t) ≡ w(α, t) − w(α, t + log a) + 2 log a = 0,

(6.63)

where w(α, t) is the solution to (6.39). This parameter α can be complex, so that the first assertion is reduced to the differentiability of t + (α) with respect to α. To this end, we make use of the implicit function theorem. and wα = 𝜕w , to deduce formally We write ẇ = 𝜕w 𝜕t 𝜕α ̇ t + (α)) − w(t ̇ + (α) + log a)} {w(α,

dt + + {wα (α, t + (α)) − wα (t + (α) + log a)} = 0 dα

by (6.63). Therefore, the identity w (α, t + (α)) − wα (α, t + (α) + log a) dt + =− α ̇ t + (α)) − w(α, ̇ t + (α) + log a) dα w(α,

(6.64)

̇ t+ (α)) < 2 < w(α, ̇ t + (α) + log a). follows with the differentiability of t + (α), from w(α, Now we show d d ̇ t + (α)) t + (α) < 0, w(α, t + (α)) = wα (α, t + (α)) + w(α, dα dα

α≫1

to assure the uniqueness of a nonminimal solution to (6.36) for 0 < λ ≪ 1. In fact, first, equality (6.64) implies d w(α, t + (α)) dα ̇ t + (α)) − w(α, ̇ t + (α) + log a)} = wα (α, t + (α)){w(α,

̇ t + (α)) − w(α, ̇ t + (α) + log a)} {w(α,

̇ t + (α)){wα (α, t + (α)) − wα (α, t + (α) + log a)} − w(α,

̇ t + (α))wα (α, t + (α) + log a) = w(α,

̇ t + (α) + log a)wα (α, t + (a)). − w(α,

(6.65)

̇ t) solves the linearized equation of (6.39), Second, the function v = v1 (t) ≡ w(α, v̈ + (n − 2)v̇ + 2(n − 1)ew v = 0,

−∞ < t < +∞.

(6.66)

Then (6.41) guarantees v1 (t) = 2 + α(n − 2)e−(n−2)t + o(et ), v̇1 (t) = −α(n − 2)2 e−(n−2)t + o(et ),

t ↓ −∞.

(6.67)

6.3 Bifurcation from symmetry | 385

Similarly, v2 (t) ≡ wα (α, t) solves (6.66) and v2 (t) = −e−(n−2)t + o(1),

v̇2 (t) = (n − 2)e−(n−2)t + o(1),

t ↓ −∞.

These asymptotics coincide with those formally derived from (6.39), by differentiating with respect to t and α. The right-hand side of (6.65) is equal to −v1 (t + (α) + log a)v(t + (α)) for v(t) = v2 (t) −

v2 (t + (a) + log a) v (t). v1 (t + (α) + log a) 1

(6.68)

̇ t + (α) + log a) > 2, while We have, furthermore, v1 (t + (α) + log a) = w(α, ̇ t + (α)) − w(α, ̇ t + (α) + log a) < 0 w(α, on the left-hand side of (6.65). Thus we obtain sgn{

d w(α, t + (α))} = sgn{v(t + (α))} dα

(6.69)

for v = v(t) in (6.68), a solution of the linearized equation (6.66) satisfying v(t + (α) + ̇ + (α) + log a), we note that the function log a) = 0. To calculate v(t t

v3 (t) = v1 (t) ∫ −∞

e−(n−2)s ds v1 (s)2

is well-defined as long as v1 (s) ≠ 0 in s ∈ (−∞, t], solving the Wronskii equation v1 v̇3 − v̇1 v3 = e−(n−2)t , and therefore, is a solution to (6.66) linearly independent of v1 (t). We also obtain v3 (t) =

1 2e(n−2)t + 2 + o(et ), 2 α(n − 2) α (n − 2)3

v̇3 (t) =

2e(n−2)t + o(et ), α2 (n − 2)2

by (6.67), which implies v2 (t) = −

1 v (t) + 2(n − 2)v3 (t) α(n − 2) 1 t

1 e−(n−2)s = v1 (t){− + 2(n − 2) ∫ ds} α(n − 2) v1 (s)2 −∞

as long as v1 (s) ≠ 0 in s ∈ (−∞, t]. Turning to (6.68), we reach t

̇ t) v(t) = 2(n − 2)w(α,

∫ t + (α)+log a

e−(n−2)s ds ̇ s)2 w(α,

t ↓ −∞

386 | 6 Equations on annulus ̇ s) ≠ 0 in s ∈ (−∞, t], which implies as long as w(α, e−(n−2)(t (α)+log a) > 0. ̇ t + (α) + log a) w(α, +

̇ + (α) + log a) = 2(n − 2) v(t

Under the transformation r 󳨃→ t in § 2.2.3, this v = v(r), r = |x|, satisfies −Δv = λeu v in A = {a < |x| < 1} ⊂ Rn and v|r=a = 0, vr |r=a > 0. The Sturm–Liouville theory concerning the number of zeros of eigenfunctions, therefore, assures iR = #{t0 ∈ (t + (α) + log a, t + (α)] | v(t0 ) = 0}.

(6.70)

Regarding (6.69), vr |r=a > 0, and (6.70), we end up with sgn{

d w(α, t + (α))} = sgn{(−1)iR }. dα

(6.71)

̇ t) solve the same Proof of Theorem 6.38. We use equality (6.70). Since v(t) and w(α, linearized equation (6.66), a comparison theorem of Sturm implies that the zeros of v and ẇ separate each other. Given 𝒪α , we take the first two points on the axis ẇ = 0 as t increases, denoted by Ra1 and Rα2 in order. We call 𝒪̃ α the portion of 𝒪α cut by Rα1 and Rα2 . The comparison theorem indicates that there exists a unique Qα1 ∈ 𝒪̃ α , the corresponding time t of which coincides with the zero point of v(t). The portion 𝒪̃ α is furthermore divided into two parts by Qα1 , that is, the arcs Ra1 Qα1 − , and Qα1 Rα2 , respectively, denoted by 𝒪̀ α0 and 𝒪̃ σ1 . Let 𝒵0 = ⋃α>0 𝒪̃ α0 , 𝒵1 = ⋃α>0 𝒪α1 + ̃ and 𝒵 = ⋃α>0 𝒪α . By Proposition 6.28, the outer boundary point Pα lies in 𝒵 for α ≫ 1. If Pα+ ∈ 𝒵0 , we have iR = 0 by (6.70). This case, however, does not arise for α ≫ 1 from the proof of Theorem 2.14. This means that Pα+ lies eventually on 𝒵1 and hence iR = 1. Theorem 6.40 ([238]). The number of bendings in 𝒞a with respect to λ axis is finite. Proof. The result follows from (6.71).

6.4 Structure of radial solutions 6.4.1 Summary We review the result on (6.36) for n ≥ 3. If the annulus A is replaced by the ball B ≡ {|x| < 1}, every classical solution is radial, and the total set of solutions is known in details. The result indicates that the feature is different between 2 < n < 10 and n ≥ 10. Concerning (6.36), it is suspected that b(a) is monotone in 0 < a < 1.5 Here we show the following facts: 5 This conjecture is due to S. S. Lin.

6.4 Structure of radial solutions | 387

1. The total set of radial solutions 𝒞a = {(λ, u(|x|)) | classical solutions to (6.36)}

is homeomorphic to R, starting from (λ, u) = (0, 0) and absorbed into the hyperplane λ = 0. 2. The number of bendings of 𝒞a with respect to λ, denoted by b(a), is finite for each a ∈ (0, 1) and satisfies lima↓0 b(a) = +∞ if 2 < n < 10. We have also b(a) = 1 for (n − 1)−1/(n−2) < a < 1, valid to a class of nonlinearities [207]. Here we show the following theorems. Theorem 6.41. If b(a1 ) = 1, then b(a) = 1 for any a ∈ (a1 , 1). Theorem 6.42. If n > 10, we have b(a) = 1 for any a ∈ (0, 1). For this purpose, we use radial Morse index denoted by iR , the number of negad2 n−1 d u(|x|) tive eigenvalues of the ordinary differential operator − dr with ⋅|r=a = 2 − r dr − λe ⋅|r=1 = 0 for (λ, u(r)) ∈ 𝒞a , r = |x|. This iR is a nonnegative integer and does not vary except for the bending point of 𝒞a , and therefore, iR∗ ≡ max𝒞a iR < +∞ by b(a) < +∞. Regarding iR∗ as a function of a ∈ (0, 1), we have the following theorem which implies Theorem 6.41. Theorem 6.43. The function iR∗ (a) is monotone. It takes any positive integer value if 2 < n < 10. Given a ∈ (0, 1), we observe iR along 𝒞a from (λ, u) = (0, 0). What is expected is that this number increases from 0 one by one at each bending point until getting to be iR∗ , and afterwards decreases similarly to 1. If this property arises, it holds that b(a) = 2iR∗ (a) − 1,

(6.72)

which ensures the monotonicity of b(a). If A is replaced by B, the number i∗ (R) increased one by one at each bending point [238]. We require several facts for the proof of these theorems established in preceding sections: 1. Any α ≥ 0 admits a unique solution w(t) = w(α, t) to (6.39). ̇ t)) | −∞ < t < +∞} and the foliation {𝒪α }α≥0 in the w–ẇ 2. Orbit 𝒪α = {(w(α, t), w(α, plane satisfy 𝒪α ∩ 𝒪β = 0, α ≠ β, and ⋃α≥0 𝒪α = R2 \ {(0, 0)}. Each 𝒪α approaches the origin (0, 0) as t ↑ +∞ spirally and nonspirally according to 2 < n < 10 and n ≥ 10, respectively. 3. Given a ∈ (0, 1) and α > 0, there exists a unique root t = t+ (α, a) of F(α, a, t) ≡ w(α, t) − w(α, t + log a) + 2 log a = 0.

(6.73)

388 | 6 Equations on annulus It holds that ̇ t+ (α, a)) < 2 < w(α, ̇ t− (α, a)) w(α,

(6.74)

and w(α, t− (α, a)) = w(α, t+ (α, a)) + 2 log a,

t− (α, a) = t+ (α, a) + log a.

(6.75)

̇ t+ (α, a))) | 0 < α < +∞} through λ = 4. We have 𝒞a ≈ 𝒦a = {(w(α, t+ (α, a)), w(α, w(α,t+ (α,a)) and 2(n − 2)e u(r) = w(α, log r + t+ (α, a)) − w(α, t+ (α, a)) − 2 log r. In particular, radial Morse index iR is a function of α and a, iR = iR (α, a). 5. The values t = t+ (α, a) and t = t− (α, a) correspond to the boundaries of the annulus, r = 1 and r = a, respectively. We define outer and inner boundary points by P± (α, a) = ̇ t± (α, a))). The crossing point of ẇ = 2 and 𝒪α corresponds to the (w(α, t± (α, a)), w(α, zero of ur , that is, the maximal point of u. 6. Each a ∈ (0, 1) admits lim w(α, t+ (α, a)) = −∞, α↓0

̇ t+ (α, a)) = 2 lim w(α, α↓0

̇ t+ (α, a)) < 0 for α ≫ 1. and also w(α, t+ (α, a)), w(α, 7. We have ⋃ 𝒦a = {(w, w)̇ ∈ R2 | ẇ < 2} \ 𝒪0 \ {(0, 0)}.

0 a2 > 0, 𝒦a2 lies along the right direction of 𝒦a1 . 8. Let v(t) = v(α, a, t) be the solution to the linearized equation v̈ + (n − 2)v̇ + 2(n − 2)ew(α,t) v = 0,

−∞ < t < ∞

(6.77)

with v|t=t− (α,a) = 0,

v|̇ t=t− (α,a) = 1.

(6.78)

Then, iR (α, a) is the number of zeros of v(α, a, t) in t ∈ (t− (α, a), t+ (α, a)). d w(α, t+ (α, a))} = (−1)iR (α,a) except for the bending point of 𝒦a , and 9. We have sgn{ dα therefore, iR is even and odd as P+ (α, a) is moving right and left with α increasing, respectively.

6.4 Structure of radial solutions | 389

6.4.2 Γ-curves If 2 < n < 10, each 𝒪α approaches the origin spirally. Hence it crosses the line ẇ = 0 infinitely many times. Let these crossing points be Rα1 , Rα2 , and so on, as t increases. The portion of 𝒪α cut by Rαℓ and Rαℓ+1 is denoted by 𝒪̃ αℓ . Let 𝒵ℓ = ⋃α>0 𝒪̃ αℓ for ℓ = 1, 2, . . . This 𝒵ℓ is actually determined only by 𝒪0 . The case n ≥ 10 is similar, but only 𝒵1 and 𝒵2 appear, because each 𝒪0 with α > 0 crosses ẇ = 0 just twice. ̇ t) also solves (6.77). It is linearly independent of v because it The function w(α, does not have a zero at t = t− (α, a) by (6.74). Then Sturm’s comparison theorem implies iR (α, a) = ℓ or ℓ − 1

if P+ (α, a) ∈ 𝒵ℓ .

(6.79)

More precisely, let the ℓth zero of v(α, a, t) in t > t− (α, a), if it exists, be tℓ (α, a). Since it ̇ tℓ (α, a))) | 0 < α < +∞} forms a is nondegenerate, the set Γℓ (a) = {(w(a, tℓ (α, a)), w(a, smooth curve in 𝒵ℓ . Then, iR = ℓ or iR = ℓ−1 as P+ (α, a) has passed Γℓ (a) an odd or even number of times as α increases, respectively. It holds that Γℓ (a) ⊂ R2± if sgn{(−1)ℓ } = ±1, where R2± = {±ẇ > 0}. For ℓ ≥ 1, the point ̇ tℓ (α, a))) ∈ 𝒪α ∩ 𝒵ℓ Qℓ (α, a) = (w(a, tℓ (α, a)), w(a, on Γℓ (a) approaches 𝒪̃ 0ℓ as α ↓ 0. Given αj ↓ 0, therefore, we have a subsequence denoted by the same symbol and t∗ ∈ R such that tℓ (αj , a) → t∗ , which implies ̇ t∗ )) ∈ 𝒪0 . Qℓ (αj , a) → Q∗ℓ (a) = (w(0, t∗ ), w(0, ̃ Let v(α, a, t) be the solution to (6.77) with v|t=tℓ(α,a) = 0, v|̇ t=tℓ(α,a) = −1, and v∗ (t) such that v̈∗ + (n − 2)v̇∗ + 2(n − 2)ew(0,t) v∗ = 0

(6.80)

with v∗ |t=t∗ = 0 and v̇∗ |t=t∗ = −1. It holds that ṽ → v∗ locally uniformly in t as α ↓ 0, while equation (6.80) is nonoscillatory as t ↓ −∞ [88]. Hence v∗ (t) has exactly ℓ zeros in (−∞, t∗ ) again by the nondegeneracy of tℓ ’s, which means that the solution (u, λ) to −Δu = λeu

in B = {|x| < 1} ⊂ Rn ,

u|𝜕B = 0,

corresponding to Q∗ℓ ∈ 𝒪0 is degenerate with the radial Morse index ℓ − 1. Hence we obtain Q∗ℓ (a) = R0ℓ [238]. In other words, each Γℓ (a), if it exists, forms a smooth curve in 𝒵ℓ with an endpoint R0ℓ : ̇ tℓ (α)) = R0ℓ . lim(w(α, tℓ (α)), w(α, α↓0

The other end point is expected to be R0ℓ−1 if ℓ ≥ 2: ̇ tℓ (α)) = R0ℓ−1 . lim (w(α, tℓ (α)), w(α,

α↑+∞

(6.81)

It is obvious that Γℓ (a) exists for ℓ = 1, 2, . . . if 2 < n < 10, while Γ2 (a) does not if n > 10. If Γ2 (a) appears for n = 10, multibendings can arise to 0 < a ≪ 1.

390 | 6 Equations on annulus 6.4.3 Radial Morse indices revisited Proof of Theorem 6.43. We recall that iR (α, a) is equal to the number of zeros of v(α, a, t) in t ∈ (t− (α, a), t+ (α, a)). Since each zero of v is nondegenerate, the integer iR (α, a) varies one by one with a, and therefore, the same is true for iR∗ (a) = maxα>0 iR (α, a). To prove the monotonicity of iR∗ (a), we differentiate (6.73) formally with respect to a, to obtain ̇ t+ (α, a)) w(α,

𝜕t 𝜕t+ 1 2 ̇ t− (α, a))( + + ) + = 0. − w(α, 𝜕a 𝜕a a a

Inequality (6.74) actually assures the existence of ̇ t− (α, a)) − 2 w(α, 𝜕t+ 1 = ⋅ 0. ̇ t+ (α, a)) − w(α, ̇ t− (α, a)) 𝜕a 𝜕a a a w(α, For 1 > a1 > a2 > 0, therefore, it holds that t− (α, a2 ) < t− (α, a1 ) < t+ (α, a1 ) < t+ (α, a2 ).

(6.82)

Since v(α, a, t) is a solution to (6.77), the coefficients of which are independent of a, we obtain iR (α, a1 ) ≤ iR (α, a2 ) by (6.78), (6.82), iR (α, a) being the number of zeros of v(α, a, t) in t ∈ (t− (α, a), t+ (α, a)), and Sturm’s comparison theorem. Hence the monotonicity iR∗ (a1 ) ≤ iR∗ (a2 ) follows. The fact that iR∗ (a) takes any nonnegative integer value for 2 < n < 10, on the other hand, is obtained by (6.76) and (6.79). Henceforth, we fix a ∈ (0, 1) and drop it as an index. Since λ = 2(n − 2)ew(α,t+ (α,a)) , the bending of 𝒞 with respect to λ corresponds to that of 𝒦 with respect to w. Hence d there exist b roots of the equation dα w(α, t+ (α)) = 0 for α, denoted by 0 < α1 < α2 < ⋅ ⋅ ⋅ < αb < +∞. Then, ̇ k , t+ (αk ))), Nk = (w(αk , t+ (αk )), w(α

1≤k≤b

denotes the set of bending points of 𝒦 with respect to w so that there exists an integer ℓ(k) ≥ 1 such that Nk ∈ Γℓ(k) .

(6.83)

We have, on the other hand, that iR (α) does not vary except for Nk on 𝒦 for 1 ≤ k ≤ b. In fact, if this property is not the case, radial solutions must bifurcate from 𝒞 because of

6.4 Structure of radial solutions | 391

the degree theory in § 2.1.5, that is, bifurcation from the eigenvalues of odd multiplicity, a contradiction. ̇ t) > 0 for −∞ < t < rα1 Let t = rα1 be the time corresponding to Rα1 , that is, w(α, ̇ rα1 ) = 0. Then, it holds that t− (α) < rα1 . Here we have and w(α, v(α, t) = (positive constant) ⋅ (v1 (t− (α))v2 (t) − v2 (t− (α))v1 (t)) for −∞ < t < rα1 , where v1 = ẇ and t

v2 (t) = v1 (t) ∫ −∞

It holds also that v2 =

ẇ 2(n−2)2 α

+

wα 6 . 2(n−2)

e−(n−2)s ds. v1 (s)2

(6.84)

Then we obtain

̇ t− (α))wα (α, t) − wα (α, t− (α))w(α, ̇ t)). v(α, t) = (positive constant) ⋅ (w(α, Without loss of generality, we take ̇ t− (α))wα (α, t) − wα (α, t− (α))w(α, ̇ t). v(α, t) = w(α, Lemma 6.44. It holds that v(α, t+ (α)) = [

d ̇ t− (α)) − w(α, ̇ t+ (α))). w(α, t+ (α))](w(α, dα

(6.85)

Proof. Equation (6.73) implies dt+ (α) wα (α, t+ (α)) − wα (α, t− (α)) =− ̇ t+ (α)) − w(α, ̇ t− (α, t− (α))) w(α, dα by the implicit function theorem, and hence ̇ t− (α)) − w(α, ̇ t− (α)))(w(α, ̇ t− (α)) v(α, t+ (α)) = (w(α,

dt+ (α) + wα (α, t− (α))). dα

Then it holds that ̇ t− (α)) w(α,

dt+ (α) d d + wα (α, t− (α)) = w(α, t− (α)) = w(α, t+ (α)) dα dα dα

by (6.75), and hence (6.85) follows. Lemma 6.45. At the bending point Nk , 1 ≤ k ≤ b, of 𝒦 with respect to w, the radial Morse index changes, subject to the following rule, where Nk ∈ 𝒵ℓ(k) : 6 This equality is derived because wα solves the linearized equation (6.77) and has the asymptotic behavior of w(α, t) as t → −∞. See [238] for details.

392 | 6 Equations on annulus 1. 2.

If ℓ(k) ≡ k (mod 2), the value iR increases 1. If ℓ(k) ≢ k (mod 2), the value iR decreases 1.

̇ k , t+ (αk )) < 0 and v(αk + ϵ, t+ (αk + ϵ)) < 0 for 0 < ϵ ≪ 1, the value iR (α) Proof. If v(α increases by 1 at α = αk so that iR (αk + ϵ) = iR (αk ) + 1 for 0 < ϵ ≪ 1. Since iR (α, a) is ̇ k , t+ (αk ))} = the number of zeros of v(α, a, t) in t ∈ (t− (α, a), t+ (α, a)), we obtain sgn{v(α d iR (αk )+1 (−1) by (6.78), while (6.85) implies ±v(α, t+ (α)) > 0 if and only if ± dα w(α, t+ (α)) > 0 for any α > 0. Since sgn{

d w(α, t+ (α))} = (−1)k , dα

αk < α < αk+1 ,

0 ≤ k ≤ b − 1,

under the agreement of α0 = 0, one has sgn{v(α, t+ (α))} = (−1)k ,

0 < α − αk ≪ 1,

1 ≤ k ≤ b.

Thus we obtain the following: (I) If k is odd and iR (αk ) is even, the value iR (α) increases by 1 at α = αk . The following facts are proven similarly: (II) If k is odd and iR (αk ) is odd, the value iR (α) decreases by 1 at α = αk . (III) If k is even and iR (αk ) is even, the value iR (α) decreases by 1 at α = αk . (IV) If k is even and iR (αk ) is odd, the value iR (α) increases by 1 at α = αk . Thus we obtain iR (αk ) = ℓ(k) − 1 if Nk ∈ 𝒵ℓ(k) . Proof of Theorem 6.41. By Lemma 6.45, the radial Morse index changes one by one at each bending point so that b(a) ≥ 2iR∗ (a) − 1 follows. If b(a1 ) = 1, therefore, one gets iR∗ (a1 ) = 1, and hence i∗ (a) = 1 for any a ∈ (a1 , 1). Since the radial Morse index is 0 if and only if it is a strict minimal solution, two more bendings imply iR∗ (a) ≥ 2. Hence iR∗ (a) = 1 implies b(a) = 1. It is obvious that Theorem 6.42 is a consequence of the following lemma. Lemma 6.46. The curve Γ2 does not appear if n > 10. Proof. Equality (6.84) follows from the fact that e(n−2)t (v1 v̇ − v1̇ v) is constant for the ̇ t) to (6.77), and hence solutions v = v(α, t) and v1 = w(α, v1 v̇ − v1̇ v = Ce−(n−2)t ,

C = −v1̇ v|t=rα2 > 0,

(6.86)

with t = rα2 standing for the time of Rα2 ∈ 𝒪α . Since the existence of Γ2 implies the positivity of v(t) for t ≫ 1, we show its negativity as t ↑ +∞.

6.4 Structure of radial solutions | 393

̇ t) ↓ 0 as t ↑ +∞, and therefore, Given α > 0, we have w(α, t) ↑ 0 and v1 = w(α, method of super/subsolution is valid to v̈ + (n − 2)v̇ + 2(n − 2)v = 0

(6.87)

and v̈ + (n − 2)v̇ + {2(n − 2) − δ}v = 0,

0 < δ ≪ 1.

(6.88)

More precisely, sub- and supersolutions to (6.77) are taken by (6.87) and (6.88), respectively, and therefore, (6.77) has a solution v+ satisfying v+ (t) = e−μ+ t{1−o(1)} as t ↑ +∞, > 0. where μ+ = (n−2)+√(n−2)(n−10) 2 The other solution is obtained from

t

v(t) = (constant) ⋅ v− (t) ∫ t0

e−(n−2)s ds v+ (s)2

for t0 sufficiently large. Thus we have a solution v− (t) linearly independent of v+ (t) satisfying v− (t) = e−μ− t{1+o(1)} as t ↑ +∞, where μ− = (n − 2) − μ+ =

(n − 2) − √(n − 2)(n − 10) > 0. 2

These asymptotics imply v̇± (t) = −μ± e−μ± t{1+o(1)} as t ↑ +∞ by t

̇ = e−(n−2)(t−to ) v(t ̇ 0 ) − 2(n − 2)e−(n−2)t ∫ ew(s)+(n−2)s v(s) ds v(t) t0

derived from (6.77). Here we note In particular,

−2(n−2) n−2−μ±

v1 (t) = C1 e−μ± t{1∓o(1)} ,

= −μ± and n − 2 > μ± . v̇1 (t) = −μ± C1 e−μ± t{1∓o(1)}

(6.89)

as t ↑ +∞ with C1 > 0, where ± indicates the possibility of either case. Substituting (6.89) into (6.86), we obtain ̇ − μ± v(t) = (positive constant) ⋅ e−μ∓ t{1∓o(1)} v(t)

(6.90)

since −(n − 2) + μ± = −μ∓ . Because v(t) has the same asymptotics as that of v± (t), equality (6.90) implies v(t) = C2 e−μ∓ t{1∓o(1)} ,

̇ = −μ∓ C2 e−μ∓ t{1∓o(1)} v(t)

with a constant C2 . Here we have C2 < 0 from μ+ +μ− = n−2 > 0, and therefore, v(t) < 0 holds for t ≫ 1.

394 | 6 Equations on annulus 6.4.4 Zone property Assume b ≥ 3 for 2 < n < 10. We have iR (α) = 1 for α ≫ 1, while Nb ∈ 𝒵1 ∪ 𝒵2 by (6.79). Since b is odd, Nb ∈ 𝒵1 implies iR = 0 before Nb by Lemma 6.45. This property is a contradiction, and hence Nb ∈ 𝒵2 and iR = 2 at Nb . By the location of 𝒵0 , 𝒵1 , 𝒵2 , . . . , the condition Nk ∈ 𝒵ℓ implies Nk ∈ 𝒵ℓ−1 ∪ 𝒵ℓ ∪ 𝒵ℓ+1 . Let Nj ∈ 𝒵j for 1 ≤ j ≤ b1 with some b1 ≥ 1. Then Nb1 +1 ∈ 𝒵b1 −1 is impossible. In fact, if it happens, then iR = b1 − 1 at Nb1 by Lemma 6.45 so that iR = b1 after Nb1 +1 , which contradicts (6.79). In other words, {Nk } cannot come outside before two continuing points stay in the same zone, and hence there is b2 ≥ 1 such that Nj ∈ 𝒵j for 1 ≤ j ≤ b2 and Nb2 +1 ∈ 𝒵b2 . What is expected is that, once this happens, Nk shifts back from one zone to another successively, so that b2 =

b+1 , 2

N b+1 +j ∈ 𝒵 b+1 −j+1 , 2

2

1≤j≤

b−1 . 2

(6.91)

The radial Morse index at Nk is simultaneously determined by Lemma 6.45, which ensures (6.72). We call (6.91) the zone property. Since Qℓ (α) → R0ℓ−1 as α ↓ 0, we have αj ↑ +∞ such that Qℓ (αj ) → R0ℓ−1 by (6.76) if ℓ ≥ 2. Property (6.91), therefore, follows if 𝒦 meets each Γℓ at most twice successively. Since iR is even and odd as P+ (α, a) is moving right and left with α increasing, d respectively; this property follows if the value dα w(α, tℓ (α)) changes sign only once as α > 0 varies for any ℓ ≥ 2. The case n = 10 has not yet be clarified. If (6.91) is true, Γ2 (a) exists, and stays away from 𝒪0 as a ↓ 0. Hence b(a) = 3 for 0 < a ≪ 1 besides its monotonicity in a. Finally, if (6.81) is true and each Γℓ is convex, one gets (6.91).

7 Hardy spaces and BMO This chapter is devoted to the regularity of the solutions. In the first section we describe Moser’s iteration scheme applied to p-harmonic functions. This scheme is already used in § 4.4, but a basic tool used here arises in accordance with the BMO estimate, which is described in the second section. Then, Hardy spaces emerge with the duality, and a regularity theorem is obtained via the compensated compactness principle in the third section. The last two sections are devoted to the formation of bubbles made by harmonic maps and Smoluchowski–Poisson equation. We thus conclude with the quantized blow-up mechanism associated with recursive hierarchy, that is, the control of the point vortex Hamiltonian, in the kinetic level, observed in Boltzmann–Poisson equation in the static level in former chapters.

7.1 Regularity of weak solutions 7.1.1 Dirichlet principle If Δu = 0 holds in a bounded domain Ω ⊂ Rn , such u is called harmonic. Given f ∈ C(𝜕Ω), the Laplace equation seeks a harmonic function taking f on the boundary: Δu = 0

in Ω,

u=f

on 𝜕Ω.

(7.1)

It is a fundamental problem in mathematical physics and the classical theory assures the following theorem. Theorem 7.1. Any f ∈ C(𝜕Ω) admits a solution u ∈ C 2 (Ω) ∩ C(Ω) to (7.1) if and only if every point of 𝜕Ω is regular. Here we say that ξ ∈ 𝜕Ω is regular if there exists a barrier w(x) at this point, that is, a superharmonic, continuous in Ω, positive in Ω\{ξ } function, and such that w(ξ ) = 0. Moreover, the superharmonicity of a continuous function is defined by the mean value property: 1 |Br (x0 )|

∫ w dx ≥ w(x0 ),

∀x0 ∈ Ω,

0 < ∀r ≪ 1.

(7.2)

𝜕Br (x0 )

Recall that Br (x0 ) denotes the ball of radius r with center x0 and also that such u ∈ C 2 (Ω) ∩ C(Ω) is called a classical solution to (7.1). A criterion using capacity is known for the existence of the barrier in Theorem 7.1.1 An easier condition is the existence of 1 A sufficient condition is the existence of a triangle with lower dimension in the outer region of Ω (Kuran’s criterion [194]). https://doi.org/10.1515/9783110556285-007

396 | 7 Hardy spaces and BMO a circumscribed ball. The uniqueness of the solution, on the other hand, always holds by the weak maximum principle for harmonic functions. Thus we obtain the following result. Corollary 7.2. Let Ω ⊂ Rn be a bounded domain, and suppose that each ξ ∈ 𝜕Ω admits a ball B ≠ 0 such that B ∩ Ω = {ξ }. Then for any f ∈ C(𝜕Ω), there is a unique classical solution u = u(x) to (7.1). Theorem 7.1 is proved by the Perron’s method. First, we define the subharmonicity of w = w(x) in Ω by the reverse inequality in (7.2). Then, we put 𝒮 (f ) = {v ∈ C(Ω) | v is subharmonic in Ω and v ≤ f on 𝜕Ω} to define u(x) = sup{v(x) | v ∈ 𝒮 (f )},

x ∈ Ω.

This u = u(x) is harmonic in Ω and if ξ ∈ 𝜕Ω is a regular point then it holds that limx∈Ω→ξ u(x) = f (ξ ). The proof begins with the representation formula of the solution to (7.1) in a ball domain, that is, the Poisson integral. Then after using the mean value theorem, Harnack’s inequality, strong and weak maximum principles, and the Harnack principle, here comes a natural expansion to super- and subharmonic functions, and then, harmonic lifting, and barrier. This argument of Poincaré reminds us partly of the complex analysis, where an integral formula is the origin of the whole story, but partly of the real analysis when a local deformation is performed in harmonic lifting, and also the use of order structure in the construction of the solution from subsolutions. The overall argument is highly elegant, beautiful, and suggestive.2 However, (7.1) has been “solved” by another method, that is, the Dirichlet principle which says that the solution u = u(x) to (7.1) attains the minimum of J(v) = 21 ∫Ω |∇v|2 dx for v = v(x) satisfying v=f

on 𝜕Ω.

(7.3)

It is thus a variational problem where a functional, a function of a function, is given, so that our task is to seek an appropriate function space. This principle is based on the widely accepted belief that the physical phenomenon mimimizes or retains a given functional. An intuitive explanation, actually, assumes that u(x) is a minimum of this variational problem, then for any function v = v(x) one gets v=0

on 𝜕Ω,

and hence t 󳨃→ J(u + tv) attains a minimum at t = 0. Since 0=

d J(u + tv)|t=0 = ∫ ∇u(x) ⋅ ∇v(x) dx, dt Ω

2 See [5, 139, 341].

(7.4)

7.1 Regularity of weak solutions | 397

integrating by parts and from (7.4), we obtain ∫ Δu ⋅ v dx = 0,

(7.5)

Ω

and therefore, Δu = 0 from the arbitrariness of v = v(x). The condition u = f on 𝜕Ω is readily required in the constraint (7.3). Nowadays, what is left in this argument is well-known. Actually, it was first set up by Dirichlet, and then used by Riemann. After a keen criticism of Weierstrass, it was eventually recovered by an insight of Hilbert. These problems are summarized as follows: 1. Is there any function which minimizes the functional? 2. Is the regularity of this minimizer sufficient to execute integration by parts? These two problems are strongly related. In fact, to solve the first problem, we needed to use a function space equipped with the topology which makes the functional continuous and the minimizing sequence compact. As a result, this minimizer must recover stronger regularity than that just in the function space.

7.1.2 Existence of weak solutions To answer the first question, we use the H 1 (Ω) = W 1,2 (Ω), recalling § 1.3.1. This function space is the set of v ∈ L2 (Ω) such that 𝜕v/𝜕xj ∈ L2 (Ω) for 1 ≤ j ≤ n in the sense of distributions. Recall also, the Lp space for 1 ≤ p ≤ ∞ with the standard norm ‖ ⋅ ‖p and integration taken in the sense of Lebesgue. Thus, v ∈ L2 (Ω) belongs to H 1 (Ω) if there is vj ∈ L2 (Ω) for 1 ≤ j ≤ n such that ∫v Ω

𝜕ϕ dx = − ∫ vj ϕ dx, 𝜕xj Ω

∀ϕ ∈ C0∞ (Ω),

(7.6)

where C0∞ (Ω) denotes the set of arbitrary many times differentiable functions in Ω with compact support. By (7.6), vj = vj (x) is determined uniquely except for a set of measure zero. Hence we write vj for 𝜕v/𝜕xj and call the xj -derivative of v in the sense of distributions. Then ∫Ω |∇v|2 is defined for v ∈ H 1 (Ω), and H 1 (Ω) becomes a Banach space equipped with the norm 1/2

‖v‖H 1 = {∫ |∇v|2 + v2 dx} . Ω

In particular, any Cauchy sequence in H 1 (Ω) converges. Moreover, since there is an inner product comparable to this norm, this H 1 (Ω) is a Hilbert space. Hence it is a

398 | 7 Hardy spaces and BMO reflexive Banach space, in particular, and every bounded sequence has a subsequence converging weakly.3 The first trouble to use this function space is the realization of the boundary condition (7.3). Fortunately, if Ω satisfies the limited cone property then the mapping γ : v ∈ C(Ω) 󳨃→ v|𝜕Ω ∈ C(𝜕Ω)

(7.7)

is uniquely extended to a bounded linear operator from H 1 (Ω) to L2 (𝜕Ω). This γ(v) ∈ L2 (𝜕Ω), called the trace to the boundary of v ∈ H 1 (Ω), can cast its boundary value [4]. The second issue prepared in advance is a reverse problem. We have to extend f (ξ ) given on 𝜕Ω to Ω to execute our approach. To avoid a significant argument, just assume that f (ξ ) is originally given as a trace of elements in H 1 (Ω) to the boundary, for simplicity. Then we can regard (7.3) as an equality of traces of both v and f to the boundary. Lipschitz domain is a natural category where the trace of H 1 (Ω)/H01 (Ω) to the boundary is an isomorphism to H 1/2 (𝜕Ω) [244]: v ∈ H01 (Ω) if and only if v ∈ H 1 (Ω) and γv = 0.4 Note that H01 (Ω) stands for the Hilbert space formed by the closure of C0∞ (Ω) in H 1 (Ω). Lemma 7.3 (Poincaré). If Ω ⊂ Rn is a bounded domain, there is a constant C = C(Ω) > 0 such that v ∈ H01 (Ω).

‖v‖2 ≤ C‖∇v‖2 ,

(7.8)

Proof. It suffices to show (7.8) for v ∈ C0∞ (Ω). In fact, for general v ∈ H01 (Ω), we have {vk } ⊂ C0∞ (Ω) such that ‖vk − v‖H 1 → 0. Then it holds that ‖∇vk ‖2 → ‖∇v‖2 , and ‖vk ‖2 → ‖v‖2 , and hence (7.8) for this v(x). Since Ω is bounded, there is ℓ > 0 such that Ω ⊂ {x = (x1 , x2 , . . . , xn ) ∈ Rn | 0 < x1 < ℓ}. Given v = v(x) ∈ C0∞ (Ω), we take its zero extension outside Ω. It holds that x1

v(x) = ∫ 0

x = (x1 , x2 , . . . , xn ) ∈ Rn ,

𝜕v (t, x2 , . . . , xn ) dt, 𝜕x1

which implies 2

ℓ ℓ 󵄨󵄨 󵄨󵄨2 󵄨󵄨 𝜕v 󵄨󵄨 𝜕v 󵄨󵄨 󵄨󵄨2 󵄨󵄨 󵄨󵄨 󵄨 󵄨 (t, x2 , . . . , xn )󵄨󵄨 dt} ≤ ℓ ⋅ ∫󵄨󵄨󵄨 (t, x2 , . . . , xn )󵄨󵄨󵄨 dt 󵄨󵄨v(x)󵄨󵄨 ≤ {∫󵄨󵄨 󵄨󵄨 𝜕x1 󵄨󵄨 󵄨󵄨 𝜕x1 󵄨󵄨 0

0

3 See [47, 373, 295] for these fundamentals of real and functional analysis. 4 If the boundary is C 1 then the trace has better properties, for instance, the extension of boundary value to the inner domain is characterized.

7.1 Regularity of weak solutions | 399

and hence 󵄨2 󵄨 ∫ 󵄨󵄨󵄨v(x1 , x2 , . . . , xn )󵄨󵄨󵄨 dx2 ⋅ ⋅ ⋅ dxn

Rn−1

󵄨󵄨 𝜕v 󵄨󵄨2 󵄨 󵄨 ≤ ℓ ∫ dt ∫ 󵄨󵄨󵄨 (t, x2 , . . . , xn )󵄨󵄨󵄨 dx2 ⋅ ⋅ ⋅ dxn . 󵄨󵄨 𝜕x1 󵄨󵄨 n−1 ℓ

0

R

From ℓ

󵄨 󵄨2 󵄨 󵄨2 ‖v‖22 = ∫ 󵄨󵄨󵄨v(x)󵄨󵄨󵄨 dx = ∫ dx1 ∫ 󵄨󵄨󵄨v(x1 , x2 , . . . , xn )󵄨󵄨󵄨 dx2 ⋅ ⋅ ⋅ dxn 0

Rn

Rn−1

ℓ 󵄨󵄨 𝜕v 󵄨󵄨󵄨2 󵄨 ≤ ℓ2 ∫ dt ∫ 󵄨󵄨󵄨 (t, x2 , . . . , xn )󵄨󵄨󵄨 dx2 ⋅ ⋅ ⋅ dxn 󵄨󵄨 𝜕x1 󵄨󵄨 n−1 0

R

󵄩󵄩 𝜕v 󵄩󵄩2 󵄩󵄩 󵄩 2 2 = ℓ2 󵄩󵄩󵄩 󵄩󵄩 ≤ ℓ ‖∇v‖2 , 󵄩󵄩 𝜕x1 󵄩󵄩2 we obtain (7.8) for C = ℓ and v ∈ C0∞ (Ω). To show the existence of the solution to (7.1), let Ω be a Lipschitz domain and f = f (ξ ) the trace to the boundary of a function in H 1 (Ω). An answer to the first question in the previous subsection is given by the following theorem. This u = u(x) is called a weak solution to (7.1). If it is C 2 in Ω, then (7.5) holds for all v ∈ C0∞ (Ω) and hence u(x) is harmonic in Ω. At this moment, however, u(x) is just in E = {v ∈ H 1 (Ω) | v = f on 𝜕Ω}. The first question of the existence of a weak solution, thus, induces the second question of its regularity, that is, Hilbert’s 19th problem. Theorem 7.4. The minimum j of J(v) = which satisfies

1 ∫ |∇v|2 dx 2 Ω

∫ ∇u ⋅ ∇v dx = 0,

for v ∈ E is attained by some u ∈ E,

v ∈ E.

(7.9)

Ω

Proof of Theorem 7.4. If u ∈ E satisfies E(u) = j ≡ infE J, then we obtain (7.4) by the argument described in §7.1.1. To show the existence of such u, we take a minimizing sequence {vk }, considering j ≥ 0: vk ∈ E, J(vk ) → j. It is obvious that ‖∇vk ‖2 = O(1). Since vk − f ∈ H01 (Ω), 󵄩 󵄩 ‖vk − f ‖2 ≤ C 󵄩󵄩󵄩∇(vk − f )󵄩󵄩󵄩2

(k = 1, 2, . . . )

with C = C(Ω) > 0, and hence 󵄩 󵄩2 ‖vk − f ‖2H 1 (Ω) = ‖vk − f ‖22 + 󵄩󵄩󵄩∇(vk − f )󵄩󵄩󵄩2 = O(1).

400 | 7 Hardy spaces and BMO This {vk − f } is thus bounded in H01 (Ω), and there is a subsequence, denoted by the same symbol, which converges weakly to some w ∈ H01 (Ω): vk − f ⇀ w in H01 (Ω).5 Then it holds that v = w + f ∈ E. We employ ‖∇⋅‖2 as a norm in H01 (Ω) by (7.8). The lower semicontinuity of the norm with respect to weak convergence then implies 󵄩 󵄩 󵄩 󵄩 lim inf󵄩󵄩󵄩∇(vk − f )󵄩󵄩󵄩2 ≥ ‖∇w‖2 = 󵄩󵄩󵄩∇(v − f )󵄩󵄩󵄩2 . k→∞

(7.10)

Since the functional w 󳨃→ T(w) = (∇w, ∇f ) is bounded linear on H01 (Ω), on the other hand, it holds that (∇(vk − f ), f ) → (∇(v − f ), ∇f ).

(7.11)

Using (7.10), (7.11), and 󵄩󵄩 󵄩2 2 2 󵄩󵄩∇(vk − f )󵄩󵄩󵄩2 = ‖∇vk ‖2 − 2(∇vk , ∇f ) + ‖∇f ‖2 = ‖∇vk ‖22 − 2(∇(vk − f ), ∇f ) − ‖∇f ‖22 , we obtain 󵄩 󵄩2 lim inf ‖∇vk ‖22 ≥ 󵄩󵄩󵄩∇(v − f )󵄩󵄩󵄩2 + 2(∇(v − f ), ∇f ) + ‖∇f ‖22 = ‖∇v‖22 k→∞ and hence

1 1 j = lim inf ‖∇vk ‖2 ≥ ‖∇v‖22 = J(v) ≥ inf J = j. E k→∞ 2 2

We thus observe that j = infE J is attained by this v. 7.1.3 p-harmonic functions The Sobolev space W 1,p (Ω), 1 ≤ p ≤ ∞, is the set of functions which belong to Lp (Ω) together with all of their derivatives of the first order. It forms a Banach space equipped with the norm ‖v‖W 1,p = ‖∇v‖p + ‖v‖p . If 1 ≤ p < ∞, its dual space W 1,p (Ω)󸀠 is identified

with W 1,p (Ω) for (1/p) + (1/p󸀠 ) = 1, and hence W 1,p (Ω) is reflexive if 1 < p < ∞. If 1 ≤ p < ∞ and Ω is a Lipschitz domain, γ in (7.7) is uniquely extended to an isomorphism from W 1,p (Ω)/W01,p (Ω) to L1/p (𝜕Ω), where W01,p (Ω) is the closure of C0∞ (Ω) in W 1,p (Ω), and, in particular, v ∈ W01,p (Ω) if and only if v ∈ W 1,p (Ω) and γv = 0. Poincaré inequality is also valid in the form of 󸀠

‖v‖p ≤ C‖∇v‖p ,

v ∈ W01,p (Ω),

and as a closed subspace of a reflexive Banach space, W01,p (Ω), 1 < p < ∞, is reflexive. 5 Weak convergence is denoted by ⇀.

7.1 Regularity of weak solutions | 401

In what follows, Ω is a Lipschitz domain and 1 < p < ∞ as in the previous subsection. We assume that f is the trace to the boundary of a function in W 1,p (Ω), denoted by the same symbol, and put E = {v ∈ W 1,p (Ω) | v = f on 𝜕Ω},

J(v) =

1 ∫ |∇v|p . p Ω

The argument in § 7.1.2 is valid for the variational problem j = infE J. Given the minimizing sequence {vk } ⊂ E, the family {vk − f } is bounded in W01,p (Ω) and there exists a subsequence, denoted by the same symbol, which converges weakly to some w ∈ W01,p (Ω). To deduce lim inf ‖∇vk ‖p ≥ ‖∇v‖p , k→∞

v = w + f,

(7.12)

and j = J(v), we apply Rellich–Kondrachov’s theorem in the form of Theorem 1.15. It follows that vk − f → w = v − f in Lp (Ω), and, in particular, (7.13)

‖vk ‖p → ‖v‖p . Since vk ⇀ v in W 1,p (Ω), it holds also that lim inf ‖vk ‖W 1,p ≥ ‖v‖W 1,p , k→∞

and then (7.12) follows from (7.13). Hence j = infE J is attained by this v. Here we change the notation, and let u = u(x) be the solution to this variational problem. Then, for any v ∈ E it holds that d J(u + tv)|t=0 = 0. dt In fact, since 1 d 󵄨󵄨 󵄨p 󵄨 󵄨p−2 󵄨∇(u + tv)󵄨󵄨󵄨 = 󵄨󵄨󵄨∇(u + tv)󵄨󵄨󵄨 ∇(u + tv) ⋅ ∇v p dt 󵄨

a. e.,

the dominated convergence implies ∫ |∇u|p−2 ∇u ⋅ ∇v = 0,

v ∈ E.

(7.14)

Ω

We call such u ∈ E is a weak solution to Δp u ≡ div(|∇u|p−2 ∇u) = 0

in Ω,

and call the nonlinear operator Δp the p-Laplacian.

u=f

on 𝜕Ω,

(7.15)

402 | 7 Hardy spaces and BMO 1,p (Ω), The first equality in (7.15) has a meaning in the sense of distributions if u ∈ Wloc that is,

∫ |∇u|p−2 ∇u ⋅ ∇ϕ = 0, Ω

ϕ ∈ C0∞ (Ω).

(7.16)

1,p Such u = u(x) is called a p-harmonic function. Here, Wloc (Ω) denotes the set of measurable functions v = v(x) satisfying ϕv ∈ W 1,p (Ω) for any ϕ ∈ C0∞ (Ω). If u ∈ W 1,p (Ω)

in (7.16), we can extend ϕ to W01,p (Ω) there, because |∇u|p−2 ∇u ∈ Lp (Ω) in this case. In case p ≠ 2, the coefficient a(x) = |∇u|p−2 ≥ 0 in Δp u is unknown, and it may be the case that a(x) = 0 somewhere in Ω. Thus the method of converting an elliptic boundary value problem to an integral equation, efficient in the linear theory, does not work to assure the regularity of this weak solution u(x). In fact, any 2-harmonic distribution is real-analytic,6 while in the case of p ≠ 2, it is known that C 1,α -regularity is best possible. Here and henceforth, 0 < α < 1 is an exponent, and for a nonnegative integer m, C m,α (Ω) denotes the space of C m -functions in Ω having mth order derivatives Hölder continuous with the exponent α. This section is devoted to the proof of C α -regularity of this weak solution in the case of 1 < p < n.7 Actually, if p > n, Morrey’s theorem 1.16 is applicable to ϕ ⋅ u for 1,p ϕ ∈ C0∞ (Ω), which guarantees u ∈ C α (Ω) for u ∈ Wloc (Ω). For 1 < p < n, on the contrary, we have the Sobolev inequality in the form of 󸀠

‖v‖ np ≤ C‖∇v‖p , n−p

v ∈ W01,p (Ω).

(7.17)

We have already mentioned that p = n is an extremal case of the embedding of W 1,p (Ω). This property is valid for the original variational problem because of the scaling invariance of the functional J(v) =

1 󵄨󵄨 󵄨p ∫󵄨󵄨∇v(x)󵄨󵄨󵄨 dx, p

p = n,

Ω

under the transformation y = Rx. 7.1.4 Local maximum principle Several notions in the theory of harmonic functions are still valid for p-harmonic functions. For example, a p-subharmonic function satisfies a variant of weak maximum principle. This notion of subharmonicity, however, is defined in the sense of distribu6 This is Weyl’s lemma, which is derived from C 2,α regularity. 7 More detailed properties, including the case p = n, are discussed in [157].

7.1 Regularity of weak solutions | 403

1,p (Ω) p-subharmonic if it satisfies tions as in (7.16). Thus we call u ∈ Wloc

∫ |∇u|p−2 ∇u ⋅ ∇ϕ ≤ 0, Ω

0 ≤ ϕ ∈ C0∞ (Ω).

(7.18)

The test function ϕ ≥ 0 is then extended to be in W 1,p (Ω) with compact support. In fact, given 0 ≤ ϕ ∈ W 1,p (Ω) with supp ϕ ⊂ Ω, its mollifier {ϕk } approximating ϕ in W 1,p (Ω) can be 0 ≤ ϕk ∈ C0∞ (Ω). Since |∇u|p−2 ∇u ∈ Lploc (Ω), the limit of 󸀠

∫ |∇u|p−2 ∇u ⋅ ∇ϕk ≤ 0 Ω

as k → ∞ realizes (7.18). 1,p Let u ∈ Wloc (Ω), 1 < p < n, be a p-subharmonic function. Assume 0 ∈ Ω without loss of generality, and B2R = B2R (0) ⊂⊂ Ω.8 Given 0 < ρ < r ≤ 2R, we can take 0 ≤ η = η(|x|) ∈ C0∞ (Br ), satisfying9 η=1

on Bρ ,

|∇η| ≤ C/(r − ρ).

(7.19)

1,p 1,p As v ∈ Wloc (Ω) implies |v| ∈ Wloc (Ω) and

∇v, { { ∇|v| = { 0, { { −∇v,

v > 0, v = 0, v < 0,

a. e.,

1,p we obtain ut± ≡ (u± ∧ t) ∨ t −1 ∈ Wloc (Ω) for t > 1, where a ∧ b = min{a, b} and a ∨ b = max{a, b}. Each α ∈ R, furthermore, admits α+1 p

∇((ut± )

α

η ) = (α + 1)(ut± ) ηp ∇ut± + p(ut± )

α+1 p−1

η

∇η

a. e.

(7.20)

By (7.20) and for t −1 ≤ ut± ≤ t, it holds that 0 ≤ ϕ = (ut± )α+1 ηp ∈ W 1,p (Ω) and supp ϕ ⊂ Ω. Substituting this ϕ into (7.18), we obtain α

α+1

(α + 1) ∫ |∇u|p−2 (ut± ) ηp ∇u ⋅ ∇ut± ≤ p ∫ |∇u|p−1 (ut± ) Ω

|∇η|ηp−1 .

Ω

The monotone convergence theorem now implies α

∫ |∇u|p−2 (ut± ) ηp ∇u ⋅ ∇ut± =

Ω

∫ |∇u|p−1 (ut± )

Ω

α+1

∫ t −1 0. Let α > −1 in (7.22). Since p 󵄨󵄨 α α α 󵄨󵄨󵄨p α 󵄨p 󵄨 󵄨 ∫󵄨󵄨󵄨∇(|u| p u) ⋅ η󵄨󵄨󵄨 = ∫󵄨󵄨󵄨( + 1)|u| p ∇u󵄨󵄨󵄨 ηp = ( + 1) ∫ |∇u|p |u|α ηp 󵄨󵄨 p 󵄨󵄨 p Ω

Ω

Ω

and 󵄨󵄨 α 󵄨󵄨p−1 α+p α α 󵄨 󵄨p−1 α+p 󵄨 󵄨 ∫󵄨󵄨󵄨∇(|u| p u) ⋅ η󵄨󵄨󵄨 |u| p |∇η| = ∫󵄨󵄨󵄨( + 1)|u| p ∇u ⋅ η󵄨󵄨󵄨 |u| p |∇η| 󵄨󵄨 p 󵄨󵄨

Ω

Ω

=(

α + 1) p

p−1

∫ |∇u|p−1 |u|α+1 |∇η|ηp−1 , Ω

there exists C(α) > 0 such that α α 󵄨p 󵄨p−1 α+p 󵄨 󵄨 ∫󵄨󵄨󵄨∇(|u| p u) ⋅ η󵄨󵄨󵄨 ≤ C(α) ∫󵄨󵄨󵄨∇(|u| p u) ⋅ η󵄨󵄨󵄨 |u| p |∇η|.

Ω

Ω

Young’s inequality, ap bp + , 󸀠 p p 󸀠

ab ≤

a, b ≥ 0,

now implies that the right-hand side of (7.23) is estimated above by α 1 󵄨󵄨 󵄨p ∫󵄨∇(|u| p u) ⋅ η󵄨󵄨󵄨 + C 󸀠 (α) ∫ |u|α+p |∇η|p , 2 󵄨

Ω

Ω

1 1 + = 1, p p󸀠

the first term of which is absorbed by the left-hand side of (7.23): α 󵄨 󵄨p ∫󵄨󵄨󵄨∇(|u| p u) ⋅ η󵄨󵄨󵄨 ≤ 2C 󸀠 (α) ∫ |u|α+p |∇η|p .

Ω

Ω

(7.23)

7.1 Regularity of weak solutions | 405

Then we obtain α α 󵄨 󵄨p 󵄨p 󵄨 ∫󵄨󵄨󵄨∇(|u| p uη)󵄨󵄨󵄨 ≤ Cp ∫󵄨󵄨󵄨∇(|u| p u) ⋅ η󵄨󵄨󵄨 + |u|α+p |∇η|p dx

Ω

Ω

≤ C(α) ∫ |u|α+p |∇η|p ,

(7.24)

Ω

using the same symbol to indicate different generic constant. Sobolev’s inequality (7.17) applied to the left-hand side of (7.24) now guarantees {∫(|u|

α +1 p

η)

np n−p

n−p n

}

≤ C(α) ∫ |u|α+p |∇η|p .

Ω

Letting θ =

n n−p

Ω

> 1, β = α + p, and α > −1, now we obtain {∫ |u|βθ ηpθ }

1/θ

≤ C(α) ∫ |u|β |∇η|p ,

Ω

(7.25)

Ω 10

where C(α) is a rational function of α.

Proposition 7.5 (Local maximum principle). Given 1 < p < n and γ > p − 1, we have 1,p C = C(n, p, γ) > 0 such that any p-subharmonic function u ∈ Wloc (Ω) satisfies ‖u‖L∞ (BR ) ≤ C{

1 ∫ |u|γ } |B2R |

1/γ

,

B2R ⊂ Ω.

(7.26)

B2R

Proof. Applying (7.25) to (7.19), we obtain 1/θ

{

1 ∫ |u|βθ } |Bρ |

≤ C(α)(r − ρ)−p ⋅ r n ⋅ ρ−n/θ

Bρ

1 ∫ |u|β . |Br |

(7.27)

Br

Given γ > p − 1, let γ = β0 ≡ α0 + p. It holds that α0 > −1. Let βi = β0 θi ↑ +∞ and Ri = R(1 + 2−i ) ∈ (0, 2R) for i = 0, 1, 2, . . . , and apply (7.27) for β = βi , r = Ri , and ρ = Ri+1 . First, the inequality n

(r − ρ)−p ⋅ r n ⋅ ρ−n/θ = 2(i+1)p ⋅ (1 + 2−i ) ⋅ (1 + 2−i−1 )

−n/θ

≤C

follows from r − ρ = 2−i−1 R and −p + n − (n/θ) = 0. Second, since C(α) in (7.27) is a rational function of α > −1, there is m ≫ 1 such that C(αi ) ≤ θ(i+1)m for i = 0, 1, 2, . . . , where αi = βi − p ↑ ∞. Thus we obtain C = C(n, p, γ) > 1 satisfying {

1 |BRi+1 |

∫ |u|βi+1 } BRi+1

1/θ

≤

C i+1 ∫ |u|βi . |BRi | BRi

10 The argument used for the following proposition is called Moser’s iteration scheme [230].

(7.28)

406 | 7 Hardy spaces and BMO This inequality means ϕi+1 ≤ C (i+1)/βi ϕi for ϕi ≡ { |B1 | ∫B |u|βi }1/βi , and hence it Ri

follows that

i

ϕi ≤ C ∑ℓ=0 (ℓ+1)/βℓ ϕ0 ≤ C ∑ℓ=0 (ℓ+1)/(β0 θ ) ϕ0 = C 󸀠 { ∞

ℓ

Ri

1 ∫ |u|γ } |BR |

1/γ

.

BR

Here, since 2R > Ri > R and βi → +∞, we obtain ϕi = {

1/βi

1 ∫ |u|βi } |BRi |

1/βi

≥{

BRi

1 ∫ |u|βi } |B2R |

→ ‖u‖L∞ (BR ) ,

BR

and hence (7.26). 7.1.5 Local minimum principle A p-superharmonic function is defined similarly: ∫ |∇u|p−2 ∇u ⋅ ∇ϕ ≥ 0, Ω

0 ≤ ϕ ∈ C0∞ (Ω).

(7.29)

A delicate argument ensures the local minimum principle for p-superharmonic func1,p tion, 0 ≤ u ∈ Wloc (Ω). We assume B8R ⊂⊂ Ω and take 0 ≤ η = η(|x|) ∈ C0∞ (Br ) satisfying (7.19) for 0 < ρ < r ≤ 4R. Substituting 0 ≤ ϕ = uα+1 ηp into (7.29), we obtain a reverse inequality of (7.22), (α + 1) ∫ |∇u|p uα ηp ≥ −p ∫ |∇u|p−1 uα+1 |∇η|ηp−1 , Ω

Ω

which implies ∫ |∇u|p uα ηp ≤ Ω

p ∫ |∇u|p−1 uα+1 |∇η|ηp−1 −(α + 1)

(7.30)

Ω

for α < −1. We repeat the arguments in §7.1.4 based on Young’s and Sobolev’s inequalities, using θ = n/(n − p) > 1, β = α + p, and α < −1. We reach {∫ uβθ ηpθ }

1/θ

≤ C(α) ∫ uβ |∇η|p , Ω

Ω

and hence (7.19) implies {

1 ∫ uβθ } |Bρ | Bρ

1/θ

≤ C(α)(r − ρ)−p r n ρ−n/θ

1 ∫ uβ |Br | Br

(7.31)

7.1 Regularity of weak solutions | 407

Given γ < 0, we put γ = β0 ≡ α0 + p, βi = β0 θi , and Ri = R(1 + 3 ⋅ 2−i ) for i = 0, 1, 2, . . . ,11 to apply (7.31) for β = βi , r = Ri , and ρ = Ri+1 . Then, inequality (7.28) is replaced by {

1 |BRi+1 |

∫ uβi+1 }

1/θ

≤

BRi+1

C i+1 ∫ uβi , |BRi |

i = 0, 1, 2, . . .

BRi

for C = C(n, p, γ) > 1, which means ϕi+1 ≤ C (i+1)/(−βi ) ϕi ,

ϕi = {

1 −β ∫ (u−1 ) i } |BRi |

1/(−βi )

.

BRi

Then we obtain ϕi ≤ C 󸀠 ϕ0 = C 󸀠 {

1 ∫ uγ } |B4R |

−1/γ

B4R

similarly, while ϕi → ‖u−1 ‖L∞ (BR ) = (ess.infBR u)−1 as i → ∞. The following lemma thus holds with σ = −γ > 0. Lemma 7.6. There is C = C(n, p, σ) > 0 determined by 1 < p < n and σ > 0 such that for 1,p any nonnegative p-superharmonic function u ∈ Wloc (Ω) and B4R ⊂⊂ Ω it holds that C ⋅ ess.infBR u ≥ {

1 ∫ u−σ } |B4R |

−1/σ

(7.32)

.

B4R

Inequality (7.31) is still valid even for 0 < β < p − 1. Given 2 ≤ ℓ < m ≤ 4, therefore, we have C(β, ℓ, m) > 0 such that 1/(βθ)

{

1 ∫ uβθ } |BℓR |

≤ C(β, ℓ, m){

BℓR

1 |BmR |

∫ uβ }

1/β

.

BmR

We now apply this inequality finitely many times. Since { |B1 | ∫B uβ }1/β is monotone 4R 4R increasing in β > 0, we obtain 1/(γθ)

{

1 ∫ uγθ } |B2R |

≤ C{

B2R

1 ∫ uσ } |B4R |

1/σ

B4R

for 0 < γ < p − 1 and σ > 0 with C = C(n, p, γ, σ) > 0. Here is a key lemma. 11 Note α0 < −1, β0 < 0, βi ↓ −∞, and 0 < Ri ≤ 4R.

(7.33)

408 | 7 Hardy spaces and BMO Lemma 7.7. Each 1 < p < n admits C = C(n, p) > 0 and σ0 = σ0 (n, p) > 0 such that 1,p for any p-superharmonic function 0 ≤ u ∈ Wloc (Ω) and B8R ⊂⊂ Ω there is σ > σ0 satisfying {

1 ∫ uσ } |B4R |

1/σ

1/σ

1 ∫ u−σ } |B4R |

⋅{

B4R

≤ C.

(7.34)

B4R

If inequalities (7.32) and (7.33) are valid with σ = σ1 > 0 then they are valid with any σ ≥ σ1 . Hence we may assume σ > σ0 there. Then inequality (7.34) implies the following estimate. Proposition 7.8 (Local minimum principle). Given 1 < p < n and 0 < γ
0 such that any p-superharmonic function 0 ≤ u ∈ Wloc (Ω) admits

{

1 ∫ uγ } |B2R | B2R

1/γ

≤ C ⋅ ess.infBR u,

(7.35)

provided that B8R ⊂ Ω. 7.1.6 BMO, Harnack inequality, and C α regularity 1,p To prove (7.34), let 0 ≤ u ∈ Wloc (Ω) be a p-superharmonic function and put α = −p in (7.30). By Young’s inequality, we obtain

∫ |∇ log u|p ηp = ∫ |∇u|p u−p ηp Ω

Ω

≤

p ∫ |∇u|p−1 u−p+1 |∇η|ηp−1 p−1 Ω

=

p ∫ |∇ log u|p−1 ηp−1 |∇η| p−1 Ω

≤

1 ∫ |∇ log u|p ηp + Cp ∫ |∇η|p , 2 Ω

Ω

and hence ∫Ω |∇ log u|p ηp ≤ C ∫Ω |∇η|p . Then (7.19) implies ∫ |∇ log u|p ≤ C(r − ρ)−p |Br |, Bρ

7.1 Regularity of weak solutions | 409

and therefore, putting r = 2ρ, we obtain 1 ∫ |∇ log u|p ≤ Cρ−p , |Bρ |

0 < ρ < R.

(7.36)

Bρ

Here we use the following inequality proven in § 7.2. Proposition 7.9 (Poincaré–Sobolev). Each 1 ≤ p < n admits C = C(n, p) > 0 such that ∗ 1 { ∫ |v − vBr (x) |p } |Br (x)|

1/p∗

≤ Cr{

Br (x)

(7.37)

Br (x)

for any x ∈ Rn , r > 0 and v ∈ W 1,p (Rn ), where vBr (x) =

1/p

1 ∫ |∇v|p } |Br (x)|

1 p∗

=

1 p

−

1 n

and

1 ∫ v. |Br (x)| Br (x)

The left-hand side of (7.37) is estimated below by this inequality to v = log u, then, we obtain

1 |v ∫ |Br (x)| Br (x)

1 1 󵄨 󵄨 ∫ 󵄨󵄨log u − (log u)Bρ 󵄨󵄨󵄨 ≤ Cρ{ ∫ |∇ log u|p } |Bρ | 󵄨 |Bρ | Bρ

1/p

− vBr (x) |. Applying

≤C

(7.38)

Bρ

by (7.36). A measurable function v = v(x) in a domain Ω ⊂ Rn is said to be BMO (bounded mean oscillation) if there is C > 0 such that 1 ∫ |v − vB | < +∞. B⊂⊂Ω |B|

(7.39)

‖ log u‖BMO ≤ C.

(7.40)

‖v‖BMO = sup

B

Hence (7.38) means

Any L∞ -function is BMO, but the converse is not always true. A typical example is log |x|. This gap is represented by the growth of distribution function.12 The following fact is also proven in § 7.2. Henceforth, kB denotes the concentric ball of B = Br (x0 ) with the radius kr. Proposition 7.10 (John–Nirenberg). For c1 = c1 (n) > 0 and c2 = c2 (n) > 0 determined by the dimension n, it holds that 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨{x ∈ B | 󵄨󵄨󵄨v(x) − vB 󵄨󵄨󵄨 > t}󵄨󵄨󵄨 ≤ c1 |B| exp(−c2 t/‖v‖BMO )

(7.41)

for any B such that √nB ⊂⊂ Ω and t > 0. 12 Introduction of BMO function and the proof of C α regularity of the weak solution to the uniformly elliptic equation are given by [169] and [230], respectively.

410 | 7 Hardy spaces and BMO Proof of Lemma 7.7. For the BMO function v = log u and a ball B = B4R ⊂⊂ Ω, we put μ(t) = |{x ∈ B | |v(x) − vB | > t}|. Taking 0 < s < c2 , we have 1 s 󵄨 󵄨 ⋅ 󵄨󵄨v(x) − vB 󵄨󵄨󵄨) dx ∫ exp( |B| ‖v‖BMO 󵄨 B

∞

1 s = ⋅ t) d(−μ(t)) ∫ exp( |B| ‖v‖BMO 0

t=∞

=

1 s ⋅ t)μ(t)] [− exp( |B| ‖v‖BMO t=0 ∞

1 s s ⋅ ⋅ t)μ(t) dt + ∫ exp( ‖v‖BMO |B| ‖v‖BMO 0

∞

c1 s c −s ⋅ t) dt ∫ exp(− 2 ‖v‖BMO ‖v‖BMO

≤1+

0

= 1 + c1 s[−

t=∞

c −s 1 ⋅ t)] exp(− 2 c2 − s ‖v‖BMO t=0

c1 s ≡ β, c2 − s

=1+

(7.42)

and therefore, 1 ∫ exp(±σ(v(x) − ‖v‖BMO )) dx ≤ β, |B|

σ = s/‖v‖BMO .

B

We thus obtain 1 1 ∫ eσv ⋅ ∫ e−σv ≤ β2 |B| |B| B

B

and hence 1/σ

{

1 ∫ uσ } |B| B

1/σ

⋅{

1 ∫ u−σ } |B|

‖ log u‖BMO /s

≤ (β2 )

≤C

B

by v = log u. This inequality means (7.34). Since (7.40) holds, this σ can be σ > σ0 for σ0 = σ0 (n, p) > 0. n The requirements for the exponents are γ > p − 1 and 0 < γ < n−p ⋅ (p − 1) for the local maximum and minimum principles, respectively. Hence Propositions 7.5 and 7.8 imply the following fact.

7.1 Regularity of weak solutions | 411

Corollary 7.11 (Harnack inequality). If 1 < p < n, we have C = C(n, p) > 1 such that any 1,p (Ω) satisfies p-harmonic function 0 ≤ u ∈ Wloc ess.supBR u ≤ C ess.infBR u

(7.43)

if B8R ⊂ Ω. The conclusion of this section is the following result Theorem 7.12. Each 1 < p < n admits 0 < α < 1 such that if E ⊂⊂ Ω is a compact set 1,p and 0 ≤ u ∈ Wloc (Ω) is p-harmonic then it holds that 󵄨󵄨 󵄨 α 󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 ≤ C|x − y| ,

x, y ∈ E

with C > 0. 1,p Proof. We write inf and sup for ess.inf and ess.sup, respectively. Let 0 ≤ u ∈ Wloc (Ω) 1 be p-harmonic, and take x0 ∈ Ω and 0 < ρ < 16 dist(x0 , Ω). We put

Mr = sup u, Br (x0 )

mr = inf u, Br (x0 )

0 < r < 16ρ.

Note that u ∈ L∞ loc (Ω), derived from the local maximum principle. Since u(x) − m16ρ ≥ 0 is p-harmonic in B8ρ (x0 ) ⊂⊂ Ω, it holds that sup {u − m16ρ } = Mρ − m16ρ ≤ C inf {u − m16ρ } Bρ (x0 )

Bρ (x0 )

= C(mρ − m16ρ )

(7.44)

by (7.43). The function M16ρ − u(x) has the same property, and therefore, M16ρ − mρ ≤ C(M16ρ − Mρ ).

(7.45)

Adding (7.44) to (7.45), we have (C + 1)(Mρ − mρ ) ≤ (C − 1)(M16ρ − m16ρ ), which means ω(ρ) ≤ θω(16ρ) for ω(ρ) ≡ Mρ − mρ and 0 < θ = (C − 1)/(C + 1) < 1. Then an iteration ensures ω(ρ ⋅ 16−i+1 ) ≤ θi ω(16ρ),

i = 1, 2, . . .

Writing θ = 16−α , we obtain α > 0 and ω(ρ ⋅ 16−i+1 ) ≤ (16−i )α ω(16ρ). For r2 = 16ρ and r1 = ρ ⋅ 16−i+1 , furthermore, this inequality means α

r ω(r1 ) ≤ ( 1 ) ω(r2 ), r2

0 < r1 < r2 < dist(x0 , 𝜕Ω).

For r2 = dist(x0 , 𝜕Ω)/2 and x ∈ 𝜕Br1 (x0 ), we obtain 1 󵄨󵄨 󵄨 α 󵄨󵄨u(x) − u(x0 )󵄨󵄨󵄨 ≤ ω(r1 ) ≤ |x − x0 | ω(r2 ), r2

x ∈ Br2 (x0 ).

Then a standard covering argument guarantees the result.

(7.46)

412 | 7 Hardy spaces and BMO

7.2 BMO 7.2.1 Poincaré–Sobolev inequality To complete the proof of Theorem 7.12, we show (7.37) and (7.41) in this section. First, inequality (7.37) follows from Sobolev’s inequality (7.17) and the following lemma. Lemma 7.13. Each 1 ≤ p < n admits C = C(n, p) > 0 such that 󵄨p 󵄨p 󵄨 󵄨 ∫ 󵄨󵄨󵄨v(y) − v(z)󵄨󵄨󵄨 dy ≤ Cr n+p−1 ∫ 󵄨󵄨󵄨∇v(y)󵄨󵄨󵄨 |y − z|1−n dy

(7.47)

Br (x)

Br (x)

for z ∈ Br (x) ⊂ Rn and v ∈ C 1 (Br (x)). Proof. Since 1

t

v(y) − v(z) = ∫ 0

d v(z + t(y − z)) dt = ∫ ∇v(z + t(y − z)) dt ⋅ (y − z), dt 0

it holds that 1

󵄨󵄨 󵄨p 󵄨p p 󵄨 󵄨󵄨v(y) − v(z)󵄨󵄨󵄨 ≤ |y − z| ∫󵄨󵄨󵄨∇v(z + t(y − z))󵄨󵄨󵄨 dt. 0

n

n−1

Let dL and dH be the n-dimensional Lebesgue measure and (n − 1)-dimensional Hausdorff measure, respectively [112].13 Given s > 0, we have ∫ Br (x)∩𝜕Bs (z) 1 p

󵄨󵄨 󵄨p n−1 󵄨󵄨v(y) − v(z)󵄨󵄨󵄨 dH (y)

≤ s ∫ dt 0

∫ Br (x)∩𝜕Bs (z)

󵄨󵄨 󵄨p n−1 󵄨󵄨∇v(z + t(y − z))󵄨󵄨󵄨 dH (y).

(7.48)

1 dH n−1 (w), while y, z ∈ Br (x) and Letting w = z + t(y − z), one has dH n−1 (y) = t n−1 y ∈ 𝜕Bs (z) imply w ∈ Br (x) and |w − z| = ts, respectively. Hence the right-hand side of (7.48) is estimated above by p

1

s ∫ 0

dt t n−1 1

= sp ∫ 0

∫ Br (x)∩𝜕Bts (z)

dt

t n−1 1

∫ Br (x)∩𝜕Bts (z)

= sn+p−1 ∫ dt 0

󵄨󵄨 󵄨p n−1 󵄨󵄨∇v(w)󵄨󵄨󵄨 dH (w) 󵄨󵄨 󵄨p 1−n n−1 n−1 󵄨󵄨∇v(w)󵄨󵄨󵄨 |w − z| dH (w) ⋅ (ts)

∫ Br (x)∩𝜕Bts (z)

󵄨󵄨 󵄨p 1−n n−1 󵄨󵄨∇v(w)󵄨󵄨󵄨 |w − z| dH (w).

(7.49)

13 In what follows we use dH n−1 for an area element of an (n − 1)-dimension spherical surface.

7.2 BMO

If g : Rn → [0, ∞] is measurable, it holds that ∞

∫ g dLn = ∫ dρ ∫ g dH n−1 0

Rn

𝜕Bρ (z)

for any z ∈ Rn . Using ts = ρ, therefore, we reach 1

∫ dt 0

g dH n−1

∫ Br (x)∩𝜕Bts (z) ∞

= s−1 ∫ χ(0,s) (ρ) dρ ∫ χBr (x) g dH n−1 0

𝜕Bρ (z)

= s−1 ∫ χBr (x) g dLn = s−1 Bs (z)

∫

g dLn .

Br (z)∩Br (x)

Hence the right-hand side of (7.49) is estimated above by sn+p−2

∫ Br (x)∩Bs (z)

󵄨󵄨 󵄨p 1−n n 󵄨󵄨∇v(w)󵄨󵄨󵄨 |w − z| dL (w),

which results in ∫ Br (x)∩𝜕Bs (z)

󵄨󵄨 󵄨p n−1 󵄨󵄨v(y) − v(z)󵄨󵄨󵄨 dH (y)

≤ sn+p−2

∫ Br (x)∩Bs (z)

󵄨󵄨 󵄨p 1−n n 󵄨󵄨∇v(w)󵄨󵄨󵄨 |w − z| dL (w).

r

Applying ∫0 ⋅ ds, we thus obtain r

∫ ds 0

∫ Br (x)∩𝜕Bs (z)

󵄨󵄨 󵄨p n−1 󵄨󵄨v(y) − v(z)󵄨󵄨󵄨 dH (y)

󵄨 󵄨p = ∫ 󵄨󵄨󵄨v(y) − v(z)󵄨󵄨󵄨 dLn (y) Br (x) r

≤ ∫ sn+p−2 ds 0 r

∫ Br (x)∩Bs (z)

󵄨󵄨 󵄨p 1−n n 󵄨󵄨∇v(w)󵄨󵄨󵄨 |w − z| dL (w)

󵄨 󵄨p ≤ ∫ sn+p−2 ds ∫ 󵄨󵄨󵄨∇v(w)󵄨󵄨󵄨 |w − z|1−n dLn (w) 0

Br (x)

󵄨 󵄨p = Cr n+p−1 ∫ 󵄨󵄨󵄨∇v(w)󵄨󵄨󵄨 |w − z|1−n dLn (w). Br (x)

| 413

414 | 7 Hardy spaces and BMO Proof of Proposition 7.9. It suffices to show (7.37) for v ∈ C 1 (Br (x)). The Sobolev embed∗ ∗ ding W01,p (Ω) ⊂ Lp (Ω) in (7.17) takes the form W 1,p (Ω) ⊂ Lp (Ω) by the extension operator if the boundary 𝜕Ω is Lipschitz continuous, that is, ‖v‖p∗ ≤ C‖v‖W 1,p , v ∈ W 1,p (Ω), for C = C(n, p, Ω) > 0. In particular, there is C1 > 0 such that 1/p∗

󵄨p∗ 󵄨 { ∫ 󵄨󵄨󵄨g(x)󵄨󵄨󵄨 dx}

󵄨p 󵄨p 󵄨 󵄨 ≤ C1 { ∫ 󵄨󵄨󵄨∇g(x)󵄨󵄨󵄨 + 󵄨󵄨󵄨g(x)󵄨󵄨󵄨 dx}

1/p

B1 (0)

B1 (0)

for any g ∈ W 1,p (B1 (0)). Put y = rx and f = rg, and observe 1/p∗

󵄨 󵄨p∗ { ∫ 󵄨󵄨󵄨g(x)󵄨󵄨󵄨 dx} B1 (0)

1/p∗

∗ 󵄨 󵄨p∗ = { ∫ r −p 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 r −n dy}

Br (0)

1/p∗

∗ 1 = Cr { ∫ |f |p dy} |Br (0)|

−1

Br (0)

and 󵄨 󵄨p 󵄨 󵄨p { ∫ 󵄨󵄨󵄨∇g(x)󵄨󵄨󵄨 + 󵄨󵄨󵄨g(x)󵄨󵄨󵄨 dx}

1/p

B1 (0)

󵄨 󵄨p 󵄨 󵄨p = { ∫ (r p ⋅ r −p 󵄨󵄨󵄨∇f (y)󵄨󵄨󵄨 + r −p 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 ) ⋅ r −n dy} Br (0)

= C{

1/p

1/p

1 ∫ |∇f |p + r −p |f |p dy} |Br (0)|

.

Br (0)

It then follows that ∗ 1 { ∫ |f |p dy} |Br (x)|

1/p∗

≤ C{

Br (x)

1/p

1 ∫ r p |∇f |p + |f |p dy} |Br (x)|

(7.50)

Br (x)

for f ∈ W 1,p (Br (x)). By Lemma 7.13, on the other hand, we have 1 ∫ |v − vBr (x) |p dy |Br (x)| Br (x)

=

󵄨󵄨 1 󵄨󵄨p 1 󵄨 󵄨 ∫ 󵄨󵄨󵄨 ∫ v(y) − v(z) dz 󵄨󵄨󵄨 dy 󵄨󵄨 |Br (x)| 󵄨󵄨 |Br (x)| Br (x)

≤

1 |Br (x)|2

Br (x)

∬ Br (x)×Br (x)

󵄨󵄨 󵄨p 󵄨󵄨v(y) − v(z)󵄨󵄨󵄨 dy dz

(7.51)

7.2 BMO |

415

C 󵄨p 󵄨 ∫ r p−1 dz ∫ 󵄨󵄨󵄨∇v(y)󵄨󵄨󵄨 |y − z|1−n dy |Br (x)|

≤

Br (x)

Br (x)

1 󵄨p 󵄨 = C ∫ 󵄨󵄨󵄨∇v(y)󵄨󵄨󵄨 dy ⋅ r p−1 ∫ |y − z|1−n dz |B (x)| r

Br (x)

(7.52)

Br (x)

for v ∈ C 1 (Br (x)). Since y, z ∈ Br (x) implies z ∈ B2r (y), the right-hand side of (7.52) is estimated above by a constant multiple of 1 󵄨 󵄨p ∫ 󵄨󵄨󵄨∇v(y)󵄨󵄨󵄨 dy ⋅ r p−1 ⋅ n ∫ |y − z|1−n dz r

Br (x)

B2r (y)

󵄨 󵄨p = Cr (p−1)+(1−n) ∫ 󵄨󵄨󵄨∇v(y)󵄨󵄨󵄨 dy. Br (x)

We thus end up with Cr p 1 󵄨 󵄨p ∫ |v − vBr (x) |p dy ≤ ∫ 󵄨󵄨󵄨∇v(y)󵄨󵄨󵄨 dy. |Br (x)| |Br (x)| Br (x)

(7.53)

Br (x)

Then inequality (7.37) follows from (7.50) for f = v − vBr (x) and (7.53): 1/p∗

∗ 1 { ∫ |v − vBr (x) |p } |Br (x)|

Br (x)

≤ C{

1/p

1 ∫ r p |∇v|p + |v − vBr (x) |p dy} |Br (x)| Br (x)

≤ C{

1/p

1 ∫ r p |∇v|p } |Br (x)|

= Cr{

Br (x)

p

1 ∫ |∇v|p } . |Br (x)| Br (x)

7.2.2 Calderón–Zygmund decomposition The existence of κ > 0 such that 󵄨 󵄨 󵄨 󵄨 μ(t) = 󵄨󵄨󵄨{x ∈ B | 󵄨󵄨󵄨v(x) − vB 󵄨󵄨󵄨 > t}󵄨󵄨󵄨 ≤ c1 |B| exp(−κt),

∀t > 0,

B ⊂⊂ Ω

implies v ∈ BMO(Ω). In fact, similarly to (7.42), one has the inequality 1 ∫ eρ|v−vB | ≤ β󸀠 = β󸀠 (n), |Ω|

ρ = κ/2.

B

Then Jensen’s inequality implies κ 1 2 1 ∫ |v − vB | ≤ ∫ e 2 |v−vB | ≤ 2β󸀠 /κ. |B| κ |B|

B

B

416 | 7 Hardy spaces and BMO The original definition of a BMO function [64] uses cube Q for ball B in (7.39). This formulation takes advantage of the dyadic decomposition [322]. Lemma 7.14. Let Q be a cube such that the minimum ball B containing Q satisfies B ⊂ Ω. Then it holds that 1 ∫ |v − vQ | ≤ c0 (n)‖v‖BMO |Q|

(7.54)

Q

with c0 (n) > 0 determined by the dimension n. Proof. First, there is c1 (n) > 0 such that |B| ≤ c1 (n)|Q|. Second, we have 1 1 ∫ |v − vQ | ≤ ∫ |v − vB | + |vB − vQ |. |Q| |Q| Q

Q

Third, vB = (vB )Q implies 1 󵄨 󵄨 |vB − vQ | = 󵄨󵄨󵄨(v − vB )Q 󵄨󵄨󵄨 ≤ ∫ |v − vB |. |Q| Q

We thus obtain (7.54) from 2c (n) 1 2 ∫ |v − vQ | ≤ ∫ |v − vB | ≤ 1 ∫ |v − vB | ≤ 2c1 (n)‖v‖BMO . |Q| |Q| |B| Q

B

Q

Lemma 7.15 (Calderón–Zygmund). Let Q0 ⊂ Rn be a cube, v ∈ L1 (Q0 ), and

1 ∫ |Q0 | Q0

|v| ≤ s.

Then there is a countable family of disjoint cubes {Qk } such that |v| ≤ s a. e. in Q0 \ ∞ n −1 ⋃∞ k=1 Qk , |vQk | ≤ 2 s, ∀k, and ∑k=1 |Qk | ≤ s ∫Q |v|. 0

n

Proof. We decompose Q0 into 2 cubes of the same size, to distinguish them according to the average integral of |v|, that is, greater than or equal to s and smaller than s, denoted by {Q1k } and {Q󸀠1k }, respectively. It holds that s|Q1k | ≤ ∫ |v| ≤ 2n |Q1k | ⋅ Q1k

1 ∫ |v| ≤ 2n s|Q1k |. |Q0 | Q0

Then we divide each of {Q󸀠1k } into 2n cubes of the same size. We distinguish them according to the average integral of |v|, to get {Q2k } and {Q󸀠2k }, similarly. Since each Q2k is contained in some Q󸀠1k󸀠 , one obtains s|Q2k | ≤ ∫ |v| ≤ 2n |Q2k | ⋅ Q2k

1 ∫ |v| ≤ 2n s|Q2k |. |Q󸀠1k󸀠 | Q󸀠

1k 󸀠

7.2 BMO

| 417

Repeating this procedure, we get a countable family of disjoint cubes {Qmk }, which is relabeled by {Qk }. It thus holds that s|Qk | ≤ ∫ |v| ≤ 2n s|Qk |, Qk

which implies |vQk | ≤ 2n s. We have, furthermore, s ∑∞ k=1 |Qk | ≤ ∫⋃ hence

∑∞ k=1 |Qk |

k

≤ s ∫Q |v|. −1

Qk

|v| ≤ ∫Q |v|, and 0

0

If x0 ∈ ̸ ⋃k Qk , finally, there is a family of shrinking cubes {Q󸀠k } with the volume reduced by 1/(2n ) each time, such that x0 ∈ Q󸀠k and |Q1󸀠 | ∫Q󸀠 |v| < s. Then 1 󵄨 󵄨 lim ∫ |v| = 󵄨󵄨󵄨v(x0 )󵄨󵄨󵄨 ≤ s k→∞ |Q󸀠 | k

k

k

a. e. x0 ∈ ̸ ⋃ Qk

Q󸀠k

k

by Lebesgue differentiation theorem.14 7.2.3 John–Nirenberg’s inequality Let B = B(Q) be the minimal ball containing the cube Q. Regarding (7.54), we show the following lemma. Lemma 7.16. If Q is a cube such that B ⊂ Ω for B = B(Q), it holds that 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨{x ∈ Q | 󵄨󵄨󵄨v(x) − vQ 󵄨󵄨󵄨 > t}󵄨󵄨󵄨 ≤ c3 |Q| exp(−c4 t/‖v‖BMO ), t > 0, where c3 = c3 (n) > 0 and c4 = c4 (n) > 0 are constants. Proof. Using c0 (n) in (7.54), we define a > 0 by ‖w‖BMO =

1 c0 (n)

(7.55)

for v(x) = aw(x). Since

󵄨 󵄨 󵄨 󵄨 {x ∈ Q | 󵄨󵄨󵄨v(x) − vQ 󵄨󵄨󵄨 > t} = {x ∈ Q | 󵄨󵄨󵄨w(x) − wQ 󵄨󵄨󵄨 > t/|a|} and ‖v‖BMO = |a|‖w‖BMO , inequality (7.55) is reduced to that for v = w(x). Hence we assume 1 (7.56) ‖v‖BMO = c0 (n) without loss of generality. Let SQ (t) = {x ∈ Q | |v(x) − vQ | > t}, t > 0, and 󵄨 󵄨 F(t) = inf{C > 0 | 󵄨󵄨󵄨SQ (t)󵄨󵄨󵄨 ≤ C ∫ |v − vQ |, Q is a cube such that B(Q) ⊂ Ω}. Q

Since |SQ (t)| ≤

1 ∫ |v t Q

− vQ |, it holds that F(t) ≤ 1/t.

14 This {Q󸀠k } shrinks nicely to x0 for almost all x0 ∈ ̸ ⋃k Qk [118].

(7.57)

418 | 7 Hardy spaces and BMO Given t ≥ 2n and s ∈ [1, 2−n t], we have 1 ∫ |v − vQ | ≤ 1 ≤ s |Q| Q

by (7.56). Applying Lemma 7.15 to Q and v(x) − vQ for Q0 and v, respectively, we obtain a countable family of cubes {Qk } satisfying the requirements. Let S(t) = S0 (t) and Sk (t) = SQk (t) for k = 1, 2, . . . For t > s and |v − vQ | ≤ s a. e. in Q \ ⋃∞ k=1 Qk , we get ∞

S(t) ⊂ S(s) ⊂ ⋃ Qk k=1

except for a set of measure zero. We have, on the other hand, |(v −vQ )Qk | ≤ 2n s for k ≥ 1, and therefore, 󵄨󵄨 󵄨 󵄨 󵄨 n 󵄨󵄨v(x) − vQk 󵄨󵄨󵄨 = 󵄨󵄨󵄨(v(x) − vQ ) − (v(x) − vQ )Qk 󵄨󵄨󵄨 > t − 2 s,

x ∈ S(t) ∩ Qk .

These relations imply ∞

∞

k=1

k=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨 n 󵄨 󵄨󵄨S(t)󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨S(t) ∩ Qk 󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨{x ∈ Qk | 󵄨󵄨󵄨v(x) − vQk 󵄨󵄨󵄨 > t − 2 s}󵄨󵄨󵄨. By the definition of F(t) and (7.56), on the other hand, we obtain 󵄨󵄨 󵄨 󵄨 n 󵄨 n 󵄨󵄨{x ∈ Qk | 󵄨󵄨󵄨v(x) − vQk 󵄨󵄨󵄨 ≥ t − 2 s}󵄨󵄨󵄨 ≤ F(t − 2 s) ∫ |v − vQk | Qk

≤ F(t − 2n s)|Qk | and hence ∞

󵄨󵄨 󵄨 n −1 n 󵄨󵄨S(t)󵄨󵄨󵄨 ≤ F(t − 2 s) ∑ |Qk | ≤ s F(t − 2 s) ∫ |v − vQ | k=1

Q

by the third requirement of {Qk } in Lemma 7.15. This inequality means 󵄨󵄨 󵄨 −1 n 󵄨󵄨SQ (t)󵄨󵄨󵄨 ≤ s F(t − 2 s) ∫ |v − vQ |, Q

or F(t) ≤ s−1 F(t − 2n s), because Q is arbitrary. To complete the proof, let

2n e e−1

≤t≤

F(t) ≤

2n e e−1

1 ≤ s ≤ 2−n t,

t ≥ 2n ,

(7.58)

+ 2n e and apply (7.57) to deduce

12 −n −αt ⋅ 2 ⋅ e ≡ Ae−αt 10

(7.59)

7.3 Hardy spaces | 419

for α = 1/(2n e). Then inequality (7.58) for s = e implies F(t + 2n e) ≤ We thus obtain (7.59) for any t ≥

2n e e−1

n 1 −αt Ae = Ae−α(t+2 e) . e

≡ t0 , and hence

󵄨 󵄨󵄨 −αt 1 ∫ |v − vQ0 | ≤ Ae−αt |Q|, 󵄨󵄨S0 (t)󵄨󵄨󵄨 ≤ Ae |Q0 |

t ≥ t0 .

Q0

Combining this inequality with 󵄨󵄨 󵄨 αt −αt 󵄨󵄨S0 (t)󵄨󵄨󵄨 ≤ |Q0 | ≤ e 0 ⋅ e |Q|,

0 < t < t0 ,

we obtain (7.55) with c3 = max{A, eαt0 } and c4 = α. Proof of Proposition 7.10. Given B with √nB ⊂ Ω, let Q be the minimal cube containing B. Then it follows that B(Q) ⊂ Ω. Since 󵄨 󵄨 󵄨 󵄨 {x ∈ B | 󵄨󵄨󵄨v(x) − vB 󵄨󵄨󵄨 > t} ⊂ {x ∈ Q | 󵄨󵄨󵄨v(x) − vQ 󵄨󵄨󵄨 > t − |vB − vQ |} and from Lemma 7.16, we have 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨{x ∈ B | 󵄨󵄨󵄨v(x) − vB 󵄨󵄨󵄨 > t}󵄨󵄨󵄨

≤ c3 |Q| exp(−c4 max{0, t − |vB − vQ |}/‖v‖BMO )

≤ c3 |Q| exp(−c4 t/‖v‖BMO ) ⋅ exp(c4 |vB − vQ |/‖v‖BMO ). Then Lemma 7.14 implies 1 󵄨 󵄨 |vB − vQ | = 󵄨󵄨󵄨(v − vQ )B 󵄨󵄨󵄨 ≤ ∫ |v − vQ | |B| B

c (n) ≤ 1 ∫ |v − vQ | ≤ c1 (n)c0 (n)‖v‖BMO , |Q| Q

and hence the result.

7.3 Hardy spaces 7.3.1 Maximal theorem We have established C α regularity of p-harmonic functions by Moser’s iteration method. There is, however, a wide class of variational problems which do not admit even weak solutions. Given a minimization problem, for example, we extract a

420 | 7 Hardy spaces and BMO minimizing sequence. There may be a case that this sequence is bounded in an appropriate function space and hence converges weakly, passing to a subsequence. This weak limit may not be a weak solution because of the lack of lower semicontinuity or compactness of the functional. Since the strong convergence assures these properties quite often, there arises the case that a minimizing sequence or a sequence of approximate solutions exhibits a subsequence which converges weakly but not strongly. In some of these cases, the center of these functions moves to infinity or concentrates at several spots. In other cases, they vanish from the local area or oscillate frequently. Such phenomena are selected by the structure and are controlled in advance in a variety of nonlinear problems.15 This section is devoted to the method of real analysis to approach these phenomena. The maximal theorem of Hardy–Littlewood is a basic tool. We state Vitali’s covering lemma in the following form.16 Lemma 7.17 (Vitali). If ℱ is a family of closed balls in Rn with the radius uniformly bounded, there exists a countable 𝒢 ⊂ ℱ composed of disjoint balls satisfying ⋃B∈ℱ B ⊂ ⋃B∈𝒢 B,̂ where B̂ denotes the concentric ball, the radius of which is a five times that of B. Lemma 7.18. For f ∈ L1 (Rn ) and t > 0 it holds that n 󵄨󵄨 󵄨 5 n ∫ |f | 󵄨󵄨{x ∈ R | Mf (x) > t}󵄨󵄨󵄨 ≤ t

(7.60)

Rn

where Mf (x) ≡ supr>0

1 |f | ∫ |Br (x)| Br (x)

∈ [0, +∞] stands for the maximal function of f (x).

Proof. Let E(t) = {x ∈ Rn | Mf (x) > t}. Each x ∈ E(t) admits r(x) > 0 such that 1 |Br(x) (x)|

∫ |f | > t. Br(x) (x)

Then Lemma 7.17 assures a countable family of disjoint closed balls {Bk } ⊂ {Br(x) (x) | ̂ x ∈ Rn } such that ⋃x∈E(t) Br(x) (x) ⊂ ⋃∞ k=1 Bk , and, in particular, ∞

E(t) ⊂ ⋃ B̂ k , k=1

1 ∫ |f | > t. |Bk | Bk

It thus holds that 5n 5n 󵄨󵄨 󵄨 n ∑ ∫ |f | ≤ ∫ |f |, 󵄨󵄨E(t)󵄨󵄨󵄨 ≤ ∑ |B̂ k | = 5 ∑ |Bk | ≤ t k t k k Bk

Rn

and hence (7.60) follows. 15 These profiles are called concentration and compensated compactness [326, 110]. 16 The proof is in [112]. See also [321, 322] for the maximal theorem, BMO, and Hardy spaces.

7.3 Hardy spaces | 421

Proposition 7.19 (Hardy–Littlewood). Each f ∈ Lp (Rn ) in 1 < p ≤ ∞ admits Mf (x) < +∞ for a. e. x ∈ Rn , and it holds that ‖Mf ‖p ≤ Ap ‖f ‖p

(7.61)

with a constant Ap > 0. Proof. Since the assertion is obvious for p = ∞ due to 0 ≤ Mf (x) ≤ ‖f ‖∞ , let 1 < p < ∞. Fix t > 0, and put f (x), |f (x)| > t/2, 0, otherwise.

f1 (x) = {

Since |f (x)| ≤ |f1 (x)| + 2t implies Mf (x) ≤ Mf1 (x) + 2t , it holds that E(t) ≡ {x ∈ Rn | Mf (x) > t} ⊂ {x ∈ Rn | Mf1 (x) > t/2}. Then Lemma 7.18 ensures n 2 ⋅ 5n 󵄨 󵄨 󵄨 󵄨 2⋅5 μ(t) ≡ 󵄨󵄨󵄨E(t)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨{x ∈ Rn | Mf1 (x) > t/2}󵄨󵄨󵄨 ≤ ‖f1 ‖1 = t t

∫ |f |, |f |>t/2

and therefore, ∞

∞

0

0

‖Mf ‖pp = ∫ Mf (x)p dx = − ∫ t p dμ(t) ≤ p ∫ t p−1 μ(t) dt Rn

∞

󵄨 󵄨 ≤ p ⋅ 2 ⋅ 5n ∫ t p−2 dt ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx 0

|f |>t/2

2|f (x)|

󵄨 󵄨 = p ⋅ 2 ⋅ 5n ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx ⋅ ∫ t p−2 dt 0

Rn

p ⋅ 2 ⋅ 5n 󵄨󵄨 p ⋅ 2 ⋅ 5n 󵄨󵄨 󵄨 󵄨 󵄨p 󵄨p−1 = ∫ 󵄨󵄨f (x)󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨2f (x)󵄨󵄨󵄨 dx = ∫ 󵄨󵄨f (x)󵄨󵄨󵄨 dx. p−1 p−1 Rn

ℛn

7.3.2 Div–Rot lemma Given 0 ≤ h ∈ C0∞ (Rn ) in supp h ⊂ B1 (0) and ‖η‖1 = 1, let ht = we take 0 < p ≤ ∞ to define p

n

p

n

1 h(⋅/t) tn

for t > 0. Then

n

ℋ (ℛ ) = {f ∈ 𝒮 (R ) | sup |ht ∗ f | ∈ L (R )} 󸀠

t>0

and ‖f ‖ℋp = ‖supt>0 |ht ∗ f |‖p , where 𝒮 󸀠 (Rn ) denotes the set of tempered distributions in Rn and convolution ht ∗ f is given by (ht ∗ f )(x) = ∫ ht (x − y)f (y) dy Rn

if f ∈ L1loc (Rn ).17 This ℋp (Rn ) is independent of h and is called the Hardy space. 17 The set of tempered distributions is the space where Fourier transformation is active [373]. In the following discussion, however, we apply this transformation only for integrable functions.

422 | 7 Hardy spaces and BMO We have ℋp (Rn ) = Lp (Rn ) for 1 < p ≤ ∞, while it holds that ℋ1 (Rn ) ⊂ L1 (Rn ) and H (Rn ) ≠ L1 (Rn ). Actually, Fefferman’s duality theorem indicates ℋ1 (ℛn )󸀠 ≅ BMO(Rn ), while we have L∞ (Rn ) ⊂ BMO(Rn ) and L∞ (Rn ) ≠ BMO(Rn ) [115, 322]. This discrepancy between L1 (Rn ) and ℋ1 (Rn ) takes a role in nonlinear problems. We show a condition for a function in the former space to belong to the latter. Let 1 < p < ∞ and assume 1

E ∈ Lp (Rn ),

B = ∇π ∈ Lp (Rn , Rn ) in Rn 󸀠

∇ ⋅ E = 0,

(7.62)

with a scalar field π ∈ 𝒟󸀠 (Rn ). It is obvious that E ⋅ B ∈ L1 (ℛn ), but this inner product lies in ℋ(Rn ) because of the div–rot structure in (7.62).18 Lemma 7.20. If E and π are smooth in (7.62) it holds that 1/α

1 󵄨󵄨 󵄨 ∫ |E|α } 󵄨󵄨ht ∗ (E ⋅ B)(x)󵄨󵄨󵄨 ≤ C{ |B (x)| t

1/β

⋅{

Bt (x)

1 ∫ |B|β } |Bt (x)|

(7.63)

Bt (x)

for any x ∈ Rn and t > 0, where 1 < α < p, 1 < β < p󸀠 , and

1 α

+

1 β

= 1 + n1 .

Proof. Since ∇ ⋅ E = 0 implies ∇ ⋅ (πE) = ∇π ⋅ E + π∇ ⋅ E = B ⋅ E, it holds that ht ∗ (E ⋅ B)(x) = ht ∗ (∇ ⋅ (πE))(x) = ∫ ∇h( Rn

= ∫ ∇h( Rn

x−y 1 ) n+1 ⋅ (Eπ)(y) dy t t

x−y 1 1 ) n+1 ⋅ E(y){π(y) − ∫ π} dy t |Bt (x)| t Bt (x)

again due to ∇ ⋅ E = 0. We thus obtain 󵄨󵄨 󵄨 ‖∇h‖∞ 󵄨 󵄨 ∫ 󵄨󵄨󵄨E(y)󵄨󵄨󵄨 ⋅ 󵄨󵄨ht ∗ (E ⋅ B)(x)󵄨󵄨󵄨 ≤ tn Bt (x)

≤ ‖∇h‖∞ c(n){

⋅{

󵄨󵄨 1 󵄨󵄨󵄨󵄨 1 󵄨 ∫ π 󵄨󵄨󵄨 dy 󵄨󵄨π(y) − 󵄨󵄨 t 󵄨󵄨 |Bt (x)| 1/α

Bt (x)

1 ∫ |E|α } |Bt (x)| Bt (x)

󵄨󵄨 π − 1 󵄨 ∫ 󵄨󵄨󵄨 󵄨󵄨 |Bt (x)|

󸀠 󸀠 1 π 󵄨α 1/α ∫ |Bt (x)| Bt (x) 󵄨󵄨

Bt (x)

t

󵄨󵄨 } 󵄨󵄨 󵄨

.

Then, Poincaré–Sobolev inequality (7.37) ensures 1/β 󵄨󵄨 󵄨󵄨β 1/β 1 1 1 󵄨 󵄨 { ≤ Ct{ ∫ 󵄨󵄨󵄨π − ∫ π 󵄨󵄨󵄨 } ∫ |∇π|β } 󵄨󵄨 󵄨󵄨 |Bt (x)| |Bt (x)| |Bt (x)| ∗

Bt (x)

Bt (x)

∗

Bt (x)

18 Poincaré lemma assures that if B is identified with a 1-form satisfying dB = 0, there is a 0-form ω such that B = dω.

7.3 Hardy spaces | 423

for

1 β∗

=

1 β

−

1 n

=1−

1 α

=

1 , α󸀠

and hence

󵄨󵄨 π − 1 󵄨 { ∫ 󵄨󵄨󵄨 󵄨󵄨 |Bt (x)|

󸀠 󸀠 1 π 󵄨α 1/α ∫ |Ω| Bt (x) 󵄨󵄨

Bt (x)

󵄨󵄨 } 󵄨󵄨 󵄨

t

≤ C{

1/β

1 ∫ |∇π|β } |Bt (x)| Bt (x)

= C{

1/β

1 ∫ |B|β } |Bt (x)|

.

Bt (x)

Then inequality (7.63) follows. Theorem 7.21 ([90]). Assume (7.62) for 1 < p < ∞. Then it holds that E ⋅ B ∈ ℋ1 (Rn ) and ‖E ⋅ B‖ℋ1 ≤ C‖E‖p ‖B‖p󸀠 . Proof. By using a standard mollifier, the inequality ‖E ⋅ B‖ℋ1 ≤ C‖E‖p ‖B‖p󸀠 is reduced to the case when E and π are smooth. Considering 1 < p < ∞, we take 1 < α < p and 1 < β < p󸀠 satisfying α1 + β1 = 1 + n1 . We obtain 1/α 1/β 󵄨 󵄨 sup󵄨󵄨󵄨ht ∗ (E ⋅ B)(x)󵄨󵄨󵄨 ≤ C(M|E|α (x)) ⋅ (M|B|β (x)) t>0

by (7.63), recalling Mf (x) = supt>0

1 |f |, ∫ |Bt (x)| Bt (x)

and therefore,

󵄩󵄩 1/α 󵄩 1/β 󵄩 󵄨 󵄨󵄩󵄩 󵄩 󵄩 ‖E ⋅ B‖ℋ1 = 󵄩󵄩󵄩sup󵄨󵄨󵄨ht ∗ (E ⋅ B)󵄨󵄨󵄨󵄩󵄩󵄩 ≤ C 󵄩󵄩󵄩(M|E|α ) 󵄩󵄩󵄩p ⋅ 󵄩󵄩󵄩(M|B|β ) 󵄩󵄩󵄩p󸀠 󵄩 t>0 󵄩1 1/β 󵄩1/β 󵄩 󵄩1/α 󵄩 ⋅ ‖|B|β ‖ p󸀠 = C 󵄩󵄩󵄩M|E|α 󵄩󵄩󵄩 p ⋅ 󵄩󵄩󵄩M|B|β 󵄩󵄩󵄩 p󸀠 ≤ C‖|E|α ‖1/α p α α β

= C‖E‖p ⋅ ‖B‖p󸀠

β

by (7.61). 7.3.3 Jacobian estimate Given u = (u1 , . . . , un ) ∈ L1loc (Rn , Rn ) in ∇ui ∈ Ln (Rn , Rn ) for 1 ≤ i ≤ n, its Jacobian 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 J(u) = 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨

𝜕u1 𝜕x1

.. .

n

𝜕u 𝜕x1

...

𝜕u1 𝜕xn

...

𝜕un 𝜕xn

.. .

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨

is defined as a function belonging to L1loc (Rn ). Corollary 7.22. Under the above assumption, it holds that J(u) ∈ ℋ1 (Rn ) and n

󵄩󵄩 󵄩 󵄩 j󵄩 󵄩󵄩J(u)󵄩󵄩󵄩ℋ1 ≤ C ∏󵄩󵄩󵄩∇u 󵄩󵄩󵄩n . i=1

424 | 7 Hardy spaces and BMO Proof. Expansion with respect to the first row yields J(u) = ∇u1 ⋅ σ for σ = (σ 1 , . . . , σ n ), using the (1, i) cofactor matrix 󵄨󵄨 󵄨󵄨 󵄨󵄨 i i+1 󵄨󵄨󵄨 σ = (−1) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨

𝜕u2 𝜕x1

.. .

𝜕un 𝜕x1

...

𝜕u2 𝜕xi−1

𝜕u2 𝜕xi+1

...

𝜕un 𝜕xi−1

...

𝜕u2 𝜕xn

𝜕un 𝜕xi+1

...

𝜕un 𝜕xn

𝜕u2 𝜕x2

...

𝜕u2 𝜕xn

𝜕un 𝜕x2

...

𝜕un 𝜕xn

.. .

.. .

.. .

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 , 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨

1 ≤ i ≤ n.

To apply Theorem 7.21, we use 󵄨󵄨 󵄨󵄨 󵄨 n i 𝜕σ 𝜕 󵄨󵄨󵄨󵄨 ∇⋅σ =∑ = 󵄨 𝜕xi 𝜕x1 󵄨󵄨󵄨󵄨 i=1 󵄨󵄨 󵄨󵄨

󵄨󵄨 󵄨󵄨 󵄨 𝜕 󵄨󵄨󵄨󵄨 − 󵄨 𝜕x2 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

.. .

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨

.. .

𝜕u2 𝜕x1

𝜕u2 𝜕x3

...

𝜕u2 𝜕xn

𝜕un 𝜕x1

𝜕un 𝜕x3

...

𝜕u n 𝜕xn

.. .

.. .

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕 󵄨󵄨 + ⋅ ⋅ ⋅ + (−1)n 𝜕xn 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

𝜕u2 𝜕x1

.. .

n

𝜕u 𝜕x1

.. .

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨

...

𝜕u2 𝜕xn−1

...

𝜕un 𝜕xn−1

.. .

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 . 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨

Here, the derivative of the first column of the first term cancels that of the second term on the right-hand side. Similarly, the derivative of the second column of the first term cancels that of the third term on the right-hand side. One thus obtains ∇ ⋅ σ = 0. Since n |σ| ≤ C ∏nj=2 |∇uj | and ∇uj ∈ Ln (Rn , Rn ), on the other hand, it holds that σ ∈ L n−1 (Rn , Rn ). n = n󸀠 . Then Theorem 7.21 implies the result for n−1

7.4 Harmonic map revisited 7.4.1 Quantized blow-up mechanism A quantized blow-up mechanism is widely observed in nonlinear problems derived from physical principles and phenomena. Energy quantization is a typical case described by the Dirichlet norm, where energy identity with bubbling occurs to the noncompact solution sequence. This quantization arises if the problem is provided with the scaling invariance of energy. Harmonic map, semilinear elliptic equation involving the critical Sobolev exponent, and H-system are typical examples. The result in § 3.3 may be called mass quantization. There, blow-up occurs only to the quantized values of mass, realized as the eigenvalue λ because of residual vanishing, the disappearance of the regular part of the limit measure. Besides, the global structure of

7.4 Harmonic map revisited | 425

the bubbling is described in accordance with the domain, particularly, the Hamiltonian which excludes multiple bubbles, prescribing the location of blow-up points. Boltzmann–Poisson equation is a typical example, but Ginzburg–Landau equation obeys a similar profile [343]. To define a harmonic map, let (Σ, g) and N 󳨅→ Rn be a compact surface and compact Riemannian manifold, respectively. Put H 1 (Σ, N) = {u ∈ H 1 (Σ, Rn ) | u(x) ∈ N a. e. x}, 󵄨 󵄨2 E(u) = EΣ (u) = ∫󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 dvΣ (x), Σ

where dvΣ = dvΣ (x) denotes the area element of (Σ, g). We say that u : Σ → N is a harmonic map if it is a solution to the Euler–Lagrange equation of E = E(u) defined for u ∈ H 1 (Σ, N), i. e., − Δu = A(u)(∇u, ∇u),

(7.64)

where A(u)(⋅, ⋅) denotes the second fundamental form of N 󳨅→ Rn . Henceforth, S2 denotes the standard two-dimensional sphere. There is a quantized blow-up mechanism of the harmonic map sequence defined on a two-dimensional domain [171, 284, 103, 285, 233]. Theorem 7.23. Let {uk } be a harmonic map sequence satisfying supk E(uk ) < +∞. Passing to a subsequence, we assume uk ⇀ u in H 1 (Σ, N) weakly to some map u ∈ H 1 (Σ, N). This u is a harmonic map, and there exist – p-sequences of points {xk1 }, . . . , {xkp } in Σ, – p-sequences of positive numbers {δk1 }, . . . , {δkp } converging to 0, and – p-nonconstant harmonic maps {ω1 }, . . . , {ωp } : S2 → N satisfying the following: p

lim E(uk ) = E(u) + ∑ E0 (ωj ),

k→∞

δi lim max{ kj , k→∞ i=j̸ δk

j=1

j δk , δki

|xki δki

− +

j xk | } j δk

E0 (ω) = ∫ |∇ω|2 dvS2 , S2

= +∞,

󵄩󵄩 󵄩󵄩 p 󵄩󵄩 󵄩󵄩 j j j j 󵄩 lim 󵄩󵄩uk − u − ∑{ω ((⋅ − xk )/δk ) − ω (∞)}󵄩󵄩󵄩 = 0. 󵄩󵄩 1 k→∞󵄩 󵄩󵄩 j=1 󵄩H (Σ,N)

(7.65)

The first equality of (7.65) is an energy identity, which says that there is no unaccounted energy loss during the iterated rescaling process near the point of singularity, sometimes referred to as the bubbling process, and that the only reason for failure of strong convergence to the weak limit is the formation of several bubbles due to the

426 | 7 Hardy spaces and BMO nonconstant harmonic maps ωj : S2 → N, j = 1, . . . , p. There is a possibility that j some of {xk }, 1 ≤ j ≤ p, converge to the same point, and this process is classified into two cases, the separated bubbles and the bubbles on bubbles [171]. This feature is extended for the sequence of approximate harmonic maps satisfying − Δuk = A(uk )(∇uk , ∇uk ) + fk

(7.66)

with ‖fk ‖L2 ≤ C, when N is a standard sphere [284]. The general case of N is also known [361, 103]. The elliptic problem associated with the critical Sobolev exponent also has this property. Recall that a form of the Sobolev imbedding theorem is formulated as ∗ 2m H01 (Ω) 󳨅→ L2 (Ω), where Ω ⊂ Rm is an open set with m ≥ 3 and 2∗ = m−2 . If Ω is bounded and 1 < q < 2∗ − 1, then we obtain a compact imbedding H01 (Ω) 󳨅→ Lq+1 (Ω), which results in the existence of infinitely many solutions to −Δu = |u|p−1 u in Ω with u|𝜕Ω = 0, while there is no solution if q ≥ 2∗ − 1 and Ω is star-shaped [274, 288, 250]. Several existence and nonexistence results of the solution to − Δu = λu + |u|2

∗

−2

u

in Ω,

u|𝜕Ω = 0

(7.67)

are also known in accordance with the shape of the domain and the value λ ∈ R [54, 17, 250]. It is the Euler–Lagrange equation of ∗ 1 λ 1 Eλ (u) = ∫ |∇u|2 − |u|2 − ∗ |u|2 dx, 2 2 2

Ω

u ∈ H01 (Ω),

whereby the control of Palais–Smale sequence is a key ingredient of the study. We say that {uk } ⊂ H01 (Ω) is a Palais–Smale sequence to Eλ if Eλ󸀠 (uk ) → 0 in H −1 (Ω) = H01 (Ω)󸀠 , that is, −Δuk = λuk + |uk |2

∗

−1

uk + fk

in Ω,

uk |𝜕Ω = 0,

lim ‖fk ‖H −1 (Ω) = 0.

k→∞

Theorem 7.24 ([323]). If {uk } ⊂ H01 (Ω) is a Palais–Smale sequence to the above dej j fined Eλ , there are p ∈ N, δk ↓ 0, and xk ∈ Ω, 1 ≤ j ≤ p, such that, passing to a subsequence, p

lim Eλ (uk ) = Eλ (u0 ) + ∑ E0 (ωj ),

k→∞

j=1

󵄩󵄩 󵄩󵄩 n−2 p 󵄩󵄩 󵄩 j − j j 󵄩 lim 󵄩󵄩󵄩uk − u0 − (δk ) 2 ∑ ωj ((⋅ − xk )/δk )󵄩󵄩󵄩 = 0, 󵄩󵄩 1 k→∞󵄩 󵄩󵄩 j=1 󵄩H (Ω) ∗ 1 1 E0 (ω) = ∫ |∇ω|2 − ∗ |ω|2 dx, 2 2 Rm

7.4 Harmonic map revisited | 427

where ωj = ωj (x), 1 ≤ j ≤ p, and u0 = u0 (x) are solutions to −Δω = |ω|2

∗

−2

ω

in Rm ,

ω ∈ L2 (Rm ),

∇ω ∈ L2 (Rm , Rm )

∗

and (7.67), respectively. A parametric representation of a surface of constant mean curvature is described by u : B = {(x, y) | x 2 + y2 < 1} → R3 satisfying Δu = 2Hux × uy ,

|ux |2 − |uy |2 = ux ⋅ uy = 0

in B,

u|𝜕B monotonic parametrization of γ fixing three points,

where γ ⊂ R3 is a given oriented rectifiable Jordan curve, H is a constant, and | ⋅ |, ×, and ⋅ are three-dimensional vector length, outer product, and inner product, respectively. This system is an Euler–Lagrange equation of EH = EH (u) defined for u ∈ H 1 (B) ∩ C(B) that is a monotonic parametrization of γ fixing three points, where 1 2H EH (u) = ∫ |∇u|2 + u ⋅ ux × uy dz, 2 3

dz = dx dy.

B

The existence of a solution arises in accordance with that of the Dirichlet problem Δu = 2Hux × uy + f

in B,

u|𝜕B = 0,

which exhibits an energy quantization of the noncompact Palais–Smale sequence [48, 325]. Here we define EH (u) for u ∈ H01 (B, R3 ), and call {uk } ⊂ H01 (B, R3 ) a Palais–Smale sequence if Δuk = 2Hukx × uky + fk

in B,

uk |𝜕B = 0,

‖fk ‖H −1 (B,R3 ) → 0.

Theorem 7.25 ([49]). If {uk } ⊂ H01 (B, R3 ) is a Palais–Smale sequence to the above dej j fined EH satisfying supk ‖∇uk ‖22 < +∞, there are p ∈ N, δk ↓ 0, and xk ∈ B, 1 ≤ j ≤ p, such that 󵄩󵄩 󵄩󵄩 p 󵄩󵄩 󵄩 j j 󵄩 lim 󵄩󵄩󵄩uk − u0 − ∑ ωj ((⋅ − xk )/δk )󵄩󵄩󵄩 = 0, 󵄩󵄩 1 k→∞󵄩 󵄩󵄩 j=1 󵄩H (B) p

󵄨 󵄨2 󵄨 󵄨2 lim ∫ |∇uk |2 dz = ∫󵄨󵄨󵄨∇u0 󵄨󵄨󵄨 dz + ∑ ∫ 󵄨󵄨󵄨∇ωj 󵄨󵄨󵄨 dz, k→∞ B

B

j=1

p

R2

lim V(uk ) = V(u0 ) + ∑ V0 (ωj ),

k→∞

j=1

1 V(u) = ∫ u ⋅ ux × uy dz, 3 B

V0 (ω) =

1 ∫ ω ⋅ ωx × ωy dz, 3 R2

428 | 7 Hardy spaces and BMO where ωj = ωj (x), 1 ≤ j ≤ p, and u0 = u0 (x) are solutions to Δω = 2Hωx × ωy

in R2 ,

ω(∞) = 0,

∫ |∇ω|2 dz < +∞, R2

and Δu = 2Hux × uy in B with u|𝜕B = 0, respectively. The Palais–Smale sequence is defined also for the harmonic map. It is formulated by {uk } satisfying (7.66) with fk → 0 in H −1 (Σ, N). Then, its energy gap is eliminated under the additional assumption ‖fk ‖2 = O(1) [271]. Differently from Theorems 7.24 and 7.25, however, each α > 0 admits a sequence of smooth maps uk : S2 → S2 , k = 1, 2, . . ., satisfying E 󸀠 (uk ) → 0, the first two relations of (7.65) with p = 1, and E(uk ) → E(u) + E(ω1 ) + α [283, 103, 361]. 7.4.2 Monotonicity formula The proof of Theorems 7.23–7.25 shares a common structure, and here we describe the harmonic map case, reformulating the problem in more generality. Thus, (M, g) denotes an m-dimensional compact Riemannian manifold with a smooth boundary 𝜕M, and N is a compact Riemannian manifold without boundary. By Nash’s theorem, this N is isometrically imbedded in Rn for large n. We define the Sobolev space composed of a class of the mappings from M to N provided with finite energy as in the previous section, i. e., H 1 (M, N) = {u ∈ H 1 (M, Rn ) | u(x) ∈ N a. e. x}, 󵄨 󵄨2 E(u) = EM (u) = ∫󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 dvM (x), M

where dvM is a volume element of (M, g). The harmonic map defined in the previous section is indicated by the weakly harmonic map in this general formulation. More precisely, a map u ∈ H 1 (M, N) is called a weakly harmonic map if d 󵄨 E(Π(u + εϕ))󵄨󵄨󵄨ε=0 = 0 dε for any ϕ ∈ C0∞ (M, Rn ), where Π : U → N is a smooth nearest point projection from some tubular neighborhood U of N to N. This is equivalent to saying that u is a weak solution of the Euler–Lagrange equation (7.64) on M, sometimes called the harmonic map equation (7.64), where Δ and A(y)(⋅, ⋅) denote the (negative) Laplace–Beltrami operator on (M, g) and the second fundamental form of the imbedding N 󳨅→ Rn at y ∈ N, respectively. If N is the round sphere Sn , it takes the form − Δu = |∇u|2 u.

(7.68)

7.4 Harmonic map revisited | 429

A map u ∈ H 1 (M, N) is called a minimizing harmonic map if E(u) ≤ E(v) for any v ∈ H 1 (M, N) satisfying u = v on 𝜕M. We call u ∈ H 1 (M, N), on the other hand, a stationary harmonic map if it is weakly harmonic and d E(u ∘ Φt )|t=0 = 0 dt

(7.69)

for any smooth family {Φt }|t|≪1 of diffeomorphisms of M such that Φ0 = Id. Any minimizing harmonic map or weakly C 2 harmonic map is stationary. If M = Ω ⊂ Rm is a bounded domain, we can take Φt (x) = x + tξ (x) for ξ ∈ ∞ C0 (Ω, Rm ). Then, (7.69) implies m

∫ ∑ (δij |∇u|2 − 2Di uDj u)Di ξ j dx = 0.

Ω i,j=1

(7.70)

Putting ξ (x) = x − z in (7.70), we obtain the monotonicity formula in the form of R2−m ∫ |∇u|2 dx − r 2−m ∫ |∇u|2 dx BR (z)

=2

Br (z)

∫

|x − z|2−m |∇u|2 dx ≥ 0,

0 0 and θ ∈ (0, 1) determined by (Σ, g) and N such that if u : Σ → N is harmonic and ∫D |∇u|2 dvg ≤ ε0 , it holds that 1

2

∫ |∇u| dvg ≤ θ ∫ |∇u|2 dvg , D(x,r)

x ∈ D1/2 ,

0 < r < 1/4.

D(x,2r)

This estimate, combined with Morrey’s Dirichlet growth theorem [139], guarantees ε-regularity, so that E(2r) = ∫ |∇u|2 dvg ≤ ε0 󳨐⇒ sup |∇u|2 ≤ Cr −2 E(2r) D2r

(7.73)

Dr

with C > 0 independent of u and r ∈ (0, 1). If {uk } is a harmonic map sequence satisfying ∫D |∇uk |2 dvg ≤ ε0 , therefore, we obtain a subsequence, denoted by the same 1

symbol, and a harmonic map u : D1 → N such that uk → u strongly in H 1 (Dr , N) and C 0 (Dr , N) for any r ∈ (0, 1), recalling (7.64). This ε-compactness guarantees the energy gap lemma. Lemma 7.29. It holds that inf{∫ |∇ω|2 dvg | ω : Σ → N, nonconstant harmonic map} > 0. Σ

This ε-compactness guarantees also a rough estimate in the following form. Theorem 7.30 ([296]). Under the assumption of Theorem 7.23, u : Σ → N is a harmonic map. Passing to a subsequence, we have uk → u locally uniformly in Σ \ {x1 , . . . , xℓ } and |∇uk |2 dvg ⇀ |∇u|2 dvg + ∑ℓj=1 mj δxj in the sense of measure, where {x1 , . . . , xℓ } ⊂ Σ and mj ≥ ε0 with an ε0 > 0. Proof. Given δ > 0, let 𝒜δ,k = {x ∈ Σ | ∫D(x,δ) |∇uk |2 dvg ≥ ε0 }. By Lemma 7.17, there is

𝒜󸀠δ,k ⊂ 𝒜δ,k such that {D(x, δ) | x ∈ 𝒜󸀠δ,k } is a disjoint family and 𝒜δ,k ⊂ ⋃x∈𝒜󸀠 D5δ (x) ≡ δ,k

432 | 7 Hardy spaces and BMO Ω(δ). Then we obtain ε0 ⋅ ♯𝒜󸀠δ,k ≤

|∇uk |2 dvg ≤ E0 ≡ sup E(uk ),

∫

k

⋃x∈𝒜󸀠

δ,k

and hence #𝒜󸀠δ,k ≤ E0 /ε0 . Passing to a subsequence, therefore, we obtain 𝒜󸀠δ,k = j

{xk1 , . . . , xkℓ } with ℓ ≤ E0 /ε0 and xk → xj ∈ Σ for 1 ≤ j ≤ ℓ. We have, on the other hand,

x ∈ ̸ Ω(δ) 󳨐⇒ ∫ |∇uk |2 dvg < ε0 , Dδ (x)

and therefore, there is a subsequence satisfying uk → u in C 1 (Σ \ Ω(δ)) by the above ε-compactness. From the diagonal argument to δk ↓ 0, we have a subsequence, still denoted by the same symbol, such that uk → u in C 1 (Σ \ {x1 , . . . , xℓ }) for ⋂k Ω(δk ) = {x1 , . . . , xℓ }. This u is harmonic in Σ \ {x1 , . . . , xℓ } and satisfies ∫Σ |∇u|2 dvg < +∞ by Fatou’s lemma, and therefore, {x1 , . . . , xℓ } are removable singular points of u by Lemma 7.27. We obtain, furthermore, |∇uk |2 dvg ⇀ |∇u|2 dvg + ∑ℓj=1 mj δxj in the sense of measure. In case of mj < ε0 , therefore, there is 0 < r0 ≪ 1 such that ∫

|∇uk |2 dvg ≤

∫

|∇u|2 dvg + mj + o(1) < ε0 ,

k ≫ 1,

D(xj ,2r0 )

D(xj ,2r0 )

which implies that xj is not a blow-up point of {uk } by (7.73). The result thus holds with mj ≥ ε0 . 7.4.5 Bubble towers We proceed to the proof of Theorem 7.23. Since 𝜕Σ = 0, there is no boundary effect on the blow-up behavior of the solution sequence. Furthermore, the blow-up profile has been localized around each concentration point. The scaling limit is described by Σ̃ = S2 with standard metric dvS2 , ∞ ∈ ̸ 𝒮 = {x1 , . . . , xℓ }, and N = Sn , where ∞ denotes the north pole. First, the following lemma is obtained by the above mentioned Hardy–BMO structure in § 7.4.3, where 0 < ε0 ≪ 1 is an absolute constant and DR is the geodesic disc with the radius R > 0 and Ω ⊂⊂ DR is an open set. Lemma 7.31. If u : DR → Sn is a harmonic map in ∫D |∇u|2 dvg ≤ ε0 and v ∈ H 1 (Ω, Rn+1 ) R is a solution to Δv = 0 in Ω with v = u on 𝜕Ω, then ∫ |∇u|2 dvg ≤ C ∫ |∇v|2 dvg Ω

with C > 0 independent of Ω.

Ω

7.4 Harmonic map revisited | 433

Proof. Since the target space of this harmonic map is a sphere, we obtain n+1

Δui = ∑ ∇uj ⋅ Eji , j=1

1 ≤ i ≤ n + 1,

Eji = uj ∇ui − ui ∇uj .

Subtract the equation of v, n+1

n+1

j=1

j=1

Δ(u − v)i = ∑ ∇(u − v)j ⋅ Eji + ∑ ∇vj ⋅ Eji ,

(7.74)

and define wi by n+1

Δwi = ∑ ∇(u − v)j ⋅ Eji j=1

in Ω,

wi |𝜕Ω = 0.

(7.75)

1 The right-hand side of (7.75) is in a local Hardy space ℋloc (R2 ) and hence n+1

󵄩 󵄩 ‖∇wi ‖2 ≤ C ∑ 󵄩󵄩󵄩∇(u − v)j 󵄩󵄩󵄩2 ⋅ ‖Eji ‖2 j=1

(7.76)

with C > 0 independent of Ω [41, 74]. Multiplying (7.74) by (u − v)i , we have 󵄨 󵄨2 ∫󵄨󵄨󵄨∇(u − v)󵄨󵄨󵄨 dvg

Ω

n+1

n+1

= ∑ ∫ ∇wi ⋅ ∇(u − v)i dvg − ∑ ∫(∇vj ⋅ Eji )(u − v)i dvg i,j=1 Ω

i=1 Ω

󵄨2

󵄨 ≤ C ∫ ε01/2 󵄨󵄨󵄨∇(u − v)󵄨󵄨󵄨 + ∫ |∇u| ⋅ |∇v| dvg Ω

Ω

since ‖Eji ‖2 ≤ C‖∇u‖2 ≤ Cε01/2 and ‖u − v‖∞ ≤ C, derived from the maximum principle for harmonic functions. Then we obtain the result for 0 < ε ≪ 1, recalling 󵄨 󵄨2 ∫ |∇u|2 dvg ≤ 2 ∫󵄨󵄨󵄨∇(u − v)󵄨󵄨󵄨 + |∇v|2 dvg .

Ω

Ω

The proof of Theorem 7.30 ensures 2

𝒮 = ⋂ {z ∈ Σ | lim inf ∫ |∇uk | dvg > ε0 }. r>0

k→∞

Dr (z)

Henceforth, we apply a hierarchical argument, without mentioning the process of subtracting subsequences unless otherwise stated. Recall {x1 , . . . , xℓ } in Theorem 7.30.

434 | 7 Hardy spaces and BMO Step 1. We extract the principal bubble from each blowup point. Let δ = i ≠ j}, and define the concentration function of {uk } at x1 ∈ 𝒮 : Q1k (t) = sup

z∈Dδ (x1 )

1 2

min{|xi − xj | |

∫ |∇uk |2 dvg . Dt (z)

It is continuous, monotone increasing in t ≥ 0, and satisfies Q1k (0) = 0 and Q1k (δ) > ε0 for k ≫ 1, and therefore, there exist xk1 ∈ Dδ (x1 ) and δk1 ∈ (0, δ) such that Q1k (δk1 ) =

∫

|∇uk |2 dvg =

Dδ1 (xk1 )

ε0 . 2

k

Then that xk1 → x1 and δk1 → 0 as k → ∞. The rescaled ũ 1k (x) = uk (δk1 x + xk1 ) is a harmonic map satisfying ε 󵄨 󵄨2 ∫ 󵄨󵄨󵄨∇ũ 1k 󵄨󵄨󵄨 dṽg ≤ 0 , 2

∀z ∈ D̃ 1δ (xk ) = (δk1 ) {Dδ (x1 ) − xk1 } −1

D1 (z)

with the equality for z = 0, where dṽg denotes the rescaled metric of dvg . From 1 0 ε-compactness of this {ũ 1k }, we obtain ũ 1k → ω1 in Hloc ∩ Cloc (S2 \ {∞}, Sn ) with ω1 2 extended to a harmonic map, S → N, satisfying ε 󵄨 󵄨2 ∫ 󵄨󵄨󵄨∇ω1 󵄨󵄨󵄨 dvS2 = 0 , 2

D1

󵄨 󵄨2 ∫󵄨󵄨󵄨∇ω1 󵄨󵄨󵄨 dvS2 ≤ sup E(vk ) < +∞. k

S2

This ω1 is called a bubble at the blow-up point x1 . Repeating this procedure, we obtain j j – sequences of points {xk } ⊂ Dδ (xj ) satisfying limk→∞ xk = xj , – –

j

sequences of positive numbers {δk } converging to 0, and nonconstant harmonic map ωj : S2 → Sn

j

j

j

j

0 1 (S2 \ {∞}, Sn ), 1 ≤ j ≤ ℓ, for ũ k (x) = uk (δk x + xk ). such that ũ k → ωj in Hloc ∩ Cloc

Step 2. We estimate the energy difference between uk and the first bubbles, putting ℓ

vk (x) = uk (x) − ∑[ωj ( j=1

j

x − xk j δk

) − ωj (∞)].

(7.77)

It is easy to see that vk ⇀ u weakly in H 1 (Σ, Sn ). Recall that Ω ⊂ Σ is an arbitrary open set. Lemma 7.32. It holds that ∫ |∇vk |2 dvg = ∫ |∇uk |2 dvg − ∑ ∫ |∇ωj |2 dvS2 + o(1). Ω

Ω

xj ∈Ω

S2

(7.78)

7.4 Harmonic map revisited | 435

Proof. Given 1 ≤ j ≤ ℓ, we take Rj > 0 and put ∫D ∫D

j j (xk ) k

Rj δ

δ (xj )

|∇vk |2 dvg and II = ∫D

j δ (xj )\DR δj (xk ) j k

{∞}, Sn ), we have

|∇vk |2 dvg = I + II where I = j

1 |∇vk |2 dvg . Since ũ k → ωj strongly in Hloc (S2 \

I = ∫ |∇ũ k |2 dṽg − ∫ |∇ωj |2 dvS2 + o(1) DRj

=

DRj

|∇uk |2 dvg − ∫ |∇ωj |2 dvS2 + o(1),

∫ D

S2

j j (xk ) k

Rj δ

while II = ∫D

j δ (xj )\DR δj (xk ) j k

|∇uk |2 dvg + o(1) follows with Rj ≫ 1. We thus obtain

∫ |∇vk |2 dvg = ∫ |∇uk |2 dvg − ∫ |∇ωj |2 dvS2 + o(1). Dδ (xj )

Dδ (xj )

S2

We have, on the other hand, uk → u strongly in H 1 (Ω \ ⋃ℓj=1 Dδ (xj )), and hence (7.78) follows. Step 3. To search for further bubbles, first, we assume lim inf ∫ |∇vk |2 dvg = ∫ |∇u|2 dvg . k→∞

Σ

Σ

In this case, vk converges strongly to u in H 1 (Σ, Sn ), and the energy identity holds by j (7.78). Moreover, since xk → xj and |xi − xj | > δ for i ≠ j, the second equation of (7.65) also holds, and thus, the proof of Theorem 7.23 is complete with p = ℓ. We have, otherwise, lim infk→∞ ∫Σ |∇vk |2 dvg > ∫Σ |∇u|2 dvg , and define the concentration set of {vk } by 2

𝒮 = ⋂ {z ∈ Σ | lim inf ∫ |∇vk | dvg > ε0 }. 󸀠

k→∞

r>0

Dr (z)

Since (7.78) implies ∫D (z) |∇vk |2 dvg < ∫D (z) |∇uk |2 dvg , we obtain 𝒮 󸀠 ⊂ 𝒮 and assume r

r

𝒮 󸀠 = {x1 , . . . , xℓ󸀠 } with 1 ≤ ℓ󸀠 ≤ ℓ.

We now detect the second bubble for x1 ∈ 𝒮 󸀠 . First, we obtain lim inf lim r↓0

k→∞

∫ |∇vk |2 dvg ≥ ε0 > 0. Dr (x1 )

2 Use the concentration function of vk : Qℓ+1 k (t) = supz∈Dδ (x1 ) ∫D (z) |∇vk | dvg , and choose

– –

t

a sequence of points {xkℓ+1 } ⊂ Dδ (x1 ) satisfying limk→∞ xkℓ+1 = x1 and a sequence of positive numbers {δkℓ+1 } converging to 0

436 | 7 Hardy spaces and BMO such that ℓ+1 Qℓ+1 k (δk ) =

|∇vk |2 dvg =

∫ Dδℓ+1 (xkℓ+1 )

ε0 . 2

k

Here, we obtain δkℓ+1 > δk1 by (7.78).

We use the rescaled map ṽkℓ+1 (x) = vk (δkℓ+1 x + xkℓ+1 ) defined for x ∈ D̃ ℓ+1 δ (xk ) = ℓ+1 −1 ℓ+1 (δk ) (Dδ (x1 ) − xk ) similarly as in Step 1. It holds that ε 󵄨 󵄨2 ∫ 󵄨󵄨󵄨∇ṽkℓ+1 󵄨󵄨󵄨 dṽg ≤ 0 , 2

∀z ∈ R2

D1 (z)

1 with the equality when z = 0, and ṽkℓ+1 ⇀ ∃ωℓ+1 weakly in Hloc (S2 \ {∞}, Sn ). Although ℓ+1 ℓ+1 these {vk }k and {ṽk }k are not harmonic, we can derive a variant of ε-compactness to the latter. δj

j ℓ+1 δkℓ+1 |xk −xk |

k , Lemma 7.33. It holds that max{ δℓ+1

,

δkj

k

δkj +δkℓ+1

} → ∞ for 1 ≤ j ≤ ℓ. j

Proof. The assertion is obvious for j ≠ 1, because limk→∞ |xk − xkℓ+1 | = |xj − x1 | > δ. If this is not the case for j = 1, there exists R > 1 such that R−1 ≤

which implies

ε0 ℓ+1 = Qℓ+1 k (δk ) = 2

δkℓ+1 δk1

≤ R and

|xk1 −xkℓ+1 | δk1 +δkℓ+1

≤R

|∇vk |2 dvg

∫ Dδℓ+1 (xkℓ+1 ) k

≤

∫ DLδ1 (xk1 )

󵄨 󵄨2 |∇vk | dvg = ∫ 󵄨󵄨󵄨∇(ũ 1k − ω1 )󵄨󵄨󵄨 dṽg → 0 2

k

DL

for some L = L(R) > 0 (L(R) = 2R + R2 would suffice) by H 1 -strong local convergence of ũ 1k → ω1 , a contradiction. Lemma 7.34. The above defined ωℓ+1 is a nonconstant harmonic map: S2 → Sn , and 1 ṽkℓ+1 → ωℓ+1 strongly in Hloc (S2 \ {∞}, Sn ). Proof. We distinguish two cases of “separated bubbles” (case 1) and “bubbles on bubbles” (case 2), regarding δkℓ+1 > δk1 .20 Case 1. There exists R > 1 such that R−1 ≤ and

|xk1 −xkℓ+1 | δk1 +δkℓ+1

δkℓ+1 δk1

≤ R and

|xk1 −xkℓ+1 | δk1 +δkℓ+1

→ +∞ or

δkℓ+1 δk1

→ +∞

→ +∞. In this case, two bubbles ω1 and ωℓ+1 are geometrically separated

20 See Figures 1 and 2 of [283].

7.4 Harmonic map revisited | 437

in spite of limk→∞ xk1 = limk→∞ xkℓ+1 = x1 , and therefore, there is L ≫ 1 such that Dδ1 L (xk1 ) ∩ Dδℓ+1 L (xkℓ+1 ) = 0 for k ≫ 1, regardless of δkℓ+1 ∼ δk1 or δkℓ+1 ≫ δk1 . Then we k

obtain DL ∩ (

k

xkℓ+1 −xk1 δk1

+

δkℓ+1 DL ) δk1

= 0, and hence

2 󵄨󵄨 xℓ+1 − xk1 − δkℓ+1 ⋅ 󵄨󵄨󵄨 󵄨 󵄨󵄨 dṽg = ) ∫ 󵄨󵄨󵄨∇ω1 ( k 󵄨󵄨 󵄨󵄨 δk1 󵄨

DL

󵄨 󵄨2 ≤ ∫ 󵄨󵄨󵄨∇ω1 󵄨󵄨󵄨 dvS2 = o(1),

󵄨󵄨 1 󵄨󵄨2 󵄨󵄨∇ω 󵄨󵄨 dvg

∫ x ℓ+1 −x 1 k k δ1 k

+

δℓ+1 k δ1 k

DL

L ≫ 1,

S2 \DL

xkℓ+1 −xk1 −δkℓ+1 ⋅ 1 ) → ω1 (∞) strongly in Hloc (S2 \ δk1 j Dδj L (xk ) ∩ Dδℓ+1 L (xkℓ+1 ) = 0, and therefore, k

which implies (up to a subsequence) ω1 (

{∞}, Sn ). If j ≠ 1, it is obvious that ωj (

xkℓ+1 −xkj −δkℓ+1 ⋅

implies

δkj

k

1 ) → ωj (∞) strongly in Hloc (S2 \ {∞}, Sn ), similarly. Here, equality (7.77)

ℓ+1 ℓ+1 ℓ+1 ℓ+1 ũ ℓ+1 k (x) = uk (δk x + xk ) = vk (δk x + xk ) ℓ

+ ∑[ωj (

j

δkℓ+1 x + xkℓ+1 − xk j

δk

j=1

) − ωj (∞)],

and then ε-compactness implies the lemma. Case 2. There exists M > 0 such that δkℓ+1 δk1

→ +∞,

|xk1 − xkℓ+1 | δkℓ+1

≤ M.

(7.79)

In this case, we have 2 󵄨󵄨 xℓ+1 − xk1 − δkℓ+1 ⋅ 󵄨󵄨󵄨 󵄨 󵄨󵄨 dvS2 ) ∫ 󵄨󵄨󵄨∇ω1 ( k 󵄨󵄨 󵄨󵄨 δk1 󵄨 2

S \Dα

=

x ℓ+1 −x 1 S2 \( k 1 k δ k

since

δkℓ+1 δk1

δℓ+1 + k1 δ k

k→∞

Dα )

→ +∞, where α > 0 is arbitrary. For j ≠ 1, on the other hand, there is strong

1 Hloc convergence ∇ωj (

defined

󵄨󵄨 1 󵄨󵄨2 󵄨󵄨∇ω 󵄨󵄨 dvS2 = o(1),

∫

{ũ ℓ+1 k }k ,

xkℓ+1 −xkj −δkℓ+1 ⋅ δkj

it follows that

) → 0. Since ε-compactness is applicable to the above

ũ ℓ+1 k

1 → ωℓ+1 strongly in Hloc (S \ ({∞} ∪ Dα )).

438 | 7 Hardy spaces and BMO From the first bubbling process, it follows that ∫D

βδ1 k

2 󵄨󵄨 ⋅ − xℓ+1 󵄨󵄨󵄨 󵄨 ∫ 󵄨󵄨󵄨∇ωℓ+1 ( ℓ+1k )󵄨󵄨󵄨 dvg ≤ 󵄨󵄨 󵄨󵄨 δk 1

Dβδ1 (xk ) k

∫ D

βδ1 k δℓ+1 k

(xk1 −xkℓ+1 )

(xk1 )

|∇vk |2 dvg = o(1) and

󵄨󵄨 ℓ+1 󵄨󵄨2 󵄨󵄨∇ω 󵄨󵄨 dvS2 = o(1)

for β ≫ 1. Hence it follows that 󵄨 󵄨2 ∫ 󵄨󵄨󵄨∇(ṽk − ωℓ+1 )󵄨󵄨󵄨 dṽg = ∫ + ∫

DR

Dα

=

DR \Dα

2 󵄨󵄨 ⋅ − x ℓ+1 󵄨󵄨󵄨 󵄨 ∫ 󵄨󵄨󵄨∇(vk − ωℓ+1 ( ℓ+1n ))󵄨󵄨󵄨 dvg + o(1), 󵄨󵄨 󵄨󵄨 δk

∀R ≫ 1,

Ak (α,β)

where Ak (α, β) = Dαδℓ+1 (xk1 ) \ Dβδ1 (xk1 ) is called the neck region, connecting two bubbles ω1 and ωℓ+1 . Since assume

k

k

δkℓ+1 δk1

→ +∞ is supposed, this region always appears, but we can

2 󵄨󵄨 ⋅ − xℓ+1 󵄨󵄨󵄨 󵄨 ∫ 󵄨󵄨󵄨∇(vk − ωℓ+1 ( ℓ+1k ))󵄨󵄨󵄨 dvg ≤ 󵄨󵄨 󵄨󵄨 δk

Ak (α,β)

∫ |∇uk |2 dvg < ε0 Ak (α,β)

by taking β ≫ 1 ≫ α > 0. Then Lemma 7.31 implies ∫ |∇uk |2 dvg ≤ C Ak (α,β)

∫ |∇wk |2 dvg ,

(7.80)

Ak (α,β)

where Δwk = 0 in Ak (α, β), wk = uk on 𝜕Ak (α, β). To estimate the right-hand side of (7.80), we define fk and gk by Δfk = 0

in Ak (α, β),

fk |𝜕D

= uk − ω1 (∞),

βδ1 k

(xk1 )

fk |𝜕D

αδℓ+1 k

(xk1 )

= uk − ωℓ+1 (P), (7.81)

and Δgk = 0

in Ak (α, β),

gk |𝜕D

= ω1 (∞),

βδ1 k

(xk1 )

respectively, where P = limk→∞

xk1 −xkℓ+1 . δkℓ+1

gk |𝜕D

αδℓ+1 k

(xk1 )

= ωℓ+1 (P), (7.82)

Since wk = fk + gk , we have

∫ |∇wk |2 dvg ≤ 2 ∫ |∇fk |2 dvg + 2 ∫ |∇gk |2 dvg , Ak (α,β)

Ak (α,β)

Ak (α,β)

7.4 Harmonic map revisited | 439

and it holds that ∫ |∇fk |2 dvg = o(1), Ak (α,β)

∫ |∇gk |2 dvg ≤ C

|ω1 (∞) − ωℓ+1 (P)|2 log(

Ak (α,β)

Inequalities (7.83) imply ∫A

k (α,β)

|∇uk |2 dvg ≤ C ∫A

k (α,β)

αδkℓ+1 ) βδk1

(7.83)

.

|∇wk |2 dvg = o(1), and hence

󵄨 󵄨2 ∫ 󵄨󵄨󵄨∇(vk (xkℓ+1 + δkℓ+1 ⋅) − ωℓ+1 )󵄨󵄨󵄨 dvg = o(1),

DR

which completes the proof of Lemma 7.34. To prove (7.83), we abbreviate 𝜕Akα = 𝜕Dαδl+1 (xk1 ) and 𝜕Akβ = 𝜕Dβδ1 (xk1 ). First, strong k

k

ℓ+1 1 convergence of ũ ℓ+1 in Hloc (S2 \ ({∞} ∪ Dα )) and of ũ 1k → ω1 in H 1 (Dβ ) implies k →ω ℓ+1 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ℓ+1 ⋅ − x = o(1), 󵄩󵄩uk − ω ( ℓ+1k )󵄩󵄩󵄩 󵄩󵄩W 1/2,2 (𝜕Ak ) 󵄩󵄩 δk α 1 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 ⋅−x = o(1). 󵄩󵄩uk − ω ( 1 k )󵄩󵄩󵄩 󵄩󵄩 δk 󵄩󵄩W 1/2,2 (𝜕Akβ )

We have, on the other hand, ωℓ+1 (

xk1 −xkℓ+1 ) δkℓ+1

→ ωℓ+1 (P) and ω1 (

⋅−xk1 ) δk1

→ ω1 (∞) by (7.79),

and hence the first equality of (7.83) by the elliptic estimate to (7.81). Second, since the boundary value of gk is a constant in (7.82), it follows that gk = Ak log |x| + Bk for Ak =

ωℓ+1 (P) − ω1 (∞) log(

αδkℓ+1 βδk1

)

,

Bk =

ω1 (∞) log αδkℓ+1 − ωℓ+1 (P) log βδk1 log(

αδkl+1 ) βδk1

,

which implies R2

R 󵄨 󵄨2 ∫ |∇gk | dvg ≤ C ∫ 󵄨󵄨󵄨g 󸀠 (r)󵄨󵄨󵄨 r dr = CA2k log( 2 ) R1 2

Ak (α,β)

R1

for R2 = αδkl+1 and R1 = βδk1 , and hence the second inequality of (7.83). Proof of Theorem 7.23. We have subtracted the principal and the second bubbles {ωj }1≤j≤ℓ and {ωj }ℓ+1≤j≤ℓ󸀠 , respectively. If there is no energy gap between these bubbles, the proof if finished. Otherwise, we apply the hierarchical argument and detect a third set of bubbles by the bubbling process of Steps 2 and 3. By Lemma 7.29, each ωi has the energy bounded from below, and therefore, this process terminates in a finite time. Eventually, we obtain the energy identity and the proof is complete.

440 | 7 Hardy spaces and BMO

7.5 L∞ -estimates 7.5.1 BMO estimate of the Green function Duality between Hardy space and BMO ensures critical estimates in partial differential equations, not obtained by the standard method [90]. The function log |x| ∈ BMO \ L∞ is a constant multiple of the fundamental solution to the two-dimensional Laplacian −Δ. Moreover, the variational problem for this operator taking a principal part is based on the H 1 Sobolev space. The associated Sobolev inequality becomes also critical in two space dimensions. Here we show that the fundamental solution to the second order elliptic operator belongs to BMO [74]. Then we obtain several estimates by using the results in the previous subsection, which ensures a condition for a function in L1 to belong to ℋ1 .21 Let L = 𝜕i (aij (x)𝜕j ) be a linear elliptic partial differential operator in divergence form with the real-valued measurable functions aij (x) = aji (x), i, j = 1, 2, such that 2

λ−1 |ξ |2 ≤ ∑ aij (x)ξi ξj ≤ λ|ξ |2 , i,j=1

x, ξ ∈ R2

for λ > 0. The Green function Gx = Gx (y) ∈ L1loc (R2 ) for L is defined for x, y ∈ R2 and satisfies −LGx = δx (y) in the sense of distribution of y, where δx (y) denotes the Dirac measure concentrated at y = x. This Green’s function is normalized by infB2 (x) Gx = 0, and we have the following fact analogous to L1 -estimate [56]. Proposition 7.35 (Stampacchia). Any 1 ≤ p < 2 and 1 ≤ q < ∞ admit A1 (p, λ) > 0 and A2 (q, λ) > 0 such that ‖∇G0 ‖Lp (B2 (0)) ≤ A1 (p, λ),

‖G0 ‖Lq (B2 (0)) ≤ A2 (q, λ).

(7.84)

̃ Let G = G0 for simplicity, and put G(x) = G(Rx) − infx∈B2 (0) G(Rx), ã ij (x) = aij (Rx), ̃ ̃ and L = 𝜕i (aij (x)𝜕j ) for R > 0. It holds that −L̃ G̃ = δ ≡ δ0 and infB2 (0) G̃ = 0, and hence ̃ ‖∇G(x)‖ Lp (B2 (0)) ≤ A1 (p, λ) for 1 ≤ p < 2 by Proposition 7.35. Putting y = Rx, we obtain R{

1/p

1 ∫ |∇G|p } |B2R (0)|

≤ (4π)−1/p A1 (p, λ),

1≤p 3R.

(7.87)

Proposition 7.36 (Cacciopoli). There exists C = C(n, λ) > 0 such that 1 u ∈ Hloc (Ω),

in Ω ⊂ Rn ,

Lu = 0

B 3 R (x0 ) ⊂ Ω

(7.88)

2

implies |∇u|2 ≤

∫ BR/2 (x0 )

C R2

∫ |u|2 . BR (x0 )

Proposition 7.37 (Harnack inequality). There exists C = C(n, λ) > 0 such any u = u(x) ≥ 0 satisfying (7.88) admits sup u ≤ C inf u. BR (x0 )

BR (x0 )

Lemma 7.38. Inequality (7.86) is valid for any B = BR (x0 ) ⊂ R2 and 1 ≤ p < 2. Proof. Turning to (7.87), we note R{

1 ∫ |∇G|p } |B| B

1/p

≤ R{

2

1 ∫ |∇G|1/2 } = π 1/2 { ∫ |∇G|2 } |B| B

≤ π 1/2 {

BR (x0 )

∫

B |x0 | (x0 ) 2

1/2

󵄨󵄨 󵄨2 󵄨󵄨∇G(y)󵄨󵄨󵄨 dy}

1/2

(7.89)

442 | 7 Hardy spaces and BMO for B = BR (x0 ) and 1 ≤ p < 2. Then, put ̃ G(x) = G(|x0 |x) − inf G(|x0 |x), x∈B2 (0)

ã ij (x) = aij (|x0 |x),

L̃ = 𝜕i (ã ij (x)𝜕j ).

x

We obtain |σ| = 1 and 0 ∈ ̸ B 3 (σ) ⊂ B2 (0) for σ = |x0 | . Then also L̃ G̃ = 0 and G ≥ 0 in 0 4 B 3 (σ). Hence Proposition 7.36 is applicable, which ensures 4

1/2

{ ∫ |∇G|̃ 2 }

1/2

≤ C1 { ∫ |G|̃ 2 } .

B 1 (σ)

B 3 (σ)

2

4

Using Proposition 7.37, now we estimate the right-hand side above by22 ̃ C2 ∫ |G|̃ ≤ C2 ∫ |G|, B 3 (σ)

B2 (0)

4

and then, the second inequality of (7.84) implies 1/2

{ ∫ |∇G|̃ 2 }

≤ C3 .

(7.90)

B 1 (σ) 2

Using the transformation of variables x = R{

1 ∫ |∇G|p } |B|

1/p

y |x0 |

≤ π 1/2 {

B

in (7.89), we obtain

∫

B |x0 | (x0 )

󵄨󵄨 󵄨2 󵄨󵄨∇G(y)󵄨󵄨󵄨 dy}

2

= C{ ∫ |∇G|̃ 2 }

1/2

1/2

≤C

B 1 (σ) 2

by (7.90), and hence all the cases of (2.29) follow. Theorem 7.39 ([74]). There is C(λ) > 0 such that ‖Gx ‖BMO(R2 ) ≤ C(λ).

(7.91)

Proof. We apply p = 1 to (7.86) and use the Poincaré–Sobolev inequality (7.37). It holds 1 that |B| ∫B |Gx − (Gx )B | ≤ C, and hence (7.91). 22 It is the reverse Hölder inequality which implies C α -regularity; see also [133].

7.5 L∞ -estimates | 443

7.5.2 Wente’s inequality Lemma 7.40. For u, v ∈ H 1 (R2 ) with compact supports, the function ψ(x) = ∫ Gx (y)(ux1 vx2 − ux2 vx1 )(y) dy R2

satisfies ‖ψ‖∞ + ‖∇ψ‖2 ≤ C‖∇u‖2 ⋅ ‖∇v‖2 .

(7.92)

Proof. Having Corollary 7.22 and Theorem 7.39, we apply Fefferman’s duality theorem in the form of ‖f ⋅ g‖L1 (Rn ) ≤ C‖f ‖BMO(Rn ) ⋅ ‖g‖ℋ1 (Rn ) to estimate the first term on the left-hand side of (7.92). To estimate the second term on the left-hand side of (7.92), we take η ∈ C0∞ (R2 ) such that η = 1 in BR , η = 0 on R2 \ B2R , and 0 ≤ η ≤ 1.23 Since n = 2, we have C > 0 independent of R such that ‖∇η‖2 ≤ C. Since −Lψ = ux1 vx2 − ux2 vx1 , we have ∑ ∫ aij (x)ψxi (η2 ψ)x = ∫ (ux1 vx2 − ux2 vx1 )ψη2 , i,j

j

ℛ2

ℛ2

and hence ∑ ∫ η2 aij (x)ψxi ψxj = ∑ ∫ (ux1 vx2 − ux2 vx1 )ψη2 − 2ψηaij ψxi ηxj dx. i,j

i,j

R2

R2

It thus holds that λ−1 ∫ η2 |∇ψ|2 ≤ ∫ (|ux1 vx2 − ux2 vx1 | + Cλ|∇ψ|η ⋅ |∇η|)|ψ| R2

R2

1 ≤ ∫ |ux1 vx2 − ux2 vx1 ||ψ| + λ−1 η2 |∇ψ|2 + C(λ)|∇η|2 |ψ|2 dx. 2 R2

An estimate of the first term on the left-hand side of (7.92) implies ∫ η2 |∇ψ|2 ≤ C(λ)(‖∇u‖2 ‖∇v‖2 ‖ψ‖∞ + ‖∇η‖22 ‖ψ‖2∞ ) ≤ C‖∇u‖22 ‖∇v‖22 ,

R2

and therefore, ‖∇ψ‖L2 (BR ) ≤ C‖∇u‖2 ⋅ ‖∇v‖2 . This C > 0 is independent of R and hence the desired result follows. 23 Since W 1,n (Rn ) ⊂ BMO(Rn ), we have ℋ1 ⊂ W −1,n , which gives an alternative proof of (7.92).

444 | 7 Hardy spaces and BMO The following inequality for L = Δ is used in constructing the surface of constant mean curvature by the variational method [366]. This surface stands for the soap bubble and is represented by H system described in § 7.4.1. Let B = B1 (0) ⊂ R2 . Theorem 7.41 (Wente). Any u, v ∈ H 1 (B) and ϕ ∈ W01,1 (B) satisfying − Lϕ = ux1 vx2 − ux2 vx1

in B,

ϕ|𝜕B = 0

(7.93)

admit ϕ ∈ C(B) ∩ H01 (B) and ‖ϕ‖L∞ (B) + ‖∇ϕ‖L2 (B) ≤ C‖∇u‖L2 (B) ⋅ ‖∇v‖L2 (B) .

(7.94)

Proof. It is obvious that ux1 vx2 − ux2 vx1 ∈ L1 (B). By the L1 -theory [56], therefore, (7.93) is uniquely solvable in ϕ ∈ W01,1 (Ω). We show, thus, (7.94) for smooth u, v. We use Poincaré–Wirtinger’s inequality 1 ∫ u, |B|

‖u‖L2 (B) ≤ C0 ‖∇u‖L2 (B) ,

u=u−

‖v‖L2 (B) ≤ C0 ‖∇v‖L2 (B) ,

1 v=v− ∫ v, |B|

B

B

where C0−1/2 denotes the second eigenvalue of −Δ under the Neumann condition. There is an extension operator Φ : H 1 (B) → Hc1 (R2 ) = {v ∈ H 1 (R2 ) | supp v is compact}, satisfying ‖Φv‖H 1 (R2 ) ≤ C1 ‖v‖H 1 (B) and Φv|B = v a. e. in B. Hence U = Φu and V = Φv satisfy ‖∇U‖L2 (R2 ) ≤ C, ‖∇u‖L2 (B) , and ‖∇V‖L2 (R2 ) ≤ C‖∇v‖L2 (B) , respectively. Then, ψ(x) = ∫ Gx (y)(Ux1 Vx2 − Ux2 Vx1 )(y) dy R2

satisfies −L(ϕ − ψ) = 0 in B, and the maximum principle implies24 ‖ϕ − ψ‖L∞ (B) ≤ ‖ψ − ϕ‖L∞ (𝜕B) = ‖ψ‖L∞ (𝜕B) ≤ ‖ψ‖L∞ (B) . By Lemma 7.40, we obtain ‖ϕ‖L∞ (B) ≤ ‖ϕ − ψ‖L∞ (B) + ‖ψ‖L∞ (B) ≤ 2‖ψ‖L∞ (B)

≤ C‖∇U‖L2 (R2 ) ‖∇V‖L2 (R2 ) ≤ C‖∇u‖L2 (B) ‖∇v‖L2 (B) ,

while the inequality ‖∇ϕ‖L2 (B) ≤ C‖∇u‖L2 (B) ‖∇v‖L2 (B) follows similarly as in this lemma. Hence the result follows. Since ‖ϕ‖∞ and ‖∇ϕ‖2 are invariant under the conformal transformation, Riemann mapping theorem extends (7.94) for B to that for a simply-connected domain Ω.25 24 This weak maximum principle is valid for the weak solution to the Dirichlet boundary problem for the elliptic operator in divergence form having bounded coefficients [189]. 25 By the argument [41] using the coarea formula, even this simply-connectedness of Ω is removed [74].

7.5 L∞ -estimates | 445

7.5.3 Brezis–Merle’s inequality revisited Let Ω ⊂ R2 be a bounded domain and f ∈ H −1 (Ω) = H01 (Ω)󸀠 . The boundary value problem − Lu = f

in Ω,

u|𝜕Ω = 0

(7.95)

admits a unique solution u ∈ H01 (Ω), and if f ∈ L1 (Ω) then u ∈ W 1,p (Ω) for 1 ≤ p < 2 [56].26 We have a sharper form and generalization of the result in § 5.3.3. Theorem 7.42 (Brezis–Merle). It holds that 󵄨 󵄨 ∫ exp(C1 󵄨󵄨󵄨u(x)󵄨󵄨󵄨/‖f ‖1 ) dx ≤ C2 (diam Ω)2

(7.96)

Ω

for Ci = Ci (λ), i = 1, 2. Proof. We may assume Ω ⊂ BR for R = side Ω, and put

1 2

diam Ω. Take the zero extension of f (x) out-

󵄨 󵄨 󵄨 󵄨 ϕ(x) = ∫ Gx (y)󵄨󵄨󵄨f (y)󵄨󵄨󵄨 dy = ∫ Gx (y)󵄨󵄨󵄨f (y)󵄨󵄨󵄨 dy.

(7.97)

BR

R2

It holds that −Lϕ = |f | in R2 and also |u(x)| ≤ ϕ(x) by the maximum principle. Hence (7.96) is deduced from ∫ exp(c1 ϕ(x)/‖f ‖1 ) dx ≤ C2 (diam Ω)2 . Ω

Using y = x/R, furthermore, we may assume R = 1. This theorem is thus reduced to the following proposition. Proposition 7.43. There are σ = σ(λ) > 0 and C = C(λ) > 0 such that (7.97) for R = 1 ensures ∫ eσϕ(x)/‖f ‖1 dx ≤ C. B1

Lemma 7.44. It holds that ‖ϕ‖BMO ≤ C‖f ‖1 . Proof. Given a disc B ⊂ R2 , we have ϕ(x) − ϕB = ∫ (Gx (y) −

1 󵄨 󵄨 ∫ Gz (y) dz)󵄨󵄨󵄨f (y)󵄨󵄨󵄨 dy, |B| B

R2

26 In a general n, this inclusion follows for 1 ≤ p
2, this u belongs to the Lorentz space

446 | 7 Hardy spaces and BMO and therefore, 󵄨 󵄨󵄨 1 1 1 󵄨 󵄨󵄨 󵄨 󵄨 ∫ |ϕ − ϕB | ≤ ∫ dy ∫ dx ⋅ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨Gx (y) − ∫ Gz (y) dz 󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 |B| |B| |B| B

B

ℛ2

B

Here we use Gx (y) = Gy (x) derived from the divergence form of L. Theorem 7.39 implies ‖g‖BMO ≤ C for g(x) = Gx (y), and hence 󵄨󵄨 1 󵄨󵄨󵄨󵄨 1 1 󵄨 ∫󵄨󵄨Gx (y) − ∫ Gz (y) dz 󵄨󵄨󵄨 dx = ∫ |g − gB | ≤ C. 󵄨󵄨 |B| 󵄨󵄨 |B| |B| B

Then we obtain

1 ∫ |ϕ |B| B

B

B

− ϕB | ≤ C‖f ‖1 .

Proof of Proposition 7.43. Let B = B1 (0). By John–Nirenberg’s inequality (7.41), there exists σ > 0 such that 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨{x ∈ B | 󵄨󵄨󵄨ϕ(x) − ϕB 󵄨󵄨󵄨 > t}󵄨󵄨󵄨 ≤ C1 |B| exp(−C2 t/‖ϕ‖BMO ) ≤ C1 |B| exp(−σt/‖f ‖1 ), t > 0, which implies 1 ∫ eσ|ϕ(x)−ϕB |/‖f ‖1 dx ≤ C, |B| B

similarly to (7.42). Here we have |ϕB | ≤

1 1 󵄨 󵄨 ∫ ∫ Gx (y)󵄨󵄨󵄨f (y)󵄨󵄨󵄨 dy dx ≤ sup ∫ Gx (y) dy ⋅ ‖f ‖1 , |B| |B| y∈B B B

where the Stampacchia estimate (7.84) applies, supy∈B obtain the result.

B

1 ∫ G (y) dx |B| B x

≤ C(λ). Then we

7.5.4 System of chemotaxis The Smoluchowski–Poisson equation ut = ∇ ⋅ (∇u − u∇v), −Δv = u in Ω × (0, T), 󵄨󵄨 𝜕u 𝜕v 󵄨 ( − u , v)󵄨󵄨󵄨 = 0, u|t=0 = u0 (x) ≥ 0 󵄨󵄨𝜕Ω 𝜕ν 𝜕ν

(7.98)

is a simplified system of chemotaxis introduced in § 1.1.1, where Ω ⊂ Rn is a bounded domain which a smooth boundary 𝜕Ω, and ν denotes the outer normal unit vector. Local-in-time unique existence of the solution u = u(x, t) ≥ 0 is assured for a regular initial value, and T ∈ (0, +∞] denotes its maximal existence time [371]. The case T < +∞ is studied in the context of self-organization. It is actually expected in mathematical biology that u(x, t) approaches a sum of δ-functions in case n = 2 [243, 85].

7.5 L∞ -estimates | 447

In this simplified form, the variable u is subject to the Smoluchowski equation, the fundamental equation of mass transport in the canonical setting [335], which implies the total mass conservation 󵄩 󵄩󵄩 󵄩󵄩u(⋅, t)󵄩󵄩󵄩1 = ‖u0 ‖1 ≡ λ.

(7.99)

The second equation is the Poisson equation, where the chemical gradient ∇v is created by u. The symmetry of the Green function G = G(x, x󸀠 ), indicated by G(x 󸀠 , x) = G(x, x󸀠 ), is due to the action–reaction law, which results in the decrease of the free energy, ℱ (u) = ∫ u(log u − 1) dx − Ω

1 ∬ u ⊗ u dx dx󸀠 2 Ω×Ω

27

for u ⊗ u = u(x, t)u(x , t). This property takes the form 󸀠

d 󵄨 󵄨2 ℱ + ∫ u󵄨󵄨󵄨∇(log u − v)󵄨󵄨󵄨 = 0, dt

(7.100)

Ω

and then the stationary state is defined by u > 0,

log u − v = constant,

‖u‖1 = λ,

which results in the Boltzmann–Poisson equation28 − Δv =

λev ∫Ω ev

in Ω,

v|𝜕Ω = 0

(7.101)

with u=

λev . ∫Ω ev

In connection with real analysis developed in this section, if n = 2, we have [334] T < +∞ 󳨐⇒ lim ∫ u log u = +∞, t↑T

(7.102)

Ω

1 for a measurable function u ≥ 0 in Rn , and recalling that u log u ∈ L1loc means u ∈ ℋloc that

󵄩 󵄩 󵄩 󵄩 lim sup ∫ u log u < +∞ 󳨐⇒ 󵄩󵄩󵄩∇v(t)󵄩󵄩󵄩2 + 󵄩󵄩󵄩v(t)󵄩󵄩󵄩∞ ≤ C, t↑T

0 ≤ t < T.

Ω

27 Poisson equation with the Neumann boundary condition is treated similarly, except for the possible boundary blow-up the magnitude of which takes a half of that of the inner blow-up. Equation (4.166) is the form if Ω is a compact Riemannian surface without boundary. 28 See [304] for the other case of the Poisson part.

448 | 7 Hardy spaces and BMO There is actually a quantized blow-up mechanism for the family of solutions to (7.101) for n = 2 under the control of ℋℓ (x1 , . . . , xℓ ) =

1 ℓ ∑ R(xi ) + ∑ G(xi , xj ), 2 i=1 1≤i −∞,

(7.105)

t↑+∞

furthermore, the limit measure μ(dx, t) in (7.104) is independent of t, and it holds that ℓ

μ(dx, t) = 8π ∑ δx∗ (dx) i=1

i

with x∗ = (x1∗ , . . . , xℓ∗ ) ∈ 𝒦ℓ . Blow-up in finite time, indicated by T < +∞, has also a profile of quantized blowup mechanism, with nonvanishing regular part denoted by f (x) dx and with possible collision of subcollapses indicated by m(x0 ) ≥ 2, below. Here, the backward self-similar transformation of u, z(y, s) = (T − t)u(x, t),

y = (x − x0 )/(T − t)1/2 ,

s = − log(T − t)

(7.106)

takes a role. More precisely, it accumulates in C∗ ((−∞, +∞), ℳ(R2 )) as s ↑ +∞, the space of ∗-weakly continuous mappings t ∈ (−∞, +∞) 󳨃→ μ(dy, s) ∈ ℳ(R2 ), where ℳ(R2 ) = C0 (R2 )󸀠 stands for the set of Radon measures in R2 . Thus we put C0 (R2 ) = {φ ∈ C(R2 ∪ {∞}) | φ(∞) = 0}, where R2 ∪ {∞} denotes the one-point compactification of R2 . We also use the fundamental solution of −Δ, Γ(y) =

1 1 log , 2π |y|

which satisfies −ΔΓ = δ0 . Theorem 7.46 ([336, 337]). Let n = 2, and assume T < +∞ in (7.98). Then u(x, t) dx defined in Ω × [0, T) is extended as μ(dx, t) ∈ C∗ ([0, T], ℳ(Ω)). It holds that μ(dx, T) = 8π ∑ m(x0 )δx0 (dx) + f (x) dx, x0 ∈𝒮

(7.107)

where 𝒮 = {x0 ∈ Ω | ∃tk ↑ T, ∃xk → x0 such that u(xk , tk ) → +∞}

stands for the blow-up set. We have 𝒮 ⊂ Ω, m(x0 ) ∈ N for x0 ∈ 𝒮 , and 0 < f = f (x) ∈ L1 (Ω) ∩ C(Ω \ 𝒮 ).

(7.108)

If limt↑T ℱ (u(⋅, t)) > −∞, furthermore, it holds that m(x0 ) = 1 for any x0 ∈ Ω. Given x0 ∈ 𝒮 , define z = z(y, s) by (7.106). Then, any sk ↑ +∞ admits a subsequence, denoted by the same symbol, such that z(y, s + sk ) dy ⇀ ζ (dy, s) in C∗ ((−∞, +∞), ℳ(R2 )).

450 | 7 Hardy spaces and BMO The scaling back A(dy󸀠 , s󸀠 ) of this ζ (dy, s), defined by ζ (dy, s) = e−s A(dy󸀠 , s󸀠 ),

y󸀠 = e−s/2 y,

s󸀠 = −e−s ,

satisfies A(dy, s) = 8πδ0 (dy),

−∞ < s < 0

if m ≡ m(x0 ) = 1, and m

A(dy, s) = 8π ∑ δyi (s) (dy), j=1

−∞ < s < 0

if m ≥ 2. In the latter case, y = (y1 (s), . . . , ym (s)), made by the support of A(dy, s), is subject to dyj ds

0 = 8π∇yj ℋm (y1 , . . . , ym ),

−∞ < s < 0,

1 ≤ j ≤ m,

where 0

ℋm (y1 , . . . , ym ) =

∑ Γ(yi − yj ).

1≤i 0 for any x0 ∈ 𝒦1 , according to the state of the initial value u0 .31 Then (7.103) holds, on the other hand, provided that either E8π = 0 or ℱ (u0 ) < infu∗ ∈F8π ℱ (u∗ ). In this case, a dual form of the Trudinger–Moser inequality guarantees (7.105), and therefore, the limit measure μ(dx, t) in (7.104) takes the form μ(dx, t) = 8πδx∗ (dx) with x∗ ∈ 𝒦1 . Conditions (7.103) and (7.109) imply λ = 8πℓ with 𝒦ℓ ≠ 0 for some ℓ ∈ N, and Eλ ≠ 0, respectively. If Ω is simply-connected, it holds that Eλ = 0 for λ ≫ 1, and therefore, we have T < +∞, provided that λ ∈ ̸ 8πN and λ ≫ 1. This property of Eλ = 0 for λ ≫ 1 does not arise any more if Ω is multiply-connected [82]. If Ω is convex and λ > 8π, on the other hand, one gets either T < +∞ or (7.109) because 𝒦ℓ = 0 holds for ℓ ≥ 2 by Theorem 3.19. In the latter case we have Eλ ≠ 0 similarly, and therefore, if Ω is convex and λ ≫ 1, it holds always that T < +∞. If Ω is convex, λ = 8π, and E8π = 0, we have u(x, t) dx ⇀ 8πδx∗ (dx)

in ℳ(Ω)

as t ↑ T for x∗ ∈ 𝒦1 , because of the Trudinger–Moser inequality and #𝒦1 = 1.

31 From the topological degree calculation and the fact that #E8π ≤ 1, it follows that E8π = 0 in the other case, D(x0 ) > 0, ∀x0 ∈ 𝒦1 .

Bibliography [1] [2]

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

U. Abresch, Constant mean curvature tori in terms of elliptic functions, J. Reine Angew. Math. 374 (1987), 169–192. A. Acker, L. E. Payne, and G. Philippin, On the convexity of level lines of the fundamental mode in the clamped membrane problem, and the existence of convex solutions in a related free boundary problem, J. Appl. Math. Phys. 32 (1981), 683–694. R. A. Adams, Sobolev Spaces, Academic Press, Boston, 1978. S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand-Reinhold, Princeton, 1965. L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1978. K. Akô, On the Dirichlet problem for quasilinear elliptic differential equations of the second order, J. Math. Soc. Jpn. 13 (1961), 45–62. G. Alessandrini, Cmtical points of solutions of elliptic equations in two variables, Ann. Sc. Norm. Super. Pisa IV 14 (1987), 229–256. A. D. Alexandroff, Uniqueness theorems for surfaces in the large, Trans. Am. Math. Soc. 21 (1955), 412–416. N. D. Alikakos, Lp bounds of solutions of reaction-diffusion equations, Commun. Partial Differ. Equ. 4 (1979), 827–868. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620–709. H. Amann and M. G. Crandall, On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J. 27 (1978), 779–790. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. F. V. Atkinson and L. A. Peletier, Ground states of −Δu = f (u) and the Emden-Fowler equation, Arch. Ration. Mech. Anal. 96 (1986), 147–165. Th. Aubin, Equations différentielles non lineaires et problém de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), 269–293. J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, 1984. L. Baccardo, F. Murat, and J. P. Puel, Résults d’existence pur certains problémes elliptiques quasilinéaires, Ann. Sc. Norm. Super. Pisa 11 (1984), 213–235. A. Bahri, Critical Points at Infinity in Some Variational Problems, Longman, New York, 1988. A. Bahri and P. L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Commun. Pure Appl. Math. 45 (1992), 1205–1215. A. Bahri and P. H. Rabinowitz, A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal. 82 (1989), 412–428. I. J. Bakelman, Convex Analysis and Nonlinear Geometric Elliptic Equations, Springer-Verlag, Berlin, 1994. C. Bandle, Mean value theorems for functions satsifying the inequality Δu + Keu ≥ 0, Arch. Ration. Mech. Anal. 51 (1973), 70–84. C. Bandle, Existence theorems, qualitive results and a priori bounds for a class of nonlinear Dirichlet problems, Arch. Ration. Mech. Anal. 58 (1975), 219–238. C. Bandle, Isoperimetric inequalities for a nonlinear eigenvalue problem, Proc. Am. Math. Soc. 56 (1976), 243–246. C. Bandle, On a differential inequality and its applications to geometry, Math. Z. 147 (1976), 253–261. C. Bandle, Isoperimetric Inequalities and Applications, Pitmann, Boston, 1980.

https://doi.org/10.1515/9783110556285-008

454 | Bibliography

[26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44]

[45] [46] [47] [48] [49] [50]

S. Baraket, Construction of singular limits for a strongly perturbed two-dimensional Dirichlet problem with exponential nonlinearity, Bull. Sci. Math. 123 (1999), 255–284. S. Baraket and Y. Dong, Singular limit solutions for two-dimensional elliptic problems with exponentially dominated nonlinearity, Chin. Anal. Math. 22B3 (2001), 287–296. S. Baraket and F. Pacard, Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. 6 (1998), 1–38. D. Bartolucchi and C.-S. Lin, Existence and uniqueness for mean field equation on multiply connected domain at the critical parameter, Math. Ann. 359 (2014), 1–44. J. Bartz, M. Struwe, R. Ye, A new approach to the Ricci flow on S2 , Ann. Sc. Norm. Super. Pisa C1. Sci. IV 21 (1994) 475–482. L. Battaglia, A. Jevnikar, A. Malchiodi, and D. Ruiz, A general existence result for the Toda system on compact surfaces, Adv. Math. 285 (2015), 937–979. J. Bebernaes and D. Eberly, A description of self-similar blow-up for dimensions n ≥ 3, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5 (1988), 1–21. J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Springer, New York, 1989. J. Bebernes, A. Bressan, and D. Eberly, A description of blowup for the solid ignition mdedia, Indiana Univ. Math. J. 36 (1987), 295–305. H. Bellout, A criterion for bloiv-up of solutions to semilinear heat equations, SIAM J. Math. Anal. 18 (1987), 722–727. P. Benilan, H. Brezis, and M. G. Crandall, A semilinear equation in L1 (ℛN ), Ann. Sc. Norm. Super. Pisa IV 2 (1975), 523–555. H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding methos, Bol. Soc. Bras. Mat. 22 (1991), 1–37. M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977. L. Bers, Local behavior of solution of general linear elliptic equations, Commun. Pure Appl. Math. 8 (1955), 473–496. F. Bethuel, On the singular set of stationary harmonic maps, Manuscr. Math. 78 (1993), 417–443. F. Bethuel and J. M. Ghidaglia, Improved regularity of solutions to elliptic equations involving Jacobians and applications, J. Math. Pures Appl. 72 (1993), 441–474. J. Beyon, Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three dimensional annului, J. Differ. Equ. 136 (1997) 136–165. A. I. Bobenko, All constant mean curvature tori in R3 , S3 , H3 in therms of theta-functions, Math. Ann. 290 (1991), 209–245. H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an appliation to the diffusion equation. A. Bressan, Stable blow-up patterns, J. Differ. Equ. 98 (1992), 57–75. H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dan les Espace de Hilbert, North-Holland, Amsterdam, 1973. H. Brezis, Analyse Fonctionnelle, Masson, Paris, 1983. H. Brezis and M. Coron, Multiple solutions of H-systems and Rellich’s conjecture, Commun. Pure Appl. Math. 37 (1984), 149–187. H. Brezis and M. Coron, Convergence of solutions of H-systems or how to blow bubbles, Arch. Ration. Mech. Anal. 89 (1985), 21–56. H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. 58 (1979), 137–151.

Bibliography | 455

[51] [52]

[53] [54] [55]

[56] [57] [58] [59] [60] [61] [62]

[63]

[64] [65] [66] [67] [68] [69] [70] [71] [72]

[73]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc. 88 (1983), 486–490. H. Brezis and P. L. Lions, A Note on Isolated Singularities for Linear Elliptic Equations, Mathematical Analysis and Applications, Part A, no. 7, Advances in Mathematics, Supplementary Studies, Academic Press, 1981, pp. 263–266. H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of −Δu = V (x)eu in two dimensions, Commun. Partial Differ. Equ. 16 (1991), 1223–1253. H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math. 36 (1983), 437–477. H. Brezis and L. A. Peletier, Asymptotics for elliptic equations involving critical growth, In: Partial Differential Equations and Calculus of Variations (I. F. Colombini, A. Marino, L. Modica, and S. Spagnolo, eds.), Birhäuser, Boston, 1989, pp. 149–192. H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in L1 , J. Math. Soc. Jpn. 25 (1973), 565–590. H. Brezis, Y. Y. Li, and I. Shafrir, A sup + inf inequality for some nonlinear elliptic equations involving exponential nonlinearities, J. Funct. Anal. 115 (1993), 344–358. H. Brezis, T. Cazenave, Y. Martel, and A. Ramiandrisoa, Blow up for ut − Δu = g(u) revisited, Adv. Differ. Equ. 1 (1996), 73–90. Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer, 1988. L. A. Caffarelli and A. Friedman, Asymptotic estimates for the plasma problem, Duke Math. J. 47 (1980), 705–742. L. A. Caffarelli and A. Friedman, Convexity of solutions of semilinear elliptic equations, Duke Math. J. 52 (1985), 431–456. E. Cagilioti, P. L. Lions, C. Marchioro, and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Commun. Math. Phys. 143 (1992), 501–525. E. Caglioti, P.-L. Lions, C. Marchioro, and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Part II, Commun. Math. Phys. 174 (1995) 229–260. A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–129. P. Cardaliaguet and R. Tahraoui, On the strict concavity of the harmonic radius in dimention N ≥ 3, J. Math. Pures Appl. 81 (2002), 223–240. L. Carleson, Selected Problems on Exceptional Sets, Van Nostrand, Princeton, 1967. L. Carleson and S. Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. 110 (1986), 113–127. A. Castro and R. Shivaji, Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric, Commun. Partial Differ. Equ. 14 (1989), 1091–1100. D. Chae and O. Imanuvilov, The existence of non-topological multivorted solutions in relativistic selfdual Chen-Simons theory, Commun. Math. Phys. 215 (2000), 119–142. S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover, New York, 1957. S. Y.-A. Chang and P.-C. Yang, Conformal deformation of metrices on S2 , J. Differ. Geom. 27 (1988), 259–296. S.-Y. A. Chang, C.-C. Chen, and C.-S. Lin, Extremal functions for a mean field equations in two dimensions, In: Lectures on Partial Differential Equations (S.-Y. A. Chang, C.-S. Lin, and S.-T. Yau, eds.), International Press, New York, 2003, pp. 61–93. S. Chanillo and M. Kiessling, Rotational symmetry of solutions of some nonlinear problems in statistical mechanics in geometry, Commun. Math. Phys. 160 (1994) 217–238.

456 | Bibliography

S. Chanillo and Y. Y. Li, Continuity of solutions of uniformly elliptic equations in R2 , Manuscr. Math. 77 (1992), 415–433. [75] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984. [76] X. Chen, Remarks on the existence of branch bubbles on the blowup analysis of equation −Δu = e2u in dimension two, Commun. Anal. Geom. 7 (1999), 295–302. [77] J. T. Chen and W. A. Huang, Convexity of capillary surfaces in the outer space, Invent. Math. 67 (1982), 253–259. [78] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615–622. [79] W. Chen and C. Li, Prescribing Gaussian curvatures on surfaces with conical singularities, J. Geom. Anal. 1 (1991), 359–372. [80] C. C. Chen and C.-S. Lin, On the symmetry of blowup solutions to a mean field equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 18 (2001), 271–296. [81] C.-C. Chen and C.-S. Lin, Sharp estimates for soluitions to multi-bubbles in compact Riemann surfaces, Commun. Pure Appl. Math. 55 (2002), 728–771. [82] C.-C. Chen and C.-S. Lin, Topological degree for a mean field equation on Riemann surfaces, Commun. Pure Appl. Math. 56 (2003), 1667–1727. [83] S. Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), 43–55. [84] K.-S. Cheng and C.-S. Lin, On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in R2 , Math. Ann. 308 (1997) 119–139. [85] S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci. 56 (1981), 217–237. [86] B. Chow, The Ricci flow on the 2-spheres, J. Differ. Geom. 33 (1991) 325–334. [87] B. Chow and D. Knopf, The Ricci Flow: An Introduction, American Mathematical Society, Providence RI, 2004. [88] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGrow-Hill, New York, 1955. [89] C. V. Coffman, A non-linear boundary value problem with many positive solutions, J. Differ. Equ. 54 (1984), 429–437. [90] R. Coifman, P. L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. 72 (1993), 247–286. [91] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, Wiley, New York, 1962. [92] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. II, Wiley, New York, 1962. [93] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340. [94] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Ration. Mech. Anal. 52 (1973), 161–180. [95] M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of non-linear elliptic eigenvalue problems, Arch. Ration. Mech. Anal. 58 (1975), 207–218. [96] E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differ. Equ. 74 (1988), 120–156. [97] E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, II, J. Differ. Equ. 88 (1990), 316–339. [98] H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, 1962. [99] D. G. De Figueiredo, P. L. Lions, and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61 (1882), 41–63. [100] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.

[74]

Bibliography | 457

[101] M. del Pino, M. Kowalczyk, and M. Musso, Singular limits in Liouville-type equations, Calc. Var. 24 (2005), 47–81. [102] W. Y. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. Anal. 91 (1986), 283–308. [103] W. Ding, and G. Tian, Energy identity for a class of approximate harmonic maps from surfaces, Commun. Anal. Geom. 4 (1995), 543–554. [104] W. Ding, J. Jost, J. Li, and G. Wang, Existence results for mean field equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16 (1999), 653–666. [105] O. Druet and P.-D. Thizy, Multi-bumps analysis for Trudinger-Moser nonlinearities quantificationa dn location of concentration points, J. Eur. Math. Soc. (to appear). [106] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. [107] P. Esposito and F. Morlando, On a quasilinear mean field equation with an exponenstial nonlinearity, J. Math. Pures Appl. 104 (2015), 354–382. [108] P. Esposito, M. Grossi, and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22 (2005), 227–257. [109] P. Esposito, M. Musso, and A. Pistoia, Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Differ. Equ. 227 (2006), 29–68. [110] L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, 1990. [111] L. C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Ration. Mech. Anal. 116 (1991), 101–113. [112] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties for Functions, CRC Press, Boca Raton, 1992. [113] G. L. Eyink, Onsager and the theory of hydrodynamics turbulence, Rev. Mod. Phys. 78 (2006), 87–135. [114] G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitz. ber. bayer Akad. Wiss. (1923), 169–172. [115] C. Fefferman, Characterization of bounded mean oscillation, Bull. Am. Math. Soc. 77 (1971), 587–588. [116] W. Fleming and R. I. Rishel, An integral formula for total gradient variations, Arch. Math. 11 (1960), 218–222. [117] M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helv. 67 (1992), 471–497. [118] G. B. Folland, Real Analysis - Modern Techniques and Theit Applications, Wiley, New York, 1984. [119] L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv. 68 (1993), 415–454. [120] A. Friedman, Partial Differential Parabolic Type, Prentice-Hall, 1964. [121] A. Friedman, Variational Principles and Free-boundary Problems, Wiley, New York, 1982. [122] H. Fujita, On the nonlinear equations Δu + eu = 0 and 𝜕v/𝜕t = Δu + ev , Bull. Am. Math. Soc. 75 (1969), 132–135. [123] N. Fukagai, T. Kusano, and K. Yoshida, Some remarks on the supersolution-sub solution method for superlinear elliptic equations, J. Math. Anal. Appl. 123 (1987), 131–141. [124] E. Gagliardo, Poperietá di alcune lassi di funzioni in piu variabili, Ric. Mat. 7 (1958), 102–137. [125] P. R. Garabedian, Partial Differential Equations, Celsa, New York, 1964. [126] X. Garaizar, A symmetry-breaking problem in the annulus, Appl. Anal. 90 (1991), 211–224. [127] N. Garofalo and J. L. Lewis, A symmetry result related to some overdetermined boundary value problems, Am. J. Math. (1989), 9–33.

458 | Bibliography

[128] N. Garofalo and F. H. Lin, Monotonicity propertier of variational integrals, Ap weights and unique continuation, Indiana Univ. Math. J. 35 (1986), 245–268. [129] N. Garofalo and F. H. Lin, Unique continuation for elliptic operators: a geometric - variational approach, Commun. Pure Appl. Math. 40 (1987), 347–366. [130] I. M. Gel’fand, Some problems in the theory of quasilinear equations, Trans. Am. Math. Soc. 29 (1963), no. 2, 295–381. [131] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge, 1993. [132] N. Ghoussoub and D. Preiss, A mountain pass principle for locating and classifying the mountain pass theorem, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6 (1989), 209–243. [133] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univercity Press, Princeton, 1983. [134] Giaquinta, M., Introduction to Regularity Theory for Nonlinear Elliptic Systems, Birkhäuser, Basel, 1993. [135] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math. 34 (1981), 525–598. [136] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differ. Equ. 6 (1981), 883–901. [137] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys. 68 (1979), 209–243. [138] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn , Adv. Math. Suppl. Stud. 7A (1981), 369–402. [139] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, second ed., Springer, Berlin, 1983. [140] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984. [141] F. Gladiali and M. Grossi, Some results on the Gel’fand problem, Commun. Partial Differ. Equ. 29 (2004), 1335–1364. [142] F. Gladiali and M. Grossi, On the spectrum of a nonlinear planar problem, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009), 191–222. [143] F. Gladiali, M. Grossi, H. Ohtsuka, and T. Suzuki, Morse index of multiple blow-up solution to the two-dimensional Gel’fand problem, Commun. Partial Differ. Equ. 39 (2014), 2028–2063. [144] M. Grossi and F. Takahashi, Nonexistence of multi-bubble solutions to some elliptic equations on convex domains, J. Funct. Anal. 259 (2010), 904–917. [145] M. Grossi, H. Ohtsuka, and T. Suzuki, Asymptotic non-degeneracy of the multiple blow-up solutions to the Gel’fand problem in two space dimensions, Adv. Differ. Equ. 16 (2011), 145–164. [146] M. Grüter and K.-O. Widman, The Green function for uniformly elliptic equations, Manuscr. Math. 37 (1982), 303–342. [147] R. D. Gulliver II, Regularity of minimizing surfaces of prescribed mean curvature, Ann. Math. (1973), 275–305. [148] B. Gustafsson, On the motion of a vortex in two-dimensional flow of an ideal fluid in simply connected domains, Tech. report, Department of Math., Royal Institute of Technology, Stockholm, 1979. [149] H. R. Haegi, Extremalprobleme und Ungleichungen Konformer Gebietsgrössen, Compos. Math. 8 (1951), 81–111. [150] A. Hamatake and T. Suzuki, Singular solutions for a nonlinear Dirichlet problem, Commun. Appl. Anal. 4 (2000), 411–432. [151] R. Hamilton, The Ricci flow on surfaces, In: Mathematics and General Relativity, Contem. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237–262.

Bibliography | 459

[152] Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 8 (1991), 159–174. [153] R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations, J. Differ. Geom. 30 (1989), 505–522. [154] P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math. 115 (1966), 271–310. [155] P. Hartman and A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations, Am. J. Math. 75 (1953), 449–476. [156] P. Hartman and A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations, III, Am. J. Math. 77 (1955), 453–474. [157] J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford, 1992. [158] F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une sphére, C. R. Acad. Sci. Paris Sér. I. Math. 311 (1990), 519–524. [159] F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 591–596. [160] N. Hirano and N. Mizoguchi, Nonradial solutions of semilinear elliptic equations on annului, J. Math. Soc. Jpn. 46 (1994), 111–117. [161] H. Hofer, A note on the topological degree at a critical point of mountain-pass type, Proc. Am. Math. Soc. 90 (1984), 309–315. [162] H. Hofer, A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem, J. Lond. Math. Soc. 31 (1985), 566–570. [163] M. Holzmann and H. Kielhöfer, Uniqueness of global positive branches of nonlinear elliptic problems, Math. Ann. 300 (1994), 221–241. [164] C. W. Hong, A best constant and the Gaussian curvature, Proc. Am. Math. Soc. 97 (1987), 737–747. [165] H. Hopf, Differential Geometry in the Large, no. 1000, Lecture Notes in Math., Springer, 1983. [166] R. Ikehata and T. Suzuki, Semilinear parabolic equations involving critical Sobolev exponent: local and asymptotic behavior of solutions, Differ. Integral Equ. 13 (2000), 869–901. [167] T. Itoh, Blow-up of Solutions for Semilinear Parabolic Equations, In: Solutions for Non-linear Elliptic Equations, Kokyuroku RIMS, vol. 679, Kyoto Univ., 1989, pp. 127–139. [168] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Am. Math. Soc. 329 (1992), 819–824. [169] F. John and L. Nirenberg, On functions of bounded mean oscillation, Commun. Pure Appl. Math. 14 (1961), 415–426. [170] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Ration. Mech. Anal. 49 (1973), 241–269. [171] J. Jost, Two-Dimensional Geometric Variational Problems, John Wiley and Sons, Chichester, 1991. [172] J. Jost, G. Wang, D. Ye, and C. Zhou, The blow up analysis of solutions of the elliptic sinh-Gordon equation, Calc. Var. 31 (2008), 263–276. [173] R. Kajikiya, Non-radial least energy solutions of the generalized Hénon equation, J. Differ. Equ. 252 (2012), 1987–2003. [174] R. Kajikiya, Least energy solutions of the generalized Hénon equation in reflectionally symmetric or point symmetric solutions, J. Differ. Equ. 253 (2012), 1621–1646. [175] R. Kajikiya, Multiple positive solutions of the Emden-Fowler equation in hollow thin symmetric domains, Calc. Var. 52 (2015), 681–704. [176] S. Kallel, On the contractiblity of spaces of formal barycenters and symmetric smash products, preprint.

460 | Bibliography

[177] T. Kan, Global structure of the solution set for a semilinear elliptic problem related to the Liouville equation on an annulus, In: Geometric Properties for Parabolic and Elliptic PDE’s (R. Magnanini and S. Sakaguchi, eds.), Springer Verlag Italia, Roma, 2013. [178] S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Commun. Pure Appl. Math. 16 (1963), 327–330. [179] H. G. Kapper, M. K. Kwong, and Y. Li, Symmetry results for reaction-diffusion equations, Differ. Integral Equ. 6 (1993), 1045–1056. [180] T. Kato, Perturbation Theory for Linear Operators, second ed., Springer, Berlin, 1976. [181] N. Kavallaris and T. Suzuki, An analytic approach to the normalized Ricci flow-like equation, Nonlinear Anal. 72 (2010), 2300–2317. [182] N. Kavallaris and T. Suzuki, An analytic approach to the normalized Ricci flow-like equation: revisited, Appl. Math. Lett. 14 (2015), 30–33. [183] B. Kawohl, Rearrangements and Convexity of Leves Sets in PDE, vol. 1150, Lecture Notes in Math., Springer, 1985. [184] J. L. Kazdan and F. W. Warner, Remarks on some quasilinear elliptic equations, Commun. Pure Appl. Math. 28 (1975), 567–597. [185] J. P. Keener and H. B. Keller, Positive solutions of convex nonlinear eigenvalue problems, J. Differ. Equ. 16 (1974), 103–125. [186] H. B. Keller and D. S. Cohen, Some positive problems suggested by nonlinear heat generation, J. Math. Mech. 16 (1967), 1361–1376. [187] C. E. Kenig and W. M. Ni, On the elliptic equation Lu − k + K exp[2u] = 0, Ann. Sc. Norm. Super. Pisa IV 12 (1985), 191–224. [188] A. U. Kennington, Power concavity and boundary value problems, Indiana Univ. Math. J. 34 (1985), 687–704. [189] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. [190] A. Kohda and T. Suzuki, Blow-up criteria for semilinear parabolic equations, J. Math. Anal. Appl. 243 (2000), 127–139. [191] N. J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 32 (1983), 603–614. [192] E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94 (1924), 97–100. [193] T. Kuo, On C ∞ sufficiency of sets of potential functions, Topology 8 (1969), 167–171. [194] U. Kuran, A new criterion of Dirichlet regularity via the quasi-boundedness of the fundamental superharmonic function, J. Lond. Math. Soc. (1) 19 (1979), 301–311. [195] K. Kurata, A unique continuation theorem for uniformly elliptic equations with strongly singular potentials, Commun. Partial Differ. Equ. 18 (1993), 1161–1189. [196] A. A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math. 43 (1983), 1350–1366. [197] T. Laetsch, On the number of solutions of boundary value problems with convex nonlinearities, J. Math. Anal. Appl. 35 (1971), 389–404. [198] A. C. Lazer and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, Nonlinear Anal. 12 (1988), 761–775. [199] J. M. Lee and T. M. Parker, The Yamabe problem, Bull. Am. Math. Soc. 17 (1987), 37–91. [200] Y. Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differ. Equ. 83 (1990), 348–367. [201] Y. Y. Li, Harnack type inequality: the method of moving planes, Commun. Math. Phys. 200 (1999) 421–444.

Bibliography | 461

[202] Y. Y. Li and I. Shafrir, Blow up analysis for solutions of −Δu = Veu in dimension two, Indiana Univ. Math. J. 43 (1994), 1255–1270. [203] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math. 118 (1983), 349–374. [204] C. S. Lin, On the second eigenfunctions of the Laplacian in R2 , Commun. Math. Phys. 111 (1987), 161–166. [205] F. H. Lin, Une remarque sur l’application x/ |x|, C. R. Acad. Sci. Paris Sér. I. Math. 305 (1987), 529–531. [206] S. S. Lin, On non-radially symmetric bifurcation in the annulus, J. Differ. Equ. 80 (1989), 251–279. [207] S. S. Lin, Positive radial solutions and non-radial bifurcation for semilinear elliptic equations in annular domains, J. Differ. Equ. 86 (1990), 367–391. [208] S. S. Lin, Existence of many positive nonradial solutions for nonlinear elliptic equations on annulus, J. Differ. Equ. 103 (1993), 338–349. [209] C. S. Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in R2 , Manuscr. Math. 84 (1994), 13–19. [210] C.-S. Lin, Uniqueness of solutions to the mean field equations for the spherical Onsager vortex, Arch. Ration. Mech. Anal. 153 (2000) 153–176. [211] C. S. Lin, An expository survey on recent development of mean field equation, Discrete Contin. Dyn. Syst., Ser. A 19 (2007), 387–410. [212] C.-S. Lin and M. Lucia, Uniqueness of solutions for a mean field equation on torus, J. Differ. Equ. 229 (2006), 172–185. [213] F. H. Lin, and T. Rivière, Energy quantization for harmonic maps, Duke Math. J. 111 (2002), 177–192. [214] C. S. Lin, W. M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equ. 72 (1988), 1–27. [215] P.-L. Lions, The concentration-compactness principle in the calculus of varition. The locally compact case. Part I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1 (1984), 109–145. [216] J. Liouville, Sur l’équation aux dérivées partielles 𝜕2 log λ/𝜕u𝜕v ± 2λa2 = 0, J. Math. 18 (1853), 71–72. [217] M. Lucia, A deformation lemma with an application to a mean field equation, Topol. Methods Nonlinear Anal. 30 (2007), 113–138. [218] M. Lucia and M. Nolasco, SU(3) Chern-Simons vortex theory and toda systems, J. Differ. Equ. 184 (2002), 443–474. [219] L. Ma and J. C. Wei, Convergence of Liouville equation, Comment. Math. Helv. 76 (2001), 506–514. [220] A. Malchiodi, Morse theory and a scalar field equation on compact surfaces, Adv. Differ. Equ. 13 (2008), 1109–1129. [221] G. Mancini and R. Musina, A free boundary problem involving limiting Sobolev exponents, Manuscr. Math. 58 (1987), 77–93. [222] V. Maz’ja, Sobolev Spaces, Springer, Berlin, 1985. [223] J. B. McLeod and L. A. Peletier, Observation on Moser’s inequality, Arch. Ration. Mech. Anal. 106 (1989), 261–285. [224] F. Mignot and J. P. Puel, Solution radiale singuliére de −Δu = λeu , C. R. Acad. Sci. Paris 307 (1988), 379–382. [225] F. Mignot, F. Murat, and J. P. Puel, Variation d’un point de retourment par rappart au domaine, Commun. Partial Differ. Equ. 4 (1979), 1263–1297. [226] N. Mizoguchi and T. Suzuki, Equations fo gass combustions: S-shaped bifurcation and mushrooms, J. Differ. Equ. 134 (1997), 1830215.

462 | Bibliography

[227] S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press, Cambridge, 1973. [228] J. L. Moseley, Asymptotic solutions for a Dirichlet problem with an exponential non-linearity, SIAM J. Math. Anal. 14 (1983), 719–735. [229] J. L. Moseley, A two-dimensional Dirichlet problem with an exponential nonlinearity, SIAM J. Math. Anal. 14 (1983), 934–946. [230] J. Moser, On Harnack’s theorem for elliptic differential equations, Commun. Pure Appl. Math. 14 (1961), 577–591. [231] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 11 (1971), 1077–1092. [232] J. Mossino, Inégalités Isopérimétriques et Applications en Physique, Hermann, Paris, 1984. [233] L. Mou, and C. Wang, Bubbling phenomena of Palais-Smale-like sequences of m− harmonic type systems, Calc. Var. 4 (1996), 341–367. [234] T. Nagai, T. Senba, and T. Suzuki, Concentration behavior of blow-up solutions for a simplified system of chemotaxis, RIMS Kokyuroku 1181 (2001), 140–176. [235] K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially-dominated nonlinearities, Asymptot. Anal. 3 (1990), 173–188. [236] K. Nagasaki and T. Suzuki, Radial and nonradial solutions for the nonlinear eigenvalue problem Δu + λeu = 0 on annului in R2 , J. Differ. Equ. 87 (1990), 144–168. [237] K. Nagasaki and T. Suzuki, Radial solutions for Δu + λeu = 0 on annului in higher dimensions, J. Differ. Equ. 100 (1992), 137–161. [238] K. Nagasaki and T. Suzuki, Spectral and related properties about the Emden-Fowler equation −Δu = λeu on circular domains, Math. Anal. 299 (1994) 1–15. [239] Y. Naito and T. Suzuki, Radial symmetry of positive solutions for semilinear elliptic equations on the unit ball in Rn , Funkc. Ekvacioj 41 (1998), 215–234. [240] Y. Naito, T. Nishimoto, and T. Suzuki, Radial symmetry of positive solutions for semilinear ellitpic equations in a disc, Hiroshima Math. J. 26 (1996), 531–545. [241] Y. Naito, T. Suzuki, and K. Yoshida, Self-similar solutions to a parabolic system modeling chemotaxis, J. Differ. Equ. 184 (2002), 386–421. [242] S. Nakane, A bifurcation phenomenon for a Dirichlet problem with an exponential nonlinearity, J. Math. Anal. Appl. 161 (1991), 227–249. [243] V. Nanjudiah, Chemotaxis, signal relaying, and aggregation morphology, J. Theor. Biol. 42 (1973), 63–105. [244] J. Nečas, Introduction to the Theory of Nonlinear Elliptic Equations, Teuber, Leipzig, 1983. [245] Z. Nehari, Conformal Mappong, McGraw-Hill, New York, 1952. [246] Z. Nehari, On the principal frequency of a membrane, Pac. J. Math. 8 (1958), 285–293. [247] Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Am. Math. Soc. 95 (1960),101–123. [248] Z. Nehari, Characteristic values associated with a class of nonlinear second-order differential equations, Acta Math. 105 (1961), 141–175. [249] W. M. Ni, Recent progress in semilinear elliptic equations, In: Solutions for Nonlinear Elliptic Equations, Kokyuroku RIMS, vol. 679, Kyoto Univ., 1989, pp. 1–39. [250] W. M. Ni, Qualitive properties of solutions to elliptic problems, In: Handbook of Differential Equations, Stationary Partial Differential Equations 1 (M. Chipot and P. Quittner, eds.), Elsevier, Amsterdam, 2004, pp. 157–233. [251] W. M. Ni and R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of Δu + f (u, r) = 0, Commun. Pure Appl. Math. 38 (1985), 67–108. [252] W. M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator - inhibitator type, Trans. Am. Math. Soc. 297 (1986), 351–368.

Bibliography | 463

[253] W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Commun. Pure Appl. Math. 44 (1991), 819–851. [254] W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247–281. [255] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa 13 (1959), 116–162. [256] M. Obata, The conjecture on conformal transformations of Riemannian mantfolds, J. Differ. Geom. 6 (1971), 247–253. [257] T. Ogawa and T. Suzuki, Microscopic asymptotics for solutions of some semilinear elliptic equations, Nagoya Math. J. 138 (1995), 33–50. [258] T. Ogawa and T. Suzuki, Nonlinear elliptic equations with critical growth related to the Trudinger inequality, Asymptot. Anal. 10 (1995), 25–40. [259] H.Ohtsuka, An approach to regularize the vortex model, In: Proc. 5th East Asia PDE Conference (H. J. Choe, C.-S. Lin, T. Suzuki and J. Wei, eds.), Gakuto International Series on Mathematical Science and Applications, Gakkotosyo, Tokyo, 2005, pp. 245–264. [260] H. Ohtsuka and T. Suzuki, Palais-Smale sequence relative to the Trudinger-Moser inequality, Calc. Var. 17 (2003), 235–255. [261] H. Ohtsuka and T. Suzuki, Blow-up analysis for Liouville type equation in self-dual gauge field theories, Commun. Contemp. Math. 7 (2005), 177–205. [262] H. Ohtsuka and T. Suzuki, Mean field equation for the equilibrium turbulence and a related functional inequality, Adv. Differ. Equ. 11 (2006), 281–304. [263] H. Ohtsuka, T. Senba, and T. Suzuki, Blowup in infinite time in the simplified system of chemotaxis, Adv. Math. Sci. Appl. 17 (2007), 445–472. [264] H. Ohtsuka, T. Sato, and T. Suzuki, Asymptotic non-degeneracy of multiple blowup solution to the Liouville Gel’fand problem with non-constant coefficient, J. Math. Anal. Appl. 398 (2013), 692–706. [265] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, 1986. [266] E. Onofri, On the positivity of the effective action in a theory of random surfaces, Commun. Math. Phys. 86 (1982), 321–336. [267] R. Osserman, The isoperimetric inequality, Bull. Am. Math. Soc. 84 (1978), 1182–1238. [268] M. Ôtani and T. Teshima, On the first eigenvalue of some quasilinear elliptic equations, Proc. Jpn. Acad., Ser. A 64 (1988), 8–10. [269] F. Pacard, Radial and non-radial solutions of −Δu = λf (u) on an annulus of Rn , n ≥ 3, J. Differ. Equ. 101 (1993), 103–138. [270] R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys. 69 (1979), 19–30. [271] T. H. Parker, Bubble tree convergence for harmonic maps, J. Differ. Geom. 44 (1996), 595–633. [272] L. E. Payne, Isoperimetric inequalities and their applications, SIAM Rev. 9 (1967), 453–488. [273] A. Pleijel, Remarks on Courant’s nodal line theorem, Commun. Pure Appl. Math. 9 (1956), 543–550. [274] S. I. Pohozaev, Eigenfunctions of the equation Δu + λf (u) = 0, Sov. Math. Dokl. 6 (1965), 1408–1411. [275] S. I. Pohozaev, On the eigenfunctions of quasilinear elliptic problems, Math. USSR Sb. 11 (1970), 171–188. [276] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton Univ. Press, Princeton, 1951. [277] M. H. Protter, Can one hear the shape of a drum? Revised, SIAM Rev. 29 (1987), 185–197. [278] M. H. Protter and H. F. Weinberger. Maximum Principles in Differential Equations, Prentice Hall, Englewood, 1967.

464 | Bibliography

[279] P. Pucci and J. Serrin, Extensions of the mountain pass theorem, J. Funct. Anal. 59 (1984), 185–210. [280] P. Pucci and J. Serrin, A mountain pass theorem, J. Differ. Equ. 60 (1985), 142–149. [281] P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), 681–703. [282] P. Pucci and J. Serrin, The structure of the critical set in the mountain pass theorem, Trans. Am. Math. Soc. 299 (1987), 115–132. [283] J. Qing, On singularities of the heat flow for harmonic maps from surfaces into spheres, Commun. Anal. Geom. 3 (1995), 297–315. [284] J. Qing, Harmonic maps from surfaces, In: Nonlinear Functional Analysis and Applications to Differential Equations (A. Ambrosetti, K. C. Chang, and I. Ekeland, eds.), World Sci. Publ., River Edge, 1998, pp. 203–236. [285] J. Qing and G. Tian, Bubbling of the heat flows for harmonic maps from surfaces, Commun. Pure Appl. Math. 50 (1997), 295–310. [286] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513. [287] P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mt. J. Math. 3 (1973), 161–202. [288] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, American Mathematical Society, Providence, 1986. [289] F. Rellich, Darstellung der Eigcnwerte von Δu + λu = 0 durch ein Randintegral, Math. Z. 46 (1940), 635–636. [290] O. Rey, The role of the Green’s function in a non-linear elliptic equation involving the critial Sobolev exponent, J. Funct. Anal. 89 (1990), 1–52. [291] T. Ricciardi and R. Takahashi, Blowup behavior for a degenerate elliptic sinh-Poisson equation with variable intensities, Calc. Var. 55 (2016), 152. [292] S. Richardson, Vortices, Liouville’s equation and the Bergmann kernel equation, Mathematika 27 (1980), 321–334. [293] T. Rivière, Everywhere discontinuous harmonic maps into spheres, Acta Math. 175 (1995), 197–226. [294] T. Rivière, Bubbling and regularity issues in geometric non-linear analysis, In: Proceedings of the International Congress of Mathematics (Beijing, 2002), vol. III, Higher Ed. Press, Beijing, 2002, pp. 197–208. [295] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987. [296] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. Math. 113 (1981), 1–24. [297] S. Sakaguchi, Uniqueness of the critical point of the solutions to some semilinear elliptic boundary value problems in R2 , Trans. Am. Math. Soc. 319 (1990), 179–190. [298] S. Sakaguchi and T. Suzuki, Interior imperfect ignition can not occur on a set of positive measure, Arch. Ration. Mech. Anal. 142 (1998), 143–153. [299] T. Sato and T. Suzuki, Asymptotic non-degeneracy of the solution to the Liouville-Gel’fand problem in two dimensions, Comment. Math. Helv. 82 (2007), 353–369. [300] T. Sato and T. Suzuki, Morse indices of the solutions to the Liouville-Gel’fand problem with variable coefficients, Funkc. Ekvacioj 61 (2018), 229–265. [301] T. Sato, T. Suzuki, and F. Takahashi, p-capacity of singular set of p-harmonic function is zero, Electron. J. Differ. Equ. 2011 67 (2011), 1–15. [302] K. Sawada and T. Suzuki, Derivation of the equilibrium mean field equations of point vortex system and vortex filament system, Theor. Appl. Mech. Jpn. 56 (2008), 285–290. [303] R. Schoen, and K. Uhlenbeck, A regularity theory for harmonic maps, J. Differ. Geom. 17 (1982), 307–335.

Bibliography | 465

[304] T. Senba and T. Suzuki, Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl. 10 (2000), 191–224. [305] J. Serrin, Removable singularities of solutions of elliptic equations, Arch. Ration. Mech. Anal. 17 (1964), 67–78. [306] J. Serrin, Local behavior of solutions of quasilinear equations, Acta Math. 111 (1964), 247–302. [307] J. Serrin, Isolated singularities of solutions of quasi-linear equations, Acta Math. (1965), 219–240. [308] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal. 43 (1971), 304–318. [309] J. Serrin and H. F. Weinberger, Isolated singularities of linear elliptic equ ations, Am. J. Math. 88 (1960), 258–272. [310] I. Shafrir, A sup + inf inequality for the equation −Δu = Veu , C. R. Acad. Sci. Paris, Sér. I 315 (1992) 159–164. [311] M. C. Shaw, Eigenfunctions of the nonlienar equation Δu + νf (x, u) = 0 in R2 , Pac. J. Math. 129 (1987), 349–356. [312] R. Shivaji, Uniqueness results for a class of positive problems, Nonlinear Anal. 7 (1983), 223–230. [313] J. A. Smoller and A. G. Wasserman, Symmetry-breaking for positive solutions of semilinear elliptic equations, Arch. Ration. Mech. Anal. 95 (1986), 217–225. [314] J. A. Smoller and A. G. Wasserman, Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions, Commun. Math. Phys. 105 (1986), 415–441. [315] J. A. Smoller and A. G. Wasserman, Bifurcation and symmetry breaking, Invent. Math. 100 (1990), 63–95. [316] J. A. Smoller and A. G. Wasserman, Symmetry, degeneracy and universality in semilinear elliptic equations, J. Funct. Anal. 89 (1990), 364–409. [317] R. Sperb, Maximum Principles and Their Applications, Academic Press, New York, 1981. [318] J. Spruck, The elliptic Sinh Gordon equation and the construction of toroidal soap bubbles, In: Calculus of Variations and Partial Differential Equations, vol. 1340, Lecture Notes in Math., Springer, 1988, pp. 275–301. [319] P. N. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7 (1990), 107–112. [320] G. Stampacchia, Le problème de Dirichlet pour les équation elliptiques du second ordre à coefficients discontinuous, Ann. Inst. Fourier 15 (1965), 189–258. [321] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. [322] E. M. Stein, Harmonic Analysis, Princeton University Press, Princeton, 1993. [323] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984), 511–517. [324] M. Struwe, Critical points of H01,n into Orlicz spaces, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5 (1988), 425–464. [325] Struwe, M., Plateau’s Problem and the Calculus of Variations, Princeton Univ. Press, Princeton, 1988. [326] M. Struwe, Variational Methods. Applications to Partial Differential Equations and Hamiltonian Systems, third edition, Springer, Berlin, 2000. [327] M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons-Higgs gauge theory, Boll. Unione Math. Ital. Sez. B(8) 1 (1998), 109–121. [328] T. Suzuki, Radial and nonradial solutions for semilinear elliptic equations on circular domains, In: Geometry of Solutions of Partial Differential Equations (G. Talenti, ed.), Academic Press, 1989, pp. 153–174.

466 | Bibliography

[329] T. Suzuki, Introduction to geometric potential theory, In: Functional Analytic Methods in Partial Differential Equations (H. Fujita, T. Ikeba, and S. T. Kuroda, eds.), Lecture Notes in Math., vol. 1450, Springer Verlag, 1990, pp. 83–103. [330] T. Suzuki, Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 9 (1992), 367–398. [331] T. Suzuki, Some remarks about singular perturbed solutions for Emden-Fowler equation with exponential nonlinearity, In: Functional Analysis and Related Topics (H. Fujita, H. Komatsu, and S. T. Kuroda, eds.), Lecture Notes in Math., vol. 1540, Springer Verlag, 1993, pp. 341–360. [332] T. Suzuki, Semilinear Elliptic Equations, Gakko Tosho, Tokyo, 1994. [333] T. Suzuki, Positive solutions for semilinear ellipitic equations on expanding annului: mouintain pass approach, Funkc. Ekvacioj 39 (1996) 143–164. [334] T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005. [335] T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, second edition, Atlantis Press, Paris, 2015. [336] T. Suzuki, Chomeotaxis, Reaction, Network - Mathematics for Self-Organization, World Scientific, Singapore, 2018. [337] T. Suzuki, Liouville’s Theory in Linear and Nonlinear Partial Differential Equations, Interaction of Analysis, Geometry, Physics, preprint. [338] T. Suzuki and K. Nagasaki, Lifting of local subdifferentiations and elliptic boundary value problems on symmetric domains, I, Proc. Jpn. Acad., Ser. A 64 (1988), 1–4. [339] T. Suzuki and K. Nagasaki, Lifting of local subdifferentiations and elliptic boundary value problems on symmetric domains, II, Proc. Jpn. Acad., Ser. A 64 (1988), 29–32. [340] T. Suzuki and K. Nagasaki, On the nonlinear eigenvalue problem Δu + λeu = 0, Trans. Am. Math. Soc. 309 (1988), 591–608. [341] T. Suzuki and T. Senba, Applied Analysis, Mathematical Methods in Natural Science, second edition, Imperial College Press, London, 2011. [342] T. Suzuki and F. Takahashi, Capacity estimate for the blow-up set of parabolic equations, Math. Z. 259 (2008), 867–878. [343] T. Suzuki and F. Takahashi, Nonlinear eigenvalue problem with quantization, In: Handbook of Differential Equations, Stationary Partial Differential Equations 5 (M. Chipot, ed.), Elsevier, Amsterdam, 2008, pp. 277–300. [344] T. Suzuki and R. Takahashi, Critical blowup exponent to a class of semilinear elliptic equations with constrants in higher dimensions - local properties, Ann. Mat. Pura Appl. 195 (2016), 1123–1151. [345] G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Ration. Mech. Anal. 3 (1953), 343–356. [346] A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problem, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986), 77–109. [347] G. Talenti, Elliptic equations and rearrangements, Ann. Sc. Norm. Sup. Pisa IV 3 (1976), 697–718. [348] G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. 120 (1977), 159–184. [349] H. Tanabe, Equations of Evolution, Pitman, London, 1979. [350] G. Tarantello, Analytic aspects of Liouville-type equations with singular sources, In: Handbook of Differential Equations, Stationary Partial Differential Equations, vol. 1 (M. Chipot and P. Quittner, eds.), Elsevier, Amsterdam, 2004, pp. 491–592. [351] W. C. Troy, Symmetry properties in systems of semilinear elliptic eqations, J. Differ. Equ. 42 (1981), 400–413.

Bibliography | 467

[352] N. S. Trudinger, On imbedding into Orlicz space and some applications, J. Math. Mech. 17 (1967), 473–484. [353] N. S. Trudinger, On imbedding into Orlicz space and some applications, J. Math. Mech. 17 (1967) 473–484. [354] N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sc. Norm. Super. Pisa 22 (1968), 265–274. [355] M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, Holden-Day, San Francisco, 1964. [356] R. C. A. M. van der Vorst, Variational identities and applications to differential systems, Arch. Ration. Mech. Anal. 116 (1991), 375–398. [357] A. Vanderbauwhende, Local Bifurcation and Symmetry, Pitman, Boston, 1982. [358] J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Univ. Press, Oxford, 2006. [359] L. Véron, Singularities of Solutions of Second Order Quasilinear Equations, Addison-Wesley-Longman, New York, 1996. [360] C. Viterbo, Indice de Morse des points critiques obtenus par minimax, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5 (1988), 212–225. [361] C. Y. Wang, Bubble phenomena of certain Palais-Smale sequences from surfaces to general targets, Houst. J. Math. 22 (1996), 559–590. [362] G. Wang and D. Ye, On a nonlinear elliptic equation arising in a free boundary problem, Math. Z. 244 (2003), 531–548. [363] J. Wei, Existence and stability of spikes for the Gierer-Meinhardt system, In: Handbook of Differential Equations, Stationary Partial Differential Equations 5 (M. Chipot, ed.), Elsevier, Amsterdam, 2008, pp. 487–585. [364] H. F. Weinberger, An isoperimetric inequality for the N-dimensional free membrane problem, J. Ration. Mech. Anal. 5 (1956), 533–536. [365] H. F. Weinberger, Remark on the preceeding paper of Serrin, Arch. Ration. Mech. Anal. 43 (1971), 319–320. [366] H. C. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969), 318–344. [367] H. C. Wente, The differential equation Δx = 2H(xu ∧ xv ) with vanishing boundary values, Proc. Am. Math. Soc. 50 (1975), 131–137. [368] H. C. Wente, Counter example to a conjecture of H. Hopf, Pac. J. Math. 121 (1986), 193–243. [369] H. C. Wente, Complete immersions of constant mean curvature, Proc. Symp. Pure Math. 54 (1993), 497–512. [370] V. H. Weston, On the asymptotic solution of a partial differential equation with an exponential nonlinearity, SIAM J. Math. Anal. 9 (1978), 1030–1053. [371] A. Yagi, Norm behavior of solutions to the parabolic system of chemotaxis, Math. Jpn. 45 (1997), 241–265. [372] D. Ye, Uneq remarque sur le comportement asymptotique des solutions de −Δu = λf (u), C. R. Acad. Sci. Paris, Sér. I 325 (1997), 1279–1282. [373] K. Yosida, Functional Analysis, Springer, Berlin, 1964. [374] E. Zeidler, Nonlinear Functional Analysis and Its Applications, III, Springer, Berlin. 1984. [375] W. P. Ziemer, Weakly Differentiable Functions; Sobolev Spaces and Functions of Bounded Variation, Springer-Verlag, New York, 1989.

Index sup + inf inequality 284 L1 estimate 33 Lp estimate 32 Alexandroff inequality 284 Aronson–Benilan inequality 236 Arrhenius law 60 associated Legendre equation 126 Benilan–Crandall inequality 236 blowup analysis 265 blowup pattern 300 Bol inequality 123, 198 Boltzmann–Poisson equation 121, 227, 228, 276 bootstrap argument 33 Brezis–Kato theorem 38 Brezis–Merle inequality 279 Brezis–Peletier equality 147 bubbled Harnack principle 203 capacity 191 Carleson theorem 191 Carleson–Chang theorem 330 Chang–Yang inequality 226 Chanillo–Li theorem 442 Christoffel symbol 111 co-area formula 186 Codazzi–Mainardi equation 112 comparison principle 66 concentration function 245 conformal parametrization 113 convection–diffusion equation 4 De Giorgi inequality 187 decreasing rearrangement 184 deformation lemma 43 Dini theorem 69 Dirichlet principle 396 div rot theorem 423 Duhamel principle 61 dyadic decomposition 416 Ekeland lemma 45 Emden transformation 83 Euler characteristics 231 extremal minimum solution 74

Faber–Krahn inequality 183 Fenchel–Moreau duality 47 first fundamental form 109 Fleming–Rishel formula 187 Fontana inequality 228, 243 Fréchet derivative 27 Fréchet–Kolmogorov theorem 24 free boundary problem 289 fundamental solution 61 Gauss curvature 110 Gauss formula 111 Gauss–Bonnet formula 231 geneous 231 Green function 130 Hardy space 421 harmonic map 425 Harnack inequality 411 Hartman–Wintner theorem 175 Hausdorff dimension 195 Hausdorff measure 191 heat equation 60 Hopf lemma 85, 87 implicit function theorem 30 improved Trudinger–Moser inequality 345 Jacobi methods 20 John–Nirenberg inequality 409 Kantrovic–Rubinstein metric 348 Kaplan method 34 Keller–Segel system 3 Kuo lemma 178 Laplace–Beltrami operator 126 Legendre transformation 47 Lieb lemma 36 Liouville integral 115, 358 Liouville property 266 logarithmic diffusion 235 Lucia lemma 336 maximal cap 88 maximal symmetry 103 maximal theorem 421 mean curvature 110 mini-max principle 51

470 | Index

minimum solution 64 Morrey theorem 25 Moser inequality 227 mountain pass situation 41 Nehari inequality 198 Nehari variation 256 nonlinear eigenvalue problem 59 normalized Ricci flow 230 Obata relation 134 Onofri inequality 228 optimal cap 88 Pólya–Szegö–Weinberger inequality 184 Palais–Smale condition 41 Perron methos 396 Pohozaev identity 6 Poincaré conjecture 231 Poincaré inequality 19, 398, 400 Poincaré metric 273 Poincaré–Sobolev inequality 409 Poincaré–Wirtinger inequality 248 point vortex Hamiltonian 143, 145 principal curvature 110 quenching 235 Rayleigh principle 18 Rellich–Kondrachov theorem 19, 25 Riccati equation 116 Robin function 130 Schauder estimate 32 Schwarz symmetrization 184 Schwarzian derivative 117 Serrin lemma 94 singular limit 82

Smoluchowski–Poisson equation 446 Sobolev inequality 20 Sobolev space 18 spherical derivative 117 spherically decreasing rearrangement 212, 216 spherically subharmonic 200 spherically superharmonic 205 Stampacchia estimate 440 strictly minimum solution 72 strong maximum principle 87 Struwe theorem 426 sub-differential 48 super/subsolutions 64 symmetric criticality 105 Talenti function 35 Talenti theorem 189 Theorem Egregium 112 Toland duality 243 topological degree 76, 332 trace 398 Trudinger–Moser inequality 226 uniformization theorem 231 unique continuation theorem 16 Vitali lemma 420 Vitali theorem 246 weak maximum principle 84, 85 Weingarten matrix 111 Wente inequality 444 Weyle formula 183 Y. Y. Li estimate 280 Yamabe equation 147 Ye theorem 283 Young inequality 404

De Gruyter Series in Nonlinear Analysis and Applications Volume 34 Tomasz W. Dłotko, Yejuan Wang Critical Parabolic-Type Problems, 2020 ISBN 978-3-11-059755-4, e-ISBN (PDF) 978-3-11-059983-1, e-ISBN (EPUB) 978-3-11-059868-1 Volume 33 Cyril Tintarev Concentration Compactness, 2020 ISBN 978-3-11-053034-6, e-ISBN (PDF) 978-3-11-053243-2, e-ISBN (EPUB) 978-3-11-053058-2 Volume 32 Peter I. Kogut, Olha P. Kupenko Approximation Methods in Optimization of Nonlinear Systems, 2019 ISBN 978-3-11-066843-8, e-ISBN (PDF) 978-3-11-066852-0, e-ISBN (EPUB) 978-3-11-066859-9 Volume 31 Pablo Blanc, Julio Daniel Rossi Game Theory and Partial Differential Equations, 2019 ISBN 978-3-11-061925-6, e-ISBN (PDF) 978-3-11-062179-2, e-ISBN (EPUB) 978-3-11-061932-4 Volume 30 Lucio Damascelli, Filomena Pacella Morse Index of Solutions of Nonlinear Elliptic Equations, 2019 ISBN 978-3-11-053732-1, e-ISBN (PDF) 978-3-11-053824-3, e-ISBN (EPUB) 978-3-11-053824-3 Volume 29 Rafael Ortega Periodic Differential Equations in the Plane. A Topological Perspective, 2019 ISBN 978-3-11-055040-5, e-ISBN (PDF) 978-3-11-055116-7, e-ISBN (EPUB) 978-3-11-055042-9

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