Linear and Nonlinear Non-Fredholm Operators: Theory and Applications 9811998795, 9789811998799

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Messoud Efendiev

Linear and Nonlinear Non-Fredholm Operators Theory and Applications

Linear and Nonlinear Non-Fredholm Operators

Messoud Efendiev

Linear and Nonlinear Non-Fredholm Operators Theory and Applications

123

Messoud Efendiev Institute of Computational Biology Helmholtz Zentrum München Neuherberg, Bayern, Germany

ISBN 978-981-19-9879-9 ISBN 978-981-19-9880-5 https://doi.org/10.1007/978-981-19-9880-5

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

This book is dedicated to the memory of the late Heydar Aliyev, on the 100th anniversary of his birth, who played an important role both in educational and scientific careers of a whole generation and beyond.

Preface

This book is devoted to linear and nonlinear non-Fredholm operators and their applications. Fredholm operators are named in honour of Erik Ivar Fredholm and constitute one of the most important classes of linear maps in mathematics. They were introduced around 1900 in the study of integral operators and, by definition, they share many properties with linear maps between finite dimensional spaces (i.e. matrices). The Fredholm property implies the solvability condition: the nonhomogeneous operator equation Lu = f is solvable if and only if the right-hand side f is orthogonal to all solutions of the homogeneous adjoint problem L∗ v = 0. The orthogonality is understood in the sense of duality in the corresponding spaces. Indeed, solvability conditions play an important role in the analysis of nonlinear problems.These properties of Fredholm operators are widely used in many methods of linear and nonlinear analysis. At present linear and nonlinear Fredholm operator theory and solvability of corresponding equations are quite well understood (see the book of [20] and the references therein). However, in the case of non-Fredholm operators the usual solvability conditions may not be applicable and solvability relations are, in general, not known. In spite of some progress on linear/nonlinear non-Fredholm operator theory these questions and related topics are not systematically studied in the mathematical literature and to the best of our knowledge are not well understood. The aim of this book is to attempt to close such a gap and initiate as well as stimulate readers to make contributions to this fascinating subject. My work on this subject, that is on solvability and well-posedness of a linear/nonlinear non-Fredholm operator equation, started in 2015 during my visit to York University (Canada) as an Alexander von Humboldt Fellow for experienced scientists. One of the seminars there was given by Dr. V. Vougalther (University of Toronto). Both during the talk and afterwards, I asked him many questions to most of them I received the answer either not known or not studied. This was a point of departure for my study of this and vii

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Preface

related questions. Indeed, I found this subject very fascinating and taking into account my ”alte liebe” on Fredholm operator theory (see my book [20]), I started with much enthusiasm to work on this topic. Later in my visit to the Fields Institute and the University of Toronto as a Deans Distinguished Visiting Professor (2018) and as a James D. Murray Distinguished Professor at the University of Waterloo (2019) together with my colleagues of the Universities mentioned above working on ”What has Biology and Medicine done for Mahematics”, I also continued to work with Dr.V. Vougalther on linear and nonlinear non-Fredholm operators and its applications to the various classes of non-Fredholm elliptic equations. One of the main question which will be considered in this book is the solvability of linear and nonlinear equations related to non-Fredholm operators. We present the explicit form of the solvability conditions and establish the existence of solutions of the non-Fredholm equations considered in this book. In particular, we address it in the following setting. Let A : E → F be the operator corresponding to the left side of equation Au = f . Assume that this operator fails to satisfy the Fredholm property (throughout this book in each chapter starting from the second one we will present a quite large class of pseudo-differential elliptic equations for which this will be the case). Let fn be a sequence of functions in the image of the operator A, such that fn → f in F as n → ∞. Denote by un a sequence of functions from E such that Aun = fn , n ∈ N. Since the operator A does not satisfy the Fredholm property, the sequence un may not be convergent. Let us call a sequence un the solution in the sense of sequences of the equation Au = f if Aun → f . If such a sequence converges to a function u∗ in the norm of the space E, then u∗ is a solution of this equation in the usual sense. A solution in the sense of sequences in this case is equivalent to the usual solution. However, in the case of non-Fredholm operators, this convergence may not hold or it can occur in some weaker sense. In such a case, a solution in the sense of sequences may not imply the existence of the usual solution. In this book we find the sufficient conditions for the equivalence of solutions in the sense of sequences and the usual solutions, that is, the conditions on sequences fn under which the corresponding sequences un are strongly convergent. In the case of elliptic integro-differential equations that fail to satisfy the Fredholm property (which can also be formulated in term of the location of the essential spectrum), we prove existence of solution in the sense of sequences in terms of the kernel of the given integro-differential elliptic operator. This book consists of five chapters and in particular includes our results that have been published in the leading journals of mathematical societies of the world. Chapter 1 has more of a teaching aid character and consists of eight sections and is dedicated, in particular, to some basic concepts concerning Sobolev

Preface

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spaces and embedding theorems, linear elliptic boundary value problems, linear Fredholm operators and its properties, properties of superposition operators in Sobolev and H¨older spaces, the Fourier transform and related quantities, fractional Laplacian as a pseudo-differential operator, as well as the properties of generalized Fourier transform in terms of the functions of continuous spectrum of the Schr¨odinger operators with shallow and short-range potential. Chapter 1 is not self-sufficient, since it is intended as auxiliary material for other chapters. Chapter 2 is devoted to the well-posedness of a class of stationary nonlinear integro-differential equations containing the classical Laplacian and a drift term for which the Fredholm property may not be satisfied. Here we formulate solvability conditions in terms of iterated kernels of a nonlinear integral operator which is related to the equation under consideration. Chapter 2 consists of four subsections. In sections 2.1 and 2.2 we consider a class of stationary nonlinear integro-differential scalar equation containing classical Laplacian and drift term on the whole line and on a finite interval respectively. In sections 2.3 and 2.4 we consider the same questions for systems of integro-differential equations containing classical Laplacian and drift term. We emphasize that the study of the system case (sections 2.3 and 2.4) is more difficult than of the scalar case (sections 2.1 and 2.2) and requires some more cumbersome technicalities to be overcome. In population dynamics the integro-differential equations describe models with intra-specific competition and nonlocal consumption of resources. On the other hand the studies of the solutions of the integro-differential equations with the drift term are relevant to the understanding of the emergence and propagation of patterns in the theory of speciation. Chapter 3 deals with the existence in the sense of sequences of solutions for some integro-differential type equations containing the drift term and the square root of the one dimensional negative Laplacian (so-called superdiffusion) on the whole real line, and on a finite interval with periodic boundary conditions in the corresponding H 2 spaces. The argument for proving existence of solutions in the sense of sequences in this chapter relies on fixed point techniques when the elliptic equations involve first order pseudodifferential operators (nonlocal) with and without the Fredholm property. Chapter 3 consists of four subsections. Sections 3.1 and 3.2 deal with scalar equations on the whole real line and a finite interval respectively. In sections 3.3 and 3.4 we consider the analogous problem for a system of equations, the study of which has additional difficulties and needs new ideas compared with the scalar case. Superdiffusion can be described as a random process of particle motion characterized by the probability density distribution of the jump length. The moments of this density distribution are finite in the case of the normal diffusion, but this is not the case for superdiffusion. Asymptotic behavior at infinity of the probability density function determines the value of the power of the negative Laplace operator (for the details see chapter 3).

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Preface

In chapter 4 we establish the existence in the sense of sequences of solutions for certain nonlinear integro-differential type equations in two dimensions involving normal diffusion in one direction and anomalous diffusion in the other direction in H 2 (R2 ) via the fixed point technique. The elliptic equation contains a second order differential operator without the Fredholm property. It is proved that, under some reasonable technical conditions, the convergence in L1 (R2 ) of the integral kernels implies the existence and convergence in H 2 (R2 ) of the solutions. Such anisotropy in the diffusion term (local versus nonlocal) make our analysis extremely difficult because in order to derive the desired estimates requires new ideas and cumbersome techniques. Chapter 4 consists of two sections. Section 4.1 is devoted to scalar nonlinear equations in the presence of the mixed-diffusion type mentioned above. These models are new and not much is understood about them, especially in the context of the integro-differential equations. We use the explicit form of the solvability conditions and establish the existence of solutions of such nonlinear equation. In section 4.2 we consider the analogous problem for a system of equations. The novelty of this section is that in each diffusion term we add the standard negative Laplacian in the x1 variable to the minus Laplacian in the x2 variable raised to a fractional power. Such anisotropy coming from a different fractional order in each equation of the system make our analysis more difficult than in scalar case and requires both new ideas and requires rather sophisticated techniques. It is important to study the equations of this kind in unbounded domains from the point of view of the understanding of the spread of the viral infections , since many countries have to deal with pandemics. In chapter 5 we consider two classes of non-Fredholm (4th and 2nd order) Schr¨odinger type operators and establish the solvability conditions in the sense of sequences for the equations involving them. To this end, we use the methods of the spectral and scattering theory for Schr¨ odinger type operators, the potential functions V (x) of which are assumed to be shallow and shortrange with a few extra regularity conditions. In this chapter, in contrast to previous ones, the coefficients of the operators are no longer constants and we cannot use the Fourier transform directly to obtain solvability conditions similar to those for the operators considered in a previous chapters. Instead we use the generalized Fourier transform which is based on replacing the Fourier harmonics by the functions of the continuous spectrum of the operator −∆ + V (x), which are the solutions of the Lippmann-Schwinger equation (for the details see chapter 5). Chapter 5 consists of four sections: 5.1-5.4. In section 5.1 we consider problems which contain the squares of the sums of second order non-Fredholm differential operators of Schr¨odinger type, that is 4th order operators. In sections 5.2 and 5.3 we deal with the solvability in the sense of sequences of the operator equation consisting of the squares of the sums of second order nonFredholm differential operators of Schr¨odinger type with a single potential

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both in the regular and singular cases. The sum of free negative Laplacian and the Schr¨odinger type operator has the meaning of the cumulative Hamiltonian of the two non-interacting quantum particles, one of these particles moves freely and the other interacts with an external potential. The last section of chapter 5, that is section 5.4, is devoted to the solvability of generalised Poisson type equations with a scalar potential. I would like to thank many friends and colleagues who gave me helpful suggestions, advice and support. In particular, I wish to thank G. Akagi, H. Berestycki, N. Dancer, Y. Du, Y. Enatsu, F. Hamel, F. Hamdullahpur, M. Otani, S. Sivagolonathan, C.A. Stuart, E. Valdinochi, V. Vougalther, J.R.L. Webb, W.L. Wendland, J. Wu, A. Zaidi. Furthermore, I am greatly indebted to my colleagues at the Institute of Computational Biology in the Helmholtz Center Munich and Technical University of Munich, Marmara University in Istanbul, Alexander von Humboldt Foundation, as well as the Springer book series for their efficient handling of publication. I started to write this book when I visited the Fields Institute with a Fields Research Fellowship. I would like to express my sincere gratitude to the Fields Institute for providing an excellent and unique scientific atmosphere. In particular, my thanks go to my colleagues, friends and staffs in the Fields Institute, namely to Kumar Murthy, Esther Berzunza, Miriam Schoeman, Bryan Eelhart and Tyler Wilson. Last but not least, I wish to thank my family for constantly encouraging me during the writing of this book.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1

Auxiliary Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1 Functional spaces and embedding theorems . . . . . . . . . . . . . . . .

1

1.2 Linear Elliptic Boundary value problems . . . . . . . . . . . . . . . . . .

4

1.3 Superposition operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.4 Pseudodifferential operators. Definitions and examples . . . . . . 17 1.5 Linear Fredholm operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Fourier transform and related topics . . . . . . . . . . . . . . . . . . . . . . 36 1.7 On the necessary conditions for preserving the nonnegative cone: double scale anomalous diffusion . . . . . . . . . . . . . . . . . . . . 44 1.8 The Lippman-Schwinger equation: the generalized Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2

Solvability in the sense of sequences: non-Fredholm operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.1 Non-Fredholm equations with normal diffusion and drift in the whole line: scalar case: scalar case . . . . . . . . . . . . . . . . . . . . . 66 2.2 Non-Fredholm equations in a finite interval with normal diffusion and drift: scalar case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.3 Non-Fredholm systems with normal diffusion and drift in the whole line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.4 Non-Fredholm systems in a finite interval with normal diffusion and drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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3

Contents

Solvability of some integro-differential equations with drift and superdiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1 The whole real line case: scalar equation . . . . . . . . . . . . . . . . . . . 100 3.2 The problem on the finite interval: scalar equation . . . . . . . . . . 104 3.3 The whole real line case: system case . . . . . . . . . . . . . . . . . . . . . . 123 3.4 The problem on the finite interval: system case . . . . . . . . . . . . . 127

4

Existence of solutions for some non-Fredholm integro-differential equations with mixed diffusion . . . . . . . . 143 4.1 Mixed-diffusion: scalar case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.2 Mixed-diffusion: system case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5

Non-Fredholm Schr¨ odinger type operators . . . . . . . . . . . . . . . . 177 5.1 Solvability in the sense of sequences with two potentials . . . . . 183 5.2 Solvability in the sense of sequences with Laplacian and a single potential: regular casel: regular case . . . . . . . . . . . . . . . . . 187 5.3 Solvability in the sense of sequences with Laplacian and a single potential: singular case: singular case . . . . . . . . . . . . . . . . 190 5.4 Generalized Poisson type equation with a potential . . . . . . . . . 196

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Chapter 1

Auxiliary Materials

1.1 Functional spaces and embedding theorems We shall use the following notation. We shall denote by R, C, Z and N the sets of real, complex, integer and natural numbers respectively; Z+ = {x ∈ Z | x ≥ 0} is the set of nonnegative integers. Rn is the standard real vector space of dimension n. We denote by Di the operator of partial differentiation with respect to xi , ∂u Di u = (i = 1, . . . n). (1.1) ∂xi As usual, we use multi-index notation to denote higher order partial derivatives, Dγ = D1γ1 · · · Dnγn , |γ| = γ1 + · · · + γn (1.2) is a partial derivatives of order |γ|, for a given γ = (γ1 , · · · , γn ), γi ∈ Z+ . Let u : Ω ⊂ Rn be a real function defined on a bounded domain Ω. The space ¯ is denoted by C(Ω); ¯ the of continuous functions over the bounded domain Ω ¯ is defined in a standard way: norm in C(Ω) ¯ ∥u∥C(Ω) ¯ = sup{|u(x)| | x ∈ Ω}.

(1.3)

The space C m (Ω) consists of all real functions on Ω which have continuous ¯ iff (abpartial derivatives up to order m. By definition, u belongs to C m (Ω) breviation for if and only if) u ∈ C m (Ω) and u and all its partial derivatives ¯ up to order m can be extended continuously to Ω. k,γ ¯ Let 0 < γ < 1 and k ∈ Z+ . By definition C (Ω) denotes the H¨ older space of functions u : Ω → R such that, Dα u : Ω → R exists and is uniformly continuous when |α| = k and such that

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Efendiev, Linear and Nonlinear Non-Fredholm Operators, https://doi.org/10.1007/978-981-19-9880-5_1

1

2

1 Auxiliary Materials

 |u|k,γ ≡ sup

 |Dα u(x) − Dα u(y)| x, y ∈ Ω, x ̸= y, |α| ≤ k |x − y|γ

(1.4)

¯ we set is finite. For u ∈ C k,γ (Ω), ∥u∥k,γ = |u|k,γ +

X

¯ max{|Dα u(x)| | x ∈ Ω}.

(1.5)

|α|≤k

We also have ¯ with u|∂Ω = φ}, C k,γ (∂Ω) = {φ : ∂Ω → R | there exists u ∈ C k,γ (Ω) and for φ ∈ C k,γ (∂Ω) we set ¯ ∥φ∥k,γ = inf{∥u∥k,γ : u|∂Ω = φ; u ∈ C k,γ (Ω)}.

(1.6)

In cases when it is clear from the context where the function under consideration is defined, we shall sometimes simply write u ∈ C k instead of, for example, u ∈ C k (Rn ). In several examples we shall use the spaces of functions that are 2π-periodic in every variable xi (i = 1, · · · , n). We shall consider such functions as being defined on the n-dimensional torus T n = Rn /(2πZ)n . We denote by Lp (Ω), 1 ≤ p < ∞, the space of measurable functions with the finite norm Z 1/p ∥u∥0,p = ∥u∥Lp = |u(x)|p dx . (1.7) Ω

We denote by L∞ (Ω) the space of almost everywhere bounded functions, ∥u∥0,∞ = ∥u∥L∞ = vrai sup{|u(x)| | x ∈ Ω}

(1.8)

¯ (for continuous functions this norm coincides with the norm of C(Ω)). The norm in the Sobolev space W l,p (Ω), l ∈ Z+ , 1 ≤ p < ∞, is defined by the formula  1/p X ∥u∥l,p =  ∥Dα u∥pLp  . (1.9) |α|≤l

In the case p = 2 this Sobolev space is a Hilbert space and is denoted by H l (Ω), H l (Ω) = W l,2 (Ω). The scalar product in H l (Ω) is defined by the formula X Z (u, v)l = Dα u(x) · Dα v(x)dx. (1.10) |α|≤l

Ω

The space W l,p (Ω) is the completion of C l (Ω) with respect to the norm (1.9).

1.1 Functional spaces and embedding theorems

3

The norms C k,γ (T n ) and W l,p (T n ) are defined by (1.4) and (1.9) with Ω = (]0, 2π[). The scalar product and the norm in H l (T n ), which are equivalent to those defined by (1.10), are defined in terms of Fourier coefficients, X (u, v)l = u ˆ(ξ) · vˆ(ξ) · (1 + |ξ|2 )l ; ∥u∥2l = ⟨u, u⟩l , (1.11) where the summation is over ξ ∈ Zn ; the bar denotes complex conjugation; u ˆ(ξ) and vˆ(ξ) are the Fourier coefficients, Z u ˆ(ξ) = (2π)−n u(x)e−ix·ξ dx. (1.12) Here the integral is over [0, 2π]n ; x · ξ = x1 ξ1 + · · · + xn ξn .

(1.13)

The formula (1.11) defines the norm in H l (T n ) for l ∈ R as well as l ∈ Z+ . \ ¯ the space ¯ by C ∞ (Ω) the set of functions We denote by C ∞ (Ω) C k (Ω); 0 k≥0

¯ which vanish on a neighbourhood of the boundary ∂Ω. We shall from C ∞ (Ω) use also spaces of functions which vanish on ∂Ω. In this case we shall denote the corresponding space as follows: ¯ ∩ {u|∂Ω = 0}, C k,γ (Ω)

W l,p (Ω) ∩ {u|∂Ω = 0}.

(1.14)

We denote the completion of C0∞ (Ω) with respect to the norm of H l (Ω) by H0l (Ω) and with respect to the norm of W 1,p (Ω) by W01,p (Ω). It is well-known that H01 (Ω) = H1 (Ω) ∩ {u|∂Ω = 0};

W01,p (Ω) = W 1,p ∩ {u|∂Ω = 0}.

(1.15)

The Sobolev spaces H ρ (Ω) with noninteger ρ ≥ 0, ρ = k+β, k ∈ Z, 0 ≤ β < 1 are endowed with the norm Z ∥u∥2ρ = ∥u∥2k + ∥u(x + y) − u(x)∥2k · |y|−n−2β dy (1.16) |y|≤δ

(u(x) is extended over a δ-neighbourhood of the boundary, see [50]). By S(Rn ) we denote the class of rapidly decreasing (at ∞) functions u(x) ∈ C ∞ (Rn ), with (1 + |x|)k |Dα u(x)| ≤ Ck,α for each α = (α1 , · · · , αn ) ∈ Zn+ and k ∈ Z+ , where Ck,α are constants.

4

1 Auxiliary Materials

Recall that an operator j : X → Y between Banach spaces with X ⊆ Y is an embedding iff j(x) = x for all x ∈ X. The operator j is continuous iff ∥x∥Y ≤ constant ∥x∥X for all x ∈ X. Further, j is compact iff j is continuous, and every bounded set in X is relatively compact in Y . If the embedding X ,→ Y is compact, then each bounded sequence {xn } in X has a subsequence {xn′ } which is convergent in Y. We shall widely use Sobolev’s embedding theorems formulated below. Theorem 1.1 Let Ω be a bounded domain in Rn , with smooth boundary ∂Ω and 0 ≤ k ≤ m − 1 (See [50]). Then W m,p (Ω) ,→ W k,q (Ω), W m,p (Ω) ,→ W k,q (Ω), ¯ W m,p (Ω) ,→ C k,δ (Ω),

1 1 m−k ≥ − > 0, q p n 1 m−k if q < ∞, = , p n n if < m − (k + δ), 0 < δ < 1. p if

The first embedding is compact if compact.

1 q

>

1 p

(1.17) (1.18) (1.19)

− m−k n . The last two embeddings are

Theorem 1.2 Let 0 ≤ β < α ≤ 1 or α, β ∈ Z with 0 ≤ β < α (see [63]). Then the embedding ¯ ,→ C β (Ω) ¯ is compact C α (Ω)

(1.20)

and for k + β < m + α, with 0 ≤ α, β ≤ 1, m ≥ k ≥ 0 the embeddings ¯ ,→ C k,β (Ω) ¯ are compact. C m,α (Ω)

(1.21)

1.2 Linear Elliptic Boundary value problems Notation. Let Ω be a bounded domain in Rn . For α = (α1 , · · · , αn ) an n-tuple of nonnegative integers, recall that Dα = α i n  n n Y X Y ∂ , |α| = αi and let ξ α = (ξi )αi if ξ ∈ Cn . ∂x i i=1 i=1 i=1 Every linear differential operator L of order 2m (m ∈ N) has the form X Lu = aα (x) · Dα u. (1.22) |α|≤2m

1.2 Linear Elliptic Boundary value problems

5

All coefficients aα (x) are assumed to be real. The partial differential operator defined by (1.22) is called elliptic of order 2m if its principal symbol , X p0 (x, ξ) = aα (x) · ξ α |α|=2m

has the property that p0 (x, ξ) ̸= 0 for all x ∈ Ω, ξ ∈ Rn \ {0}. The differential operator L defined by (1.22) is called uniformly elliptic in Ω, if there is some c > 0, such that X (−1)m aα (x)ξ α ≥ C|ξ|2m for every x ∈ Ω, ξ ∈ Rn \ {0}. (1.23) |α|=2m

Throughout we assume that ∂Ω is a smooth (n − 1)-manifold. Suppose now that L is elliptic and of order 2m. Let {mi , 1 ≤ i ≤ m} be distinct integers with 0 ≤ mi ≤ 2m − 1, and suppose that for 1 ≤ i ≤ m we prescribe a differential operator Bi of order mi on ∂Ω, by X Bi u(x) = bα,i (x)Dα u(x), i = 1, · · · , m. (1.24) |α|≤mi

The family of boundary operators B = {B1 , · · · , Bm } is said to satisfy the Shapiro-Lopatinski covering condition with respect to L provided that the ⃗ ∈ Rn \ {0} following algebraic condition is satisfied. For each x ∈ ∂Ω, N ⃗ ⟩ = 0, consider the (m + 1) normal to ∂Ω at x and ξ ∈ Rn \ {0} with ⟨ξ, N polynomials of a single complex variable ⃗ ), τ 7−→ p0 (x, ξ + τ N X ⃗ )α ≡ p0,i (x, ξ, τ ), τ 7−→ bα,i (x) · (ξ + τ N

1 ≤ i ≤ m.

(1.25)

|α|=mi + ⃗ ) which have positive Let τ1+ , · · · , τm be the m complex zeros of p0 (x, ξ + τ N imaginary part. Then {p0,i (τ )}m are assumed to be linearly independent i=1 m Y ⃗ , τ ), i.e., after division by M + (x, ξ, N, τ ) modulo (τ − τi+ ) = M + (x, ξ, N i=1

all the various remainders are linearly independent. In other words, let ⃗ , τ) = p′0,i (x, ξ, N

m−1 X

⃗ ) · τ k, bi,k (x, ξ, N

i = 1, · · · , m

k=0

⃗ , τ ). Then the condition of the be the remainders after division by M + (x, ξ, N Shapiro-Lopatinski implies that

6

1 Auxiliary Materials

⃗ )∥ = D(x, ξ, N ) = det ∥bik (x, ξ, N ̸ 0

(1.26)

⃗ ∈ Rn \ {0} normal to ∂Ω at x and ξ ∈ Rn \ {0} for all x ∈ ∂Ω, and for all N ⃗ with ⟨ξ, N ⟩ = 0. Definition 1.1 We say that (L, B1 , · · · , Bm ) defines an elliptic boundary value problem of order (2m, m1 , · · · , mm ) if L given by (1.22), is uniformly elliptic and of order 2m, each Bi given by (1.24) has order mi , 0 ≤ mi ≤ 2m − 1, the mi ’s are distinct, ∂Ω is non characteristic to Bi at each point and {Bi }m i=1 satisfy the Shapiro-Lopatinski condition with respect to L (see [47]). We have the following lemma (see [4, 33]). Lemma 1.1 Let (L, B1 , · · · , Bm ) define an elliptic boundary value problem of order (2m, m1 , · · · , mm ). Then   ∂u l l l ˜ ˜ L ◦ △ , B1 ◦ △ , · · · , Bm ◦ △ , L ◦ ⃗ ∂N defines an elliptic boundary value problem of order (2k+2l, m1 +2l, · · · , mm + ˜ is the Laplace-Beltrami operator, l ∈ N . 2l, 2m + 1) where △ Proof. The principal symbol of L ◦ △l is |ξ|2l · p0 (x, ξ), so it is clear that L ◦ △l is uniformly elliptic. ⃗ ∈ Rn \ {0}, with ⟨ξ, N ⃗ ⟩ = 0 and N ⃗ normal to ∂Ω at x. Let x ∈ ∂Ω and ξ, N ˜ l and L◦ ∂ at ξ +τ N ⃗ It is obvious that the principal symbol operators Bi ◦ △ ⃗ ∂N

⃗ ) and τ p0 (x, ξ + τ N ⃗ ) respectively, where ψl (ξ) ̸= 0. are ψl (ξ) · p0i (x, ξ + τ N + ⃗ ) = 0 having positive imaginary If τ1+ , · · · , τm are the m roots of p0 (x, ξ + τ N 2 ⃗ ⃗ ) = 0 with positive part, then the m + 1 roots of |ξ + τ N | · p0 (x, ξ + τ N + imaginary part are given by τ1+ , · · · , τm , i·

⃗| |N |ξ| .

We must show that if λ1 , · · · , λm+1 ∈ C and h(τ ) is a polynomial with ψl (ξ)

m X

⃗ ) + λm+1 τ p0 (x, ξ + τ N ⃗) λi · p0i (x, ξ + τ N

i=1

= h(τ ) ·

⃗| i|N τ− |ξ|

! ·

m Y

(τ − τi+ )

(1.27)

i=1

then λi = 0, 1 ≤ i ≤ m + 1 and h(τ ) ≡ 0. Due to the assumption that (B1 , · · · , Bm ) satisfy the covering condition it is not difficult to see that λ1 = · · · = λm = 0. But then the right-hand side of (1.27) has more roots with positive imaginary part than does the left-hand side, so that λm+1 = 0 and h(τ ) ≡ 0. ⊔ ⊓

1.2 Linear Elliptic Boundary value problems

7

With appropriate smoothness conditions on the coefficients (see Lemma 2.2 below), elliptic boundary value problems induce linear Fredholm operators in Sobolev spaces. Here the spaces W 2m+k−mi −1/p,p (∂Ω) with the fractional differentiation order 2m + k − mi − p1 play a decisive role. Before giving a precise definition we wish to point out a priori the most important property of these spaces, i.e. the surjective boundary operator ¯ → C ∞ (∂Ω) T : C ∞ (Ω) ¯ its classical boundary value T u which assigns to each function u ∈ C ∞ (Ω) on ∂Ω, can be extended uniquely to a continuous linear surjective operator T : W 2m+k,p (Ω) → W 2m+k−mi −1/p,p (∂Ω). Here k ≥ 0 and m ≥ 1 are integers, and 1 < p < ∞ (we are mainly interested in the case p = 2, W 2m,2 (Ω) = H 2m (Ω)). Then T u is described naturally as the generalized boundary value of u ∈ W 2m+k,p (Ω). These functions u have generalized derivatives Dα u up to order 2m + k on Ω. The functions Dα u with |α| ≤ mi have generalized boundary values which all lie in W 2m+k−mi −1/p,p (∂Ω), since mi < 2m. Consequently, Bi u ∈ W 2m+k−mi −1/p,p (∂Ω) also. The differential operators L and the boundary operator Bi are thus to be understood in the space of generalized derivatives on Ω and as generalized boundary values respectively. Definition of the space W m−1/p,p (∂Ω). Let Ω be an open subset of Rn with sufficiently smooth boundary and {Ui }li=1 ¯ with diffeomorphisms φi : Ui → Rn , φi ∈ C m (Ui ), be an open covering of Ω such that φi (Ui ) = V1 = {y ∈ Rn | |y| < 1} if Ui ⊂ Ω, and ¯ = V + = {y ∈ Rn | |y| < 1, yn ≥ 0}, φi (Ui ∩ Ω) 1 φi (Ui ∩ ∂Ω) = V˜1 = {y ∈ Rn | |y| < 1, yn = 0} if Ui ∩ ∂Ω ̸= ∅. Let χi (x) be a partition of unity subordinated to {Ui }li=1 and let λi (y) := χi (φ−1 i (y)). For each u(x) ∈ C m (∂Ω), 0 < δ < 1, p > 1 we define the norm:

8

1 Auxiliary Materials

∥u∥′m−δ,p,∂Ω =

X X

Z

′

i∈I ′

X

|α|≤m−1

Z

′

V˜1

|α| =m−1

·

Z V˜1

V˜1

|Dyα (λi (y) · ui (y))|p dy ′ +

|Dyα (λi (y) · ui (y)) − Dzα (λi (z) · ui (z)) |p

dy ′ dz ′ ′ |y − z ′ |n+p−1−δp

 p1 ,

(1.28)

′ ′ where ui (y) =Pu(φ−1 i (y)), y = (y1 , · · · , yn−1 ), I ⊂ {1, · · · , l} such that Ui ∩ ′ ∂Ω ̸= ∅ and implies that the sum is taken over those α for which αn = 0, α = (α1 , · · · , αn ). 1

By definition, the norm in W m− p ,p (∂Ω), p > 1 is defined as the norm ∥ · ∥′m− 1 ,p,∂Ω . For more details see [50]. p

Let us return to the discussion of elliptic boundary value problems . We first recall some results regarding linear Fredholm operators. Let X and Y be real Banach spaces. By L(X, Y ) we denote the Banach space of bounded linear operators from X to Y . An operator T in L(X, Y ) is called Fredholm if the kernel (nullspace) Ker T := {x ∈ X | T x = 0} has finite dimension and the image (range) of T , R(T ) := {T x | x ∈ X} is of finite codimension in Y , that is codim R(T ) = dim Y /R(T ) < ∞. For a Fredholm operator T : X → Y , the numerical Fredholm index of T , ind(T ) is defined by ind(T ) = dim Ker T − codim(R(T )). Lemma 1.2 Let Ω ⊂ Rn be open and bounded with ∂Ω smooth. Suppose that s > n/2, aα ∈ H s (Ω) if |α| ≤ 2m, while bα,i ∈ H s+2m−mi (∂Ω) and i = 1, · · · m. Then the following three assertions are equivalent: (i)

The operator A = (L, B1 , · · · , Bm ) A : H s+2m (Ω) −→ H s (Ω) ×

m Y

H s+2m−mi (∂Ω)

(1.29)

i=1

is an elliptic boundary value problem of order (2m, m1 , · · · , mm ) (ii) The operator A = (L, B1 , · · · Bm ) is Fredholm (iii) There is some c > 0, such that if u ∈ H s+2m (Ω), then " ∥u∥2m+s ≤ c ∥Lu∥s +

m X

# ∥Bi (x, D)u∥2m+s−mi − 12 + ∥u∥s .

(1.30)

i=1

Proof. If each aα ∈ C s (Ω) and each bα,i ∈ C 2k+s−mi (Ω), then a priori estimate (1.30) is contained in [1]. It is not difficult to see that (1.30) also

1.3 Superposition operators

9

holds under the present smoothness conditions. Thus, in fact a priori estimate (1.30) and equivalence (i) and (iii) follows from [1]. Equivalence (i) and (iii) to (ii) can be proved analogously to [2]. ⊔ ⊓ Remark 1.1 Of course, the Fredholm index of (L, B1 , · · · , Bm ) need not  i−1 be equal to 0. If L is uniformly elliptic and Bi u(x) = ∂∂N⃗ u(x) for 1 ≤ i ≤ m, then the index A = (L, B1 , · · · , Bm ) : H 2m+s (Ω) → H s (Ω) ×

m Y

1

H 2m+s−mi − 2 (∂Ω)

i=1

is 0 (see [4, 46]). Remark 1.2 (C γ -theory) The a priori estimates(1.30) remain valid if we choose the following B- spaces for 0 < γ < 1: ¯ Y = C s,γ (Ω), ¯ Z = C(Ω), ¯ Yj = C 2m+s−mi ,γ X = C 2m+s,γ (Ω), i.e. ∥u∥X ≤ constant(∥Lu∥Y +

m X

∥Bj u∥Yj + ∥u∥Z ).

(1.31)

j=1

Remark 1.3 The important fact is that the index of corresponding operators is the same in both theories. Remark 1.4 As shown in [1, 2] the terms ∥u∥s and ∥u∥Z in (1.30), (1.31) disappear if dim Ker A = {0}, where Au = (Lu, B1 u, · · · , Bm u).

1.3 Superposition operators The investigation of nonlinear equations in the following chapters relies on ¯ and properties of mappings of the form u 7→ f (u) in the spaces C α (Ω) p l L (Ω), H (Ω). Definition 1.2 Let Ω ⊂ Rn be a domain. We say that a function Ω × Rm ∋ (x, u) 7−→ f (x, u) ∈ R satisfies the Carath´eodory conditions if u 7−→ f (x, u) is continuous for almost every x ∈ Ω and

10

1 Auxiliary Materials

x 7−→ f (x, u) is measurable for every u ∈ Ω. Given any f satisfying the Carath´eodory conditions and a function u : Ω → Rm , we can define another function by composition F u(x) := f (x, u(x)).

(1.32)

The composed operator F is called a Nemytskii operator. In this section we ¯ Lp (Ω), H l (Ω) state some important results on the composition of C α (Ω), with nonlinear functions (some of them without proof [42, 63]). Proposition 1.1 Let Ω ⊂ Rn be a bounded domain and Ω × Rm ∋ (x, u) 7−→ f (x, u) ∈ R satisfy the Carath´eodory conditions. In addition, let |f (x, u)| ≤ f0 (x) + c(1 + |u|)r

(1.33)

where f0 ∈ Lp0 (Ω), p0 ≥ 1, and rp0 ≤ p1 . Then the Nemytskii operator F defined by (1.32) is bounded from Lp1 (Ω) into Lp0 (Ω), and satisfies ∥F (u)∥0,p0 ≤ C1 · (1 + ∥u∥rp1 )

(1.34)

Proof. By (1.33) and (1.7) ∥F (u)∥0,p0 ≤ ∥f0 (x)∥0,p0 + C∥1∥0,p0 + C∥|u|r ∥0,p0 Z  p1 0 ′ rp0 ≤C +C |u| dx = C ′ + ∥u∥r0,p0 r .

(1.35)

Ω

Since Ω is bounded, then by H¨older’s inequality ∥v∥0,q ≤ C(Ω)∥v∥0,p when 1 ≤ q ≤ p, v ∈ Lp (Ω) 1

(1.36)

1

where C(Ω) = mes(Ω) q − p . Inequalities (1.35) and (1.36) with q = rp0 and p = p1 imply (1.34).

⊔ ⊓

It is well-known that the notions of continuity and boundedness of a nonlinear operator are independent of one another ([42]). It turns out that the following is valid. Theorem 1.3 Let Ω ⊂ Rn be a bounded domain and let Ω × Rm ∋ (x, u) 7−→ f (x, u) ∈ R satisfy the Carath´eodory conditions. In addition, let p ∈ (1, ∞) and g ∈ Lq (Ω) (where p1 + 1q = 1) be given, and let f satisfy

1.3 Superposition operators

11

|f (x, u)| ≤ C|u|p−1 + g(x). Then the Nemytskii operator F defined by (1.32) is a bounded and continuous map from Lp (Ω) to Lq (Ω). For a more detailed treatment, the reader can consult [42, 63]. Theorem 1.4 Let Ω be a bounded domain in Rn with smooth boundary and let Ω × R ∋ (x, u) 7→ f (x, u) ∈ R satisfy the Carath´eodory conditions. Then for s > n/2, f induces 1) a continuous mapping from H s (Ω) into H s (Ω) if f ∈ C s , 2) a continuously differentiable mapping from H s (Ω) into H s (Ω) if f ∈ C s+1 . Proof. First we consider the simplest case, that is f = f (u) is independent of x. ¯ Hence we have By the Sobolev embedding theorem, we have H s (Ω) ⊂ C(Ω). ¯ for everyu ∈ H s (Ω). Moreover, if u is in C (s) (Ω), ¯ we can obtain f (u) ∈ C(Ω) the derivatives of f (u) by the chain rule, and in the general case, we can use approximation by smooth functions. Note that all derivatives of f (u) have the form of a product involving a derivative of f and derivatives of u. The ¯ while any l-th derivative of u lies in H s−l (Ω), which first factor is in C(Ω), 2n/(n−2(s−l)) imbeds into L (Ω) if s − l < n2 . We can use this fact and H¨older’s inequality to show that all derivatives of f (u) up to order s are in L2 (Ω); moreover, it is clear from this argument that f is actually continuous from H s (Ω) into H s (Ω). A proof of the differentiability in this special case is that f = f (u) is based on the relation 1

Z

fu′ (v + θ(u − v))(u − v)dθ

f (u) − f (v) = 0

and the same arguments as before. Let us now consider the general case, that is f = f (x, u). Let |α| ≤ s. We must show that u 7−→ Dα F (u) (1.37) defines a continuous map of H s (Ω) into L2 (Ω). It is not difficult to see that (1.37) is a finite sum of operators of the form u(x) 7−→ g(x, u(x)) · Dγ u(x)

(1.38)

where |γ| = γ1 +· · ·+γn ≤ s, while g is a partial derivative of f order at most s. It is obvious that Dγ is continuous from H s (Ω) into L2 (Ω) for |γ| ≤ s. ¯ implies that On the other hand, the continuous embedding of H s (Ω) in C(Ω)

12

1 Auxiliary Materials

u(x) 7−→ g(x, u(x)) ¯ Thus is continuous from H s (Ω) into C(Ω). u(x) 7−→ g(x, u(x)) · Dγ u(x) defines a continuous map of H s (Ω) into L2 (Ω) and hence so does u 7−→ Dα F (u). ⊔ ⊓ As before, let p ∈ N and p˜ denote the number of multi-indices with |α| ≤ p and let Ω be a bounded domain in Rn . Corollary 1.1 An analogous result is valid for a continuity of the operator F u(x) = f (x, u(x), · · · , Dp u(x)) : H s+p (Ω) → H s (Ω) where p, s ∈ N with s >

n 2

and f : Ω × Rp˜ → R is C s .

Corollary 1.2 Let p, s ∈ N with s >

n 2

and

f : Ω × Rp˜ → R be C s+1 . Then the operator F : H s+p (Ω) → H s (Ω) defined by F u(x) = f (x, u(x), · · · , Dp u(x)) is Fr´echet differentiable from H s+p (Ω) into H s (Ω). We will show some continuity and C 1 -differentiability results for a nonlinear differential operator of the form Au(x) = f (x, u(x), ..D 2p u(x)) in H¨ older spaces. They are based on the following Theorems 1.5 and 1.6. ¯× Theorem 1.5 Let the function f (x, y) = f (x, y1 , · · · , yp˜) be defined on Ω Rp˜ and satisfy the following conditions: 1) f (x, 0) = 0 2 ∂ f ≤ C(R), sup ∥f ∥C 1,α (Ω) 2) For any R > 0, sup ¯ ≤ C(R), where |y|≤R ∂yi ∂yj |y|≤R C(R) is constant depending on R. ¯ 0 < α < 1, ∥ui ∥C α (Ω) Let u1 (x), · · · , up˜(x) ∈ C α (Ω), ˜. ¯ ≤ R, i = 1, . . . , p Then ∥f (x, u1 (x), · · · , up˜(x))∥C α (Ω) ¯ ≤ C1 (R) ·

p˜ X i=1

Proof. Obviously,

∥ui ∥C α (Ω) ¯ .

(1.39)

1.3 Superposition operators

13

Z

1

d f (x, ty1 , . . . , typ˜)dt 0 dt Z 1 p˜ X ∂f (x, ty1 , . . . , typ˜) = yj dt ∂yj 0 j=1

f (x, y, . . . , yp˜) =

=

p˜ X

φj (x, y1 , . . . , yp˜) · yj

j=1

where

1

Z φj (x, y1 , . . . , yp˜) = 0

∂f (x, ty1 , . . . , typ˜) dt. ∂yj

Hence f (x, u1 (x), · · · , up˜(x)) =

p˜ X

φj (x, u1 (x), · · · , up˜(x)) · uj (x).

j=1

¯ 0 < α < 1 is a Banach algebra, we have Since C α (Ω), ∥f (x, u1 (x), · · · up˜(x))∥C α ≤

p˜ X

∥φj (x, u1 (x), · · · up˜(x))∥C α · ∥uj ∥C α .

j=1

Hence we have to prove that sup ∥φj (x, u1 (x), · · · , up˜(x))∥C α ≤ C1 (R). |y|≤R

Indeed |φj (x + ξ, u1 (x + ξ), · · · , up˜(x + ξ)) − φj (x, u1 (x), · · · , up˜(x))| ≤ |φj (x+ξ, u1 (x+ξ), · · · , up˜(x+ξ))−φj (x, u1 (x+ξ), · · · , up˜(x+ξ))| +|φj (x, u1 (x + ξ), · · · , up˜(x + ξ)) − φj (x, u1 (x), · · · , up˜(x))|. (1.40) The first term on the right-hand side of (1.40) is bounded by C(R) · |ξ|α . The second term is bounded by ∂φj ·|φj (x, u1 (x+ξ), · · · , up˜(x+ξ))−φj (x, u1 (x), · · · , up˜(x))| ≤ CR R|ξ|α . sup ∂y k |y|≤R (1.41) The estimates (1.40) and (1.41) yield (1.39). ⊔ ⊓ ¯× Theorem 1.6 Let the function f (x, y) = f (x, y1 , · · · , yp˜) be defined on Ω Rp˜ satisfy the following conditions: 1) f (x, 0) = 0, grady f (x, 0) = 0

14

1 Auxiliary Materials

2) For any R > 0, sup ∥f (x, y)∥C 2,α (Ω) ¯ |y|≤R

∂3f ≤ ≤ C(R) and sup ∂yi ∂yj ∂yk |y|≤R

C(R), where C(R) is constant depending on R. Let as before, u1 (x), · · · , up˜(x) ∈ ¯ with ∥ui ∥C α (Ω) C α (Ω) ˜. ¯ ≤ R, i = 1, · · · , p Then the following estimate holds. ∥f (x, u1 (x), · · · , up˜(x))∥C α (Ω) ¯ ≤ C2 (R) ·

p˜ X

∥ui ∥2C α .

(1.42)

i=1

Proof. Obviously we have f (x, y1 , · · · , up˜) =

p˜ X

gij (x, y1 , · · · , yp˜) · yi · yj

i,j=1

so we can write f (x, u1 (x), · · · , up˜(x)) =

p˜ X

gij (x, u1 (x), . . . , up˜(x)) · ui (x) · uj (x)

i,j=1

and we have ∥f (x, u1 (x), · · · , up˜(x)∥C α (Ω) ¯ ≤

p˜ X

∥gij (x, u1 (x), · · · , up˜(x)∥C α ∥ui ∥C α ∥uj ∥C α

i,j=1

(1.43) Due to Theorem 1.5 we obtain ∥gij (x, u1 (x), · · · , up˜(x)∥C α (Ω) ¯ ≤ C0 (R)

(1.44)

Hence the estimates (1.43) and (1.44) yield (1.42) ∥f (x, u1 (x), · · · , up˜(x)∥C α (Ω) ¯ ≤ C2 (R) ·

p˜ X

∥ui ∥2C α .

i=1

This completes the proof.

⊔ ⊓

We apply Theorems 1.5 and 1.6 to the operator Au(x) = f (x, u(x), · · · , D2p u(x)) where the function f (x, y1 , · · · , yp˜) satisfy conditions of Theorems 1.5 and 1.6, respectively. Hence we have

1.3 Superposition operators

15

∥Au∥C 2p,α ≤ C(R) · ∥u∥C α . Moreover as it follows from Theorem 1.6 A ∈ C 1 , A′ (0) = 0 and ∥A′ (u + h) − A′ (u)∥L(C 2p,α ,C α ) ≤ C · ∥h∥C 2p,α (Ω) ¯ . Remark 1.5 As shown in the proofs of Theorems 1.5 and 1.6, continuity and differentiability of the operator Au(x) = f (x, u(x), · · · , D2p u(x)) between ¯ and C α (Ω) ¯ remains valid under slightly weaker conditions on a C 2p,α (Ω) given function f (x, y1 , · · · , yp˜). We leave these as exercises for the reader. In the investigation of nonlinear boundary value problems related to pseudodifferential operators and in particular nonlinear Riemann-Hilbert problems we need properties of the Nemytskii operators in the spaces H s (S 1 ) or C p,α (S 1 ), where S 1 is the unit circle. We recall some of the properties which will be used often in the sequel. The norm in C α (M ) is given by ∥f ∥C α (M ) = ∥f ∥C + sup x̸=y

|f (x) − f (y)| , M = S1. |x − y|α

As before, by C k,α (M ) we denote the space of H¨ older continuous functions, which have derivatives up to order k, with Dk f ∈ C α (M ). Let F be a superposition operator defined by F u(x) = f (x, u(x)),

x ∈ M.

The following theorems are not hard to prove (although not obvious). Theorem 1.7 Let k ∈ R+ . Then the superposition operator F : E1 → E2 defined by F u(x) = f (x, u(x)) acts as a bounded operator in each of the following cases (see also [62]) 1) 2)

f ∈ C(S 1 × R, R), E1 = C(S 1 ), E2 = C(S 1 ) f ∈ C 1 (S 1 × R, R), E1 = C α (S 1 ), E2 = C α (S 1 ), 0 < α < 1

Theorem 1.8 Let k ∈ R+ , 0 < α < 1. Then the superposition operator F : E1 → E2 defined by F u(x) = f (x, u(x)) is m times continuously differentiable in each of the following cases 1) 2)

D0,j f ∈ C k (S 1 × R, R), E1 = C k (S 1 ), E2 = C k (S 1 ) D0,j f ∈ C k+1 (S 1 × R, R), E1 = C k,α (S 1 ), E2 = C k,α (S 1 ),

The j-th derivative of F is given by D0,j F (x, u(x))h1 (x) . . . hj (x) = Dj F (f )(h1 , . . . hj )(x).

16

1 Auxiliary Materials

Analogous results are valid in Sobolev spaces : Theorem 1.9 Let X = Y = H s (S 1 )(s ≥ 1) be the Sobolev space of real functions x(τ ) on the circumference of a circle, where 0 ≤ τ < 2π; f (τ, x) is a smooth real function, x ∈ R, 0 ≤ τ < 2π. Then the operator F : H s (S 1 ) → H s (S 1 ) defined by F x(τ ) = f (τ, x(τ )) is continuous. Proof. It is not difficult to see, that 

d dτ

k f (τ, x(τ )) =

X

Cp,q,r1 ...rq

p+q≤k r1 +···+rq =k−p rj ≥0

∂ p+q f (τ, x(τ )) (r1 ) x (τ ) · · · x(rq ) (τ ) ∂τ p · ∂xq

where Cp,q,r1 ...rq are some constants. l

x(τ ) If x(τ ) ∈ H s , then it follows that the derivatives { d dτ | 0 ≤ l ≤ s − 1} are l ds continuous. Therefore in dτ s f (τ, x(τ )) all terms without ones are continuous. s

x(τ ) The last term is equal to d dτ × Q(τ ) where Q(τ ) is a continuous function, s hence also square integrable.

As a consequence of these arguments we obtain continuity.

⊔ ⊓

Remark 1.6 An analogous result holds for vector functions, and also in the multidimensional case, for functions on arbitrary smooth compact manifold with boundary. Lemma 1.3 Let the function f ∈ C 2 (R, R) satisfies C1 |u|p−1 ≤ f ′ (u) ≤ C1 |u|p−1 , p > 1, with C1 and C2 some positive constants. Then, for every s ∈ (0, 1) and 1 < q ≤ ∞, we have 1/p

∥u∥W s/p,pq (Ω) ≤ Cp ∥f (u)∥W s,q (Ω) where the constant Cp is independent of u. Proof. Indeed, let f −1 be the inverse function to f . Then, due to conditions on f , the function G(v) := sgn(v)|f −1 (v)|p is nondegenerate and satisfies C2 ≤ G′ (v) ≤ C1 , for some positive constants C1 and C2 . Therefore, we have |f −1 (v1 ) − f −1 (v2 )|p ≤ Cp |G(v1 ) − G(v2 )| ≤ Cp′ |v1 − v2 |, for all v1 , v2 ∈ R. Finally, according to the definition of the fractional Sobolev spaces ,

1.4 Pseudodifferential operators. Definitions and examples ∥f −1 (v)∥pq s/p,qp W

(Ω)

17

|f −1 (v(x)) − f −1 (v(y)|pq dx dy |x − y|n+sq Z Ω Z Ω |v(x) − v(y)|q ≤C∥v∥qLq (Ω) + Cp′ dx dy = Cp′′ ∥v∥qW s,q (Ω) , n+sq Ω Ω |x − y|

:=∥f −1 (v)∥pq + Lpq (Ω)

Z Z

where we have implicitly used that f −1 (v) ∼ sgn(v)|v|1/p . Lemma 1.3 is proved. ⊔ ⊓

1.4 Pseudodifferential operators. Definitions and examples Let a(x, ξ) be a C ∞ function for all x ∈ Rnx and ξ ∈ Rnξ \{0} which is positively homogeneous of degree σ ≥ 0 in ξ: a(x, tξ) = tσ a(x, ξ),

t > 0.

(1.45)

Assume that on the sphere |ξ| = 1, a(x, ξ) has a limit a(∞, ξ) as x → ∞ and that a′ (x, ξ) = a(x, ξ) − a(∞, ξ) ∈ S(Rnx ) uniformly in ξ. In fact we assume that the same is true for all derivatives with respect to ξ, i.e., for any integers p, α, β, (1 + |x|)p · ∂xα Dξβ (a(x, ξ) − a(∞, ξ)) → 0 as |x| → ∞ ∂ ∂ uniformly in ξ, for |ξ| = 1 where ∂x = 1i ∂x , Dξ = ∂ξ . The function a(x, ξ) is called the symbol. In sections 1.4 we will mainly follow [41] specifying some details.

Definition 1.3 A (homogeneous) pseudodifferential operator A(x, D) with the symbol a(x, ξ) is defined on functions u(x) ∈ S(Rnx ) in the following way: Z A(x, D)u(x) = (2π)−n/2 eix·ξ · a(x, ξ)ˆ u(ξ)dξ. (1.46) Rn ξ

Recall that S(Rnx ) is the space of all functions on Rnx which are of class C ∞ and such that |x|k |Dα u(x)| is bounded for every k ∈ N and every multi-index α. The formula (1.46) can be rewritten as follows:

18

1 Auxiliary Materials

A(x, D)u(x) =(2π)−n/2

Z

eix·ξ a(∞, ξ)ˆ u(ξ)dξ Z + (2π)−n/2 eix·η a′ (x, η)ˆ u(η) dη,

(1.47)

where a′ (x, ξ) = a(x, ξ)−a(∞, ξ). Then it is Z not difficult to see, that (Fubini’s −n/2 ˆ formula) (Au)(ξ) = a(∞, ξ)ˆ u(ξ) + (2π) a ˆ′ (ξ − η, η)ˆ u(η) dη, where a ˆ′ (ξ, η) = Fx→ξ a′ (x, η) and u ˆ(ξ) := Fx→ξ u = (2π)−n/2

Z

e−ix·ξ u(x)dx.

The formula (1.47) is convenient from a computational point of view. Particular cases are: 1) Homogeneous differential operator A(x, D) i.e.

A(x, D)u(x) =

X

aα (x) · ∂ α u(x)

|α|=σ

=

X

aα (x) · F −1 ξ α F u(ξ)

|α|=σ

 = (2π)−n/2

Z

 X

eix·ξ 

aα (x) · ξ α  u ˆ(ξ)dξ.

(1.48)

|α|=σ

Here the symbol a(x, ξ) is the characteristic polynomial, i.e. X a(x, ξ) = aα (x) · ξ α . |α|=σ

If aα (x) = aα (∞) + a′α (x), then  ˆ Au(ξ) = aα (∞, ξ)ˆ u(ξ) + (2π)−n/2

Z

 X



a ˆ′α (ξ − η) · η α  u ˆ(η) dη

|α|=σ

which is an expression of the differential operator (1.48) in the form (1.47). 2) The operator ∧u = F −1 |ξ|F u. It is obvious that ∧2 u = − △ u is the Laplace operator . The symbol of ∧u is a(x, ξ) = |ξ|. 3) In the case when σ = 0 a homogeneous pseudodifferential operator is called a homogenous singular integral operator. We set

1.4 Pseudodifferential operators. Definitions and examples

Z

19

a(x, ξ ′ )dξ ′ .

a0 (x) = |ξ ′ |=1

Then one can show that Z A(x, D)u(x) = a0 (x)u(x) + lim

ϵ→0 |x−y|>ϵ

K(x, x − y) u(y) dy |x − y|n

(1.49)

where the function K(x, z) is positive homogeneous of degree zero in z, such that K(x, z) −1 (2π)−n/2 = Fξ→x [a(x, ξ) − a0 (x)]. |z|n Remark 1.7 By H l (Rnx ), −∞ < l < ∞ we denote the completion of S(Rnx ) in the norm (1.50) Z ∥u∥2l = (1 + |ξ|2 )l · |ˆ u(ξ)|2 dξ. (1.50) Order. A linear operator L : S(Rnx ) → S(Rnx ) is said to have order σ, if for each real s there exists a constant Cs such that ∥Lu∥s−σ ≤ Cs ∥u∥s .

(1.51)

The infimum of the set of σ ′ of L is called the true order of L. In the sequel we will abbreviate pseudodifferential operators by ψDO. Boundedness of ψDO with homogeneous symbol of degree σ. Lemma 1.4 Let A(x, D) be a homogeneous pseudodifferential operator with homogeneous symbol of degree σ. Then A(x, D) has order σ, i.e. ∥Au∥l−σ ≤ const. · ∥u∥l .

(1.52)

Proof. It is not difficult to estimate the norm of the operator A1 u = F −1 a(∞, ξ)F u Since |a(∞, ξ)| ≤ C · |ξ|σ , then

(1.53)

20

1 Auxiliary Materials

Z


l−σ 1 + |ξ|2 · |a(∞, ξ)ˆ u(ξ)|2 dξ Z
l ≤ C2 1 + |ξ|2 · |ˆ u(ξ)|2 dξ

∥A1 u∥2l−σ =

= C 2 ∥u∥2l . It remains to check (1.52) for the operator Z −n/2 ˆ Au(ξ) = (2π) a ˆ′ (ξ − η, η)ˆ u(η) dη, where a ˆ′ (ξ, η) = Fx→ξ (a(x, η) − a(∞, η)) . For simplicity assume that a(∞, ξ) ≡ 0. Since |ˆ a′ (ξ − η, η)| ≤ Cp

|η|σ (1 + |ξ − η|2 )p

for sufficiently large p, it follows that 2 l−σ

(1 + |ξ| )

ˆ 2 ≤ const. · |Au|

Z

l−σ 2 l−σ 2 (1+|η| ) 2

(1+|ξ|2 )

·

(1+|η|2 )l/2 (1+|ξ−η|2 )p

2 · |ˆ u(η)| dη

.

(1.54) Since |ξ|2 ≤ 2|ξ − η|2 + 2|η|2 it follows that 1 + |ξ|2 ≤ 2(1 + |ξ − η|2 )(1 + |η|)2 ; and analogously 1 + |η|2 ≤ 2(1 + |ξ − η|2 )(1 + |ξ|2 ). Hence, 

Choosing p =

1 + |ξ|2 1 + |η|2 n+1 2

2 l−σ

(1 + |ξ| )

+

k

≤ 2|k| (1 + |ξ − η|2 )|k| for each k ∈ R.

|l−σ| 2

(1.55)

in (1.54) we have:

ˆ 2 ≤ const. |Au|

Z

(1 + |η|2 )l/2 · |ˆ u(η)|2 (1 + |ξ − η|2 )

From the Schwarz inequality it follows that

n+1 2

!2 dη

.

(1.56)

1.4 Pseudodifferential operators. Definitions and examples

21

Z 2 Z Z φ(ξ − η)v(η) dη ≤ |φ(ξ − η)| dη · |φ(ξ − η)| · |v(η)|2 dη. Hence

Z

2 Z Z Z

φ(ξ − η)v(η) dη ≤ |φ(η)| dη · |φ(ξ − η)| · |v(η)|2 dη dξ

0 Z 2 = |φ(η)|dη · ∥v∥20 . (1.57) Therefore from (1.56) and (1.57) it follows that ∥Au∥2l−σ ≤ const. ∥u∥2l . ⊔ ⊓ Particular cases: 1) The differential operator A(x, D)u =

P

aα (x)Dα u has order σ.

|α|=σ

2) The singular integral operator S(x, D) has order 0. 3) The operator A(x, D)u(x) = a(x) · u(x) has order 0. Pseudodifferential operators of negative order. Let a(x, ξ) be a function which is positive homogeneous of degree σ < 0 in ξ. Then it has a singularity at ξ = 0, and the formula (1.46) has no meaning. By ζ(ξ) we denote a fixed C ∞ non-negative function which equals one for |ξ| > 1 and vanishes for |ξ| < 12 . By definition, a pseudodifferential operator with symbol a(x, ξ), a(x, tξ) = tσ a(x, ξ), where σ < 0 and t > 1 is Z −n/2 Aξ (x, D)u(x) = (2π) eixξ · ζ(ξ) · a(x, ξ) · u ˆ(ξ)dξ. (1.58) It is not difficult to prove, that Lemma 1.4 is valid for pseudodifferential operators Aξ (x, D). Remark 1.8 Independent of the value of σ, a difference Aζ1 (x, D) − Aζ2 (x, D) with the same symbol has true order equal to −∞, since ζ1 (ξ) − ζ2 (ξ) ∈ C0∞ . Therefore sometimes we will also call the operator Aζ (x, D) a pseudodifferential operator of order σ, σ ≥ 0. Let A(x, D) and B(x, D) (or Aζ1 (x, D), Bζ2 (x, D)) be pseudodifferential operators with symbols a(ξ) and b(ξ). Obviously A ◦ B (or Aζ1 ◦ Bζ1 ) is a pseudodifferential operator with symbol a(ξ) · b(ξ). If A(x, D) and B(x, D) are differential operators, i.e. X X A(x, D)u = aα (x)∂ α u, B(x, D)u = bβ (x) · ∂ β u |α|=σ

|β|=λ

22

1 Auxiliary Materials

then A(x, D) ◦ B(x, D) is not a homogeneous differential operator with the symbol a(x, ξ) · b(x, ξ):   X X X A ◦ Bu = aα (x)∂ α  bβ (x)∂ β u(x) = Cγ (x, D)u; |α|=σ

|ρ|=λ

|γ|≤σ+λ

where the symbol Cγ (x, ξ) of Cγ (x, D) is defined by Cγ (x, ξ) =

1 γ D a(x, ξ) · ∂xγ b(x, ξ) γ! ξ

where |γ| ≤ σ + λ, γ! = γ1 ! . . . γn ! and Dγ = result is valid for pseudodifferential operators.

∂ γ1 +...+γn . An analogous ∂ξ1γ1 . . . ∂ξnγn

Theorem 1.10 Let A(x, D) and B(x, D) be homogeneous pseudodifferential operators of orders σ and λ with the symbols a(x, ξ) and b(x, ξ). Then the following is valid: A(x, D) ◦ B(x, D) = C0 u + C1 u + . . . + Cρ−1 u + Tρ u,

(1.59)

where Ci (x, D) P are pseudodifferential operators of order λ + σ − i with sym1 bols Ci (x, ξ) = |α|=i α! Dξα a(x, ξ) · ∂xβ b(x, ξ), i = 0, . . . , ρ − 1 and Tρ u is an operator of order σ + λ − ρ, such that ∥Tρ u∥l−(σ+λ−ρ) ≤ const. ∥u∥l .

(1.60)

Proof. In order to consider operators of arbitrary order we introduce as before C ∞ non-negative functions ζi (ξ), i = 1, 2 which are equal to one for |ξ| > 1 and vanish for |ξ| < 12 . We set h(x, ξ) = ζ1 (ξ) · a(x, ξ), g(x, ξ) = ζ2 (ξ) · b(x, ξ). Instead of Aζ1 and Aζ2 we will write A(x, D), B(x, D). If A(x, D) = A1 + A2 , B(x, D) = B1 + B2 , where A1 (x, D) and B1 (x, D) are operators with symbols a(∞, ξ) and b(∞, ξ), then AB = A1 B1 + A2 B1 + A1 B2 + A2 B2 .

(1.61)

For simplicity we assume that a(∞, ξ) = b(∞, ξ) = 0. Then Z Z −n ˆ d AB(ξ) = (2π) · h(ξ − η, η) gˆ(η − τ, τ )ˆ u(τ )dτ dη  Z  Z −n/2 −n/2 ˆ = (2π) · (2π) h(ξ − η, η) · gˆ(η − τ, τ )dη u ˆ(τ )dτ (1.62)

1.4 Pseudodifferential operators. Definitions and examples

23

which follows from Fubini’s theorem. According to the Taylor formula (with respect to the second variable) ˆ − η, η) = h(ξ

X |α|≤ρ−1

1 αˆ ˆ ρ (ξ − η, η, τ ) (1.63) ∂ h(ξ − η, τ ) · (η − τ )α + h α!

ˆ ρ is the remainder. Hence, it follows from (1.63) that where h X AB = Φα + Tρ,1 |α|≤ρ−1

where Φα (x, D) = (2π)

−n/2

Z e

 1 α α D h(x, ξ) · ∂x g(x, ξ) u ˆ(ξ)dξ α! ξ

(1.64)

hρ (ξ − η, η, τ ) · gˆ(η − τ, τ )dη u ˆ(τ )dτ.

(1.65)

ixξ



and −n/2 Td ρ,1 u(ξ) = (2π)

Z Z

Consider the difference of Ci (x, D)u −

P

Φα (x, D)u, where

|α|=i



Ci (x, D)u = (2π)−n/2

Z

 X 1 eix·ξ ζ(ξ)  Dα a(x, ξ) · ∂xβ b(x, ξ) u ˆ(ξ)dξ α! ξ |α|=i

(1.66) and ζ(ξ) is a C ∞ non-negative function which equals one for |ξ| > 1 and vanishes for |ξ| < 12 . The difference of corresponding symbols is  X 1, |ξ| ≤ r/2 ζ(ξ) · Ci (x, ξ) − fα (x, ξ) = (1.67) 0, |ξ| > r |α|=i

where r is some positive number. P Hence Tρ,2 (x, D) := Ci (x, D)u− |α|=i Φα (x, D) has true order equal to −∞. Therefore ρ−1 X AB = Ci + Tρ,1 + Tρ,2 . (1.68) i=0

In order to complete the proof of Theorem 1.10, we need the estimate ∥Tρ,1 (x, D)u∥l−(σ+λ−ρ) ≤ const. ∥u∥l . Recall that

(1.69)

24

1 Auxiliary Materials

ˆ ρ (ξ − η, η, τ ) = h

X 1 ˆ − η, τ + θ(η − τ )) · (η − τ )β , ∂ β h(ξ β! η

|β|=ρ

where 0 < θ < 1. Hence for σ ≥ ρ (ρ sufficiently large) we have   σ−ρ |η − τ |ρ 2 σ−ρ 2 2 ˆ 2 |hρ (ξ − η, η, τ )| ≤ const. (1 + |τ | ) + (1 + |η| ) . (1 + |ξ − η|2 )ρ (1.70) An analogous estimate holds for σ < ρ. This is not difficult and is left to the reader. For the function gˆ we have λ

(1 + |τ |2 ) 2 |ˆ g (η − τ, τ )| ≤ const. , (1 + |η − τ |2 )ρ

(1.71)

for sufficiently large ρ. From (1.70) and (1.71) it follows that 2

(1 + |ξ| )

l−(σ+λ−ρ) 2

Z

ˆ ρ (ξ − η, η, τ )| · |ˆ |h g (η − τ, τ )|dη ≤ C

(1 + |τ |2 )l/2

n+1 . (1 + |ξ − τ |2 ) 2 (1.72) Inequality (1.69) follows from (1.72). This proves Theorem 1.10. ⊔ ⊓

Corollary 1.3 Let A(x, D), B(x, D) be pseudodifferential operators of order σ and λ respectively. Let C0 (x, D) be a pseudodifferential operator of order σ + λ with symbol a(x, ξ) · b(x, ξ). Then A(x, D) ◦ B(x, D) − C0 (x, D) has order σ + λ − 1. Corollary 1.4 Let A(x, D), B(x, D) be pseudodifferential operators of order σ and λ respectively. Then [A, B] = AB − BA has order σ + λ − 1. In particular, if A(x, D) has order σ, then ∂j A − A∂j has order σ, where ∂ ∂j = 1i · ∂x ; moreover, if φ(x), ψ(x) ∈ C0∞ (Rn ), such that φ(x) · ψ(x) = 0, j then φAψ has order σ − 1. Corollary 1.5 Let A(x, D) be a pseudodifferential operator of order σ and let Ωi , i = 1, 2 be bounded domains in Rn with Ω 1 ∩ Ω 2 = ∅. Suppose u(x) ∈ H l (Rn ) for some l, with supp u ∈ Ω2 . Then outside the support of u, A(x, D)u ∈ C ∞ . Proof. Let φ1 (x), φ2 (x) ∈ C0∞ (Rn ) be such that φi (x) = 1 in Ωi , i = 1, 2 and supp φ1 ∩ supp φ2 = ∅. For each x ∈ Ω1 we have Au(x) = φ1 (x)A [φ2 (x)u(x)] .

1.4 Pseudodifferential operators. Definitions and examples

25

Let a(x, ξ) be the symbol of A(x, D). Since Dξα φ1 = 0 for |α| > 0 it follows from Theorem 1.10, that φ1 (x)A(x, D) = A1 (x, D) + T1 , where A1 (x, D) is a pseudodifferential operator with symbol a1 (x, ξ) = φ1 (x) · a(x, ξ), and T1 has true order equal to −∞. Analogously, since Dξα a1 (x, ξ)∂xα φ2 (x) ≡ 0 for all α, then T2 φ := A1 (x, D)φ has true order equal to −∞. Thus φ1 (x)A(x, D)φ2 (x) = φ1 (x)T1 + T2 has equal to −∞. Consequently v(x) = φ1 (x)A(φ2 (x)u(x)) belongs to T order n l Hl (R ). Hence due to the Sobolev embedding theorem we obtain v(x) ∈ C ∞ (Rn ). ⊔ ⊓ In the literature Corollary 1.5 is called the pseudo-local property. Remark 1.9 In the case of A(x, D)u =

P

aα (x)Dα u, we have Au(x) ≡ 0

|α|=σ

in Ω1 . Conjugate operator. Let A(x, D) be a pseudodifferential operator of order σ defined by Z −n/2 A(x, D)u = (2π) eix·ξ ζ(ξ)a(x, ξ)ˆ u(ξ)dξ. Let

Z (u, v) =

u(x)v(x)dx

(1.73)

and a∗ (x, ξ) = a(x, ξ) the complex conjugate. Then one can show that A∗ (x, D) defined by (1.74) is the conjugate operator to A(x, D) with respect to the scalar product (1.73), Z −n/2 ∗ d A v(ξ) = (2π) e−ix·ξ a∗ (x, ξ)ζ(ξ)v(x)dx. (1.74) Lemma 1.5 Let A(x, D) be a pseudodifferential operators of order σ with symbol a(x, ξ) and let A∗ (x, D) be conjugate to A(x, D). Then, for all ρ > 0, ρ an integer, ρ−1 X A∗ (x, D) = Bi (x, D) + Tρ i=0

where Bi (x, D) are ψDO with symbol X 1 ∂ α · Dξα a∗ (x, ξ) α! x

|α|=i

26

1 Auxiliary Materials

and Tρ has order σ − ρ. We omit the details of the proof which is based on the Taylor expansions of the function a ˜′∗ (ξ − η, ξ) with respect to ξ, when ξ belongs to a small neighborhood of η. Remark 1.10 The lemma is obvious if A(x, D)u =

X

aα (x)Dα u (here we

|α|=σ

take ζ(ξ) ≡ 1). Indeed, then A∗ (x, D)u(x) =

X

Dα [a∗α (x)u(x)]

|α|=σ

where a∗α is conjugate to aα (x). Hence A∗ (x, D)u(x) −

X

a∗α (x)Dα u

|α|=σ

has order σ − 1. In section 1.4 ψDO was defined in a canonical way by formulas (1.46) and (1.58). Theorem 1.10 and Lemma 1.5 showed us, that the operations A, B 7→ A ◦ B and A 7→ A∗ do not lie in the class of ψDO which are canonically defined by (1.46) or (1.58). In other words, it is not an algebra. In order to avoid this drawback, we have to add a number of remainder terms to the definition of ψDO. Of course, we have a degree of freedom to choose these additional terms. A reasonable way to proceed is to add terms of ψDO operators decreasing orders. More precisely: Definition 1.4 Let r0 , r1 , r2 , . . . be a sequence (finite or infinite), such that r0 > r1 > . . . > rn , . . ., and rk → −∞ as k → ∞ (if the sequence is infinite). Let Ak (x, D) be a ψDO of order rk defined in a canonical way. Let A be an operator defined on S(Rnx ), such that A−

N X

Ak (x, D)

k=0

for all N has order less than rN . Then by pseudodifferential operator with asymptotic expansion A∼

∞ X

definition

Ak (x, D).

k=0

By definition the symbol of A, i.e. a(x, ξ), is the formal series

A

is

(1.75)

1.4 Pseudodifferential operators. Definitions and examples ∞ X

27

ak (x, ξ).

(1.76)

k=0

If the sequence {rk } is finite, then ψDO is defined by A =

ρ X

Ak (x, D) + T∞

k=0

where T∞ has true order equal to −∞. From the definition it follows that, in both cases ψDO has order r0 . Furthermore, ψDO with symbol a(x, ξ) ≡ 0 has true order equal to −∞, since ak (x, ξ) ≡ 0, for k = 1, 2, · · · implies Ak (x, D) ≡ 0, k = 1, 2 . . ., and vice versa; if a pseudodifferential operator has order equal to −∞, then its symbol is equal to zero (see [41]). Thus a ψDO is defined by its symbol modulo operators which have orders equal to −∞. Let Ω be an open set of Rn . The following theorem is given in [39]. Theorem 1.11 Let aj (x, ξ) be C ∞ functions in Ω × (Rnξ \{0}) which are positively homogeneous of degree rj ↘ −∞. Then there exists a ψDO with symbol ∞ X ak (x, ξ). k=0

Proof. Let φ(ξ) be a C ∞ function in Rn which equals 0 when |ξ| ≤ 12 and equals 1 when |ξ| ≥ 1. We can choose a sequence tj ↗ +∞ increasing so rapidly that     β D φ ξ aj (x, ξ) + (1 + |x|k ) ∂xα Dβ φ ξ aj (x, ξ) ξ ξ tj tj ≤ 2−j · (1 + |ξ|)rj−1 −|β|

(1.77)  

for all |α| + |β| + k ≤ j and |α| > 0. In fact, Dξβ [φ tξj aj (x, ξ)] can be computed by Leibniz’s formula. It is obvious that a term in which φ is not differentiated is homogeneous of order rj − |β| in ξ when |ξ| > tj and is equal t to 0 when |ξ| ≤ 2j . In other terms we have t−q j , when 0 < q ≤ |β| and these tj terms are equal to 0 both when |ξ| ≤ 2 and |ξ| > tj . Hence, the left hand side of (1.77) can be estimated by const. ·

1 r

tj j−1

−rj

· |ξ|rj−1 −|β|

and it is obvious that we can choose a sequence tj , such that     β D φ ξ aj (x, ξ) +(1+|x|k ) ∂xα Dβ φ ξ aj (x, ξ) ≤ 2−j ·(1+|ξ|)rj−1 −|β| . ξ ξ tj tj

28

1 Auxiliary Materials

With t0 = 1 set a(x, ξ) =

∞ X

 φ

j=0

ξ tj

 aj (x, ξ).

(1.78)

Because of our choice of the tj , the series and the differentiated series converge absolutely. We set Z A(x, D)u(x) = (2π)−n/2 eix·ξ a(x, ξ)ˆ u(ξ)dξ and Aj (x, D)u(x) = (2π)

−n/2

Z e

ix·ξ

 φ

ξ tj

 aj (x, ξ)ˆ u(ξ)dξ.

Therefore instead of the asymptotic expansion A ∼

∞ X

Aj we have

j=0 ∞ X

A(x, D)u =

Aj (x, D)u.

j=0

It is not difficult to see that A(x, D) −

k X

Aj (x, D)u

j=0

has order rk+1 . We denote by L the class ψDO defined by asymptotic expansions (1.75). Theorem 1.12 L is an algebra with involution . P∞ P∞ Proof. Let A(x, D) ∼ j=0 Aj (x, D) and B(x, D) ∼ k=0 Bk (x, D) where Aj (x, D) and Bk (x, D) are canonical ψDO’s defined by (1.46) with orders σj and λk . We have to show that A(x, D) ◦ B(x, D) ⊂ L. It is sufficient to P show that for any number q there is a finite sum of canonical operators Cj (x, D), whose symbols have strictly decreasing orders, and which differ from A(x, D) ◦ B(x, D) by an operator of order < q. Suppose the symbols of the Aj (x, D) and the Bk (x, D) has degrees r0 > r1 > . . . → −∞, and s0 > s1 > . . . → −∞, respectively. Let N be so large that σ0 + sN < q and λ0 + σN < q, and set A=

N X j=0

Aj + TN,1 ,

B=

N X

Bk + TN,2

k=0

where the orders TN,1 and TN,2 are less than σN and sN respectively. Hence

1.4 Pseudodifferential operators. Definitions and examples

29

  ! N N N X X X AB =  Aj + TN,1  Bk + TN,2 = Aj Bk + TN,3 j=0

k=0

j,k=0

where the order of TN,3 is less than q = max(σ0 +λN , λ0 +σN ). The assertion of Theorem 1.12 follows from Theorem 1.10 for Aj · Bk . Analogously, one can show that if A ∈ L, then A∗ ∈ L. Elliptic ψDO. Let A(x, D) be a ψDO with symbol a(x, ξ) ∼

∞ X

aj (x, ξ).

j=0

By definition, A is elliptic, if a0 (x, ξ) ̸= 0 when ξ ̸= 0. We denote by Id the identity operator. Theorem 1.13 Let A(x, D) be an elliptic ψDOs. Then there exists B(x, D) ∈ L, such that each operator AB − Id and BA − Id has order equal to −∞. We omit the proof (see [2, 41]). In the end of this section, let us discuss some particular class of pseudo-differential operator, so-called fractional Laplacian which will play an important role in the following chapters. It is worth to note that, many models arising in Biology, Medicine, Ecology and Finance lead to nonlinear problems driven by fractional Laplace-type operators. There are several ways of defining this operator in the whole space Rn , at least to my knowledge ten in the literature, most of them are equivalent (see [44]). I will give here that definitions which are more relevant for the content of this book. Let s ∈ (0, 1). For the convenience of the reader (see also section 1.1), we define the fractional Sobolev space H s (Rn ) := W 2,s (Rn ); Z Z   |u(x) − u(y)|2 s n 2 n H (R ) := u ∈ L (R ) dxdy < +∞ . n+2s Rn Rn |x − y| For u, v ∈ H s (Rn ), we set Z Z ⟨u, v⟩H s (Rn ) := u(x)v(x) dx + Rn

Z Rn

Rn

(u(x) − u(y))(v(x) − v(y)) dxdy. |x − y|n+2s

One can easily see that ⟨·, ·⟩H s (Rn ) is an inner product on H s (Rn ) and that the space H s (Rn ) endowed with this inner product is a Hilbert space . It is well-known that the space H s (Rn ), s ∈ (0, 1) can be defined alternatively via the Fourier transform, i.e.,

30

1 Auxiliary Materials s



n

H (R ) :=

Z 2 n u ∈ L (R )

 (1 + |ξ| )|Fx→ξ u(ξ)| | dξ < +∞ , 2s

2

Rn

where by Fx→ξ u(ξ) is defined the Fourier transform of u(x), x ∈ Rn (sometimes we used instead of u ˆ(ξ) by the notation Fx→ξ u(ξ). The first definition of the fractional Laplacian (−∆x )s , s ∈ (0, 1) is expressed via the Fourier transform as a pseudo-differential (nonlocal!) operator with 1 the symbol is equal to |ξ|2s , where |ξ| = (ξ1 + · · · + ξn ) 2 . More precisely, let u ∈ S(Rn ) (for the definition of this space see previous section). Then we have Fx→ξ u ∈ S(Rn ), but the function ξ 7→ |ξ|2s · Fx→ξ u(ξ) ∈ S(Rn ), because |ξ|2s creates singularity at ξ = 0. What is more important that, |ξ|2x Fx→ξ u(ξ) ∈ L1 (Rn ) ∩ L2 (Rn ), so we can use the inverse Fourier transform. Consequently, for any u ∈ S(Rn ) we define the operator (−∆x )s : S(Rn ) → L2 (Rn ) by −1 (−∆x )s u(x) := Fξ→x (|ξ|2x Fx→ξ u(ξ))(x), ∀x ∈ Rn .

This pseudo-differential operator is called the fractional Laplacian of order s. It is easy to see that (−∆x )0 u = u, (−∆x )1 u = −∆x u, for any 0 < s1 , s2 < 1 (−∆x )s1 ◦ (−∆x )s2 = (−∆x )s2 ◦ (−∆x )s1 for any multi-index β ∈ Rn Dxα (−∆x )s u = (−∆x )s Dxα u, so that it follows that (−∆x )s u ∈ C ∞ (Rn ) for any u ∈ S(Rn ). Remark 1.11 Actually, we can extend this and other definitions given below that is, (−∆x )s u to H s (Rn ) using the fact that the space S(Rn ) is dense in H s (Rn ) (see, e.g., [15]). The following definition of the fractional Laplacian is often used on the literature, i.e., let u ∈ S(Rn ). Z (u(x) − u(y))2 dy, ∀x ∈ Rn , (−∆x )s u(x) := C(n, s) lim ε→0 Rn \B(x,ε) |x − y|n+2s where the normalized constant C(n, s) given by C(n, s) :=

s · ns · Γ (s + n2 ) , s ∈ (0, 1) n π 2 Γ (1 − s)

1.5 Linear Fredholm operators

31

and Γ (·) is the Euler function. Remark 1.12 It is not difficult to check that, for u ∈ S(Rn ) and x ∈ Rn and s ∈ (0, 12 ) the integral Z

u(x) − u(y) dy |x − y|n+2s

is absolutely convergent, so that for s ∈ (0, 12 ) (−∆x )s u(x) := C(n, s)

Z Rn

u(x) − u(y) dy. |x − y|n+2s

The following proposition shows the relation between the ”classical” Laplacian and the fractional Laplacian (see [44]). Proposition 1.2 Let u ∈ S(Rn ). Then 1) lim+ (−∆x )s u = u s→0

2) lim− (−∆x )s u = −∆x u s→1

1.5 Linear Fredholm operators Fredholm operators are one of the most important classes of linear maps in mathematics which were introduced around 1900 in the study of integral operators. As we will see below they share many properties with linear maps between finite dimensional spaces. Let X and Y be Banach spaces, either over R or C. By L(X, Y ) we denote the space of bounded linear operators from X to Y , and consider L(X, Y ) as a Banach space with the usual norm. As we already mentioned in Section 1.2, an operator T is called Fredholm if the kernel (nullspace) Ker T = {x ∈ X | T x = 0} has finite dimension and the image (range) of T , R(T ) := {T x | x ∈ X} is of finite codimension in Y , that is, codim R(T ) := dim(Y /R(T )) < ∞. For a Fredholm operator T : X → Y , the numerical Fredholm index of T , ind T is defined by ind T := dim Ker T − codim R(T ). We denote the set of Fredholm operator as well as Fredholm operator of index m by Φ(X, Y ) and Φm (X, Y ) respectively. In the study of global solvability of linear Fredholm operator equation the subset Φ0 (X, Y ) of L(X, Y ) play an important role. It is worth to note that for T ∈ Φ0 (X, Y ) it follow that

32

1 Auxiliary Materials

dim Ker T = dim(Y /R(T )). Hence for T ∈ Φ0 (X, Y ) from global solvability T x = y for any y ∈ Y it follows uniqueness of soluation of T x = y and vice-versa. Obviously, each invertible linear operator T : X → Y belong to Φ0 (X, Y ). The following lemma holds: Lemma 1.6 Let T ∈ L(X, Y ). Then the following assertions are equivalent: 1. The operator T is Fredholm of index 0. 2. There is a compact operator K ∈ L(X, Y ) such that L + K is invertible. 3. There is an isomorphism S ∈ L(Y, X) such that S ◦ L is a compact vector field, that is, it is compact perturbation of identity. Proof. Let T ∈ Φ0 (X, Y ). Since Ker T is finite dimensional, there is a continuous projection P of X onto Ker T . Let W be a closed subspace of Y , W := Y /T X. By definition of T ∈ Φ0 (X, Y ), both Ker T and W are of the same finite dimension, so we may select S ∈ GL(Ker T, W ). We set K := S◦P and observe that K is compact and T + K is an isomorphism. Thus 1. implies 2. To verify that 2. implies 3., choose K : X → Y linear compact operator so that T + K is invertible. We set S := (T + K)−1 . Then it is easy to see that S ◦ T = I − (T + K)−1 ◦ K. Hence 2. implies 3. Finally to show that 3. implies 1. we have to remind Riesz-Schauder Theorem that states that compact linear vector field is Fredholm of index 0. Indeed, let S ◦ T = Id + K. Then T = S −1 (Id + K) and ind T = ind S −1 + ind(Id + K) = 0. ⊔ ⊓ Remark 1.13 An operator S : Y → X satisfy 3. is called a parametrix for T. The subset Φ0 (X, Y ) possesses interesting biological properties. For simplicity of presentation as well as for notational convenience we consider the case when X = Y . We denote by GL(X) the group of linear operator A : X → X. According to Kuiper’s [43], the group GL(X) on a separable Hilbert (infinite dimensional real) space is connected and even is contractible. The analogous result is also true for many Banach spaces [20]. It is not difficult to see that Φ0 (X) inherits connectedness from GL(X). Indeed, if T ∈ Φ0 (X) then t 7−→ tT + (1 − t)S −1 , 0 ≤ t ≤ 1, where S is a parametrix for T , defines a path in Φ0 (X) connecting T to GL(X). Remark 1.14 If X = Rn , then Φ0 (Rn ) = L(Rn ) is contractible and GL(Rn ) has two connected components, which are labeled by the function sgndet. In contrast to a finite dimensional case Φ0 (X) is connected when X is infinite

1.5 Linear Fredholm operators

33

dimensional space but not contractible (see [33]). Moreover, the set Φ0 (X) is open in L(X). Below we recap following well-known properties of Φ0 (X), some of them without proof (see [20],[33] and references therein). Proposition 1.3 Let A, B ∈ Φ0 (X). Then 1) AB ∈ Φ0 (X) and ind AB = ind A + ind B. 2) Let A ∈ Φ0 (X) and K : X → X is a compact linear operator. Then A + K ∈ Φ0 (X) and ind(A + K) = ind A Let X ∗ and Y ∗ be two Banach spaces with X ⊂ X ∗ and Y ⊂ Y ∗ , where X and Y also are Banach spaces. We assume that X ∗ = X ⊕Rp and Y ∗ = Y ⊕Rq . By definition A˜ ∈ L(X ∗ , Y ∗ ) is an extension of A ∈ L(X, Y ) by {p, q} dimensions ˜ = Ax for x ∈ X. if Ax Lemma 1.7 Let the operator A˜ ∈ L(X ∗ , Y ∗ ) be an extension of A ∈ L(X, Y ) by {p, q} dimension. Then ind A = ind A˜ − p + q. Proof. We define A∗ : X ∗ → Y ∗ A∗ (X + Y ) := Ax for x ∈ X, y ∈ Rp . Obviously, A∗ : X ∗ → Y ∗ is an extension of A by {p, q} dimension and for A∗ ∈ L(X ∗ , Y ∗ ) holds dim Ker A∗ = dim Ker A + p and codim A∗ = codim A + q. Hence, ind A = ind A∗ − p + q. On the other hand, dim Ker A∗ = dim Ker A + p,

codim A∗ = codim A + q.

˜ = A∗ x for all x ∈ X (they are different only by a finite-dimensional Since Ax setting), we obtain ind A˜ = ind A∗ . This completes the proof.

⊔ ⊓

The following Lemma 1.8 plays an important role and also holds for a quite large class of a nonlinear operator (see [20],[51],[63] and references therein).

34

1 Auxiliary Materials

Lemma 1.8 Let A ∈ Φ0 (X). Then the operator A is proper in any closed bounded set Ω ⊂ X. Proof. By definition the mapping A : X → X is called proper Ω ⊂ X if for any compact set K ⊂ X the preimage A−1 (K) ∩ Ω is compact on Ω. Let hn ⊂ A−1 (K) ∩ Ω, i.e., there exists un ∈ A−1 (K) ∩ Ω such that Aun = hn . Since hn ∈ K and K is compact, it follows that there exists {hnk } ⊂ K such that hnk → h ∈ K as n → ∞. Hence Aunk = hnk , unk ∈ A−1 (K) ∩ Ω. Recall that A ⊂ Φ0 (X) and consequently there exists a compact operator T : X → X such that, A = M + T , where M ∈ GL(X). Hence, M unk + T unk = hnk implies M unk = hnk −T unk . Since T is compact and unk is bounded it follows that there exists {unkj } ⊂ Ω such that T unkj → ξ as j → ∞. Thus M unkj → h − ξ as j → ∞. Taking into account M ∈ GL(X) we obtain unkj → M −1 (h − ξ) ∈ Ω as j → ∞. This provides Lemma 1.8.

⊔ ⊓

One of the important area of linear (or nonlinear) Fredholm operators is linear (nonlinear) singular integral equation and its solvability. For simplicity of presentation we restrict ourselves to linear singular integral operator (linear pseudo-differential operator order zero) and global solvability of corresponding equation. A prominent example of such example is operator A(λ, u), i.e., A(λ, u) : H s (S 1 ) → H s (S 1 ), s > 1 defined by A(λ, u) := a(τ )u(eiτ ) + b(τ )(Hλ u)(eiτ ), where a(τ ), b(τ ) are given smooth 2π-periodic functions, λ ∈ R and (Hλ u)(eiτ ) =

1 p.v. 2π

2π

Z

u(eiσ ) cot 0

τ −σ dσ + λ. 2

Note that, here the integral is taken in the sense of principla value. It is worth to note that the operator A(λ, u) which is defined above appears in the study of classical linear Riemann-Hilbert problem in unit disk D = {z ∈ C | |z| < 1}. Linear Riemann-Hilbert problem : Let D = {z ∈ C | |z| < 1} and a(τ ), b(τ ), c(τ ) are given real 2π-periodic smooth functions. We are looking for a holomorphic in D and continuous in D function w(z) := u(z) + iv(z)

1.5 Linear Fredholm operators

35

whose real imaginary part on the boundary of D satisfying a(τ )u(eiτ ) + b(τ )v(eiτ ) = c(τ ), where τ is angular coordinate on Γ = ∂D = {eiτ | 0 ≤ τ < 2π}. It is well-known that, real and imaginary part of holomorphic function w(z) = u(z) + iv(z) in D = {z | |z| < 1} on the boundary cannot be arbitrary and given by the Hilbert operator: Z 2π 1 τ −σ iτ v(e ) = p.v. u(eiσ ) cot dσ + λ =: (Hλ u)(eiτ ) 2π 2 0 and u(eiτ ) = −(Hλ v)(eiτ ), where λ is arbitrary real number. Assuming a(τ )+ ib(τ ) ̸= 0 and wind [a(τ )+ ib(τ )] = 0, where 2π

Z wind [a(τ ) + ib(τ )] :=

d arg(a(τ ) + ib(τ )), 0

we can solve linear Riemann-Hilbert problem explicitly, that is, v1 (eiτ ) := arg(b(τ ) − ia(τ )), u1 (eiτ ) := (H0 v1 )(eiτ ), u2 (eiτ ) + iv2 (eiτ ) = eu1 (e

iτ

)+iv1 (eiτ )

.

u2 (eiτ ) + iv2 (eiτ ) is a solution of the homogeneous linear Riemann-Hilbert problem : a(τ )u2 (eiτ ) + b(τ )v2 (eiτ ) = 0. Let v3 (eiτ ) :=

(u2

(eiτ )

c(τ ) , + iv2 (eiτ ))(b(τ ) + ia(τ ))

iτ

u3 (e ) = (H0 v3 )(eiτ ) + λ, u(eiτ ) + iv(eiτ ) = (u3 (eiτ ) + iv3 (eiτ ))(u2 (eiτ ) + iv2 (eiτ )), where λ ∈ R is arbitrary, (H0 u)(eiτ ) := that in this case

1 2π

p.v.

R 2π 0

u(eiσ ) cot τ −σ 2 dσ. Note

w(eiτ ) := u(eiτ ) + iv(eiτ ) = (u3 (eiτ ) + iv3 (eiτ ))(u2 (eiτ ) + iv2 (eiτ )) is a solution of the nonhomogeneous linear Riemann-Hilbert problem, i.e.,

36

1 Auxiliary Materials

a(τ )u(eiτ ) + b(τ )v(eiτ ) = c(τ ). We would like to especially emphasize that, the linear Riemann-Hilbert problem is equivalent to the following operator equation A(λ, u) = 0, A(λ, u) : R × H s (S 1 ) → H s (S 1 ) defined by A(λ, u) := a(τ )u(eiτ ) + b(τ )(Hλ u)(eiτ ). From the explicit formula of solution of the linear Riemann-Hilbert problem follows that, the operator A(λ, u) under the assumptions a(τ ) + ib(τ ) ̸= 0 and wind [a(τ ) + ib(τ )] = 0 is Fredholm operator of index equal to one, as operator R × H s (S 1 ) → H s (S 1 ) for s > 1. Hence, as a corollary we obtain that, A∗ : H s (S 1 ) → H s (S 1 ) defined by (A∗ u)eiτ := a(τ )u(eiτ ) + (H0 )(eiτ ) belong to Φ0 (H s (S 1 ), H s (S 1 )), s > 1. Remark 1.15 The operator H0 which was defined above is called Hilbert transform and very canonical pseudo-differential operator order zero with the symbol a(x, ξ) = i sgn ξ. The Hilbert transform is a bounded linear (nonlocal) operator on the spaces 1) Lp (S 1 ),

1 0 we have Na,b < ∞. In the case of a = 0, we express Z p b dG(s) b b G(p) = G(0) + ds, ds 0 such that R p dG(s) b b b G(p) G(0) ds ds 0 = + . p2 − ibp p(p − ib) p(p − ib)

(1.85)

By virtue of definition (1.79) of the standard Fourier transform, we easily estimate dG(p) 1 b ≤ √ ∥xG(x)∥L1 (R) . dp 2π Hence, we obtain R p b dG(s) 0 ds ds ∥xG(x)∥L1 (R) √ 0, bk ∈ R, bk ̸= 0 is Fredholm, non-selfadjoint, its set of eigenvalues is given by λa,b,k (n) = n2 − ak − ibk n,

n∈Z

(2.43)

einx and its eigenfunctions are the standard Fourier harmonics √ , n ∈ Z. 2π When ak = 0, we will use the similar ideas in the constrained subspace (2.36) instead of H 2 (I). Evidently, the eigenvalues of each operator La,b,k are simple, as distinct from the analogous situation without the drift term, when the eigenvalues corresponding to n ̸= 0 have the multiplicity of two. Let us first suppose that for some v(x) ∈ Hc2 (I, RN ) there exist two solutions u(1),(2) (x) ∈ Hc2 (I, RN ) of system (2.31) with Ω = I. Then the vector function w(x) := u(1) (x) − u(2) (x) ∈ Hc2 (I, RN ) will be a solution of the homogeneous system of equations −

d2 w k dwk − bk − ak wk = 0, 2 dx dx

1 ≤ k ≤ N.

But the operator La,b,k : H 2 (I) → L2 (I), ak > 0 discussed above does not possess nontrivial zero modes. Hence, w(x) vanishes in I. We choose arbitrarily v(x) ∈ Hc2 (I, RN ) and apply the Fourier transform (1.97) to system (2.31) considered on the interval I. This yields

2.4 Non-Fredholm systems in a finite interval with normal diffusion and drift

√

uk,n =

83

√ Gk,n fk,n n2 Gk,n fk,n 2 , n u = 2π , 1 ≤ k ≤ N n ∈ Z, k,n n2 − ak − ibk n n2 − ak − ibk n (2.44) := Fk (v(x), x)n . Thus, we obtain √ √ |uk,n | ≤ 2πNa,b,k |fk,n |, |n2 uk,n | ≤ 2πNa,b,k |fk,n |,

2π

with fk,n

where Na,b,k < ∞ under the given auxiliary assumptions by virtue of Lemma 2.3 of Appendix. Hence, " ∞ # N ∞ X X X 2 2 2 2 ∥u∥Hc2 (I,RN ) = |uk,n | + |n uk,n | n=−∞

k=1

2 ≤ 4πNa,b

N X

n=−∞

∥Fk (v(x), x)∥2L2 (I) < ∞

k=1

due to (2.29) of Assumption 2.2 for |v(x)|RN ∈ L2 (I). Thus, for any v(x) ∈ Hc2 (I, RN ) there exists a unique u(x) ∈ Hc2 (I, RN ) solving system (2.31) with its Fourier transform given by (1.97) and the map τa,b : Hc2 (I, RN ) → Hc2 (I, RN ) is well defined. Let us consider arbitrary v (1),(2) (x) ∈ Hc2 (I, RN ) with their images under the map discussed above u(1),(2) = τa,b v (1),(2) ∈ Hc2 (I, RN ). By applying the Fourier transform (1.97), we easily derive (1)

uk,n =

√

(1)

2π

Gk,n fk,n

(2)

n2 − ak − ibk n

, uk,n =

√

(2)

2π

Gk,n fk,n n2 − ak − ibk n

, 1 ≤ k ≤ N, n ∈ Z,

(j)

with fk,n := Fk (v (j) (x), x)n , j = 1, 2. Hence, (1)

(2)

|uk,n − uk,n | ≤

√

(1)

(2)

2πNa,b,k |fk,n − fk,n |, √ (1) (2) (1) (2) |n2 (uk,n − uk,n )| ≤ 2πNa,b,k |fk,n − fk,n |. Therefore, ∥u

(1)

−

u(2) ∥2Hc2 (I,RN )

=

N X k=1

"

∞ X

(1) |uk,n

n=−∞

2 ≤ 4πNa,b

N X

−

(2) uk,n |2

+

∞ X

# |n

2

(1) (uk,n

−

(2) uk,n )|2

n=−∞

∥Fk (v (1) (x), x) − Fk (v (2) (x), x)∥2L2 (I) .

k=1 (1),(2)

Clearly, vk (x) ∈ H 2 (I) ⊂ L∞ (I), 1 ≤ k ≤ N due to the Sobolev embedding. By virtue of (2.4) we easily arrive at √ ∥τa,b v (1) − τa,b v (2) ∥Hc2 (I,RN ) ≤ 2 πNa,b L∥v (1) − v (2) ∥Hc2 (I,RN ) ,

84

2 Solvability in the sense of sequences: non-Fredholm operators

with the constant in the right side of this estimate less than one as assumed. Thus, the fixed point theorem yields the existence and uniqueness of a vector function v (a,b) ∈ Hc2 (I, RN ), satisfying τa,b v (a,b) = v (a,b) , which is the only solution of system (2.27) in Hc2 (I, RN ). Suppose v (a,b) (x) = 0 identically in I. This gives us the contradiction to the assumption that Gk,n Fk (0, x)n ̸= 0 for some 1 ≤ k ≤ N and a certain n ∈ Z. ⊔ ⊓ We proceed to establishing the final main statement of this section. Proof of Theorem 2.8. Apparently, the limiting kernels Gk (x), 1 ≤ k ≤ N are also periodic on the interval I (see the argument of Lemma 2.4 of Appendix). Each system (2.37) has a unique solution u(m) (x), m ∈ N belonging to Hc2 (I, RN ) by virtue of Theorem 2.7 above. The limiting system of equations (2.27) admits a unique solution u(x), which belongs to Hc2 (I, RN ) by means of Lemma 2.4 of Appendix along with Theorem 2.7. Let us apply Fourier transform (1.97) to both sides of systems (2.27) and (2.37). This yields uk,n =

√

Gk,n φk,n 2π 2 , n − ak − ibk n

(m) uk,n

=

√

(m)

2π

Gk,m,n φk,n

n2 − ak − ibk n

,

(2.45)

(m)

where 1 ≤ k ≤ N, n ∈ Z, m ∈ N with φk,n and φk,n denoting the Fourier images of Fk (u(x), x) and Fk (u(m) (x), x) respectively under transform (1.97). We easily obtain the upper bound

√ Gk,m,n Gk,n

(m) |uk,n − uk,n | ≤ 2π 2 − 2

|φ |+

n − ak − ibk n n − ak − ibk n ∞ k,n l



√ Gk,m,n

(m) + 2π 2 − φk,n |.

|φ

n − ak − ibk n ∞ k,n l

Hence, (m) ∥uk

− uk ∥L2 (I) ≤

√



Gk,m,n Gk,n

2π 2 − 2

∥Fk (u(x), x)∥L2 (I) +

n − ak − ibk n n − ak − ibk n ∞ l



√ Gk,m,n

+ 2π 2

∥F (u(m) (x), x) − Fk (u(x), x)∥L2 (I) .

n − ak − ibk n ∞ k l

By virtue of bound (2.30) of Assumption 2.2, we arrive at v uN uX t ∥Fk (u(m) (x), x) − Fk (u(x), x)∥2L2 (I) ≤ L∥u(m) (x) − u(x)∥L2 (I,RN ) . k=1

(2.46)

2.4 Non-Fredholm systems in a finite interval with normal diffusion and drift

85

(m)

Note that uk (x), uk (x) ∈ H 2 (I) ⊂ L∞ (I) via the Sobolev embedding. Evidently, ∥u(m) (x) − u(x)∥2L2 (I,RN )

2 N

X Gk,m,n Gk,n

≤4π − 2

2

∥F (u(x), x)∥2L2 (I)

n − ak − ibk n n − ak − ibk n ∞ k k=1 l " #2 (m)

L2 ∥u(m) (x) − u(x)∥2L2 (I,RN ) .

+ 4π Na,b Thus, we derive

∥u(m) (x) − u(x)∥2L2 (I,RN )

2 N

4π X Gk,m,n Gk,n

≤ − 2

2

∥F (u(x), x)∥2L2 (I) .

n − ak − ibk n n − ak − ibk n ∞ k ε(2 − ε) k=1

l

Clearly, Fk (u(x), x) ∈ L2 (I), 1 ≤ k ≤ N for u(x) ∈ Hc2 (I, RN ) by means of bound (2.29) of Assumption 2.2. Lemma 2.4 below implies that u(m) (x) → u(x),

m→∞

(2.47)

in L2 (I, RN ). Obviously, (m) |n2 uk,n

2

− n uk,n | ≤

√



n2 G

n2 Gk,n

k,m,n 2π 2 − 2

|φ |+

n − ak − ibk n n − ak − ibk n ∞ k,n l



√

n2 Gk,m,n (m) + 2π 2 − φk,n |.

|φ

n − ak − ibk n ∞ k,n l

By means of (2.46), we obtain

d2 u(m) d2 u

k k −

dx2 dx2

L2 (I)

≤

√



n2 G

n2 Gk,n

k,m,n 2π 2 − 2

∥F (u(x), x)∥L2 (I) +

n − ak − ibk n n − ak − ibk n ∞ k l



√

n2 Gk,m,n + 2π 2

L∥u(m) − u∥L2 (I,RN ) .

n − ak − ibk n ∞ l

d2 u(m) d2 u → as m → ∞ dx2 dx2 2 N (m) 2 N in L (I, R ). Hence, u (x) → u(x) in the Hc (I, R ) norm as m → ∞. Lemma 2.4 of Appendix along with (2.47) give us

Suppose that u(m) (x) = 0 identically in the interval I for some m ∈ N. This gives us a contradiction to the assumption that Gk,m,n Fk (0, x)n ̸= 0 for

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2 Solvability in the sense of sequences: non-Fredholm operators

some 1 ≤ k ≤ N and a certain n ∈ Z. The analogous argument holds for the solution u(x) of the limiting system of equations (2.27). ⊔ ⊓

Appendix In the proof of the theorems of sections 2.3 and 2.4 we used several lemmas referring to Appendix. For the convenience of the reader we will present the proofs of this lemmas . Although the quantities at first glance looks like similar to the ones in section 1.6, as we will see a proof of lemmas are required additional techniques and ideas. Let Gk (x) be a function, Gk (x) : R → R, for which we denote its standard Fourier transform using the hat symbol as Z ∞ ck (p) := √1 G Gk (x)e−ipx dx, p ∈ R, (2.48) 2π −∞ such that

ck (p)∥L∞ (R) ≤ √1 ∥Gk ∥L1 (R) ∥G 2π

(2.49)

Z ∞ 1 ck (q)eiqx dq, x ∈ R. For the technical purposes we and Gk (x) = √ G 2π −∞ define the auxiliary quantities 

Na,b,k := max



p2 G ck (p) ck (p) G

, , (2.50)

p2 − ak − ibk p L∞ (R) p2 − ak − ibk p L∞ (R)

with ak ≥ 0, bk ∈ R, bk ̸= 0, 1 ≤ k ≤ N, N ≥ 2. Let N0,b,k stand for (2.50) when ak vanishes. Under the conditions of Lemma 2.1 below, quantities (2.50) will be finite. This will enable us to define Na,b := max Na,b,k < ∞. 1≤k≤N

(2.51)

The technical lemmas below are the adaptations of the ones established in section 1.6 (see also [28]) for the studies of the single integro-differential equation with drift , analogous to system (2.27). We provide them for the convenience of the readers. Lemma 2.1 Let N ≥ 2, 1 ≤ k ≤ N , bk ∈ R, bk ̸= 0 and Gk (x) : R → R, Gk (x) ∈ L1 (R) and 1 ≤ l ≤ N − 1. a) Let ak > 0 for 1 ≤ k ≤ l. Then Na,b,k < ∞.

2.4 Non-Fredholm systems in a finite interval with normal diffusion and drift

87

b) Let ak = 0 for l + 1 ≤ k ≤ N and in addition xGk (x) ∈ L1 (R). Then N0,b,k < ∞ if and only if (Gk (x), 1)L2 (R) = 0

(2.52)

holds. Proof. First of all, let us observe that in both cases a) and b) of our lemma ck (p) ck (p) G p2 G the boundedness of 2 yields the boundedness of 2 . p − ak − ibk p p − ak − ibk p ck (p) p2 G Indeed, we can write 2 as the following sum p − ak − ibk p ck (p) + ak G

p2

ck (p) ck (p) G pG + ibk 2 . − ak − ibk p p − ak − ibk p

(2.53)

Evidently, the first term in (2.53) is bounded by means of (2.49) since Gk (x) ∈ L1 (R) as assumed. The third term in (2.53) can be estimated from above in the absolute value via (2.49) as

p

ck (p)| |bk ||p||G 1 ≤ √ ∥Gk (x)∥L1 (R) < ∞. 2 2 2 2 2π (p − ak ) + bk p

ck (p) ck (p) G p2 G ∈ L∞ (R) implies that 2 ∈ L∞ (R). To ob− ak − ibk p p − ak − ibk p tain the result of the part a) of the lemma, we need to estimate Thus,

p2

p

ck (p)| |G . (p2 − ak )2 + b2k p2

(2.54)

Apparently, the numerator of (2.54) can be bounded from above by virtue of (2.49) and the denominator in (2.54) can be easily estimated below by a finite, positive constant, such that ck (p) G 2 ≤ C∥Gk (x)∥L1 (R) < ∞. p − ak − ibk p Here and further down C will denote a finite, positive constant. This yields that under the given conditions, when ak > 0 we have Na,b,k < ∞. In the case of ak = 0, we express p

Z ck (p) = G ck (0) + G 0

such that

ck (s) dG ds, ds

88

2 Solvability in the sense of sequences: non-Fredholm operators

R p dGck (s) ck (p) ck (0) ds G G = + 0 ds . 2 p − ibk p p(p − ibk ) p(p − ibk )

(2.55)

By means of definition (1.79) of the standard Fourier transform, we easily estimate dG 1 ck (p) ≤ √ ∥xGk (x)∥L1 (R) . dp 2π Hence, we derive R p c dGk (s) 0 ds ds ∥xGk (x)∥L1 (R) √ 0 for 1 ≤ k ≤ l. b) Let ak = 0 for l + 1 ≤ k ≤ N and additionally xGk,m (x) ∈ L1 (R), such that xGk,m (x) → xGk (x) in L1 (R) as m → ∞ and (Gk,m (x), 1)L2 (R) = 0, We also assume that

√ (m) 2 πNa,b L ≤ 1 − ε

m ∈ N.

(2.58)

(2.59)

holds for all m ∈ N as well with some fixed 0 < ε < 1. Then, for all 1 ≤ k ≤ N , we have

2.4 Non-Fredholm systems in a finite interval with normal diffusion and drift

89

ck (p) [ G G k,m (p) → 2 , − ak − ibk p p − ak − ibk p

m → ∞,

(2.60)

ck (p) [ p2 G p2 G k,m (p) → , p2 − ak − ibk p p2 − ak − ibk p

m→∞

(2.61)

p2

in L∞ (R), such that

G

[

k,m (p)

2

p − ak − ibk p

p2 G

[

k,m (p)

2

p − ak − ibk p

L∞ (R)



ck (p) G

→ 2

p − ak − ibk p

L∞ (R)

L∞ (R)



p2 G ck (p)

→ 2

p − ak − ibk p

L∞ (R)

Moreover,

,

m → ∞,

(2.62)

,

m → ∞.

(2.63)

√ 2 πNa,b L ≤ 1 − ε.

(2.64)

Proof. Obviously, for all 1 ≤ k ≤ N , we have 1 c [ ∥G ∥Gk,m (x) − Gk (x)∥L1 (R) → 0, k,m (p) − Gk (p)∥L∞ (R) ≤ √ 2π as assumed. Let us prove that (2.60) c [ p2 [ G k,m (p) − Gk (p)] can be written as the sum p2 − ak − ibk p " c \ [G k,m (p) − Gk (p)] + ak

implies

ck (p) \ G G k,m (p) − 2 p2 − ak − ibk p p − ak − ibk p

# + ibk p

m → ∞,

(2.61).

(2.65) Indeed,

c \ [G k,m (p) − Gk (p)] . p2 − ak − ibk p (2.66)

The first term in (2.66) tends to zero as m → ∞ in the L∞ (R) norm by virtue of (2.65). The third term in (2.66) can be bounded from above in the absolute value as c [ |p||G k,m (p) − Gk (p)| c [ |bk | p ≤ ∥G k,m (p) − Gk (p)∥L∞ (R) , (p2 − ak )2 + b2k p2 hence it converges to zero as m → ∞ in the L∞ (R) norm via (2.65) as well. Therefore, the statement of (2.60) yields (2.61). Evidently, (2.62) and (2.63) will follow from the statements of (2.60) and (2.61) respectively by virtue of the triangle inequality. First we prove (2.60) in the case a) when ak > 0. Then one needs to estimate ck (p)| [ (p) − G |G p k,m . (p2 − ak )2 + b2k p2

(2.67)

90

2 Solvability in the sense of sequences: non-Fredholm operators

Apparently, the denominator in fraction (2.67) can be bounded from below by a positive constant and the numerator in (2.67) can be estimated from above by means of (2.65). This gives us the result of (2.60) when the constant ak is positive. Then let us turn our attention to establishing (2.60) in the case b) when ak = 0. Hence, we have orthogonality relations (2.58). Let us prove that the analogous statement will hold in the limit. Indeed, |(Gk (x), 1)L2 (R) | = |(Gk (x)−Gk,m (x), 1)L2 (R) | ≤ ∥Gk,m (x)−Gk (x)∥L1 (R) → 0 as m → ∞ due to the one of our assumptions. Therefore, (Gk (x), 1)L2 (R) = 0,

1 ≤ k ≤ N.

(2.68)

We express p

Z ck (p) = G ck (0) + G 0

ck (s) dG ds, ds

p

Z [ [ G k,m (p) = Gk,m (0) + 0

[ dG k,m (s) ds, ds

where 1 ≤ k ≤ N, m ∈ N. By means of (2.58) and (2.68), we obtain ck (0) = 0, G

[ G k,m (0) = 0,

1 ≤ k ≤ N,

m ∈ N.

Thus, i R p h dG \ ck (s) dG k,m (s) G − ds c [ ds ds Gk (p) 0 k,m (p) − 2 2 = . p − ibk p p − ibk p p(p − ibk )

(2.69)

From the definition of the standard Fourier transform (1.79) we easily derive dG ck (p) dG 1 [ k,m (p) − ≤ √ ∥xGk,m (x) − xGk (x)∥L1 (R) . dp dp 2π This enables us to obtain the upper bound on the right side of (2.69) by ∥xGk,m (x) − xGk (x)∥L1 (R) √ → 0, 2π|bk |

m → ∞,

as assumed, which proves (2.60) when ak = 0. Apparently, under our conditions Na,b,k < ∞,

(m)

Na,b,k < ∞,

m ∈ N,

1 ≤ k ≤ N, ak ≥ 0, bk ∈ R, bk ̸= 0

by virtue of the result of Lemma 2.1 above. We have bounds (2.59). A trivial limiting argument using (2.62) and (2.63) yields (2.64). ⊔ ⊓

2.4 Non-Fredholm systems in a finite interval with normal diffusion and drift

91

Let the function Gk (x) : I → R, Gk (0) = Gk (2π) and its Fourier transform on the finite interval is defined as Z 2π e−inx Gk,n := Gk (x) √ dx, n ∈ Z (2.70) 2π 0 and Gk (x) =

∞ X

einx Gk,n √ . Evidently, we have the estimate from above 2π n=−∞ 1 ∥Gk,n ∥l∞ ≤ √ ∥Gk (x)∥L1 (I) . 2π

(2.71)

Similarly to the whole real line case, we will use

) (



Gk,n n2 Gk,n



Na,b,k := max 2

, 2

n − ak − ibk n ∞ n − ak − ibk n ∞ l

(2.72)

l

for ak ≥ 0, bk ∈ R, bk ̸= 0, 1 ≤ k ≤ N , N ≥ 2. Let N0,b,k stand for (2.72) when ak = 0. Under the conditions of Lemma 2.3 below, expressions (2.72) will be finite. This will allow us to define Na,b := max Na,b,k < ∞. 1≤k≤N

(2.73)

We have the following trivial statement. Lemma 2.3 Let N ≥ 2, 1 ≤ k ≤ N, bk ∈ R, bk ̸= 0 and Gk (x) : I → R, Gk (x) ∈ L∞ (I), Gk (0) = Gk (2π) and 1 ≤ l ≤ N − 1. a) Let ak > 0 for 1 ≤ k ≤ l. Then Na,b,k < ∞. b) If ak = 0 for l + 1 ≤ k ≤ N . Then N0,b,k < ∞ if and only if (Gk (x), 1)L2 (I) = 0.

(2.74)

Proof. Clearly, in both cases a) and b) of our lemma the boundedGk,n n2 Gk,n ness of 2 implies the boundedness of 2 . Indeed, n − ak − ibk n n − ak − ibk n n2 Gk,n can be easily expressed as 2 n − ak − ibk n Gk,n + a

Gk,n nGk,n + ibk 2 . n2 − ak − ibk n n − ak − ibk n

(2.75)

Obviously, the first term in (2.75) can be trivially estimated from above via (2.71) for Gk (x) ∈ L∞ (I) ⊂ L1 (I). The third term in (2.75) can be bounded from above by means of (2.71) as well, namely

92

2 Solvability in the sense of sequences: non-Fredholm operators

|n||Gk,n | 1 |bk | p ≤ |Gk,n | ≤ √ ∥Gk (x)∥L1 (I) < ∞. 2 2 2 2 2π (n − ak ) + bk n Gk,n n2 Gk,n ∞ ∈ l yields ∈ l∞ . To establish the n2 − ak − ibk n n2 − ak − ibk n statement of the part a) of the lemma, we need to consider Hence,

p

(n2

|Gk,n | . − ak )2 + b2k n2

(2.76)

Apparently, the denominator in (2.76) can be estimated from below by a positive constant and the numerator in (2.76) can be trivially treated by virtue of (2.71). Thus, Na,b,k < ∞ when ak > 0. To prove the result of the part b), we observe that G k,n (2.77) n(n − ibk ) is bounded if and only if Gk,0 = 0, which is equivalent to orthogonality relation (2.74). In this case (2.77) can be easily bounded from above by 1 ∥Gk (x)∥L1 (I) 1 ∥Gk (x)∥L1 (I) √ p ≤√ 0 for 1 ≤ k ≤ l. b) Let ak = 0 for l + 1 ≤ k ≤ N and additionally (Gk,m (x), 1)L2 (I) = 0,

m ∈ N.

(2.80)

2.4 Non-Fredholm systems in a finite interval with normal diffusion and drift

Let us also assume that

√ (m) 2 πNa,b L ≤ 1 − ε

93

(2.81)

holds for all m ∈ N as well with a certain fixed 0 < ε < 1. Then, for all 1 ≤ k ≤ N , we have Gk,m,n Gk,n → 2 , n2 − ak − ibk n n − ak − ibk n

m → ∞,

(2.82)

n2 Gk,n n2 Gk,m,n → 2 , − ak − ibk n n − ak − ibk n

m→∞

(2.83)

n2

in l∞ , such that





Gk,m,n Gk,n



2

→ 2

,

n − ak − ibk n ∞

n − ak − ibk n ∞ l

m → ∞,

(2.84)

l



n2 G

n2 Gk,n



k,m,n

2

→ 2

, m → ∞.

n − ak − ibk n ∞

n − ak − ibk n ∞ l

Moreover,

(2.85)

l

√ 2 πNa,b L ≤ 1 − ε.

(2.86)

Proof. Evidently, under the given conditions, the limiting kernel functions Gk (x), 1 ≤ k ≤ N are periodic as well. Indeed, we derive |Gk (0) − Gk (2π)| ≤ |Gk (0) − Gk,m (0)| + |Gk,m (2π) − Gk (2π)| ≤ 2∥Gk,m (x) − Gk (x)∥L∞ (I) → 0 when m → ∞ due to the one of our assumptions. Hence, Gk (0) = Gk (2π) with 1 ≤ k ≤ N holds. Apparently, 1 ∥Gk,m,n − Gk,n ∥l∞ ≤ √ ∥Gk,m − Gk ∥L1 (I) 2π √ ≤ 2π∥Gk,m − Gk ∥L∞ (I) → 0, m → ∞

(2.87)

as assumed. We observe that the statements of (2.82) and (2.83) yield (2.84) and (2.85) respectively by means of the triangle inequality. Let us show that n2 [Gk,m,n − Gk,n ] (2.82) yields (2.83). For this purpose, we write as n2 − ak − ibk n [Gk,m,n − Gk,n ] + ak

Gk,m,n − Gk,n n[Gk,m,n − Gk,n ] + ibk 2 . n2 − ak − ibk n n − ak − ibk n

(2.88)

The first term in (2.88) converges to zero in the l∞ norm as m → ∞ due to estimate (2.87). The third term in (2.88) can be estimated from above in the absolute value as

94

2 Solvability in the sense of sequences: non-Fredholm operators

|bk ||n||Gk,m,n − Gk,n | p ≤ ∥Gk,m,n − Gk,n ∥l∞ . (n2 − ak )2 + b2k n2 Therefore, it tends to zero as m → ∞ in the l∞ norm via (2.87) as well. This proves that (2.82) implies (2.83). First we prove (2.82) in the case a) when ak > 0. Let us consider the expression |G − Gk,n | p k,m,n . (2.89) 2 (n − ak )2 + b2k n2 Evidently, the denominator of (2.89) can be estimated from below by a positive constant and the numerator bounded from above by virtue of (2.87). This implies (2.82) for ak > 0. Then we turn our attention to establishing (2.82) in the case b) when ak = 0. By virtue of the one of our assumptions, we have orthogonality conditions (2.80). Let us show that the analogous relations hold in the limit. Indeed, |(Gk (x), 1)L2 (I) | = |(Gk (x) − Gk,m (x), 1)L2 (I) | ≤ 2π∥Gk,m (x) − Gk (x)∥L∞ (I) → 0,

m→∞

as assumed. Hence, (Gk (x), 1)L2 (I) = 0,

1 ≤ k ≤ N,

which is equivalent to Gk,0 = 0. Clearly, Gk,m,0 = 0, 1 ≤ k ≤ N, m ∈ N due to orthogonality relations (2.80). Then by virtue of (2.87), we arrive at √ G 2π∥Gk,m (x) − Gk (x)∥L∞ (I) k,m,n − Gk,n . ≤ n(n − ibk ) |bk | The norm in the right side of this estimate converges to zero as m → ∞. Thus, (2.82) holds when ak = 0 as well. Apparently, under the given conditions (m)

Na,b,k < ∞, Na,b,k < ∞, m ∈ N, 1 ≤ k ≤ N, ak ≥ 0, bk ∈ R, bk ̸= 0 by means of the result of Lemma 2.3 above. We have inequality (2.81). A trivial limiting argument using (2.84) and (2.85) yields (2.86). ⊔ ⊓

Chapter 3

Solvability of some integro-differential equations with drift and superdiffusion

In this chapter we establish the existence in the sense of sequences of solutions for some integro-differential type equations containing the drift term and the square root of the one dimensional negative Laplacian (super-diffusion), on the whole real line or on a finite interval with periodic boundary conditions in the corresponding H 2 spaces. The argument relies on the fixed point technique when the elliptic equations involve first order differential operators with and without Fredholm property. It is proven that, under the reasonable technical assumptions, the convergence in L1 of the integral kernels implies the existence and convergence in H 2 of solutions. This chapter consists of four sections 3.1-3.4. Sections 3.1 and 3.2 deals with the scalar equations in the whole real line and a finite interval respectively. In sections 3.3 and 3.4 we consider analogous problem for a system of equations study of which conjugated with additional difficulties and ideas than in a scalar case. More precisely, in sections 3.1 and 3.2 we consider a class of nonlinear problems, for which the Fredholm property may not be satisfied: r Z d2 du − − 2u + b + au + G(x − y)F (u(y), y)dy = 0, x ∈ Ω, (3.1) dx dx Ω where Ω is a domain on the real line, a ≥ 0, b ∈ R, b ̸= 0 are the constants. r d2 The operator − 2 is actively used, for instance in the studies of the sudx perdiffusion problems (see e.g. [60] and the references therein). Superdiffusion can be described as a random process of particle motion characterized by the probability density distribution of the jump length. The moments of this density distribution are finite in the case of normal diffusion , but this is not the case for superdiffusion. Asymptotic behavior at the infinity of the probability density function determines the value of the power of the negative Laplacian (see [49]). For the simplicity of presentation we restrict ourselves to the one dimensional case. The solvability of integer-differential equation with drift

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Efendiev, Linear and Nonlinear Non-Fredholm Operators, https://doi.org/10.1007/978-981-19-9880-5_3

95

96

3 Solvability of some integro-differential equations with drift and superdiffusion

and influx/efflux terms on the real line was studied recently in [61]. However, the method used there worked for the powers of the fractional negative Laplacian 0 < s < 14 , but it did not cover the situation when s = 12 . Formulation of the results The nonlinear part of problem (3.1) will satisfy the following regularity conditions. Assumption 3.1 Function F (u, x) : R × Ω → R is satisfying the Carath´eodory condition , so that |F (u, x)| ≤ k|u| + h(x)

f or

u ∈ R, x ∈ Ω

(3.2)

with a constant k > 0 and h(x) : Ω → R+ , h(x) ∈ L2 (Ω). Furthermore, it is a Lipschitz continuous function, so that |F (u1 , x) − F (u2 , x)| ≤ l|u1 − u2 |

f or any u1,2 ∈ R, x ∈ Ω

(3.3)

with a constant l > 0. In order to demonstrate the solvabity of equation (3.1), we will use the auxiliary problem r Z d2 du − 2u − b − au = G(x − y)F (v(y), y)dy. (3.4) dx dx Ω Let us introduce Z

f1 (x)f¯2 (x)dx,

(f1 (x), f2 (x))L2 (Ω) :=

(3.5)

Ω

with a slight abuse of notations when these functions are not square integrable, like for example those involved in orthogonality condition (3.31) further down. Indeed, if f1 (x) ∈ L1 (Ω) and f2 (x) is bounded, then the integral in the right side of (3.5) makes sense. In section 3.1 we discuss the situation on the whole real line, Ω = R, such that the appropriate Sobolev space is equipped with the norm

d2 u 2

2 2 ∥u∥H 2 (R) := ∥u∥L2 (R) + 2 . (3.6)

dx 2 L (R)

The main issue for our equation above is that in the absence of the drift term we were dealing with the self-adjoint, non Fredholm operator r d2 − 2 − a : H 1 (R) → L2 (R), a ≥ 0, dx

3 Solvability of some integro-differential equations with drift and superdiffusion

97

which was the obstacle to solve our problem (see [60]). However, the situation is different when the constant in the drift term b ̸= 0. The operator r d2 d La,b := − 2 − b − a : H 1 (R) → L2 (R), (3.7) dx dx where a ≥ 0 and b ∈ R, b ̸= 0 involved in the left side of equation (3.4) is non-selfadjoint. By virtue of the standard Fourier transform, it can be easily obtained that the essential spectrum of such operator La,b is given by λa,b (p) = |p| − a − ibp,

p ∈ R.

Evidently, when the constant a > 0 the operator La,b is satisfies the Fredholm property, because its essential spectrum does not contain the origin. But when a vanishes, our operator La,b does not satisfy the Fredholm property since the origin belongs to its essential spectrum . Let us demonstrate that under the reasonable technical assumptions equation (3.4) defines a map Ta,b : H 2 (R) → H 2 (R) with the constants a ≥ 0, b ∈ R, b ̸= 0, which is a strict contraction. Theorem 3.1 Let Ω = R, G(x) : R → R, G(x) ∈ W 1,1 (R) and Assumption 3.1 hold. √ I) When a > 0, b ∈ R, b ̸= 0 we assume that 2 πNa,b l < 1, where Na,b is introduced in (3.30). Then the map v 7→ Ta,b v = u on H 2 (R) defined by problem (3.4) has a unique fixed point va,b , which is the only solution of equation (3.1) in H 2 (R). II)When a = 0, b ∈ R, b ̸= 0 we√assume that xG(x) ∈ L1 (R), orthogonality condition (3.31) holds and 2 πN0,b l < 1. Then the map T0,b v = u on H 2 (R) defined by problem (3.4) possesses a unique fixed point v0,b , which is the only solution of equation (3.1) in H 2 (R). In both cases I and II the fixed point va,b , a ≥ 0, b ∈ R, b ̸= 0 is nontrivial provided the intersection of supports of the Fourier transforms of functions b is a set of nonzero Lebesgue measure in R. supp F\ (0, x) ∩ supp G Let us note that in the situation when a > 0 of the theorem above, as distinct from part I) of Theorem 1 of [60] describing the equation without the drift term, the orthogonality relations are not needed. We introduce the sequence of approximate equations related to problem (3.1) on the whole real line, namely r Z ∞ d2 dum − − 2 um + b + aum + Gm (x − y)F (um (y), y)dy = 0 (3.8) dx dx −∞

98

3 Solvability of some integro-differential equations with drift and superdiffusion

with the constants a ≥ 0, b ∈ R, b ̸= 0 and m ∈ N. The sequence of kernels {Gm (x)}∞ m=1 converges to G(x) as m → ∞ in the appropriate function spaces discussed further down. Let us demonstrate that, under the certain technical assumptions, each of equations (3.8) possesses a unique solution um (x) ∈ H 2 (R), the limiting problem (3.1) admits a unique solution u(x) ∈ H 2 (R), and um (x) → u(x) in H 2 (R) as m → ∞. As we already mentioned in chapter 2 this is called existence of solutions in the sense of sequences. In such case, the solvability relations can be formulated for the iterated kernels Gm . They imply the convergence of the kernels in terms of the Fourier transforms (see Appendix) and, as a consequence, the convergence of the solutions (Theorems 3.2, 3.4). Our second main proposition is as follows. Theorem 3.2 Let Ω = R, m ∈ N, Gm (x) : R → R, Gm (x) ∈ W 1,1 (R) are such that Gm (x) → G(x) in W 1,1 (R) as m → ∞. Let Assumption 3.1 hold. I) If a > 0, b ∈ R, b ̸= 0, assume that √ 2 πNa,b,m l ≤ 1 − ε for all m ∈ N with a certain fixed 0 < ε < 1 and Na,b,m defined in (3.34). Then each problem (3.8) has a unique solution um (x) ∈ H 2 (R), and limiting equation (3.1) admits a unique solution u(x) ∈ H 2 (R). II)If a = 0, b ∈ R, b ̸= 0, assume that xGm (x) ∈ L1 (R), xGm (x) → xG(x) in L1 (R) as m → ∞, orthogonality relation (3.36) is valid and √ 2 πN0,b,m l ≤ 1 − ε for all m ∈ N with some fixed 0 < ε < 1. Then each problem (3.8) admits a unique solution um (x) ∈ H 2 (R), and limiting equation (3.1) possesses a unique solution u(x) ∈ H 2 (R). In both cases I and II, we have um (x) → u(x) in H 2 (R) as m → ∞. The unique solution um (x) of each problem (3.8) does not vanish identically on the whole real line provided that the intersection of supports of the Fourier b m is a set of nonzero Lebesgue meaimages of functions supp F\ (0, x) ∩ supp G sure in R. Similarly, the unique solution u(x) of limiting equation (3.1) is b is a set of nonzero Lebesgue measure in nontrivial if supp F\ (0, x) ∩ supp G R. In section 3.2 we consider the analogous problem on the finite interval Ω = I := [0, 2π] with periodic boundary conditions, such that the appropriate functional space is given by H 2 (I) = {u(x) : I → R | u(x), u′′ (x) ∈ L2 (I), u(0) = u(2π), u′ (0) = u′ (2π)}.

3 Solvability of some integro-differential equations with drift and superdiffusion

99

For the technical purposes, we will use the following auxiliary constrained subspace H02 (I) = {u(x) ∈ H 2 (I) | (u(x), 1)L2 (I) = 0}, (3.9) which is a Hilbert space as well (see e.g. Chapter 2.1 of [38]). Similarly, H01 (I) = {u(x) ∈ H 1 (I) | (u(x), 1)L2 (I) = 0}. Let us demonstrate that equation (3.4) in this case defines a map τa,b on the above mentioned spaces with the constants a ≥ 0, b ∈ R, b ̸= 0. This map will be a strict contraction under the stated technical assumptions. Theorem 3.3 Let Ω = I, G(x) : I → R, G(x) ∈ C(I),

dG(x) ∈ L1 (I), dx

G(0) = G(2π), F (u, 0) = F (u, 2π) for u ∈ R and Assumption 3.1 hold. √ I) If a > 0, b ∈ R, b ̸= 0 we assume that 2 πNa,b l < 1 with Na,b given by (3.51). Then the map v 7→ τa,b v = u on H 2 (I) defined by problem (3.4) possesses a unique fixed point va,b , the only solution of equation (3.1) in H 2 (I). II)If a = 0, b ∈ √ R, b ̸= 0, let us assume that orthogonality condition (3.52) holds and 2 πN0,b l < 1. Then the map τ0,b v = u on H02 (I) defined by problem (3.4) has a unique fixed point v0,b , the only solution of equation (3.1) in H02 (I). In both cases I and II the fixed point va,b , a ≥ 0, b ∈ R, b ̸= 0 does vanish identically on the interval I provided the Fourier coefficients Gn F (0, x)n ̸= 0 for a certain n ∈ Z. 2 Remark 3.1 We use the constrained rsubspace H0 (I) in case II) of our theod2 d rem, so that the Fredholm operator − 2 − b : H01 (I) → L2 (I) has the dx dx trivial kernel.

To establish the existence in the sense of sequences of solutions for our integrodifferential equation on the interval I, we consider the sequence of approximate equations, analogously to the situation on the whole real line with m ∈ N, namely r Z 2π d2 dum − − 2 um + b + aum + Gm (x − y)F (um (y), y)dy = 0, (3.10) dx dx 0 where a ≥ 0, b ∈ R, b ̸= 0 are the constants. Our final main proposition is as follows.

100

3 Solvability of some integro-differential equations with drift and superdiffusion

Theorem 3.4 Let Ω = I, m ∈ N, Gm (x) : I → R, Gm (x) ∈ C(I), L1 (I), so that Gm (x) → G(x) in C(I),

dGm (x) dx

∈

dGm (x) dG(x) → in L1 (I) as m → ∞, dx dx

Gm (0) = Gm (2π), F (u, 0) = F (u, 2π) for u ∈ R. Let Assumption 3.1 hold. I) If a > 0, b ∈ R, b ̸= 0, assume that √ 2 πNa,b,m l ≤ 1 − ε for all m ∈ N with a certain fixed 0 < ε < 1 and Na,b,m defined in (3.56). Then each problem (3.10) admits a unique solution um (x) ∈ H 2 (I) and limiting equation (3.1) has a unique solution u(x) ∈ H 2 (I). II)If a = 0, b ∈ R, b ̸= 0, assume that the orthogonality condition (3.58) is valid and √ 2 πN0,b,m l ≤ 1 − ε for all m ∈ N with some fixed 0 < ε < 1. Then each problem (3.10) has a unique solution um (x) ∈ H02 (I) and limiting equation (3.1) possesses a unique solution u(x) ∈ H02 (I). In both cases I and II we have um (x) → u(x) as m → ∞ in the norms in H 2 (I) and H02 (I) respectively. The unique solution um (x) of each problem (3.10) does not vanish identically on the interval I provided that the Fourier coefficients Gm,n F (0, x)n ̸= 0 for some n ∈ Z. Similarly, the unique solution u(x) of limiting equation (3.1) is nontrivial if Gn F (0, x)n ̸= 0 for a certain n ∈ Z. Remark 3.2 In this book we deal with the real valued functions by virtue of the assumptions on F (u, x), Gm (x) and G(x) involved in the integral terms of the iterated and limiting problems discussed above. Remark 3.3 The importance of Theorems 3.2 and 3.4 above is the continuous dependence of the solutions with respect to the integral kernels.

3.1 The whole real line case: scalar equation Proof of Theorem 3.1. Let us first suppose that in the situation of Ω = R for a certain v ∈ H 2 (R) there exist two solutions u1,2 ∈ H 2 (R) of problem (3.4). Then their difference w(x) := u1 (x) − u2 (x) ∈ H 2 (R) will solve the homogeneous equation

3.1 The whole real line case: scalar equation

r −

101

d2 dw w−b − aw = 0. dx2 dx

Because the operator La,b : H 1 (R) → L2 (R) introduced in (3.7) does not possess any nontrivial zero modes, w(x) ≡ 0 on R. We choose an arbitrary v(x) ∈ H 2 (R) and apply the standard Fourier transform (1.79) to both sides of (3.4). This yields u b(p) =

√

2π

b fb(p) G(p) , |p| − a − ibp

p2 u b(p) =

√

2π

b fb(p) p2 G(p) , |p| − a − ibp

(3.11)

where fb(p) denotes the Fourier image of F (v(x), x). Clearly, we have the estimates from above √ √ |b u(p)| ≤ 2πNa,b |fb(p)| and |p2 u b(p)| ≤ 2πNa,b |fb(p)|. Note that Na,b < ∞ by means of Lemma 3.1 of Appendix without any orthogonality relations for a > 0 and under orthogonality condition (3.31) when a = 0. This enables us to obtain the upper bound on the norm 2 ∥u∥2H 2 (R) = ∥b u(p)∥2L2 (R) + ∥p2 u b(p)∥2L2 (R) ≤ 4πNa,b ∥F (v(x), x)∥2L2 (R) < ∞

due to inequality (3.2) of Assumption 3.1 above with v(x) ∈ L2 (R). Thus, for an arbitrary v(x) ∈ H 2 (R) there exists a unique solution u(x) ∈ H 2 (R) of problem (3.4) with its Fourier transform given by (3.11) and the map Ta,b : H 2 (R) → H 2 (R) is well defined. This enables us to choose arbitrarily v1,2 (x) ∈ H 2 (R) such that their images u1,2 = Ta,b v1,2 ∈ H 2 (R). According to (3.4), we have r Z ∞ d2 du1 − 2 u1 − b − au1 = G(x − y)F (v1 (y), y)dy, (3.12) dx dx −∞ r Z ∞ d2 du2 − 2 u2 − b − au2 = G(x − y)F (v2 (y), y)dy. (3.13) dx dx −∞ Let us apply the standard Fourier transform (1.79) to both sides of equations (3.12) and (3.13), which gives us u b1 (p) = u b2 (p) =

√

√

2π

b fb1 (p) G(p) , |p| − a − ibp

p2 u b1 (p) =

2π

b fb2 (p) G(p) , |p| − a − ibp

p2 u b2 (p) =

√

√

2π

b fb1 (p) p2 G(p) , |p| − a − ibp

2π

b fb2 (p) p2 G(p) , |p| − a − ibp

where fb1 (p) and fb2 (p) stand for the Fourier images of F (v1 (x), x) and F (v2 (x), x) respectively. Apparently, we have the estimates from above

102

3 Solvability of some integro-differential equations with drift and superdiffusion

|b u1 (p) − u b2 (p)| ≤

√

2πNa,b |fb1 (p) − fb2 (p)|, √ |p2 u b1 (p) − p2 u b2 (p)| ≤ 2πNa,b |fb1 (p) − fb2 (p)|. This enables us to obtain the inequality for the norms 2 ∥u1 − u2 ∥2H 2 (R) ≤ 4πNa,b ∥F (v1 (x), x) − F (v2 (x), x)∥2L2 (R) .

Clearly, v1,2 (x) ∈ H 2 (R) ⊂ L∞ (R) via the Sobolev embedding. By virtue of upper bound (3.3) we easily arrive at √ ∥Ta,b v1 − Ta,b v2 ∥H 2 (R) ≤ 2 πNa,b l∥v1 − v2 ∥H 2 (R) . (3.14) The constant in the right side of (3.14) is less than one as assumed. By means of the fixed point theorem , there exists a unique function va,b ∈ H 2 (R) with the property Ta,b va,b = va,b . This is the only solution of problem (3.1) in H 2 (R). Suppose va,b (x) is trivial on the real line. This will contradict to our assumption that the Fourier transforms of G(x) and F (0, x) do not vanish on a set of nonzero Lebesgue measure in R. ⊔ ⊓ Let us proceed to establishing the existence in the sense of sequences of the solutions for our integro-differential equation on the real line. Proof of Theorem 3.2. By virtue of the result of Theorem 3.1 above, each equation (3.8) has a unique solution um (x) ∈ H 2 (R), m ∈ N. Limiting problem (3.1) possesses a unique solution u(x) ∈ H 2 (R) by means of Lemma 3.2 below along with Theorem 3.1. Let us apply the standard Fourier transform (1.79) to both sides of (3.1) and (3.8). This yields u b(p) =

√

2π

b φ(p) G(p) b , |p| − a − ibp

u bm (p) =

√

2π

b m (p)φ G bm (p) , |p| − a − ibp

m ∈ N,

(3.15)

where φ(p) b and φ bm (p) stand for the Fourier images of F (u(x), x) and F (um (x), x) respectively. Obviously,

b m (p) b √ G(p)

G

|b um (p) − u b(p)| ≤ 2π − |φ(p)|+ b

|p| − a − ibp |p| − a − ibp ∞ L

b m (p) √

G

+ 2π

|p| − a − ibp

(R)

|φ bm (p) − φ(p)|, b L∞ (R)

so that ∥um −u∥L2 (R) ≤

√



G

b m (p) b G(p)

2π −

|p| − a − ibp |p| − a − ibp

∥F (u(x), x)∥L2 (R) + L∞ (R)

3.1 The whole real line case: scalar equation

b m (p) √

G

+ 2π

|p| − a − ibp

103

∥F (um (x), x) − F (u(x), x)∥L2 (R) . L∞ (R)

Inequality (3.3) of Assumption 3.1 above gives us ∥F (um (x), x) − F (u(x), x)∥L2 (R) ≤ l∥um (x) − u(x)∥L2 (R) .

(3.16)

Evidently, um (x), u(x) ∈ H 2 (R) ⊂ L∞ (R) due to the Sobolev embedding. Thus, we arrive at

( ) b m (p) √

G

∥um (x) − u(x)∥L2 (R) 1 − 2π l ≤

|p| − a − ibp ∞ L



b m (p) b √ G(p)

G

≤ 2π −

|p| − a − ibp |p| − a − ibp

(R)

∥F (u(x), x)∥L2 (R) . L∞ (R)

By virtue of (3.35) when a > 0 and (3.37) for a = 0, we derive √





bm (p) b G(p) 2π G

−

ε |p| − a − ibp |p| − a − ibp

∥um (x) − u(x)∥L2 (R) ≤

∥F (u(x), x)∥L2 (R) .

L ∞ ( R)

Inequality (3.2) of Assumption 3.1 above implies that F (u(x), x) ∈ L2 (R) for u(x) ∈ H 2 (R). Therefore, um (x) → u(x),

m→∞

(3.17)

in L2 (R) due to the result of Lemma 3.2 of Appendix. Evidently, p2 u b(p) =

√

2π

b φ(p) p2 G(p) b , |p| − a − ibp

p2 u bm (p) =

√

2π

b m (p)φ p2 G bm (p) , |p| − a − ibp

m ∈ N.

Hence 2

2

|p u bm (p) − p u b(p)| ≤

√



p2 G

b m (p) b p2 G(p)

2π −

|p| − a − ibp |p| − a − ibp



p2 G b m (p)

+ 2π

|p| − a − ibp √

|φ(p)|+ b L∞ (R)

|φ bm (p) − φ(p)|. b L∞ (R)

Using (3.16), we obtain

d2 u

d2 u m

−

dx2 dx2

L 2 ( R)

≤

√



p2 G

bm (p) b p2 G(p)

2π −

|p| − a − ibp |p| − a − ibp

L ∞ ( R)

∥F (u(x), x)∥L2 (R) +

104

3 Solvability of some integro-differential equations with drift and superdiffusion



p2 G b m (p)

+ 2π

|p| − a − ibp √

l∥um (x) − u(x)∥L2 (R) . L∞ (R)

By means of the result of Lemma 3.2 of Appendix along with (3.17), we d2 u d2 um derive → 2 in L2 (R) as m → ∞. Definition (3.6) of the norm gives 2 dx dx us um (x) → u(x) in H 2 (R) as m → ∞. Let us suppose that the unique solution um (x) of equation (3.10) discussed above is trivial on the whole real line for a certain m ∈ N. This will contradict to the assumption that the Fourier transforms of Gm (x) and F (0, x) do not vanish identically on a set of nonzero Lebesgue measure in R. The similar argument is valid for the unique solution u(x) of limiting problem (3.1). ⊔ ⊓

3.2 The problem on the finite interval: scalar equation Proof of Theorem 3.3. Let us present the proof of our theorem in the situation when a > 0. If the constant a vanishes, the ideas will be analogous. When a = 0, we will need to use the constrained subspace (3.9) instead of H 2 (I). The non-selfadjoint operator in the left side of problem (3.4) r d2 d La,b := − 2 − b − a : H 1 (I) → L2 (I) (3.18) dx dx satisfies the Fredholm property. Its set of eigenvalues is given by λa,b (n) = |n| − a − ibn,

n∈Z

(3.19)

einx and its eigenfunctions are the standard Fourier harmonics √ , n ∈ Z. Note 2π that the eigenvalues of the the operator La,b are simple, as distinct from the similar situation without the drift term, when the eigenvalues corresponding to n ̸= 0 are two-fold degenerate (see [60]). First suppose that for some v(x) ∈ H 2 (I) there exist two solutions u1,2 (x) ∈ H 2 (I) of problem (3.4) with Ω = I. Then the function w(x) := u1 (x) − u2 (x) ∈ H 2 (I) will be a solution of the homogeneous equation r d2 dw − 2w − b − aw = 0. dx dx Since the operator La,b : H 1 (I) → L2 (I) discussed above does not possess nontrivial zero modes, we have w(x) ≡ 0 in I.

3.2 The problem on the finite interval: scalar equation

105

We choose arbitrarily v(x) ∈ H 2 (I) and apply the Fourier transform (1.97) to problem (3.4) studied on the interval I. This yields un =

√

2π

Gn f n , |n| − a − ibn

n 2 un =

√

2π

n2 Gn f n , |n| − a − ibn

n∈Z

(3.20)

with fn := F (v(x), x)n . This enables us to derive the upper bounds √ √ |un | ≤ 2πNa,b |fn |, |n2 un | ≤ 2πNa,b |fn |. Note that Na,b < ∞ under our assumptions by virtue of Lemma 3.3 of Appendix. Thus, we arrive at ∥u∥2H 2 (I) =

∞ X n=−∞

|un |2 +

∞ X

2 |n2 un |2 ≤ 4πNa,b ∥F (v(x), x)∥2L2 (I) < ∞

n=−∞

due to inequality (3.2) of Assumption 3.1 for v(x) ∈ H 2 (I). Hence, for an arbitrary v(x) ∈ H 2 (I) there exists a unique u(x) ∈ H 2 (I), which solves equation (3.4) with its Fourier transform given by (3.20), such the map τa,b : H 2 (I) → H 2 (I) in the first case of our the theorem is well defined. Let us consider any v1,2 (x) ∈ H 2 (I), such their images under the map mentioned above u1,2 = τa,b v1,2 ∈ H 2 (I). By virtue of (3.4), we have r Z 2π d2 du1 − 2 u1 − b − au1 = G(x − y)F (v1 (y), y)dy, (3.21) dx dx 0 r Z 2π d2 du2 − 2 u2 − b − au2 = G(x − y)F (v2 (y), y)dy. (3.22) dx dx 0 By means of Fourier transform (1.97) applied to both sides of (3.21) and (3.22), we easily obtain u1,n = n2 u1,n

√

√ Gn f1,n Gn f2,n , u2,n = 2π , |n| − a − ibn |n| − a − ibn √ √ n2 Gn f1,n n2 Gn f2,n = 2π , n2 u2,n = 2π , |n| − a − ibn |n| − a − ibn 2π

n∈Z

with fj,n := F (vj (x), x)n , j = 1, 2. Hence, √ √ |u1,n −u2,n | ≤ 2πNa,b |f1,n −f2,n |, |n2 (u1,n −u2,n )| ≤ 2πNa,b |f1,n −f2,n |, so that

106

3 Solvability of some integro-differential equations with drift and superdiffusion

∥u1 − u2 ∥2H 2 (I) = ≤

∞ X

|u1,n − u2,n |2 +

n=−∞ 2 4πNa,b ∥F (v1 (x), x)

∞ X

|n2 (u1,n − u2,n )|2

n=−∞

− F (v2 (x), x)∥2L2 (I) .

Obviously, v1,2 (x) ∈ H 2 (I) ⊂ L∞ (I) due to the Sobolev embedding. By virtue of inequality (3.3) we easily arrive at √ ∥τa,b v1 − τa,b v2 ∥H 2 (I) ≤ 2 πNa,b l∥v1 − v2 ∥H 2 (I) . (3.23) The constant in the right side of (3.23) is less than one via the one of our assumptions. Then the fixed point theorem implies the existence and uniqueness of a function va,b ∈ H 2 (I) which satisfies τa,b va,b = va,b . This is the only solution of equation (3.1) in H 2 (I) in the first case of our theorem. Let us suppose that va,b (x) is trivial in I. This will contradict to the assumption that Gn F (0, x)n ̸= 0 for a certain n ∈ Z. Note that in the situation of the theorem when a > 0 our argument does not rely on any orthogonality conditions. ⊔ ⊓ We turn our attention to establishing the final main result of this section. Proof of Theorem 3.4. Evidently, the limiting kernel G(x) is also a periodic function on our interval I (see the argument of Lemma 3.4 of Appendix). Each problem (3.10) has a unique solution um (x), m ∈ N, which belongs to H 2 (I) in the case if a > 0 and to H02 (I) in the situation when a = 0 by virtue of Theorem 3.3. Limiting equation (3.1) possesses a unique solution u(x) belonging to H 2 (I) in the situation when a > 0 and to H02 (I) in the case when a vanishes via Lemma 3.4 below along with Theorem 3.3. Let us apply Fourier transform (1.97) to both sides of equations (3.1) and (3.10). This yields un =

√

2π

G n φn , |n| − a − ibn

um,n =

√

2π

Gm,n φm,n , |n| − a − ibn

n ∈ Z,

m ∈ N.

(3.24) Here φn and φm,n are the Fourier images of F (u(x), x) and F (um (x), x) respectively under transform (1.97). We easily obtain estimate of the upper bound

√ Gm,n Gn

|um,n − un | ≤ 2π −

|φn |

|n| − a − ibn |n| − a − ibn ∞ l

√ Gm,n

+ 2π − φn |.

|φ

|n| − a − ibn ∞ m,n l

Hence,

3.2 The problem on the finite interval: scalar equation

∥um − u∥L2 (I)

107



√ Gm,n Gn

≤ 2π −

∥F (u(x), x)∥L2 (I) +

|n| − a − ibn |n| − a − ibn ∞ l



√ Gm,n

+ 2π

∥F (um (x), x) − F (u(x), x)∥L2 (I) .

|n| − a − ibn ∞ l

By virtue of inequality (3.3) of Assumption 3.1, we derive ∥F (um (x), x) − F (u(x), x)∥L2 (I) ≤ l∥um (x) − u(x)∥L2 (I) .

(3.25)

Obviously, um (x), u(x) ∈ H 2 (I) ⊂ L∞ (I) via the Sobolev embedding. Clearly,

) (

√ Gm,n

∥um − u∥L2 (I) 1 − 2πl ≤

|n| − a − ibn ∞ l

√ Gm,n Gn

≤ 2π −

∥F (u(x), x)∥L2 (I) .

|n| − a − ibn |n| − a − ibn ∞ l

By means of inequalities (3.57) in the case when a > 0 and (3.59) in the situation when a vanishes, we arrive at

√

2π Gm,n Gn

∥um − u∥L2 (I) ≤ −

∥F (u(x), x)∥L2 (I) . ε |n| − a − ibn |n| − a − ibn ∞ l

Apparently, F (u(x), x) ∈ L2 (I) for u(x) ∈ H 2 (I) due to upper bound (3.2) of Assumption 3.1. The result of Lemma 3.4 of Appendix implies that um (x) → u(x),

m→∞

(3.26)

in L2 (I). Evidently, 2

2

|n um,n − n un | ≤

√



n2 G

n2 Gn

m,n 2π −

|φ |+

|n| − a − ibn |n| − a − ibn ∞ n l



√

n2 Gm,n + 2π

|φm,n − φn |.

|n| − a − ibn ∞ l

Let us use (3.25) to obtain

d2 u

d 2 um m

−

2

dx2 dx

L2 (I)

≤

√



n2 G

n2 G n m,n

2π −

∥F (u(x), x)∥L2 (I) +

|n| − a − ibn |n| − a − ibn ∞ l



√

n2 Gm,n + 2π

l∥u (x) − u(x)∥L2 (I) .

|n| − a − ibn ∞ m l

108

3 Solvability of some integro-differential equations with drift and superdiffusion

d2 um d2 u → as m → ∞ dx2 dx2 2 2 in L (I). Thus, um (x) → u(x) in the H (I) norm as m → ∞. By means of Lemma 3.4 along with (3.26), we derive

If we suppose that um (x) ≡ 0 in the interval I for some m ∈ N, then we will obtain a contradiction to the assumption that Gm,n F (0, x)n ̸= 0 for a certain n ∈ Z. The similar reasoning is valid for the solution u(x) of limiting equation (3.1). ⊔ ⊓

Appendix Attentive readers already noted that in sections 3.1 and 3.2 during the proofs of Theorems 3.1-3.4 we dealth with the completely new quantities

) (



Gn n2 Gn



Na,b := max ,

,

|n| − a − ibn ∞

|n| − a − ibn ∞ l l

) (

n2 G

G



m,n m,n Na,b,m := max

,

|n| − a − ibn ∞

|n| − a − ibn ∞ l

l

as well as (3.30) and (3.31) related to the equation (3.1) and additional regularities on the kernel G(x) and Gm (x) than in chapter 2. Therefore in Appendix (see below) we present the proofs of Lemmas 3.1-3.4 that reflects impact of superdiffusion term in (3.1) both for the case of whole line and for the case of a finite interval. Let G(x) be a function, G(x) : R → R, for which we denote as it was in a previous chapters its standard Fourier transform using the hat symbol as Z ∞ 1 b G(p) := √ G(x)e−ipx dx, p ∈ R, (3.27) 2π −∞ so that

1 and G(x) = √ 2π

1 b ∥G(p)∥ ∥G(x)∥L1 (R) L∞ (R) ≤ √ 2π Z

(3.28)

∞ iqx b G(q)e dq, x ∈ R. Clearly, (3.28) yields −∞



1

dG(x) b ∥pG(p)∥L∞ (R) ≤ √

2π dx

.

(3.29)

L1 (R)

For the technical purposes we introduce the auxiliary quantities that reflects superdiffusion nature (3.1).

3.2 The problem on the finite interval: scalar equation

n

Na,b := max

b G(p)

,

|p| − a − ibp L∞ (R)



o b p2 G(p)

,

∞ |p| − a − ibp L (R)

109

(3.30)

where a ≥ 0, b ∈ R, b ̸= 0 are the constants. Lemma 3.1 Let G(x) : R → R, G(x) ∈ W 1,1 (R). a) If a > 0, b ∈ R, b ̸= 0 then Na,b < ∞. b) If a = 0, b ∈ R, b ̸= 0 and additionally xG(x) ∈ L1 (R) then N0,b < ∞ if and only if the orthogonality condition (G(x), 1)L2 (R) = 0

(3.31)

is valid. Proof. First of all, let us observe that in both cases a) and b) of our lemma, under the given conditions the expression b p2 G(p) ∈ L∞ (R). |p| − a − ibp p is bounded on |p| − a − ibp b the whole real line and pG(p) ∈ L∞ (R) due to inequality (3.28) above. Let us turn our attention to establishing the result of the part a) of the lemma. We need to estimate the expression Indeed, it can be easily verified that the function

b |G(p)| p . (|p| − a)2 + b2 p2

(3.32)

Apparently, the numerator of (3.32) can be bounded from above by virtue of (3.28) and the denominator in (3.32) can be trivially estimated below by a finite, positive constant, such that b G(p) ≤ C∥G(x)∥L1 (R) < ∞ |p| − a − ibp due to the one of our assumptions. Here and further down C will denote a finite, positive constant. This yields that under the given assumptions, for a > 0 we have Na,b < ∞. In the situation when a = 0, we will use the identity p

Z b b G(p) = G(0) + 0

b dG(s) ds. ds

Hence R p dG(s) b b b ds G(p) G(0) = + 0 ds . |p| − ibp |p| − ibp |p| − ibp

(3.33)

110

3 Solvability of some integro-differential equations with drift and superdiffusion

By means of definition (1.79) of the standard Fourier transform, we easily arrive at dG(p) 1 b ≤ √ ∥xG(x)∥L1 (R) . dp 2π Therefore, R p b dG(s) 0 ds ds ∥xG(x)∥L1 (R) 0, b ∈ R, b ̸= 0, let √ 2 πNa,b,m l ≤ 1 − ε

(3.35)

for all m ∈ N with some fixed 0 < ε < 1. b) If a = 0, b ∈ R, b ̸= 0, let xGm (x) ∈ L1 (R), xGm (x) → xG(x) in L1 (R) as m → ∞, the orthogonality relation (Gm (x), 1)L2 (R) = 0, holds. Let in addition

m∈N

√ 2 πN0,b,m l ≤ 1 − ε

(3.36)

(3.37)

for all m ∈ N with a certain fixed 0 < ε < 1. Then b m (p) b G G(p) → , |p| − a − ibp |p| − a − ibp

m → ∞,

(3.38)

b m (p) b p2 G p2 G(p) → , |p| − a − ibp |p| − a − ibp

m→∞

(3.39)

in L∞ (R), so that

3.2 The problem on the finite interval: scalar equation



G b m (p)



|p| − a − ibp

p2 G b m (p)



|p| − a − ibp Moreover,

111

L∞ (R)



b G(p)

→

|p| − a − ibp

L∞ (R)

L∞ (R)



p2 G(p)

b

→

|p| − a − ibp

L∞ (R)

, m → ∞,

(3.40)

, m → ∞.

(3.41)

√ 2 πNa,b l ≤ 1 − ε.

(3.42)

Proof. Using inequality (3.28), we easily obtain 1 b m (p) − G(p)∥ b ∥G ∥Gm (x) − G(x)∥L1 (R) → 0, L∞ (R) ≤ √ 2π

m → ∞ (3.43)

via the one of our assumptions. Obviously, (3.40) and (3.41) will easily follow from (3.38) and (3.39) respectively by virtue of the standard triangle inequality. p Let us use the fact that the function ∈ L∞ (R) along with the |p| − a − ibp analog of estimate (3.29) to derive p2 G b m (p) b p2 G(p) b m (p) − G(p)]∥ b − ≤ C∥p[G L∞ (R) |p| − a − ibp |p| − a − ibp

C

dGm (x) dG(x) ≤√ − .

dx 1 2π dx L (R)

Hence

p2 G

b m (p) b p2 G(p)

−

|p| − a − ibp |p| − a − ibp

L∞ (R)



C

dGm (x) dG(x) ≤√ −

dx 2π dx

→0 L1 (R)

as m → ∞ due to the one of our assumptions, such that (3.39) holds. In order to establish (3.38) in the situation when a > 0, we need to consider b (p) − G(p)| b |G p m . 2 (|p| − a) + b2 p2

(3.44)

Apparently, the denominator in fraction (3.44) can be estimated from below by a positive constant and the numerator in (3.44) can be bounded from above by virtue of (3.28). Thus

G

b m (p) b G(p)

− ≤ C∥Gm (x) − G(x)∥L1 (R) → 0

|p| − a − ibp |p| − a − ibp ∞ L

(R)

112

3 Solvability of some integro-differential equations with drift and superdiffusion

as m → ∞ according to the one of the assumptions, such that (3.38) holds in the case a) of our lemma. Let us proceed to establishing (3.38) in the situation when a vanishes. In this case orthogonality relations (3.31) hold as assumed. We easily demonstrate that the analogous statement will be valid in the limit. Clearly, |(G(x), 1)L2 (R) | = |(G(x) − Gm (x), 1)L2 (R) | ≤ ∥Gm (x) − G(x)∥L1 (R) → 0 as m → ∞ via the one of our assumptions, so that (G(x), 1)L2 (R) = 0

(3.45)

holds. Evidently, we have Z b b G(p) = G(0) + b m (p) = G b m (0) + G

p

b dG(s) ds, ds 0 Z p b dGm (s) 0

ds

ds,

m ∈ N.

Formulas (3.36) and (3.31) give us b G(0) = 0,

b m (0) = 0, G

m ∈ N.

Therefore, i R p h dGbm (s) dG(s) b G 0 − ds b b ds ds (p) G(p) m − = . |p| − ibp |p| − ibp |p| − ibp

(3.46)

The definition of the standard Fourier transform (3.27) yields dG b 1 b m (p) dG(p) − ≤ √ ∥xGm (x) − xG(x)∥L1 (R) . dp dp 2π This enables us to obtain the upper bound on the right side of (3.46) given by ∥xGm (x) − xG(x)∥L1 (R) p , 2π(1 + b2 ) so that

G

b G(p)

b m (p)

−

|p| − ibp |p| − ibp

≤ L∞ (R)

∥xGm (x) − xG(x)∥L1 (R) p →0 2π(1 + b2 )

as m → ∞ due to the one of the given conditions. Thus (3.38) holds when a = 0. Apparently, under the assumptions of our lemma

3.2 The problem on the finite interval: scalar equation

Na,b < ∞,

Na,b,m < ∞,

m ∈ N,

113

a ≥ 0,

b ∈ R,

b ̸= 0

by virtue of the result of Lemma 3.1 above. We have upper bounds (3.35) when a > 0 and (3.37) if a vanishes. An easy limiting argument using (3.40) and (3.41) yields (3.42). ⊔ ⊓ Consider the function G(x) : I → R, such that G(0) = G(2π). Its Fourier transform on our finite interval is defined as Z 2π e−inx Gn := G(x) √ dx, n ∈ Z, (3.47) 2π 0 so that G(x) =

∞ X

einx Gn √ . Evidently, the inequality 2π n=−∞ 1 ∥Gn ∥l∞ ≤ √ ∥G(x)∥L1 (I) 2π

(3.48)

holds. Clearly, if our function is continuous on the interval I, we have the upper bound ∥G(x)∥L1 (I) ≤ 2π∥G(x)∥C(I) . (3.49) The estimate from above ∥nGn ∥l∞



1

dG(x) ≤√

2π dx

easily follows from (3.47). Similarly to the

(

Gn

Na,b := max

,

|n| − a − ibn ∞ l

(3.50) L1 (I)

whole real line case, we introduce

)

n2 Gn

, (3.51)

|n| − a − ibn ∞ l

where a ≥ 0, b ∈ R, b ̸= 0 are the constants. We have the following technical proposition. Lemma 3.3 Let G(x) : I → R, G(x) ∈ C(I), G(2π).

dG(x) dx

∈ L1 (I) and G(0) =

a) If a > 0, b ∈ R, b ̸= 0 then Na,b < ∞. b) If a = 0, b ∈ R, b ̸= 0 then N0,b < ∞ if and only if the orthogonality condition (G(x), 1)L2 (I) = 0. (3.52) is valid.

114

3 Solvability of some integro-differential equations with drift and superdiffusion

Proof. It can be easily verified that in both cases a) and b) of the lemma under the given assumptions we have n2 G n ∈ l∞ . |n| − a − ibn

(3.53)

n ∈ l∞ and nGn ∈ l∞ due to inequality (3.50) along with |n| − a − ibn the one of our conditions, such that (3.53) holds. Indeed,

To establish the statement of the part a) of our lemma, we need to consider the expression |Gn | p . (3.54) (|n| − a)2 + b2 n2 Apparently, the denominator in (3.54) can be trivially estimated below by a positive constant and the numerator in (3.54) can be easily bounded above by virtue of (3.48) along with (3.49). Thus, Na,b < ∞ in the situation when a > 0. To demonstrate the validity of the result of our lemma in the case when a vanishes, we note that G n (3.55) |n| − ibn is bounded if and only if G0 = 0. This is equivalent to orthogonality relation (3.52). In this case we can estimate expression (3.55) from above by √

√ ∥G(x)∥C(I) ∥G(x)∥L1 (I) 1 √ ≤ 2π √ 0, b ∈ R, b ̸= 0, let √ 2 πNa,b,m l ≤ 1 − ε

(3.57)

for all m ∈ N with some fixed 0 < ε < 1. b) If a = 0, b ∈ R, b ̸= 0, let the orthogonality relation (Gm (x), 1)L2 (I) = 0, hold. Let in addition

m∈N

(3.58)

√ 2 πN0,b,m l ≤ 1 − ε

(3.59)

for all m ∈ N with a certain fixed 0 < ε < 1. Then

Gm,n Gn → , |n| − a − ibn |n| − a − ibn

m → ∞,

n2 Gm,n n2 Gn → , m→∞ |n| − a − ibn |n| − a − ibn

(3.61)

in l∞ , so that





Gm,n Gn



→

, m → ∞,

|n| − a − ibn ∞

|n| − a − ibn ∞ l

Moreover, the estimate

(3.62)

l



n2 G

n2 Gn



m,n

→

, m → ∞.

|n| − a − ibn ∞

|n| − a − ibn ∞ l

(3.60)

(3.63)

l

√ 2 πNa,b l ≤ 1 − ε

(3.64)

is valid. Proof. Evidently, under the given conditions, the limiting kernel function G(x) will be periodic as well. Indeed, we easily derive |G(0) − G(2π)| ≤ |G(0) − Gm (0)| + |Gm (2π) − G(2π)| ≤ 2∥Gm (x) − G(x)∥C(I) → 0,

m→∞

due to the one of our assumptions. Hence, G(0) = G(2π). By means of (3.47) along with (3.48) we arrive at

116

3 Solvability of some integro-differential equations with drift and superdiffusion

√ 1 ∥Gm,n − Gn ∥l∞ ≤ √ ∥Gm − G∥L1 (I) ≤ 2π∥Gm − G∥C(I) → 0, 2π

m→∞

(3.65) via the one of the given conditions. It can be easily verified that the statements of (3.60) and (3.61) will yield (3.62) and (3.63) respectively by virtue of the triangle inequality. By means of (3.50), we arrive at the upper bound

n2 G

n2 G n

m,n −

|n| − a − ibn |n| − a − ibn ∞ l





1 n

dGm (x) dG(x) ≤√ − →0



dx 1 2π |n| − a − ibn ∞ dx l

L (I)

as m → ∞ as assumed, so that (3.61) holds. In order to demonstrate the validity of (3.60) in the situation when a > 0, we need to consider |Gm,n − Gn | p

(|n| − a)2 + b2 n2

.

(3.66)

Clearly, the denominator of (3.66) can be estimated from below by a positive constant and the numerator bounded from above by virtue of (3.65). This implies (3.60) for a > 0. Let us conclude the proof of the lemma by establishing (3.60) in the case when a = 0. According to the one of the given assumptions, we have orthogonality relation (3.58). It can be easily verified that the analogous condition is valid in the limit. Indeed, |(G(x), 1)L2 (I) | = |(G(x) − Gm (x), 1)L2 (I) | ≤ 2π∥Gm (x) − G(x)∥C(I) → 0,

m→∞

as assumed. Hence, (G(x), 1)L2 (I) = 0, which is equivalent to G0 = 0. Clearly, Gm,0 = 0, m ∈ N due to orthogonality relation (3.58). By virtue of (3.65), we derive √ G 2π∥Gm (x) − G(x)∥C(I) m,n − Gn √ . ≤ |n| − ibn 1 + b2 Because the norm in the right side of this upper bound converges to zero as m → ∞, (3.60) is valid in the situation when a = 0 as well. Apparently, under the given conditions we have Na,b < ∞,

Na,b,m < ∞,

m ∈ N,

a ≥ 0,

b ∈ R,

b ̸= 0

3.2 The problem on the finite interval: scalar equation

117

by means of the result of our Lemma 3.3. We have inequalities (3.57) for a > 0 and (3.59) when a = 0. A trivial limiting argument relying on (3.62) and (3.63) yields (3.64). ⊔ ⊓ In sections 3.3 and 3.4 below we establish the existence in the sense of sequences of solutions for certain systems of integro-differential equations which involve the drift terms and the square root of the one dimensional negative Laplace operator, on the whole real line or on a finite interval with periodic boundary conditions in the corresponding H 2 spaces. The argument again is based on the fixed point technique when the elliptic systems contain first order differential operators with and without Fredholm property. It is proven that, under the reasonable technical conditions, the convergence in L1 of the integral kernels yields the existence and convergence in H 2 of the solutions. We would like especially to emphasize that the study of the systems is more complicated than of the scalar case and requires to overcome more cumbersome technicalities. Thus in sections 3.3 and 3.4 (see below) we consider a class of stationary nonlinear systems of equations for which the Fredholm property may not be satisfied: r d2 duk − − 2 u k + bk + a k uk dx dx Z Gk (x − y)Fk (u1 (y), u2 (y), ..., uN (y), y)dy = 0, x ∈ Ω,

+

(3.67)

Ω

where ak ≥ 0, bk ∈ R, bk ̸= 0 are the constants, 1 ≤ k ≤ N, N ≥ 2 and Ω ⊆ R. Here and throughout sections 3.3 and 3.4 the vector function u := (u1 , u2 , ..., uN )T ∈ RN . (3.68) r d2 We remind that the operator − 2 is defined as a pseudodifferential operdx ator order one (see section 1.4) and is actively used, for example in the studies of the superdiffusion problems. Superdiffusion can be described as a random process of particle motion characterized by the probability density distribution of the jump length. The moments of this density distribution are finite in the case of the normal diffusion, but this is not the case for the superdiffusion. Asymptotic behavior at the infinity of the probability density function determines the value of the power of the negative Laplace operator (see [49]). For the simplicity of the presentation we restrict ourselves to the one dimensional situation. The study of the solvability of the integro-differential system (3.67) is more complicated than in the single nonlocal equation case covered in sections 3.1 and 3.2 (see also [31]). It requires the use of the Sobolev spaces for the vector functions, which is more cumbersome, especially in the situation on the finite interval with periodic boundary conditions , where we use

118

3 Solvability of some integro-differential equations with drift and superdiffusion

the constrained subspaces. Moreover, in the argument in our system case we use the auxiliary expressions (3.100), (3.106), (3.123), (3.128) depending on the additional index 1 ≤ k ≤ N , N ≥ 2, which is an extra technicality. In the situation when the drift terms are absent, namely when bk = 0, 1 ≤ k ≤ N , the system analogous to (3.67) was discussed in [60] . Formulation of the results The technical assumptions of these sections 3.3 and 3.4 will be analogical to the ones of sections 3.1 and 3.2 (see also [31]), adapted to the work with vector functions. It is also more difficult to perform the analysis in the Sobolev spaces for vector functions, especially in the system on our finite interval with periodic boundary conditions when the constraints are imposed on some of the components. The nonlinear part of problem (3.67) will satisfy the following regularity conditions. Assumption 3.2 Let 1 ≤ k ≤ N . Functions Fk (u, x) : RN × Ω → R are satisfying the Carath´eodory condition, so that v uN uX t Fk2 (u, x) ≤ K|u|RN + h(x) f or u ∈ RN , x ∈ Ω (3.69) k=1

with a constant K > 0 and h(x) : Ω → R+ , h(x) ∈ L2 (Ω). Furthermore, they are Lipschitz continuous functions, so that for any u(1),(2) ∈ RN , x ∈ Ω: v uN uX t (F (u(1) , x) − F (u(2) , x))2 ≤ L|u(1) − u(2) | N , (3.70) k k R k=1

with a constant L > 0. Here and further down the norm of a vector function given by (3.68) is: v uN uX |u|RN := t u2k . k=1

The nonlinear function involved there was allowed to have a sublinear growth (see [10]). In order to establish the solvabity of (3.67), we will use the auxiliary system with 1 ≤ k ≤ N , namely r Z d2 duk − 2 u k − bk − a k uk = Gk (x − y)Fk (v1 (y), v2 (y), ..., vN (y), y)dy, dx dx Ω (3.71) where ak ≥ 0, bk ∈ R, bk ̸= 0 are the constants. Let us denote

3.2 The problem on the finite interval: scalar equation

Z

119

f1 (x)f¯2 (x)dx,

(f1 (x), f2 (x))L2 (Ω) :=

(3.72)

Ω

with a slight abuse of notations when these functions are not square integrable, like for instance those involved in orthogonality relations (3.102) below. Indeed, if f1 (x) ∈ L1 (Ω) and f2 (x) ∈ L∞ (Ω), then the integral in the right side of (3.72) is well defined. In section 3.3 we consider the situation on the whole real line, Ω = R, so that the appropriate Sobolev space is equipped with the norm

d2 ϕ 2

2 2 . (3.73) ∥ϕ∥H 2 (R) := ∥ϕ∥L2 (R) + 2

dx 2 L (R)

Then for a vector function given by (3.68), we have ∥u∥2H 2 (R,RN )

:=

N X k=1

∥uk ∥2H 2 (R)

=

N X

( ∥uk ∥2L2 (R)

k=1



d2 u 2

k + 2

dx 2

) .

(3.74)

L (R)

We also use the norm ∥u∥2L2 (R,RN ) :=

N X

∥uk ∥2L2 (R) .

k=1

By means of Assumption 3.2 above, we are not allowed to consider the higher powers of our nonlinearities, than the first one. This is restrictive from the point of view of the applications. But this guarantees that our nonlinear vector function is a bounded and continuous map from L2 (Ω, RN ) to L2 (Ω, RN ). The main issue for our system of equations above is that in the absence of the drift terms we were dealing with the self-adjoint, non Fredholm operators r d2 − 2 − ak : H 1 (R) → L2 (R), ak ≥ 0, dx which was the obstacle to solve our system. The similar situations but in linear problems, both self- adjoint and non selfadjoint containing the differential operators without the Fredholm property have been treated extensively in recent years (see [26],[57], [58] and the references therein). However, the situation is different when the constants in the drift terms bk ̸= 0. For 1 ≤ k ≤ N , the operators r d2 d La,b,k := − 2 − bk − ak : H 1 (R) → L2 (R), (3.75) dx dx with ak ≥ 0 and bk ∈ R, bk ̸= 0 contained in the left side of the system of equations (3.71) are non-selfadjoint. By means of the standard Fourier transform, it can be trivially obtained that the essential spectra of such operators

120

3 Solvability of some integro-differential equations with drift and superdiffusion

La,b,k are given by λa,b,k (p) = |p| − ak − ibk p,

p ∈ R.

Clearly, for ak > 0 the operators La,b,k satisfy the Fredholm property, since their essential spectra do not contain the origin. But when ak = 0, our operators La,b,k fail to satisfy the Fredholm property because the origin belongs to their essential spectra. We establish that under the reasonable technical conditions system (3.71) defines a map Ta,b : H 2 (R, RN ) → H 2 (R, RN ), which is a strict contraction. Theorem 3.5 Let Ω = R, N ≥ 2, 1 ≤ l ≤ N − 1, 1 ≤ k ≤ N , bk ∈ R, bk ̸= 0 and Gk (x) : R → R, Gk (x) ∈ W 1,1 (R) and Assumption 3.2 hold. I) Let ak > 0 for 1 ≤ k ≤ l. II)Let ak = 0 for l + 1 ≤ k ≤ N , additionally xGk (x) ∈ L1 (R), orthogo√ nality conditions (3.102) hold and 2 πNa,b L < 1, where Na,b is defined in (3.101) below. Then the map v 7→ Ta,b v = u on H 2 (R, RN ) defined by system (3.71) has a unique fixed point v (a,b) , which is the only solution of the system of equations (3.67) in H 2 (R, RN ). The fixed point v (a,b) is nontrivial provided that for a certain 1 ≤ k ≤ N the intersection of supports of the Fourier transforms of functions supp F\ k (0, x)∩ ck is a set of nonzero Lebesgue measure in R. supp G Note that in the case I) of the theorem above, when ak > 0, as distinct part I) of Assumption 2 of [60] describing the problem without the drift term, the orthogonality conditions are not needed. Let us introduce the sequence of approximate systems of equations related to problem (3.67) on the whole real line, namely r (m) du d2 (m) (m) − − 2 u k + bk k + a k u k dx dx Z ∞ (m) (m) (m) + Gk,m (x − y)Fk (u1 (y), u2 (y), ..., uN (y), y)dy = 0 (3.76) −∞

with the constants ak ≥ 0, bk ∈ R, bk ̸= 0, 1 ≤ k ≤ N and m ∈ N. Each sequence of kernels {Gk,m (x)}∞ m=1 tends to Gk (x) as m → ∞ in the appropriate function spaces discussed below. We establish that, under the given technical conditions, each of systems of equations (3.76) has a unique solution u(m) (x) ∈ H 2 (R, RN ), limiting system (3.67) possesses a unique solution u(x) ∈ H 2 (R, RN ), and u(m) (x) → u(x) in H 2 (R, RN ) as m → ∞. This is the so-called existence of solutions in the sense of sequences. In such case, the solvability conditions can be formulated for the iterated kernels Gk,m . They yield the convergence of the kernels in terms of the Fourier

3.2 The problem on the finite interval: scalar equation

121

transforms (see Appendix) and, consequently, the convergence of the solutions (Theorems 3.6, 3.8). Our second main statement is as follows. Theorem 3.6 Let Ω = R, N ≥ 2, 1 ≤ l ≤ N −1, 1 ≤ k ≤ N , bk ∈ R, bk ̸= 0, m ∈ N, Gk,m (x) : R → R, Gk,m (x) ∈ W 1,1 (R), so that Gk,m (x) → Gk (x) in W 1,1 (R) as m → ∞. Let Assumption 3.2 hold. I) Let ak > 0 for 1 ≤ k ≤ l. II)Let ak = 0 for l + 1 ≤ k ≤ N . Assume that xGk,m (x) ∈ L1 (R), xGk,m (x) → xGk (x) in L1 (R) as m → ∞, orthogonality conditions (3.108) are valid along with upper bound (3.109). Then each system (3.76) possesses a unique solution u(m) (x) ∈ H 2 (R, RN ), and limiting problem (3.67) admits a unique solution u(x) ∈ H 2 (R, RN ), so that u(m) (x) → u(x) in H 2 (R, RN ) as m → ∞. The unique solution u(m) (x) of each system (3.76) is nontrivial provided that for a certain 1 ≤ k ≤ N the intersection of supports of the Fourier images b of functions supp F\ k (0, x) ∩ supp Gk,m is a set of nonzero Lebesgue measure in R. Similarly, the unique solution u(x) of limiting system (3.67) does not c vanish identically on the real line if supp F\ k (0, x)∩supp Gk is a set of nonzero Lebesgue measure in R for some 1 ≤ k ≤ N . Section 3.4 is devoted to the studies of the analogical system of equations on the finite interval Ω = I := [0, 2π] with periodic boundary conditions. The appropriate function space is given by H 2 (I) = {v(x) : I → R | v(x), v ′′ (x) ∈ L2 (I), v(0) = v(2π), v ′ (0) = v ′ (2π)}. We aim at uk (x) ∈ H 2 (I), 1 ≤ k ≤ l. For the technical purposes, we will use the following auxiliary constrained subspace H02 (I) = {v(x) ∈ H 2 (I) | (v(x), 1)L2 (I) = 0},

(3.77)

which is a Hilbert space as well (see e.g. Chapter 2.1 of [38]). The aim is to have uk (x) ∈ H02 (I), l + 1 ≤ k ≤ N . Similarly, H01 (I) = {v(x) ∈ H 1 (I) | (v(x), 1)L2 (I) = 0}. The resulting space used to establish the existence in the sense of sequences of solutions u(x) : I → RN of system (3.67) will be the direct sum of the spaces given above, namely 2 Hc2 (I, RN ) = ⊕lk=1 H 2 (I) ⊕N k=l+1 H0 (I).

The corresponding Sobolev norm is given by

122

3 Solvability of some integro-differential equations with drift and superdiffusion

∥u∥2Hc2 (I,RN ) :=

N X

{∥uk ∥2L2 (I) + ∥u′′k ∥2L2 (I) },

k=1

with u(x) : I → RN . Another useful norm is ∥u∥2L2 (I,RN ) :=

N X

∥uk ∥2L2 (I) .

k=1

We establish that system (3.71) in this case defines a map τa,b : Hc2 (I, RN ) → Hc2 (I, RN ). This map will be a strict contraction under the stated technical conditions. Theorem 3.7 Let Ω = I, N ≥ 2, 1 ≤ l ≤ N − 1, 1 ≤ k ≤ N , bk ∈ R, k (x) bk ̸= 0 and Gk (x) : I → R, Gk (x) ∈ C(I), dGdx ∈ L1 (I), Gk (0) = Gk (2π), N Fk (u, 0) = Fk (u, 2π) for u ∈ R and Assumption 3.2 hold. I) Let ak > 0 for 1 ≤ k ≤ l. II)Let √ ak = 0 for l + 1 ≤ k ≤ N , orthogonality relations (3.125) hold and 2 πNa,b L < 1 with Na,b defined in (3.124). Then the map τa,b v = u on Hc2 (I, RN ) defined by system of equations (3.71) has a unique fixed point v (a,b) , the only solution of system (3.67) in Hc2 (I, RN ). The fixed point v (a,b) is nontrivial on the interval I provided that the Fourier coefficients Gk,n Fk (0, x)n ̸= 0 for some 1 ≤ k ≤ N and a certin n ∈ Z. Remark 3.4 We use the constrained subspace H02 (I) in the direct sum of spaces Hc2 (I, RN ), such that the Fredholm operators r d2 d − 2 − bk : H01 (I) → L2 (I) for l + 1 ≤ k ≤ N have the trivial kernels. dx dx To show the existence in the sense of sequences of solutions for our integrodifferential system of equations on the interval I, we consider the sequence of approximate systems, similarly to the situation on the whole real line with m ∈ N, 1 ≤ k ≤ N and the constants ak ≥ 0, bk ∈ R, bk ̸= 0, so that r (m) du d2 (m) (m) − − 2 u k + bk k + a k u k + dx dx Z 2π (m) (m) (m) + Gk,m (x − y)Fk (u1 (y), u2 (y), ..., uN (y), y)dy = 0. (3.78) 0

The final main statement of section 3.4 is as follows. Theorem 3.8 Let Ω = I, N ≥ 2, 1 ≤ l ≤ N − 1, 1 ≤ k ≤ N , bk ∈ R, bk ̸= 0, m ∈ N,

3.3 The whole real line case: system case

123

Gk,m (x) : I → R, Gk,m (0) = Gk,m (2π), Gk,m (x) ∈ C(I),

dGk,m (x) ∈ L1 (I), dx

so that Gk,m (x) → Gk (x)

in

C(I),

dGk,m (x) dGk (x) → dx dx

in

L1 (I)

as

m → ∞,

Fk (u, 0) = Fk (u, 2π) for u ∈ RN . Let Assumption 3.2 hold. I) Let ak > 0 for 1 ≤ k ≤ l. II)Let ak = 0 for l + 1 ≤ k ≤ N . Assume that orthogonality relations (3.130) are valid along with upper bound (3.131). Then each system (3.78) possesses a unique solution u(m) (x) ∈ Hc2 (I, RN ) and the limiting system of equations (3.67) has a unique solution u(x) ∈ Hc2 (I, RN ), so that u(m) (x) → u(x) in Hc2 (I, RN ) as m → ∞. The unique solution u(m) (x) of each system of equations (3.78) does not vanish identically on the interval I provided that the Fourier coefficients Gk,m,n Fk (0, x)n ̸= 0 for some 1 ≤ k ≤ N and a certain n ∈ Z. Similarly, the unique solution u(x) of limiting system (3.67) is nontrivial on I if Gk,n Fk (0, x)n ̸= 0 for a certain 1 ≤ k ≤ N and some n ∈ Z. Remark 3.5 In sections 3.3 and 3.4 we deal with the real valued vector functions by means of the assumptions on Fk (u, x), Gk,m (x) and Gk (x) contained in the integral terms of the approximate and limiting systems of equations discussed above. Remark 3.6 The significance of Theorems 3.6 and 3.8 is the continuous dependence of the solutions of our systems with respect to the integral kernels.

3.3 The whole real line case: system case Proof of Theorem 3.5. First we suppose that in the case of Ω = R for some v ∈ H 2 (R, RN ) there exist two solutions u(1),(2) ∈ H 2 (R, RN ) of system (3.71). Then their difference w(x) := u(1) (x) − u(2) (x) ∈ H 2 (R, RN ) will satisfy the homogeneous system of equations r d2 dwk − 2 w k − bk − ak wk = 0, 1 ≤ k ≤ N. dx dx Since each operator La,b,k : H 1 (R) → L2 (R) defined in (3.75) does not have any nontrivial zero modes, w(x) vanishes identically in R. Let us choose arbitrarily v(x) ∈ H 2 (R, RN ) and apply the standard Fourier transform (3.27) to both sides of system (3.71). Thus, we obtain

124

3 Solvability of some integro-differential equations with drift and superdiffusion

ck (p)fbk (p) ck (p)fbk (p) √ p2 G G , p2 u ck (p) = 2π , 1 ≤ k ≤ N, |p| − ak − ibk p |p| − ak − ibk p (3.79) where fbk (p) stands for the Fourier image of Fk (v(x), x). Evidently, we have the upper bounds √ √ |c uk (p)| ≤ 2πNa,b,k |fbk (p)| and |p2 u ck (p)| ≤ 2πNa,b,k |fbk (p)|, 1 ≤ k ≤ N. u ck (p) =

√

2π

Note that Na,b,k < ∞ by virtue of Lemma 3.5 of Appendix without any orthogonality conditions if ak > 0 and under orthogonality relation (3.102) for ak = 0. This allows us to derive the estimate from above on the norm ∥u∥2H 2 (R,RN ) =

N X

{∥c uk (p)∥2L2 (R) + ∥p2 u ck (p)∥2L2 (R) }

k=1

≤

2 4πNa,b

N X

∥Fk (v(x), x)∥2L2 (R)

(3.80)

k=1

with Na,b defined in (3.101). Clearly, the right side of (3.80) is finite via inequality (3.69) of Assumption 3.2 above since |v(x)|RN ∈ L2 (R). Hence, for an arbitrary v(x) ∈ H 2 (R, RN ) there exists a unique solution u(x) ∈ H 2 (R, RN ) of system (3.71) with its Fourier image given by (3.79) and the map Ta,b : H 2 (R, RN ) → H 2 (R, RN ) is well defined. This allows us to choose arbitrary v (1),(2) (x) ∈ H 2 (R, RN ), so that their images u(1),(2) = Ta,b v (1),(2) ∈ H 2 (R, RN ). By means of (3.71), we have for 1 ≤ k ≤ N (1)

r −

du d2 (1) u − bk k dx2 k dx

(1)

− auk

Z

∞

= −∞

(1)

(1)

(1)

Gk (x − y)Fk (v1 (y), v2 (y), ..., vN (y), y)dy, (3.81)

r

(2)

du d2 (2) − 2 uk − b k k dx dx

(2)

− auk

Z

∞

= −∞

(2)

(2)

(2)

Gk (x − y)Fk (v1 (y), v2 (y), ..., vN (y), y)dy. (3.82)

We apply the standard Fourier transform (3.27) to both sides of systems (3.81) and (3.82). This yields for 1 ≤ k ≤ N (1) ck (p)fd √ G d (1) k (p) uk (p) = 2π , |p| − ak − ibk p

(1) ck (p)fd √ p2 G d (1) k (p) p2 uk (p) = 2π , |p| − ak − ibk p

(2) ck (p)fd √ G d (2) k (p) uk (p) = 2π , |p| − ak − ibk p

(2) ck (p)fd √ p2 G d (2) k (p) p2 uk (p) = 2π . |p| − ak − ibk p

d d (1) (2) Here fk (p) and fk (p) denote the Fourier transforms of Fk (v (1) (x), x) and Fk (v (2) (x), x) respectively. Evidently, we have the upper bounds

3.3 The whole real line case: system case

125

√ d d d d (1) (2) (1) (2) uk (p) − uk (p) ≤ 2πNa,b,k fk (p) − fk (p) , √ d d 2d d (1) (2) (1) (2) p uk (p) − p2 uk (p) ≤ 2πNa,b,k fk (p) − fk (p) , where 1 ≤ k ≤ N . This allows us to derive the inequality for the norms 2 ∥u(1) − u(2) ∥2H 2 (R,RN ) ≤ 4πNa,b

N X

∥Fk (v (1) (x), x) − Fk (v (2) (x), x)∥2L2 (R) .

k=1 (1),(2)

Obviously, vk (x) ∈ H 2 (R) ⊂ L∞ (R) due to the Sobolev embedding. By means of (3.70) of Assumption 3.2 we easily derive √ ∥Ta,b v1 − Ta,b v2 ∥H 2 (R,RN ) ≤ 2 πNa,b L∥v1 − v2 ∥H 2 (R,RN ) . (3.83) The constant in the right side of (3.83) is less than via the one of our assumptions. By virtue of the fixed point theorem, there exists a unique vector function v (a,b) ∈ H 2 (R, RN ), such that Ta,b v (a,b) = v (a,b) . This is the only solution of the system of equations (3.67) in H 2 (R, RN ). Suppose v (a,b) (x) vanishes identically in R. This will be a contradiction to our assumption that for some 1 ≤ k ≤ N the Fourier transforms of Gk (x) and Fk (0, x) are nontrivial on a set of nonzero Lebesgue measure on the real line. ⊔ ⊓ We turn our attention to establishing the existence in the sense of sequences of the solutions for our system of integro-differential equation on R. Proof of Theorem 3.6. By means of the result of Theorem 3.5 above, each system of equations (3.76) admits a unique solution u(m) (x) ∈ H 2 (R, RN ), m ∈ N. Limiting system (3.67) has a unique solution u(x) ∈ H 2 (R, RN ) by virtue of Lemma 3.6 below along with Theorem 3.5. We apply the standard Fourier transform (3.27) to both sides of problems (3.67) and (3.76). This gives us for 1 ≤ k ≤ N, m ∈ N u ck (p) =

√

2π

ck (p)c G φk (p) , |p| − ak − ibk p

[ √ G [ d k,m (p)φ k,m (p) (m) uk (p) = 2π . |p| − ak − ibk p

(3.84)

Here φ ck (p) and φ [ k,m (p) denote the Fourier transforms of Fk (u(x), x) and Fk (u(m) (x), x) respectively. Evidently,

√

ck (p) [ G G d

k,m (p) (m) ck (p) ≤ 2π − |c φk (p)|+ uk (p) − u

|p| − ak − ibk p |p| − ak − ibk p ∞ L



[ √ G (p)

k,m + 2π

|p| − ak − ibk p

|φ [ ck (p)|. k,m (p) − φ L∞ (R)

(R)

126

3 Solvability of some integro-differential equations with drift and superdiffusion

Hence, (m)

∥uk

− uk ∥ L 2 ( R) ≤

√



ck (p) \ G G

k,m (p) −

|p| − ak − ibk p |p| − ak − ibk p

∥Fk (u(x), x)∥L2 (R) +

2π

L ∞ ( R)



[ G

k,m (p) + 2π

|p| − ak − ibk p √

∥Fk (u(m) (x), x) − Fk (u(x), x)∥L2 (R) . L∞ (R)

By means of inequality (3.70) of Assumption 3.2 above we arrive at v uN uX t ∥Fk (u(m) (x), x) − Fk (u(x), x)∥2L2 (R) ≤ L∥u(m) (x) − u(x)∥L2 (R,RN ) . k=1

(3.85) (m) Obviously, uk (x), uk (x) ∈ H 2 (R) ⊂ L∞ (R) for 1 ≤ k ≤ N, m ∈ N via the Sobolev embedding. We derive ∥u(m) (x) − u(x)∥2L2 (R,RN )

2

N

X ck (p) [ G G

k,m (p) ≤4π − ∥Fk (u(x), x)∥2L2 (R)

|p| − ak − ibk p |p| − ak − ibk p ∞ k=1 L (R) " #2 (m)

+ 4π Na,b

L2 ∥u(m) (x) − u(x)∥2L2 (R,RN ) ,

so that via (3.109), we have ∥u(m) (x) − u(x)∥2L2 (R,RN )

2 N

ck (p) [ 4π X G G

k,m (p) ≤ −

|p| − ak − ibk p |p| − ak − ibk p ∞ ε(2 − ε) k=1

L

∥Fk (u(x), x)∥2L2 (R) . (R)

Upper bound (3.69) of Assumption 3.2 above gives us that Fk (u(x), x) ∈ L2 (R), 1 ≤ k ≤ N for u(x) ∈ H 2 (R, RN ). Thus, u(m) (x) → u(x),

m→∞

(3.86)

in L2 (R, RN ) by means of the result of Lemma 3.6 of Appendix. Clearly, for 1 ≤ k ≤ N, m ∈ N we have p2 u ck (p) =

√

2π

Hence, we obtain

ck (p)c p2 G φk (p) , |p| − ak − ibk p

[ √ p2 G [ d k,m (p)φ k,m (p) (m) p2 uk (p) = 2π . |p| − ak − ibk p

3.4 The problem on the finite interval: system case

127

√ p2 G

2c [ (p) p G (p) 2d

k,m k (m) ck (p) ≤ 2π − p uk (p)−p2 u

|p| − ak − ibk p |p| − ak − ibk p

p2 G

[

k,m (p) + 2π

|p| − ak − ibk p √

|c φk (p)|+ L∞ (R)

|φ [ ck (p)|. k,m (p) − φ L∞ (R)

Using inequality (3.85), we arrive at

d2 u(m) d2 uk

k −

dx2 dx2 2 L (R)

2[ ck (p) √ p2 G

p Gk,m (p)

≤ 2π − ∥Fk (u(x), x)∥L2 (R)

|p| − ak − ibk p |p| − ak − ibk p ∞ L (R)

2[ √

p Gk,m (p) + 2π L∥u(m) (x) − u(x)∥L2 (R,RN ) .

|p| − ak − ibk p ∞ L

(R)

By virtue of the result of Lemma 3.6 of Appendix along with (3.86), we establish that d2 u d2 u(m) → 2 2 dx dx

in

L2 (R, RN ),

m → ∞.

Definition (3.73) of the norm implies that u(m) (x) → u(x) in H 2 (R, RN ) as m → ∞. We suppose that the unique solution u(m) (x) of the system of equations (3.76) studied above vanishes on the whole real line for some m ∈ N. This will contradict to our assumption above that for some 1 ≤ k ≤ N the Fourier transforms of Gk,m (x) and Fk (0, x) are nontrivial on a set of nonzero Lebesgue measure on the real line. The similar reasoning holds for the unique solution u(x) of limiting system of equations (3.67). ⊔ ⊓

3.4 The problem on the finite interval: system case Proof of Theorem 3.7. Evidently, each operator contained in the left side of the system of equations (3.71) r d2 d La,b,k := − 2 − bk − ak : H 1 (I) → L2 (I), 1 ≤ k ≤ l (3.87) dx dx with the constants ak > 0, bk ∈ R, bk ̸= 0 is Fredholm, non-selfadjoint. Its set of eigenvalues is given by

128

3 Solvability of some integro-differential equations with drift and superdiffusion

λa,b,k (n) = |n| − ak − ibk n,

n ∈ Z.

(3.88)

einx Its eigenfunctions are the standard Fourier harmonics √ , n ∈ Z. When 2π ak = 0, we will exploit the analogous ideas in the constrained subspace (3.77) instead of H 2 (I). Clearly, the eigenvalues of each operator La,b,k are simple, as distinct from the analogical situation without the drift term, when the eigenvalues corresponding to n ̸= 0 are two-fold degenerate (see [60]). Let us suppose that for a certain v(x) ∈ Hc2 (I, RN ) there exist two solutions u(1),(2) (x) ∈ Hc2 (I, RN ) of system (3.71) with Ω = I. Then the vector function w(x) := u(1) (x) − u(2) (x) ∈ Hc2 (I, RN ) will satisfy the homogeneous system of equations r d2 dwk − 2 w k − bk − ak wk = 0, 1 ≤ k ≤ N. dx dx Because the operator La,b,k : H 1 (I) → L2 (I) with ak > 0 for 1 ≤ k ≤ l discussed above does not have any nontrivial zero modes, we obtain that w(x) is trivial in I. Let us choose an arbitrary v(x) ∈ Hc2 (I, RN ) and apply the Fourier transform (3.119) to system (3.71) studied on the interval I. This gives us uk,n =

√

2π

Gk,n fk,n , |n| − ak − ibk n

n2 uk,n =

√

2π

n2 Gk,n fk,n , |n| − ak − ibk n

(3.89)

for 1 ≤ k ≤ N , n ∈ Z, where fk,n := Fk (v(x), x)n . We easily obtain the estimates from above √ √ |uk,n | ≤ 2πNa,b,k |fk,n |, |n2 uk,n | ≤ 2πNa,b,k |fk,n |. Clearly, Na,b,k < ∞ under the stated conditions via the result of Lemma 3.7 of Appendix. Therefore, " ∞ # N ∞ X X X 2 2 2 2 ∥u∥Hc2 (I,RN ) = |uk,n | + |n uk,n | k=1

n=−∞

2 ≤ 4πNa,b

N X

n=−∞

∥Fk (v(x), x)∥2L2 (I)

(3.90)

k=1

with Na,b defined in (3.124). Evidently, the right side of (3.90) is finite via inequality (3.69) of Assumption 3.2 for |v(x)|RN ∈ L2 (I). Thus, for an arbitrary v(x) ∈ Hc2 (I, RN ) there exists a unique u(x) ∈ Hc2 (I, RN ), which satisfies system (3.71) and its Fourier image is given by (3.89). Therefore, the map τa,b : Hc2 (I, RN ) → Hc2 (I, RN ) is well defined.

3.4 The problem on the finite interval: system case

129

We consider any v (1),(2) (x) ∈ Hc2 (I, RN ), so that their images under the map mentioned above u(1),(2) = τa,b v (1),(2) ∈ Hc2 (I, RN ). By means of (3.71), we have for 1 ≤ k ≤ N that (1)

r −

du d2 (1) u − bk k dx2 k dx

(1)

− a k uk

2π

Z

(1)

(1)

(1)

Gk (x − y)Fk (v1 (y), v2 (y), ..., vN (y), y)dy,

= 0

(3.91)

r

(2)

du d2 (2) − 2 uk − b k k dx dx

(2)

− a k uk

2π

Z

(2)

(2)

(2)

Gk (x − y)Fk (v1 (y), v2 (y), ..., vN (y), y)dy,

= 0

(3.92)

where ak ≥ 0, bk ∈ R, bk ̸= 0 are the contants. By virtue of Fourier transform (3.119) applied to both sides of the systems of equations (3.91) and (3.92), we easily derive for 1 ≤ k ≤ N, n ∈ Z that (1)

uk,n = (1)

n2 uk,n =

√

√

(1)

2π

Gk,n fk,n |n| − ak − ibk n

(2)

√

,

uk,n =

,

n2 uk,n =

(1)

2π

n2 Gk,n fk,n |n| − ak − ibk n

(2)

(2)

2π √

Gk,n fk,n |n| − ak − ibk n

,

(2)

2π

n2 Gk,n fk,n |n| − ak − ibk n

,

(j)

with fk,n := Fk (v (j) (x), x)n , j = 1, 2. Thus, (1)

(2)

|uk,n −uk,n | ≤

√

(1)

(2)

2πNa,b |fk,n −fk,n |,

(1)

(2)

|n2 (uk,n −uk,n )| ≤

√

(1)

(2)

2πNa,b |fk,n −fk,n |.

Hence, we estimate the norm as ∥u

(1)

−u(2) ∥2Hc2 (I,RN )

=

N h X ∞ X k=1

2 ≤ 4πNa,b

N X

n=−∞

(1) (2) |uk,n −uk,n |2 +

∞ X

i (1) (2) |n2 (uk,n −uk,n )|2 ≤

n=−∞

∥Fk (v (1) (x), x) − Fk (v (2) (x), x)∥2L2 (I) .

k=1 (1),(2) vk (x)

Evidently, ∈ H 2 (I) ⊂ L∞ (I), 1 ≤ k ≤ N by means of the Sobolev embedding. Using inequality (3.70) of Assumption 3.2, we derive ∥τa,b v (1) − τa,b v (2) ∥Hc2 (I,RN ) √ ≤2 πNa,b L∥v (1) − v (2) ∥Hc2 (I,RN ) .

(3.93)

The constant in the right side of estimate (3.93) is less than due to the one of our assumptions. Then the fixed point theorem gives us the existence and uniqueness of a vector function v (a,b) ∈ Hc2 (I, RN ) satisfying τa,b v (a,b) = v (a,b) . This is the only solution of system (3.67) in Hc2 (I, RN ). If we suppose that v (a,b) (x) vanishes identically in I, we will obtain the contradiction to our condition that Gk,n Fk (0, x)n ̸= 0 for a certain 1 ≤ k ≤ N and some n ∈ Z. ⊔ ⊓

130

3 Solvability of some integro-differential equations with drift and superdiffusion

Let us turn our attention to establishing the final main result of section 3.4. Proof of Theorem 3.8. Obviously, the limiting kernels Gk (x), 1 ≤ k ≤ N are periodic as well on our interval I (see the argument of Lemma 3.8 of Appendix). Each system (3.78) admits a unique solution u(m) (x) ∈ Hc2 (I, RN ), m ∈ N by means of the result of Theorem 3.7 above. The limiting system of equations (3.67) has a unique solution u(x) ∈ Hc2 (I, RN ) due to Lemma 3.8 below along with Theorem 3.7. We apply Fourier transform (3.119) to both sides of systems (3.67) and (3.78). This gives us uk,n =

√

2π

Gk,n φk,n , |n| − ak − ibk n

(m)

uk,n =

√

(m)

2π

Gk,m,n φk,n

|n| − ak − ibk n

,

(3.94)

(m)

with 1 ≤ k ≤ N, n ∈ Z, m ∈ N. Here φk,n and φk,n stand for the Fourier images of Fk (u(x), x) and Fk (u(m) (x), x) respectively under transform (3.119). We have a trivial estimate from above

√ Gk,m,n Gk,n

(m) |uk,n − uk,n | ≤ 2π −

|φk,n |+

|n| − ak − ibk n |n| − ak − ibk n ∞ l



√ Gk,m,n

(m) + 2π − φk,n |.

|φ

|n| − ak − ibk n ∞ k,n l

Thus, (m)

∥uk

− uk ∥L2 (I) ≤

√



Gk,m,n Gk,n

−

∥Fk (u(x), x)∥L2 (I) +

|n| − ak − ibk n |n| − ak − ibk n ∞

2π

l



√ Gk,m,n

+ 2π

∥Fk (u(m) (x), x) − Fk (u(x), x)∥L2 (I) .

|n| − ak − ibk n ∞ l

Inequality (3.70) of Assumption 3.2 above implies that v uN uX t ∥Fk (u(m) (x), x) − Fk (u(x), x)∥2L2 (I) ≤ L∥u(m) (x) − u(x)∥L2 (I,RN ) . k=1 (m) uk (x), uk (x)

Evidently, embedding. Obviously,

(3.95) ∈ H (I) ⊂ L (I), 1 ≤ k ≤ N due to the Sobolev 2

∞

3.4 The problem on the finite interval: system case

131

∥u(m) (x) − u(x)∥2L2 (I,RN )

2 N

X Gk,m,n Gk,n

≤4π −

∥F (u(x), x)∥2L2 (I)

|n| − ak − ibk n |n| − ak − ibk n ∞ k k=1 l h i2 (m) 2 (m) 2 + 4π Na,b L ∥u (x) − u(x)∥L2 (I,RN ) . Hence, we arrive at ∥u(m) (x) − u(x)∥2L2 (I,RN )

2 N

4π X Gk,m,n Gk,n

≤ −

∥Fk (u(x), x)∥2L2 (I) .

|n| − ak − ibk n |n| − ak − ibk n ∞ ε(2 − ε) k=1

l

Evidently, Fk (u(x), x) ∈ L2 (I), 1 ≤ k ≤ N for u(x) ∈ Hc2 (I, RN ) via inequality (3.69) of Assumption 3.2. By means of the result of Lemma 3.8 of Appendix we derive that u(m) (x) → u(x),

m→∞

(3.96)

in L2 (I, RN ). Clearly, (m) |n2 uk,n

− n2 uk,n | ≤

√



n2 G

n2 Gk,n

k,m,n 2π −

|φk,n |+

|n| − ak − ibk n |n| − ak − ibk n ∞ l



√

n2 Gk,m,n (m) + 2π − φk,n |.

|φ

|n| − ak − ibk n ∞ k,n l

Using (3.95) we arrive at

d2 u(m)

2 d u

k k −

dx2 dx2 2 L (I)

2 √ n2 Gk,n

n Gk,m,n

≤ 2π −

∥Fk (u(x), x)∥L2 (I)

|n| − ak − ibk n |n| − ak − ibk n ∞ l

2 √ n G

k,m,n + 2π

L∥u(m) (x) − u(x)∥L2 (I,RN ) .

|n| − ak − ibk n ∞ l

d2 u(m) d2 u → as dx2 dx2 2 N (m) 2 N m → ∞ in L (I, R ). Therefore, u (x) → u(x) in the Hc (I, R ) norm as m → ∞. By virtue of Lemma 3.8 along with (3.96), we obtain that

Suppose that u(m) (x) is trivial in the interval I for a certain m ∈ N. This will yield a contradiction to our assumption that Gk,m,n Fk (0, x)n ̸= 0 for a

132

3 Solvability of some integro-differential equations with drift and superdiffusion

certain 1 ≤ k ≤ N and some n ∈ Z. The analogical reasoning holds for the solution u(x) of the limiting system of equations (3.67). ⊔ ⊓

Appendix In the proofs of theorems of sections 3.3 and 3.4 we used several lemmas referring to them to Appendix. Although the quantities look like similar to those quantities given in Appendix for sections 3.1 and 3.2, as we will see below the proofs of lemmas 3.5-3.8 are required new ideas and additional techniques. Let Gk (x) be a function, Gk (x) : R → R, for which we denote its standard Fourier transform using the hat symbol as Z ∞ ck (p) := √1 G Gk (x)e−ipx dx, p ∈ R. (3.97) 2π −∞ Clearly, ck (p)∥L∞ (R) ≤ √1 ∥Gk (x)∥L1 (R) ∥G 2π 1 and Gk (x) = √ 2π

Z

(3.98)

∞

ck (q)eiqx dq, x ∈ R. By means of (3.98), we have G −∞

ck (p)∥L∞ (R) ∥pG



1

dGk (x) ≤√

2π dx

.

(3.99)

L1 (R)

For the technical purposes we will use the auxiliary quantities

o ck (p) p2 G

,

∞ |p| − ak − ibk p L (R) (3.100) where ak ≥ 0, bk ∈ R, bk ̸= 0 are the constants, 1 ≤ k ≤ N , N ≥ 2. Under the assumptions of Lemma 3.5 below, all the quantities (3.100) will be finite, so that Na,b := max Na,b,k < ∞. (3.101) n

Na,b,k := max

ck (p) G

,

|p| − ak − ibk p L∞ (R)



1≤k≤N

The auxiliary lemmas below are the adaptations of the ones proved in sections 3.1 and 3.2 (see also [31]) in order to study the single integro-differential equation with drift and superdiffusion, analogical to system (3.67). However, as we mentioned above these adaptations are not obvious and requires additional ideas and cumbersome works. Let us provide them for the convenience of the readers.

3.4 The problem on the finite interval: system case

133

Lemma 3.5 Let N ≥ 2, 1 ≤ k ≤ N , bk ∈ R, bk ̸= 0 and Gk (x) : R → R, Gk (x) ∈ W 1,1 (R) and 1 ≤ l ≤ N − 1. a) Let ak > 0 for 1 ≤ k ≤ l. Then Na,b,k < ∞. b) Let ak = 0 for l + 1 ≤ k ≤ N and additionally xGk (x) ∈ L1 (R). Then N0,b,k < ∞ if and only if (Gk (x), 1)L2 (R) = 0

(3.102)

is valid. Proof. First of all, it can be trivially checked that in both cases a) and b) of the lemma, under our assumptions the expressions ck (p) p2 G ∈ L∞ (R), |p| − ak − ibk p

1 ≤ k ≤ N.

(3.103)

p ck (p) ∈ L∞ (R) are bounded and pG |p| − ak − ibk p via inequality (3.99) above, which yields (3.103). We turn our attention to establishing the result of the part a) of our lemma. Let us estimate the expressions ck (p)| |G p , 1 ≤ k ≤ l. (3.104) (|p| − ak )2 + b2k p2 Evidently, the functions

Clearly, the numerator of (3.104) can be bounded from above via (3.98) and the denominator in (3.104) can be trivially estimated below by a finite, positive constant, so that ck (p) G ≤ C∥Gk (x)∥L1 (R) < ∞ |p| − ak − ibk p as assumed. Here and below C will stand for a finite, positive constant. This implies that under the given conditions, if ak > 0 we have Na,b,k < ∞. In the cases of ak = 0, we will use that p

Z ck (p) = G ck (0) + G 0

ck (s) dG ds. ds

Thus, R p dGck (s) ck (p) ck (0) ds G G = + 0 ds . |p| − ibk p |p| − ibk p |p| − ibk p

(3.105)

Using definition (3.97) of the standard Fourier transform, we easily obtain

134

3 Solvability of some integro-differential equations with drift and superdiffusion

dG 1 ck (p) ≤ √ ∥xGk (x)∥L1 (R) . dp 2π Hence, R p c dGk (s) 0 ds ds ∥xGk (x)∥L1 (R) 0 for 1 ≤ k ≤ l. b) Let ak = 0 for l + 1 ≤ k ≤ N and in addition xGk,m (x) ∈ L1 (R), so that xGk,m (x) → xGk (x) in L1 (R) as m → ∞ and (Gk,m (x), 1)L2 (R) = 0,

m∈N

(3.108)

is valid. Let in addition √ (m) 2 πNa,b L ≤ 1 − ε

(3.109)

for all m ∈ N as well with a certain fixed 0 < ε < 1. Then, for all 1 ≤ k ≤ N , we have ck (p) [ G G k,m (p) → , |p| − ak − ibk p |p| − ak − ibk p

m → ∞,

(3.110)

3.4 The problem on the finite interval: system case

135

ck (p) [ p2 G p2 G k,m (p) → , |p| − ak − ibk p |p| − ak − ibk p in L∞ (R), so that



[ G

k,m (p)

|p| − ak − ibk p

p2 G

[

k,m (p)

|p| − ak − ibk p Furthermore,

m→∞

L∞ (R)



ck (p) G

→

|p| − ak − ibk p

L∞ (R)

L∞ (R)



ck (p) p2 G

→

|p| − ak − ibk p

L∞ (R)

(3.111)

,

m → ∞,

(3.112)

,

m → ∞.

(3.113)

√ 2 πNa,b L ≤ 1 − ε.

(3.114)

Proof. By means of inequality (3.98), we easily obtain for 1 ≤ k ≤ N that 1 c [ ∥G ∥Gk,m (x) − Gk (x)∥L1 (R) → 0, k,m (p) − Gk (p)∥L∞ (R) ≤ √ 2π

m→∞

(3.115) due to the one of our assumptions. Evidently, (3.112) and (3.113) will trivially follow from the statements of (3.110) and (3.111) respectively by virtue of the standard triangle inequality. p We use the fact that the functions ∈ L∞ (R) along with the |p| − ak − ibk p analog of bound (3.99). This yields p2 G ck (p) [ p2 G k,m (p) c [ − ≤ C∥p[G k,m (p) − Gk (p)]∥L∞ (R) ≤ |p| − ak − ibk p |p| − ak − ibk p

C

dGk,m (x) dGk (x) ≤√ −

dx dx 2π

. L1 (R)

Thus,

p2 G

ck (p) \ p2 G

k,m (p) −

|p| − ak − ibk p |p| − ak − ibk p

L ∞ ( R)





C dGk,m (x) dGk (x) ≤ √ −

dx dx 2π

→0

L 1 ( R)

as m → ∞ via the one of our assumptions, so that (3.111) is valid. Let us establish (3.110) in the situation a) when ak > 0. For that purpose we need to consider ck (p)| [ (p) − G |G p k,m , 1 ≤ k ≤ l. (3.116) 2 (|p| − ak ) + b2k p2 Evidently, the denominator in fraction (3.116) can be bounded from below by a positive constant and the numerator in (3.116) can be estimated from above by means of (3.115). Hence,

136

3 Solvability of some integro-differential equations with drift and superdiffusion



ck (p) [ G G

k,m (p) −

|p| − ak − ibk p |p| − ak − ibk p

≤ C∥Gk,m (x) − Gk (x)∥L1 (R) → 0 L∞ (R)

as m → ∞ due to the one of the assumptions, so that (3.110) is valid in the case a) of the lemma. Then we turn our attention to proving (3.110) in the situation b) when ak = 0. In this case orthogonality conditions (3.108) are valid as assumed. We easily derive that the analogical statements will hold in the limit. Evidently, |(Gk (x), 1)L2 (R) | = |(Gk (x)−Gk,m (x), 1)L2 (R) | ≤ ∥Gk,m (x)−Gk (x)∥L1 (R) → 0 as m → ∞ by virtue of the one of our assumptions. Thus, (Gk (x), 1)L2 (R) = 0,

l+1≤k ≤N

(3.117)

is valid. Obviously, we have p

Z ck (p) = G ck (0) + G 0

ck (s) dG ds, ds

p

Z [ [ G k,m (p) = Gk,m (0) + 0

[ dG k,m (s) ds, ds

with l + 1 ≤ k ≤ N, m ∈ N. Formulas (3.117) and (3.108) imply that ck (0) = 0, G

[ G k,m (0) = 0,

l + 1 ≤ k ≤ N,

m ∈ N.

Hence, i R p h dG \ ck (s) dG k,m (s) G − ds c [ ds ds Gk (p) 0 k,m (p) − = . |p| − ibk p |p| − ibk p |p| − ibk p

(3.118)

Using the definition of the standard Fourier transform (3.97) we easily derive dG ck (p) dG 1 [ k,m (p) − ≤ √ ∥xGk,m (x) − xGk (x)∥L1 (R) . dp dp 2π This allows us to obtain the estimate from above on the right side of (3.118) as ∥xGk,m (x) − xGk (x)∥L1 (R) p , 2π(1 + b2k ) such that

G ck (p) G

[

k,m (p) −

|p| − ibk p |p| − ibk p

≤ L∞ (R)

∥xGk,m (x) − xGk (x)∥L1 (R) p → 0, 2π(1 + b2k )

m→∞

3.4 The problem on the finite interval: system case

137

as assumed. Therefore, (3.110) is valid in the case b) of the lemma when ak = 0. Evidently, under the stated conditions we have (m)

Na,b,k < ∞, Na,b,k < ∞, m ∈ N,

1 ≤ k ≤ N, ak ≥ 0, bk ∈ R, bk ̸= 0

by means of the result of Lemma 3.5 above. We have inequalities (3.109). An trivial limiting argument using (3.112) and (3.113) gives us (3.114). ⊔ ⊓ Consider the function Gk (x) : I → R, so that Gk (0) = Gk (2π). Its Fourier transform on our finite interval is given by Z 2π e−inx Gk,n := Gk (x) √ dx, n ∈ Z, (3.119) 2π 0 such that Gk (x) =

∞ X

einx Gk,n √ . Obviously, the upper bound 2π n=−∞ 1 ∥Gk,n ∥l∞ ≤ √ ∥Gk (x)∥L1 (I) 2π

(3.120)

is valid. Evidently, if our function is continuous on the interval I, we have the estimate from above ∥Gk (x)∥L1 (I) ≤ 2π∥Gk (x)∥C(I) .

(3.121)

The upper bound ∥nGk,n ∥l∞



1

dGk (x) ≤√

2π dx

(3.122) L1 (I)

trivially comes from (3.120). Analogously to the whole real line case, we define

) (



n2 Gk,n Gk,n



Na,b,k := max , (3.123)

,

|n| − ak − ibk n ∞

|n| − ak − ibk n ∞ l

l

where ak ≥ 0, bk ∈ R, bk ̸= 0 are the constants, 1 ≤ k ≤ N, N ≥ 2. Let N0,b,k denote (3.123) when ak vanishes. Under the conditions of Lemma 3.7 below, the expressions Na,b,k will be finite. This will enable us to introduce Na,b := max Na,b,k < ∞. 1≤k≤N

(3.124)

We have the following elementary statement. Lemma 3.7 Let N ≥ 2, 1 ≤ k ≤ N , bk ∈ R, bk ̸= 0, 1 ≤ l ≤ N − 1 and

138

3 Solvability of some integro-differential equations with drift and superdiffusion

Gk (x) : I → R, Gk (x) ∈ C(I),

dGk (x) ∈ L1 (I), Gk (0) = Gk (2π). dx

a) Let ak > 0 for 1 ≤ k ≤ l. Then Na,b,k < ∞. b) If ak = 0 for l+1 ≤ k ≤ N then N0,b,k < ∞ if and only if the orthogonality relation (Gk (x), 1)L2 (I) = 0 (3.125) holds. Proof. It can be easily checked that in both cases a) and b) of our lemma under the given conditions we have n2 Gk,n ∈ l∞ , |n| − ak − ibk n

1 ≤ k ≤ N.

(3.126)

n

∈ l∞ and nGk,n ∈ l∞ via inequality (3.122) along |n| − ak − ibk n with the one of the stated assumptions. Hence (3.126) is valid. Clearly,

Let us establish the statement of the part a) of the lemma. For that purpose, we need to consider the expression p

|Gk,n | , (|n| − ak )2 + b2k n2

1 ≤ k ≤ l.

(3.127)

Evidently, the denominator in (3.127) can be easily bounded from below by a positive constant. The numerator in (3.127) can be trivially estimated from above by means of (3.120) along with (3.121). Hence, Na,b,k < ∞ in the case when ak > 0. Let us demonstrate the validity of the result of the lemma in the situation when ak = 0. Obviously, G k,n , l + 1 ≤ k ≤ N |n| − ibk n is bounded if and only if Gk,0 = 0. This is equivalent to orthogonality condition (3.125). In this case we easily arrive at for l + 1 ≤ k ≤ N that G √ ∥Gk (x)∥C(I) ∥Gk (x)∥L1 (I) 1 k,n p ≤ 2π p 0 for 1 ≤ k ≤ l. b) Let ak = 0 for l + 1 ≤ k ≤ N and in addition (Gk,m (x), 1)L2 (I) = 0, We also assume that

m ∈ N.

√ (m) 2 πNa,b L ≤ 1 − ε

(3.130)

(3.131)

is valid for all m ∈ N as well with some fixed 0 < ε < 1. Then, for all 1 ≤ k ≤ N , we have Gk,m,n Gk,n → , |n| − ak − ibk n |n| − ak − ibk n

m → ∞,

(3.132)

n2 Gk,m,n n2 Gk,n → , |n| − ak − ibk n |n| − ak − ibk n

m→∞

(3.133)

in l∞ , so that





Gk,m,n Gk,n



→

,

|n| − ak − ibk n ∞

|n| − ak − ibk n ∞ l



n2 G

n2 Gk,n



k,m,n

→

,

|n| − ak − ibk n ∞

|n| − ak − ibk n ∞ l

Furthermore, the estimate

m → ∞,

(3.134)

m → ∞.

(3.135)

l

l

140

3 Solvability of some integro-differential equations with drift and superdiffusion

√ 2 πNa,b L ≤ 1 − ε

(3.136)

holds. Proof. Obviously, under the stated assumptions, the limiting kernels Gk (x), 1 ≤ k ≤ N are periodic as well. Indeed, we easily obtain |Gk (0) − Gk (2π)| ≤ |Gk (0) − Gk,m (0)| + |Gk,m (2π) − Gk (2π)| ≤ 2∥Gk,m (x) − Gk (x)∥C(I) → 0 as m → ∞ as assumed. Thus, Gk (0) = Gk (2π), 1 ≤ k ≤ N . By virtue of (3.120) along with (3.121) we arrive at 1 ∥Gk,m,n − Gk,n ∥l∞ ≤ √ ∥Gk,m − Gk ∥L1 (I) 2π √ ≤ 2π∥Gk,m − Gk ∥C(I) → 0,

m→∞

(3.137)

due to the one of our assumptions. It can be trivially checked that the statements of (3.132) and (3.133) will imply (3.134) and (3.135) respectively via the triangle inequality. Using (3.122), we obtain the estimate from above

n2 G

n2 Gk,n

k,m,n −

|n| − ak − ibk n |n| − ak − ibk n ∞ l



dG (x) dG (x) 1 n

k,m

k − , ≤√



dx dx 1 2π |n| − ak − ibk n ∞ l

L (I)

which tends to zero as m → ∞ as assumed, so that (3.133) is valid. Let us establish (3.132) in the situation a) when ak > 0. For that purpose, we need to treat |G − Gk,n | p k,m,n , 1 ≤ k ≤ l. (3.138) (|n| − ak )2 + b2k n2 Obviously, the denominator of (3.138) can be bounded from below by a positive constant and the numerator estimated from above via (3.137). This gives us (3.132) for ak > 0. Let us demonstrate the validity of (3.132) in the case case b) when ak = 0. By means of the one of the given assumptions, we have orthogonality conditions (3.130). It can be trivially checked that the analogical relations holds in the limit. Indeed, |(Gk (x), 1)L2 (I) | = |(Gk (x) − Gk,m (x), 1)L2 (I) | ≤ 2π∥Gk,m (x) − Gk (x)∥C(I) → 0, via the one of our assumptions. Thus,

m→∞

3.4 The problem on the finite interval: system case

(Gk (x), 1)L2 (I) = 0,

141

l + 1 ≤ k ≤ N.

This is equivalent to Gk,0 = 0, l + 1 ≤ k ≤ N . Evidently, Gk,m,0 = 0, l + 1 ≤ k ≤ N, m ∈ N by virtue of orthogonality condition (3.130). Using (3.137), we easily obtain that √ G 2π∥Gk,m (x) − Gk (x)∥C(I) k,m,n − Gk,n p . ≤ |n| − ibk n 1 + b2k Since the norm in the right side of this estimate from above tends to zero as m → ∞, (3.132) holds in the case when ak = 0 as well. Clearly, under the stated assumptions we have (m)

Na,b,k < ∞, Na,b,k < ∞, m ∈ N, 1 ≤ k ≤ N, ak ≥ 0, bk ∈ R, bk ̸= 0 by virtue of the result of our Lemma 3.7 above. We assume the validity of upper bound (3.131). A simple limiting argument using (3.134) and (3.135) gives us (3.136). ⊔ ⊓

Chapter 4

Existence of solutions for some non-Fredholm integro-differential equations with mixed diffusion

We establish the existence in the sense of sequences of solutions for certain integro-differential type equations in two dimensions involving the normal diffusion in one direction and the anomalous diffusion in the other direction in H 2 (R2 ) via the fixed point technique. The elliptic equation contains a second order differential operator without the Fredholm property. It is proved that, under the reasonable technical conditions, the convergence in L1 (R2 ) of the integral kernels implies the existence and convergence in H 2 (R2 ) of the solutions. Chapter 4 consists of two sections. Section 4.1 is devoted to scalar nonlinear equations in the presence of the mixed-diffusion type mentioned above. In section 4.2, we consider the analogous problem for a system of equations.

4.1 Mixed-diffusion: scalar case In this section we treat a class of stationary nonlinear problems, for which the Fredholm property may not be satisfied: !s Z ∂2u ∂2 − − u + G(x − y)F (u(y), y)dy = 0, 0 < s < 1, (4.1) ∂x21 ∂x22 R2 where x = (x1 , x2 ) ∈ R2 , y = (y1 , y2 ) ∈ R2 . Here the operator !s ∂2 ∂2 Ls := − 2 + − : H 2 (R2 ) → L2 (R2 ), 0 < s < 1 ∂x1 ∂x22

(4.2)

is defined via section 1.4. The novelty of this section (as well as this chapter) is that in the diffusion term we add the standard minus Laplacian in the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Efendiev, Linear and Nonlinear Non-Fredholm Operators, https://doi.org/10.1007/978-981-19-9880-5_4

143

144

4 Existence of solutions for some non-Fredholm integro-differential equations

x1 variable with the negative Laplace operator in x2 raised to a fractional power. Such model is new and not much is understood about it, especially in the context of the integro-differential equations. The difficulty we have to overcome is that such problem becomes anisotropic and it is more technical to obtain the desired estimates when dealing with it. Formulation of the results The nonlinear part of equation (4.1) will satisfy the following regularity conditions. Assumption 4.1 Function F (u, x) : R × R2 → R is satisfying the Carath´eodory condition , such that |F (u, x)| ≤ k|u| + h(x)

f or

u ∈ R, x ∈ R2

(4.3)

with a constant k > 0 and h(x) : R2 → R+ , h(x) ∈ L2 (R2 ). Moreover, it is a Lipschitz continuous function, such that |F (u1 , x) − F (u2 , x)| ≤ l|u1 − u2 |

f or

any

u1,2 ∈ R,

x ∈ R2

(4.4)

with a constant l > 0. In order to study of the existence of solutions of (4.1), we introduce the auxiliary equation !s Z ∂2u ∂2 − 2+ − u= G(x − y)F (v(y), y)dy, 0 < s < 1. (4.5) ∂x1 ∂x22 R2 We denote

Z

f1 (x)f¯2 (x)dx,

(f1 (x), f2 (x))L2 (R2 ) :=

(4.6)

R2

In this section we work in the space of the two dimensions, such that the appropriate Sobolev space is equipped with the norm ∥u∥2H 2 (R2 ) := ∥u∥2L2 (R2 ) + ∥∆u∥2L2 (R2 ) .

(4.7)

In the equation above we are dealing with the operator Ls defined in (4.2). By virtue of the standard Fourier transform (1.79), it can be easily checked that its essential spectrum is given by λs (p) = p21 + |p2 |2s ,

p = (p1 , p2 ) ∈ R2 .

(4.8)

Since set (4.8) contains the origin, our operator Ls fails to satisfy the Fredholm property, which is the obstacle to solve our equation. This chapter is related to our article [28] since we also deal with the non Fredholm operator, now involved in the problem, which is not linear anymore

4.1 Mixed-diffusion: scalar case

145

and contains the nonlocal terms. Currently, as distinct from [30], the space dimension is restricted to d = 2 to avoid the extra technicalities. Theorem 4.1 Let Assumption 4.1 hold, 0 < s < 1, the function G(x) : R2 → R, such that G(x) ∈ L1 (R2 ) and x2 G(x) ∈ L1 (R2 ). Moreover, (−∆)1−s G(x) ∈ L1 (R2 ). 1 We also assume that orthogonality conditions (4.21), (4.22) hold if 0 < s ≤ 2 1 and relations (4.21), (4.22) and (4.23) are valid for < s < 1 and that 2 √ 2 2 2 2πN2,s l < 1. Then the map T2,s v = u on H (R ) defined by problem (4.5) admits a unique fixed point v2,s , which is the only solution of equation (4.1) in H 2 (R2 ). This fixed point v2,s is nontrivial provided the intersection of supports of the b is a set of nonzero Fourier transforms of functions supp F\ (0, x) ∩ supp G 2 Lebesgue measure in R . Related to equation (4.1) in the space of two dimensions, we study the sequence of approximate equations with m ∈ N !s Z ∂ 2 um ∂2 − − 2 um + Gm (x−y)F (um (y), y)dy = 0, 0 < s < 1. (4.9) ∂x21 ∂x2 R2 The sequence of kernels {Gm (x)}∞ m=1 tends to G(x) as m → ∞ in the appropriate function spaces discussed below. We will show that, under the appropriate technical conditions, each of equations (4.9) has a unique solution um (x) ∈ H 2 (R2 ), the limiting problem (4.1) admits a unique solution u(x) ∈ H 2 (R2 ), and um (x) → u(x) in H 2 (R2 ) as m → ∞. In this case, the solvability relations can be formulated for the iterated kernels Gm . They yield the convergence of the kernels in terms of the Fourier transforms (see Appendix) and, as a consequence, the convergence or the solutions (Theorem 4.2 below). Our second main proposition is as follows. Theorem 4.2 Let Assumption 4.1 hold, 0 < s < 1, m ∈ N, the functions Gm (x) : R2 → R are such that Gm (x) ∈ L1 (R2 ), x2 Gm (x) ∈ L1 (R2 ) and (−∆)1−s Gm (x) ∈ L1 (R2 ). Moreover, Gm (x) → G(x) in L1 (R2 ), x2 Gm (x) → x2 G(x) and (−∆)1−s Gm (x) → (−∆)1−s G(x) in L1 (R2 ) as m → ∞. We also assume that for all m ∈ N orthogonality conditions (4.38), (4.39) 1 hold if 0 < s ≤ and relations (4.38), (4.39) and (4.40) are valid for 2 1 < s < 1. Furthermore, we suppose that (4.41) holds for all m ∈ N with 2 a certain 0 < ε < 1.

146

4 Existence of solutions for some non-Fredholm integro-differential equations

Then each problem (4.9) admits a unique solution um (x) ∈ H 2 (R2 ), limiting equation (4.1) possesses a unique solution u(x) ∈ H 2 (R2 ) and um (x) → u(x) in H 2 (R2 ) as m → ∞. The unique solution um (x) of each problem (4.9) is nontrivial provided that the intersection of supports of the Fourier transforms of functions b m is a set of nonzero Lebesgue measure in R2 . Simsupp F\ (0, x) ∩ supp G ilarly, the unique solution u(x) of limiting equation (4.1) does not vanish b is a set of nonzero Lebesgue measure in identically if supp F\ (0, x) ∩ supp G R2 . Remark 4.1 In this book we work with real valued functions by virtue of the assumptions on F (u, x), Gm (x) and G(x) involved in the nonlocal terms of the iterated and limiting problems discussed above. Remark 4.2 The importance of Theorem 4.2 above is the continuous dependence of solutions with respect to the integral kernels. Proof of Theorem 4.1. Let us first suppose that for a certain v(x) ∈ H 2 (R2 ) there exist two solutions u1,2 (x) ∈ H 2 (R2 ) of problem (4.5). Then their difference w(x) := u1 (x) − u2 (x) ∈ H 2 (R2 ) will be a solution of the homogeneous equation !s ∂2 ∂2w − 2 + − w = 0. ∂x1 ∂x22 Because the operator Ls : H 2 (R2 ) → L2 (R2 ) defined in (4.2) does not have any nontrivial zero modes, the function w(x) vanishes in the space of two dimensions. We choose arbitrarily v(x) ∈ H 2 (R2 ). Let us apply the standard Fourier transform (1.79) to both sides of (4.5) and arrive at u b(p) = 2π

b fb(p) G(p) , p21 + |p2 |2s

p2 u b(p) = 2π

b fb(p) p2 G(p) , 2 p1 + |p2 |2s

(4.10)

where fb(p) denotes the Fourier image of F (v(x), x). Evidently, we have the estimates from above |b u(p)| ≤ 2πN2,s |fb(p)|

and

|p2 u b(p)| ≤ 2πN2,s |fb(p)|.

Note that N2,s < ∞ by means of Lemma 4.1 of Appendix under the given conditions. This enables us to obtain the upper bound on the norm 2 ∥u∥2H 2 (R2 ) = ∥b u(p)∥2L2 (R2 ) + ∥p2 u b(p)∥2L2 (R2 ) ≤ 8π 2 N2,s ∥F (v(x), x)∥2L2 (R2 ) ,

4.1 Mixed-diffusion: scalar case

147

which is finite by virtue of (4.3) of Assumption 4.1 because v(x) ∈ L2 (R2 ). Clearly, v(x) ∈ H 2 (R2 ) ⊂ L∞ (R2 ) due to the Sobolev embedding. Thus, for an arbitrary v(x) ∈ H 2 (R2 ) there exists a unique solution u(x) ∈ H 2 (R2 ) of problem (4.5), such that its Fourier image is given by (4.10). Hence, the map T2,s : H 2 (R2 ) → H 2 (R2 ) is well defined. This allows us to choose arbitrary functions v1,2 (x) ∈ H 2 (R2 ), such that their images u1,2 := T2,s v1,2 ∈ H 2 (R2 ). Clearly, (4.5) yields !s Z ∂ 2 u1 ∂2 − + − u = G(x − y)F (v1 (y), y)dy, (4.11) 1 ∂x21 ∂x22 R2 !s Z ∂ 2 u2 ∂2 − + − u = G(x − y)F (v2 (y), y)dy, (4.12) 2 ∂x21 ∂x22 R2 where 0 < s < 1. Let us apply the standard Fourier transform (1.79) to both sides of the equations of system (4.11), (4.12) above. We arrive at u b1 (p) = 2π

b fb1 (p) G(p) , p21 + |p2 |2s

u b2 (p) = 2π

b fb2 (p) G(p) . p21 + |p2 |2s

(4.13)

Here fb1 (p) and fb2 (p) stand for the Fourier images of F (v1 (x), x) and F (v2 (x), x) respectively. By means of (4.13) we derive the upper bounds |u b1 (p) − u b2 (p)| ≤ 2πN2,s |fb1 (p) − fb2 (p)|,

p 2 |u b1 (p) − u b2 (p)| ≤ 2πN2,s |fb1 (p) − fb2 (p)|,

such that ∥u1 − u2 ∥2H 2 (R2 ) = ∥b u1 (p) − u b2 (p)∥2L2 (R2 ) + ∥p2 [b u1 (p) − u b2 (p)]∥2L2 (R2 ) 2 ≤ 8π 2 N2,s ∥F (v1 (x), x) − F (v2 (x), x)∥2L2 (R2 ) .

Evidently, v1,2 (x) ∈ H 2 (R2 ) ⊂ L∞ (R2 ) via the Sobolev embedding. Condition (4.4) above implies that √ ∥T2,s v1 − T2,s v2 ∥H 2 (R2 ) ≤ 2 2πN2,s l∥v1 − v2 ∥H 2 (R2 ) and the constant in the right side of this inequality is less than one via the one of our assumptions. Thus, by means of the fixed point theorem, there exists a unique function v2,s ∈ H 2 (R2 ) with the property T2,s v2,s = v2,s , which is the only solution of problem (4.1) in H 2 (R2 ). Suppose v2,s (x) = 0 identically in the space of two dimensions. This will contradict to our assumption that the Fourier images of G(x) and F (0, x) do not vanish on a set of nonzero Lebesgue measure in R2 . ⊔ ⊓ Let us proceed to establishing the solvability in the sense of sequences for our integro-differential problem in the space of two dimensions.

148

4 Existence of solutions for some non-Fredholm integro-differential equations

Proof of Theorem 4.2. By virtue of the result of Theorem 4.1 above, each problem (4.9) has a unique solution um (x) ∈ H 2 (R2 ), m ∈ N. Limiting equation (4.1) admits a unique solution u(x) ∈ H 2 (R2 ) by means of Lemma 4.2 below along with Theorem 4.1. Let us apply the standard Fourier transform (1.79) to both sides of (4.1) and (4.9). This yields b φ(p) b m (p)φ G(p) b G bm (p) , u bm (p) = 2π 2 , 2 2s p1 + |p2 | p1 + |p2 |2s b φ(p) b m (p)φ p2 G(p) b p2 G bm (p) 2 p2 u b(p) = 2π 2 , p u b (p) = 2π , m p1 + |p2 |2s p21 + |p2 |2s u b(p) = 2π

(4.14) m ∈ N, (4.15)

where φ(p) b and φ bm (p) stand for the Fourier images of F (u(x), x) and F (um (x), x) respectively. Apparently,

G

b G(p)

b m (p)

|b um (p) − u b(p)| ≤ 2π 2 − |φ(p)|+ b

p1 + |p2 |2s p21 + |p2 |2s ∞ 2 L



G

b m (p) +2π 2

p1 + |p2 |2s

(R )

|φ bm (p) − φ(p)|. b L∞ (R2 )

Hence ∥um − u∥L2 (R2 )



G

b G(p)

b m (p)

≤ 2π 2 −

p1 + |p2 |2s p21 + |p2 |2s



G

b m (p) +2π 2

p1 + |p2 |2s

∥F (u(x), x)∥L2 (R2 ) + L∞ (R2 )

∥F (um (x), x) − F (u(x), x)∥L2 (R2 ) . L∞ (R2 )

Upper bound (4.4) of Assumption 4.1 gives us ∥F (um (x), x) − F (u(x), x)∥L2 (R2 ) ≤ l∥um (x) − u(x)∥L2 (R2 ) .

(4.16)

Note that um (x), u(x) ∈ H 2 (R2 ) ⊂ L∞ (R2 ) due to the Sobolev embedding. Thus, we arrive at

( )

G

b m (p) ∥um (x) − u(x)∥L2 (R2 ) 1 − 2π 2 l ≤

p1 + |p2 |2s ∞ 2 L



G

b G(p)

b m (p)

≤ 2π 2 −

2

p1 + |p2 |2s p1 + |p2 |2s Using (4.24), we derive

(R )

∥F (u(x), x)∥L2 (R2 ) . L∞ (R2 )

4.1 Mixed-diffusion: scalar case

∥um (x)−u(x)∥L2 (R2 )

149



b m (p) b 2π G(p)

G

≤ −

2

2 ε p1 + |p2 |2s p1 + |p2 |2s

∥F (u(x), x)∥L2 (R2 ) . L∞ (R2 )

Upper bound (4.3) of Assumption 4.1 gives us F (u(x), x) ∈ L2 (R2 ) for u(x) ∈ L2 (R2 ). Hence, we obtain that under the given conditions um (x) → u(x),

m→∞

(4.17)

in L2 (R2 ) due to the result of Lemma 4.2 of Appendix. By virtue of (4.15), we arrive at

p2 G

b m (p) b p2 G(p)

2 2 |p u bm (p) − p u b(p)| ≤ 2π 2 − |φ(p)|+ b

p1 + |p2 |2s p21 + |p2 |2s ∞ 2 L



p2 G b m (p)

+2π 2

p1 + |p2 |2s

(R )

|φ bm (p) − φ(p)|. b L∞ (R2 )

Therefore,

p2 G

bm (p) b p2 G(p)

−

2 2 2s 2s

p1 + |p2 | p1 + |p2 |

∥∆um (x) − ∆u(x)∥L2 (R2 ) ≤ 2π

∥F (u(x), x)∥L2 (R2 ) +

L ∞ ( R2 )



p2 G b m (p)

+2π 2

p1 + |p2 |2s

∥F (um (x), x) − F (u(x), x)∥L2 (R2 ) . L∞ (R2 )

Inequality (4.16) enables us to obtain the upper bound

p2 G

bm (p) b p2 G(p)

−

2 2 2s 2s

p1 + |p2 | p1 + |p2 |

∥∆um (x) − ∆u(x)∥L2 (R2 ) ≤ 2π

∥F (u(x), x)∥L2 (R2 ) +

L ∞ ( R2 )



p2 G b m (p)

+2π 2

p1 + |p2 |2s

l∥um (x) − u(x)∥L2 (R2 ) . L∞ (R2 )

By means of the result of Lemma 4.2 of Appendix along with (4.17), we derive ∆um (x) → ∆u(x) in L2 (R2 ) as m → ∞. Definition (4.7) of the norm gives us um (x) → u(x) in H 2 (R2 ) as m → ∞. Suppose the solution um (x) of problem (4.9) studied above vanishes in the space of two dimensions for a certain m ∈ N. This will contradict to the given condition that the Fourier transforms of Gm (x) and F (0, x) are nontrivial on a set of nonzero Lebesgue measure in R2 . The analogous argument is valid for the solution u(x) of limiting equation (4.1). ⊔ ⊓

150

4 Existence of solutions for some non-Fredholm integro-differential equations

Appendix Let G(x) be a function, G(x) : R2 → R, for which we denote its standard Fourier transform using the hat symbol as Z 1 b G(p) := G(x)e−ipx dx, p ∈ R2 , (4.18) 2π R2 such that b ∥G(p)∥ L∞ (R2 ) ≤

1 ∥G∥L1 (R2 ) 2π

(4.19)

Z

1 iqx b G(q)e dq, x ∈ R2 . For the technical purposes we intro2π R2 duce the auxiliary quantities

( )

G(p)

p2 G(p)

b b



N2,s := max 2 , 2 , 0 < s < 1.

p1 + |p2 |2s ∞ 2

p1 + |p2 |2s ∞ 2 and G(x) =

L

(R )

L

(R )

(4.20) Lemma 4.1 Let 0 < s < 1, the function G(x) : R2 → R, such that G(x) ∈ L1 (R2 ) and x2 G(x) ∈ L1 (R2 ). We also assume that (−∆)1−s G(x) ∈ L1 (R2 ). a) If 0 < s ≤

1 then N2,s < ∞ if and only if 2 (G(x), 1)L2 (R2 ) = 0,

(4.21)

(G(x), x1 )L2 (R2 ) = 0.

(4.22)

1 < s < 1. Then N2,s < ∞ if and only if orthogonality conditions 2 (4.21) and (4.22) along with

b) Suppose

(G(x), x2 )L2 (R2 ) = 0

(4.23)

hold. Proof. Let us first observe that in both cases a) and b) of our lemma the b b G(p) p2 G(p) boundedness of 2 yields that is bounded as well. We 2 2s p1 + |p2 | p1 + |p2 |2s easily express b b b p2 G(p) p2 G(p) p2 G(p) = χ + χ{|p|>1} . {|p|≤1} p21 + |p2 |2s p21 + |p2 |2s p21 + |p2 |2s

(4.24)

Here and further down χA will denote the characteristic function of a set A ⊆ R2 . Clearly, the first term in the right side of (4.24) can be estimated

4.1 Mixed-diffusion: scalar case

151

from above in the absolute value by

G(p)

b

2

p1 + |p2 |2s

1} 2 2(1−s) 2 2s cos θ + | sin θ|2s |p| cos θ + | sin θ| (4.26) Here and throughout the chapter C will stand for a finite, positive constant. By means of (4.25), the right side of (4.26) can be estimated from above by C ∥(−∆)1−s G(x)∥L1 (R2 ) < ∞. 2π Therefore,

p21

b p2 G(p) ∈ L∞ (R2 ) as well. Evidently, p21 + |p2 |2s b b b G(p) G(p) G(p) = 2 χ{|p|≤1} + 2 χ{|p|>1} . 2s 2s + |p2 | p1 + |p2 | p1 + |p2 |2s

(4.27)

The second term in the right side of (4.27) can be trivially bounded from above in the absolute value using (4.19) as b |G(p)|χ ∥G(x)∥L1 (R2 ) {|p|>1} ≤ ≤ C∥G(x)∥L1 (R2 ) < ∞ p2 cos2 θ + |p|2s | sin θ|2s 2π(cos2 θ + | sin θ|2s ) via the one of our assumptions. Let us express Z |p| b ∂G b b G(p) = G(0) + |p| (0, θ) + ∂|p| 0

s

Z 0

! b ∂ 2 G(|q|, θ) d|q| ds. ∂|q|2

(4.28)

Identity (4.28) allows us to write the first term in the right side of (4.27) as

152

4 Existence of solutions for some non-Fredholm integro-differential equations

∂G b |p| ∂|p| (0, θ) G(0) χ + χ{|p|≤1} + {|p|≤1} 2 2 2s p1 + |p2 | p1 + |p2 |2s b


b ∂ 2 G(|q|,θ) d|q| ds ∂|q|2 χ{|p|≤1} . p21 + |p2 |2s

R |p| R s 0

0

(4.29) Using the definition of the standard Fourier transform (4.18), we easily derive ∂ 2 G(p) 1 2 b ∥x G(x)∥L1 (R2 ) < ∞ (4.30) ≤ ∂|p|2 2π as assumed. Let us verify the following trivial statement p21

p2 χ{|p|≤1} ≤ 1, + |p2 |2s

p ∈ R2 .

(4.31)

Indeed, the left side of (4.31) can be easily estimated as

|p|2

cos2

|p|2 |p|2(1−s) χ{|p|≤1} 2(1−s) χ{|p|≤1} 2s 2s θ + |p| | sin θ| |p| − |p|2(1−s) sin2 θ + | sin θ|2s |p|2(1−s) χ{|p|≤1} |p|2(1−s) + | sin θ|2s − sin2 θ ≤ 1, ≤

such that (4.31) is valid. By means of (4.30) along with (4.31), we obtain the upper bound in the absolute value for the third term in (4.29) as ∥x2 G(x)∥L1 (R2 ) |p|2 ∥x2 G(x)∥L1 (R2 ) χ ≤ 1

≤ ∥G(x)∥L1 (R2 ) + ∥x2 G(x)∥L1 (R2 ) < ∞.

4.1 Mixed-diffusion: scalar case

153

By virtue of (4.32) along with (4.33) we easily derive b ∂G i (0, θ) = − (Q1 cos θ + Q2 sin θ). ∂|p| 2π

(4.34)

Using (4.34), we can write the sum of the first two terms in (4.29) as b G(0) i|p|{Q1 cos θ + Q2 sin θ} χ{|p|≤1} − χ{|p|≤1} . |p|2 cos2 θ + |p|2s | sin θ|2s 2π(|p|2 cos2 θ + |p|2s | sin θ|2s ) (4.35) Let us fix the polar angle θ = 0 and suppose that |p| tends to zero. Obviously, b expression (4.35) will be unbounded unless G(0) and Q1 vanish in both cases a) and b) of our lemma. This is equivalent to orthogonality conditions (4.21) and (4.22). Therefore, it remains to analyze the term −

i|p|Q2 sin θ χ{|p|≤1} . 2π(|p|2 cos2 θ + |p|2s | sin θ|2s )

(4.36)

1 Let us first consider the situation when < s < 1. We fix the polar angle 2 π θ = and let |p| → 0. Then (4.36) will be unbounded unless Q2 = 0. This 2 is equivalent to orthogonality relation (4.23) and completes the proof of the part b) of our lemma. 1 Finally, we treat the case when 0 < s ≤ . Then (4.36) can be trivially esti2 mated from above in the absolute value as |p||Q2 || sin θ| 1 1−2s χ{|p|≤1} ≤ |p| | sin θ|1−2s |Q2 |χ{|p|≤1} 2π(|p|2 cos2 θ + |p|2s | sin θ|2s ) 2π |Q2 | ≤ < ∞, 2π such that in the case a) of the lemma no any further orthogonality conditions than (4.21) and (4.22) are needed. ⊔ ⊓ For the purpose of the study of problems (4.9), we define the following technical expressions

( )

G

p2 G b m (p)

b m (p)

N2,s, m := max 2 , 2 , (4.37)

p1 + |p2 |2s ∞ 2

p1 + |p2 |2s ∞ 2 L

(R )

L

(R )

where 0 < s < 1 and m ∈ N. Our final statement is as follows. Lemma 4.2 Let 0 < s < 1, m ∈ N, the functions Gm (x) : R2 → R, such that Gm (x) ∈ L1 (R2 ), Gm (x) → G(x) in L1 (R2 ) as m → ∞. Similarly,

154

4 Existence of solutions for some non-Fredholm integro-differential equations

x2 Gm (x) ∈ L1 (R2 ), x2 Gm (x) → x2 G(x) in L1 (R2 ) as m → ∞. Moreover, (−∆)1−s Gm (x) ∈ L1 (R2 ), (−∆)1−s Gm (x) → (−∆)1−s G(x) in L1 (R2 ) as m → ∞. We also assume that for all m ∈ N (Gm (x), 1)L2 (R2 ) = 0,

(4.38)

(Gm (x), x1 )L2 (R2 ) = 0

(4.39)

1 1 if 0 < s ≤ and for < s < 1 orthogonality conditions (4.38), (4.39) along 2 2 with (Gm (x), x2 )L2 (R2 ) = 0, m ∈ N (4.40) hold. Finally, let us suppose that √ 2 2πN2,s,

m

l ≤1−ε

(4.41)

is valid for all m ∈ N with some 0 < ε < 1. Then b m (p) b G G(p) → 2 , 2s + |p2 | p1 + |p2 |2s

m → ∞,

(4.42)

b m (p) b p2 G p2 G(p) → , p21 + |p2 |2s p21 + |p2 |2s

m→∞

(4.43)

p21

in L∞ (R2 ), such that

G

b m (p)

2

p1 + |p2 |2s

L∞ (R2 )



G(p)

b

→ 2

p1 + |p2 |2s

L∞ (R2 )



p2 G b m (p)

2

p1 + |p2 |2s

L∞ (R2 )



p2 G(p)

b

→ 2

2s

p1 + |p2 |

L∞ (R2 )

Furthermore,

,

m → ∞,

(4.44)

,

m → ∞.

(4.45)

√ 2 2πN2,s l ≤ 1 − ε

(4.46)

holds. Proof. By means of inequality (4.19) along with the one of our assumptions we arrive at 1 b m (p) − G(p)∥ b ∥G ∥Gm (x) − G(x)∥L1 (R2 ) → 0, L∞ (R2 ) ≤ 2π

m → ∞. (4.47)

Note that under the given conditions by means of the result of Lemma 4.1 above we have N2,s,m < ∞. Let us use (4.38) to estimate

4.1 Mixed-diffusion: scalar case

155

|(G(x), 1)L2 (R2 ) | = |(G(x) − Gm (x), 1)L2 (R2 ) | ≤ ∥Gm (x) − G(x)∥L1 (R2 ) → 0,

m→∞

as assumed, such that orthogonality condition (4.21) holds in the limit. By virtue of (4.39) and the assumptions of the lemma we easily derive |(G(x), x1 )L2 (R2 ) | =|(G(x) − Gm (x), x1 )L2 (R2 ) | Z ≤ |Gm (x) − G(x)||x1 |dx R2 Z Z ≤ |Gm (x) − G(x)||x|dx + |x|≤1

|Gm (x) − G(x)||x|dx |x|>1

≤∥Gm (x) − G(x)∥L1 (R2 ) + ∥x2 Gm (x) − x2 G(x)∥L1 (R2 ) → 0, m → ∞. 1 < s < 1, by 2 the similar reasoning we can show that orthogonality condition (4.23) holds in the limit as well. The result of Lemma 4.1 above gives us that in both 1 1 cases 0 < s ≤ and < s < 1 we have N2,s < ∞. 2 2 Let us establish that (4.42) implies (4.43). Clearly, we have the identity

Hence, orthogonality relation (4.22) is valid in the limit. When

b m (p) − G(p)] b b m (p) − G(p)] b b m (p) − G(p)] b p2 [ G p2 [ G p2 [ G = χ + χ{|p|>1} . {|p|≤1} 2 2 2 2s 2s 2s p1 + |p2 | p1 + |p2 | p1 + |p2 | (4.48) Apparently, the second term in the right side of (4.48) can be estimated in the absolute value as b m (p) − G(p)| b b m (p) − G(p)| b |p|2(1−s) |G |p|2(1−s) |G χ{|p|>1} ≤ ≤ 2 2s 2(1−s) 2 2s cos θ + | sin θ| |p| cos θ + | sin θ| C b m (p) − G(p)]∥ b ∥(−∆)1−s [Gm (x) − G(x)]∥L1 (R2 ) ≤ C∥|p|2(1−s) [G L∞ (R2 ) ≤ 2π due to (4.25). Thus,

p2 [ G

b m (p) − G(p)] b

χ

{|p|>1} 2 2s

∞ 2 p1 + |p2 | L

(R )

C ≤ ∥(−∆)1−s [Gm (x) − G(x)]∥L1 (R2 ) → 0, m → ∞ 2π as assumed. Evidently, the first term in the right side of (4.48) can be bounded from above in the norm as

156

4 Existence of solutions for some non-Fredholm integro-differential equations



p2 [ G

b m (p) − G(p)] b

χ

{|p|≤1} 2

∞ 2 p1 + |p2 |2s L (R )

G

b b Gm (p)

m (p) ≤ 2 − 2 → 0,

p1 + |p2 |2s p1 + |p2 |2s ∞ 2 L

m→∞

(R )

assuming that (4.42) holds. Therefore, (4.43) will be valid as well. Obviously, b m (p) b b m (p) − G(p) b b m (p) − G(p) b G G(p) G G − 2 = χ{|p|≤1} + χ{|p|>1} . 2 2 2s 2s 2s + |p2 | p1 + |p2 | p1 + |p2 | p1 + |p2 |2s (4.49) The second term in the right side of (4.49) can be estimated from above in the absolute value using (4.25) as p21

p2

b m (p) − G(p)| b b m (p) − G(p)| b |G |G χ{|p|>1} ≤ 2 2s 2s 2 cos θ + |p| | sin θ| cos θ + | sin θ|2s b m (p) − G(p)∥ b ≤ C∥G L∞ (R2 ) ≤

Hence, we obtain

G

b

b m (p) − G(p)

χ

2 {|p|>1}

p1 + |p2 |2s

≤ L∞ (R2 )

C ∥Gm (x) − G(x)∥L1 (R2 ) . 2π

C ∥Gm (x) − G(x)∥L1 (R2 ) → 0, 2π

m→∞

as assumed. Similar to (4.28), we express for m ∈ N Z |p| b b m (p) = G b m (0) + |p| ∂ Gm (0, θ) + G ∂|p| 0

s

Z 0

! b m (|q|, θ) ∂2G d|q| ds. (4.50) ∂|q|2

Orthogonality conditions (4.21) and (4.38) yield b G(0) = 0,

b m (0) = 0, G

m ∈ N.

(4.51)

By means of (4.50) along with (4.28) and (4.51) the first term in the right side of (4.49) can be written as h i bm b G ∂G |p| ∂∂|p| (0, θ) − ∂|p| (0, θ) χ{|p|≤1} p21 + |p2 |2s " ! i R |p|  R s ∂ 2 Gbm (|q|,θ) ∂ 2 G(|q|,θ) b − ∂|q|2 d|q| ds ∂|q|2 0 0 +

p21 + |p2 |2s

χ{|p|≤1} .

(4.52)

4.1 Mixed-diffusion: scalar case

157

By virtue of the definition of the standard Fourier transform (4.18), we easily derive ∂2G b θ) 1 2 b m (|p|, θ) ∂ 2 G(|p|, − ∥x Gm (x) − x2 G(x)∥L1 (R2 ) . (4.53) ≤ ∂|p|2 ∂|p|2 2π Inequalities (4.53) and (4.31) enable us to obtain the upper bound in the absolute value for the second term in (4.52) given by p2 ∥x2 Gm (x) − x2 G(x)∥L1 (R2 ) 1 2 χ{|p|≤1} ≤ ∥x Gm (x) − x2 G(x)∥L1 (R2 ) . 4π(p21 + |p2 |2s ) 4π Therefore,

R |p| R s  ∂ 2 Gbm (|q|,θ) ∂ 2 G(|q|,θ)


b

− ∂|q|2 d|q| ds ∂|q|2 0

0

χ

{|p|≤1}

p21 + |p2 |2s

L∞ (R2 )

1 ≤ ∥x2 Gm (x) − x2 G(x)∥L1 (R2 ) → 0 4π as m → ∞ due to the one of the assumptions of the lemma. A trivial calculation yields bm ∂G i (0, θ) = − [(Gm (x), x1 )L2 (R2 ) cos θ + (Gm (x), x2 )L2 (R2 ) sin θ], (4.54) ∂|p| 2π b ∂G i (0, θ) = − [(G(x), x1 )L2 (R2 ) cos θ + (G(x), x2 )L2 (R2 ) sin θ]. (4.55) ∂|p| 2π 1 < s < 1. By means of or2 thogonality relations (4.39) and (4.40) along with (4.54), we obtain that bm ∂G (0, θ) = 0, m ∈ N. Similarly, (4.22) and (4.23) along with (4.55) imply ∂|p| b ∂G 1 that (0, θ) = 0. Hence, in the case of < s < 1, the first term in (4.52) ∂|p| 2 vanishes. 1 Then we turn our attention to the situation when 0 < s ≤ . Let us estimate 2 the norm Z Z ∥|x|Gm (x)∥L1 (R2 ) = |x||Gm (x)|dx + |x||Gm (x)|dx ≤ Let us consider first the situation when

|x|≤1

|x|>1

≤ ∥Gm (x)∥L1 (R2 ) + ∥x2 Gm (x)∥L1 (R2 ) < ∞ due to the conditions of our lemma. Thus, |x|Gm (x) ∈ L1 (R2 ). By the similar reasoning, we derive

158

4 Existence of solutions for some non-Fredholm integro-differential equations

Z ∥|x|Gm (x) − |x|G(x)∥L1 (R2 ) =

Z |x||Gm (x) − G(x)|dx +

|x|≤1

|x||Gm (x) − G(x)|dx ≤ |x|>1

≤ ∥Gm (x) − G(x)∥L1 (R2 ) + ∥x2 Gm (x) − x2 G(x)∥L1 (R2 ) → 0,

m→∞

as assumed, such that |x|Gm (x) → |x|G(x) in L1 (R2 ) as m → ∞. Orthogonality relation (4.39) and formula (4.54) yield bm ∂G i (0, θ) = − sin θ ∂|p| 2π

Z Gm (x)x2 dx. R2

Analogously, by virtue of (4.22) and (4.55) we arrive at b ∂G i (0, θ) = − sin θ ∂|p| 2π

Z G(x)x2 dx. R2

This allows us to estimate the first term in (4.52) from above in the absolute value as R |p|| sin θ| R2 |x||Gm (x) − G(x)|dx χ{|p|≤1} 2π(p2 cos2 θ + |p|2s | sin θ|2s ) |p|1−2s | sin θ|1−2s ≤ ∥|x|Gm (x) − |x|G(x)∥L1 (R2 ) χ{|p|≤1} 2π 1 ≤ ∥|x|Gm (x) − |x|G(x)∥L1 (R2 ) . 2π 1 we obtain 2

 ∂ Gb

 b ∂G m

|p|

∂|p| (0, θ) − ∂|p| (0, θ)

χ

{|p|≤1}

p21 + |p2 |2s

Hence, for 0 < s ≤

1 ≤ ∥|x|Gm (x) − |x|G(x)∥L1 (R2 ) → 0, 2π

L∞ (R2 )

m→∞

as discussed above. Therefore, by virtue of the argument above (4.42) holds 1 1 in both cases when 0 < s ≤ and for < s < 1. Evidently, by means of 2 2 the standard triangle inequality (4.44) and (4.45) follow easily from (4.42) and (4.43) respectively. Finally, (4.46) is valid via a simple limiting argument using (4.44) and (4.45). ⊔ ⊓

4.2 Mixed-diffusion: system case

159

4.2 Mixed-diffusion: system case We prove the existence in the sense of sequences of solutions for some system of integro-differential type equations in two dimensions containing the normal diffusion in one direction and the anomalous diffusion in the other direction in H 2 (R2 , RN ) using the fixed point technique. The system of elliptic equations contains second order differential operators without the Fredholm property. It is established that, under the reasonable technical assumptions, the convergence in L1 (R2 ) of the integral kernels yields the existence and convergence in H 2 (R2 , RN ) of the solutions. We emphasize that the study of the systems is more difficult than of the scalar case and requires to overcome more cumbersome technicalities. This section is devoted to a class of stationary nonlinear systems of mixeddiffusion equations, for which the Fredholm property may not be satisfied: ! sk ∂ 2 uk ∂2 − − uk + ∂x21 ∂x22 Z Gk (x − y)Fk (u1 (y), u2 (y), ..., uN (y), y)dy = 0, 0 < sk < 1, (4.56) R2

where 1 ≤ k ≤ N , N ≥ 2, x = (x1 , x2 ) ∈ R2 , y = (y1 , y2 ) ∈ R2 . Here and further down the vector function u := (u1 , u2 , ..., uN )T ∈ RN .

(4.57)

The nonlocal operators L sk

∂2 := − 2 + ∂x1

∂2 − ∂x22

! sk : H 2 (R2 ) → L2 (R2 ),

0 < sk < 1, 1 ≤ k ≤ N, N ≥ 2

(4.58)

are defined in section 1.4. The existence of solutions of the single equation analogous to system (4.56) was covered in section 4.1 (see also our paper [25]). The novelty of this section 4.2 is that in each diffusion term we add the standard negative Laplacian in the x1 variable to the minus Laplacian in x2 raised to a fractional power. These models are new and not much is understood about them, especially in the context of the integro-differential equations. The technical difficulty we have to overcome is that such problems become anisotropic and it is more difficult to derive the desired estimates when dealing with them. Formulation of the results

160

4 Existence of solutions for some non-Fredholm integro-differential equations

The technical conditions of this section will be analogous to the ones of section 4.1 (see also [25]), adapted to the work with vector functions. Performing the analysis in the Sobolev spaces for vector functions is more complicated. The nonlinear part of system (4.56) will satisfy the following regularity conditions. Assumption 4.2 Let 1 ≤ k ≤ N . Functions Fk (u, x) : RN × R2 → R are satisfying the Carath´eodory condition (see section 1.3 and [42]), such that v uN uX t Fk2 (u, x) ≤ K|u|RN + h(x) f or u ∈ RN , x ∈ R2 (4.59) k=1

with a constant K > 0 and h(x) : R2 → R+ , h(x) ∈ L2 (R2 ). Furthermore, they are Lipschitz continuous function, so that for any u(1),(2) ∈ RN , x ∈ R2 : v uN uX t (F (u(1) , x) − F (u(2) , x))2 ≤ L|u(1) − u(2) | N (4.60) k k R k=1

with a constant L > 0. Here and below the norm of a vector function given by (4.57) is: v uN uX |u|RN = t u2k . k=1

The solvability of a local elliptic problem in a bounded domain in RN was discussed in [10]. The nonlinear function there was allowed to have a sublinear growth. In order to establish the existence of solutions of (4.56), we introduce the auxiliary system of equations with 1 ≤ k ≤ N, N ≥ 2, namely ! sk Z ∂ 2 uk ∂2 − + − u = Gk (x − y)Fk (v1 (y), v2 (y), ..., vN (y), y)dy, k ∂x21 ∂x22 R2 0 < sk < 1. (4.61) Let us denote Z

f1 (x)f¯2 (x)dx,

(f1 (x), f2 (x))L2 (R2 ) :=

(4.62)

R2

In this section we consider the situation in the space of the two dimensions, so that the appropriate Sobolev space is equipped with the norm ∥ϕ∥2H 2 (R2 ) := ∥ϕ∥2L2 (R2 ) + ∥∆ϕ∥2L2 (R2 ) .

(4.63)

4.2 Mixed-diffusion: system case

161

Then for a vector function (4.57), we have ∥u∥2H 2 (R2 ,RN )

:=

N X

∥uk ∥2H 2 (R2 )

k=1

=

N X

{∥uk ∥2L2 (R2 ) + ∥∆uk ∥2L2 (R2 ) }. (4.64)

k=1

Let us also use the norm ∥u∥2L2 (R2 ,RN ) :=

N X

∥uk ∥2L2 (R2 ) .

k=1

By virtue of Assumption 4.2 above, we are not allowed to consider the higher powers of the nonlinearities, than the first one, which is restrictive from the point of view of the applications. But this guarantees that our nonlinear vector function is a bounded and continuous map from L2 (R2 , RN ) to L2 (R2 , RN ). In the system above we are dealing with the operators Lsk defined in (4.58). By means of the standard Fourier transform (2.48), it can be trivially obtained that the essential spectrum of Lsk is given by λsk (p) = p21 + |p2 |2sk ,

p = (p1 , p2 ) ∈ R2 ,

1 ≤ k ≤ N.

(4.65)

Clearly, each set (4.65) contains the origin. Hence, our operators Lsk do not satisfy the Fredholm property, which is the obstacle to solve our system of equations. In this section we prove that under the reasonable technical conditions system (4.61) defines a map T2,s : H 2 (R2 , RN ) → H 2 (R2 , RN ), which is a strict contraction. Theorem 4.3 Let N ≥ 2, 1 ≤ k ≤ N , 0 < sk < 1, 1 ≤ M ≤ N − 1, the functions Gk (x) : R2 → R, such that Gk (x) ∈ L1 (R2 ) and x2 Gk (x) ∈ L1 (R2 ). Furthermore, let (−∆)1−sk Gk (x) ∈ L1 (R2 ) and Assumption 4.2 hold. 1 and orthogo2 nality relations (4.82), (4.83) are valid. Moreover, for M + 1 ≤ k ≤ N we 1 have < sk < 1 and orthogonality conditions (4.82), (4.83) and (4.84) hold 2 √ and that 2 2πN2,s L < 1, where N2,s is defined in (4.81). Then the map v 7→ T2,s v = u on H 2 (R2 , RN ) defined by system (4.61) has a unique fixed point v2,s , which is the only solution of the system of equations (4.56) in H 2 (R2 , RN ). Let us also assume that for 1 ≤ k ≤ M we have 0 < sk ≤

This fixed point v2,s is nontrivial provided that for some 1 ≤ k ≤ N the intersection of supports of the Fourier transforms of functions supp F\ k (0, x)∩ 2 c supp Gk is a set of nonzero Lebesgue measure in R .

162

4 Existence of solutions for some non-Fredholm integro-differential equations

Related to system (4.56) in the space of two dimensions, we consider the sequence of the approximate systems of equations with m ∈ N, namely ! sk (m) ∂ 2 uk ∂2 (m) − − uk ∂x21 ∂x22 Z (m) (m) (m) + Gk,m (x − y)Fk (u1 (y), u2 (y), ..., uN (y), y)dy = 0 (4.66) R2

with 0 < sk < 1, 1 ≤ k ≤ N, N ≥ 2. Each sequence of kernels {Gk,m (x)}∞ m=1 converges to Gk (x) as m → ∞ in the appropriate function spaces discussed further down. Let us prove that, under the appropriate technical assumptions, each of systems (4.66) possesses a unique solution u(m) (x) ∈ H 2 (R2 , RN ), the limiting system of equations (4.56) has a unique solution u(x) ∈ H 2 (R2 , RN ), and u(m) (x) → u(x) in H 2 (R2 , RN ) as m → ∞. Similar to section 4.1 we formulate the solvability conditions in term of the iterated kernels Gk,m . They imply the convergence of the kernels in terms of the Fourier transforms (see Appendix) and, consequently, the convergence or the solutions (Theorem 4.4). Our second main result of this section is as follows. Theorem 4.4 Let m ∈ N, N ≥ 2, 1 ≤ k ≤ N , 0 < sk < 1, 1 ≤ M ≤ N − 1, the functions Gk,m (x) : R2 → R, Gk,m (x) ∈ L1 (R2 ), x2 Gk,m (x) ∈ L1 (R2 ) are such that Gk,m (x) → Gk (x), x2 Gk,m (x) → x2 Gk (x) in L1 (R2 ) as m → ∞. Moreover, (−∆)1−sk Gk,m (x) ∈ L1 (R2 ), so that (−∆)1−sk Gk,m (x) → (−∆)1−sk Gk (x) in L1 (R2 ) as m → ∞ and Assumption 4.2 hold. 1 for 1 ≤ k ≤ M and orthogonality relations (4.101), Suppose 0 < sk ≤ 2 1 (4.102) are valid, < sk < 1 for M +1 ≤ k ≤ N and orthogonality conditions 2 (4.101), (4.102) and (4.103) hold. Furthermore, we assume that (4.104) is valid for all m ∈ N with some fixed 0 < ε < 1. Then each system of equations (4.66) has a unique solution u(m) (x) ∈ H 2 (R2 , RN ), limiting system (4.56) admits a unique solution u(x) ∈ H 2 (R2 , RN ), so that u(m) (x) → u(x) in H 2 (R2 , RN ) as m → ∞. The unique solution u(m) (x) of each system of equations (4.66) does not vanish identically in our space of two dimensions provided that for some 1 ≤ k ≤ N the intersection of supports of the Fourier transforms of func2 [ tions supp F\ k (0, x) ∩ supp Gk,m is a set of nonzero Lebesgue measure in R . Analogously, the unique solution u(x) of limiting system (4.56) is nontrivial 2 c if supp F\ k (0, x) ∩ supp Gk is a set of nonzero Lebesgue measure in R for a certain 1 ≤ k ≤ N . Proof of Theorem 4.3. First we suppose that for some v(x) ∈ H 2 (R2 , RN ) there exist two solutions u(1),(2) (x) ∈ H 2 (R2 , RN ) of system (4.61). Then

4.2 Mixed-diffusion: system case

163

their difference w(x) := u(1) (x) − u(2) (x) ∈ H 2 (R2 , RN ) will satisfy the homogeneous system of equations ! sk ∂ 2 wk ∂2 − + − wk = 0, 1 ≤ k ≤ N. ∂x21 ∂x22 Evidently, each operator Lsk : H 2 (R2 ) → L2 (R2 ) defined in (4.58) does not possess any nontrivial zero modes. Hence, the vector function w(x) is trivial in the space of two dimensions. Let us choose an arbitrary v(x) ∈ H 2 (R2 , RN ) and apply the standard Fourier transform (2.48) to both sides of system (4.61). This yields u ck (p) = 2π

ck (p)fbk (p) G , p21 + |p2 |2sk

p2 u ck (p) = 2π

ck (p)fbk (p) p2 G , p21 + |p2 |2sk

1 ≤ k ≤ N, (4.67)

where fbk (p) stands for the Fourier image of Fk (v(x), x). Clearly, we have the upper bounds |c uk (p)| ≤ 2πN2,sk |fbk (p)|

and

|p2 u ck (p)| ≤ 2πN2,sk |fbk (p)|,

1 ≤ k ≤ N.

Note that all N2,sk here are finite by virtue of Lemma 4.3 of the Appendix under the stated assumptions. This allows us to derive the estimate from above on the norm ∥u∥2H 2 (R2 ,RN )

=

N X

{∥c uk (p)∥2L2 (R2 ) + ∥p2 u ck (p)∥2L2 (R2 ) } ≤

k=1

≤ 8π 2

N X

2 N2,s ∥Fk (v(x), x)∥2L2 (R2 ) . k

(4.68)

k=1

The right side of (4.68) is finite by means of (4.59) of Assumption 4.2 since v(x) ∈ H 2 (R2 , RN ). Obviously, vk (x) ∈ H 2 (R2 ) ⊂ L∞ (R2 ), 1 ≤ k ≤ N via the Sobolev embedding. Hence, for an arbitrary v(x) ∈ H 2 (R2 , RN ) there exists a unique solution u(x) ∈ H 2 (R2 , RN ) of system (4.61), so that its Fourier image is given by (4.67). Therefore, the map T2,s : H 2 (R2 , RN ) → H 2 (R2 , RN ) is well defined. This enables us to choose arbitrary vector functions v (1),(2) (x) ∈ H 2 (R2 , RN ), so that their images u(1),(2) := T2,s v (1),(2) ∈ H 2 (R2 , RN ). By virtue of (4.61), we have for 1 ≤ k ≤ N ! sk Z (1) ∂ 2 uk ∂2 (1) (1) (1) (1) − + − 2 uk = Gk (x−y)Fk (v1 (y), v2 (y), ..., vN (y), y)dy, ∂x21 ∂x2 R2 (4.69)

164

4 Existence of solutions for some non-Fredholm integro-differential equations (2)

∂ 2 uk ∂2 − + − ∂x21 ∂x22

! sk (2) uk

Z

(2)

= R2

(2)

(2)

Gk (x−y)Fk (v1 (y), v2 (y), ..., vN (y), y)dy,

(4.70) where 0 < sk < 1. We apply the standard Fourier transform (2.48) to both sides of systems (4.69), (4.70) above. This gives us for 1 ≤ k ≤ N (1) ck (p)fd G d (1) k (p) uk (p) = 2π 2 , p1 + |p2 |2sk

(1) ck (p)fd p2 G d (1) k (p) p2 uk (p) = 2π 2 , p1 + |p2 |2sk

(4.71)

(2) ck (p)fd G d (2) k (p) uk (p) = 2π 2 , p1 + |p2 |2sk

d (2) p2 uk (p)

(2) ck (p)fd p2 G k (p) = 2π 2 . p1 + |p2 |2sk

(4.72)

d d (1) (2) In the formulas above fk (p) and fk (p) designate the Fourier images of (1) (2) Fk (v (x), x) and Fk (v (x), x) respectively. Using (4.71) and (4.72), we obtain the estimates from above d d d d (1) (2) (1) (2) uk (p) − uk (p) ≤ 2πN2,sk fk (p) − fk (p) , d d 2d d (1) (2) (1) (2) p uk (p) − p2 uk (p) ≤ 2πN2,sk fk (p) − fk (p) with 1 ≤ k ≤ N . Hence, we derive the upper bound for the norm ∥u(1) − u(2) ∥2H 2 (R2 ,RN ) =

N n

2 X d

d

(1) (2)

uk (p) − uk (p)

L2 (R2 )

k=1 2 ≤8π 2 N2,s

N X

h i 2 d d

(1) (2) + p2 uk (p) − uk (p)

o

L2 (R2 )

∥Fk (v (1) (x), x) − Fk (v (2) (x), x)∥2L2 (R2 ) ,

k=1 (1),(2)

where N2,s is defined in (4.81). Clearly, vk (x) ∈ H 2 (R2 ) ⊂ L∞ (R2 ) due to the Sobolev embedding. Condition (4.60) of Assumption 4.2 above yields √ ∥T2,s v (1) − T2,s v (2) ∥H 2 (R2 ,RN ) ≤ 2 2πN2,s L∥v (1) − v (2) ∥H 2 (R2 ,RN ) . (4.73) The constant in the right side of inequality (4.73) is less than one as assumed. Therefore, by virtue of the fixed point theorem, there exists a unique vector function v2,s ∈ H 2 (R2 , RN ), so that T2,s v2,s = v2,s , which is the only solution of the system of equations (4.56) in H 2 (R2 , RN ). Suppose v2,s (x) is trivial in the whole space of two dimensions. This will contradict to the given condition that for a certain 1 ≤ k ≤ N the Fourier images of Gk (x) and Fk (0, x) do not vanish on a set of nonzero Lebesgue measure in R2 . ⊔ ⊓

4.2 Mixed-diffusion: system case

165

We proceed to establishing the solvability in the sense of sequences for our system of integro-differential equations in the space of two dimensions. Proof of Theorem 4.4. By means of the result of Theorem 4.3 above, each system of equations (4.66) admits a unique solution u(m) (x) ∈ H 2 (R2 , RN ), m ∈ N. Limiting system (4.56) possesses a unique solution u(x) ∈ H 2 (R2 , RN ) by virtue of Lemma 4.4 below along with Theorem 4.3. We apply the standard Fourier transform (2.48) to both sides of the systems of equations (4.56) and (4.66). This gives us for 1 ≤ k ≤ N , m ∈ N u ck (p) = 2π p2 u ck (p) = 2π

ck (p)c G φk (p) , p21 + |p2 |2sk

ck (p)c p2 G φk (p) , p21 + |p2 |2sk

[ G [ d k,m (p)φ k,m (p) (m) uk (p) = 2π , p21 + |p2 |2sk

(4.74)

[ p2 G [ d k,m (p)φ k,m (p) (m) p2 uk (p) = 2π . p21 + |p2 |2sk

(4.75)

Here φ ck (p) and φ [ k,m (p) designate the Fourier images of Fk (u(x), x) and Fk (u(m) (x), x) respectively. Obviously,

G ck (p) G d

[

k,m (p) (m) |uk (p) − u ck (p)| ≤ 2π 2 − 2 |c φk (p)|+

p1 + |p2 |2sk p1 + |p2 |2sk ∞ 2 L



G

[ k,m (p) +2π 2

p1 + |p2 |2sk

(R )

|φ [ ck (p)|. k,m (p) − φ L∞ (R2 )

Thus, (m) ∥uk

− uk ∥ L 2 ( R2 )



G ck (p) G

\

k,m (p) ≤ 2π 2 − 2

p1 + |p2 |2sk p1 + |p2 |2sk

∥Fk (u(x), x)∥L2 (R2 ) +

L ∞ ( R2 )



G

[ k,m (p) +2π 2

p1 + |p2 |2sk

∥Fk (u(m) (x), x) − Fk (u(x), x)∥L2 (R2 ) . L∞ (R2 )

Inequality (4.60) of Assumption 4.2 yields v uN uX t ∥Fk (u(m) (x), x) − Fk (u(x), x)∥2L2 (R2 ) ≤ L∥u(m) (x) − u(x)∥L2 (R2 ,RN ) . k=1

(4.76) (m) Clearly, uk (x), uk (x) ∈ H 2 (R2 ) ⊂ L∞ (R2 ) with 1 ≤ k ≤ N, m ∈ N by virtue of the Sobolev embedding. Hence, we obtain

166

4 Existence of solutions for some non-Fredholm integro-differential equations

∥u(m) (x) − u(x)∥2L2 (R2 ,RN )

2 N X ck (p) [ G (p) G

k,m ≤8π 2 − 2 ∥Fk (u(x), x)∥2L2 (R2 )

2

p1 + |p2 |2sk p1 + |p2 |2sk ∞ 2 k=1 L (R ) h i2 (m) + 8π 2 N2,s L2 ∥u(m) (x) − u(x)∥2L2 (R2 ,RN ) , (m)

where N2,s is defined in (4.100). Using (4.104), we derive ∥u(m) (x) − u(x)∥2L2 (R2 ,RN )

2 N ck (p) [ 8π 2 X G

G

k,m (p) ≤ − 2

2

2s 2s k k

ε(2 − ε) p1 + |p2 | p1 + |p2 | ∞ k=1

L

∥Fk (u(x), x)∥2L2 (R2 ) . (R2 )

By means of upper bound (4.59) of Assumption 4.2, we have Fk (u(x), x) ∈ L2 (R2 ), 1 ≤ k ≤ N for u(x) ∈ H 2 (R2 , RN ). Let us recall the result of Lemma 4.4 below. Therefore, under the stated conditions u(m) (x) → u(x),

m→∞

(4.77)

in L2 (R2 , RN ). Using (4.75), we arrive at



p2 G

2c [ (p) p G (p) 2d

k,m k (m) ck (p) ≤ 2π 2 − p uk (p) − p2 u

p1 + |p2 |2sk p21 + |p2 |2sk

p2 G

[ k,m (p) +2π 2

p1 + |p2 |2sk

|c φk (p)|+ L∞ (R2 )

|φ [ ck (p)|. k,m (p) − φ L∞ (R2 )

Hence, (m)

∥∆uk (x) − ∆uk (x)∥L2 (R2 )

p2 G ck (p) p2 G

[

k,m (p) ≤2π 2 − 2 ∥Fk (u(x), x)∥L2 (R2 )

p1 + |p2 |2sk p1 + |p2 |2sk ∞ 2 L (R )

p2 G

[ k,m (p) + 2π 2 ∥Fk (u(m) (x), x) − Fk (u(x), x)∥L2 (R2 ) .

p1 + |p2 |2sk ∞ 2 L

(R )

Upper bound (4.76) allows us to derive the estimate from above (m) ∥∆uk (x)−∆uk (x)∥L2 (R2 )



p2 G ck (p) p2 G

\

k,m (p) ≤ 2π 2 −

p1 + |p2 |2sk p21 + |p2 |2sk

L ∞ ( R2 )

∥Fk (u(x), x)∥L2 (R2 ) +

4.2 Mixed-diffusion: system case

167



p2 G

[ k,m (p) +2π 2

p1 + |p2 |2sk

L∥u(m) (x) − u(x)∥L2 (R2 ,RN ) . L∞ (R2 )

We recall the result of Lemma 4.4 of Appendix and use (4.77). This yields ∆u(m) (x) → ∆u(x)

L2 (R2 , RN ),

in

m → ∞.

By virtue of the definition (4.64) of the norm we establish that u(m) (x) → u(x) in H 2 (R2 , RN ) as m → ∞. Let us suppose the unique solution u(m) (x) of the system of equations (4.66) discussed above is trivial in our space of two dimensions for some m ∈ N. This will contradict to the stated condition that the Fourier transforms of Gk,m (x) and Fk (0, x) do not vanish on a set of nonzero Lebesgue measure in R2 for a certain 1 ≤ k ≤ N . The similar argument holds for the unique solution u(x) of limiting system (4.56). ⊔ ⊓

Appendix Let Gk (x) be a function, Gk (x) : R2 → R. We denote its standard Fourier transform using the hat symbol as Z ck (p) := 1 G Gk (x)e−ipx dx, p ∈ R2 . (4.78) 2π R2 Evidently, ck (p)∥L∞ (R2 ) ≤ ∥G

1 ∥Gk (x)∥L1 (R2 ) 2π

(4.79)

Z

1 ck (q)eiqx dq, x ∈ R2 . For the technical purposes we inG 2π R2 troduce the auxiliary expressions

( )

G

p2 G ck (p)

ck (p)

N2,sk := max 2 , 2 , 0 < sk < 1,

p1 + |p2 |2sk ∞ 2

p1 + |p2 |2sk ∞ 2 and Gk (x) =

L

(R )

L

(R )

(4.80) where 1 ≤ k ≤ N , N ≥ 2. Under the conditions of Lemma 4.3 below, all the quantities (4.80) will be finite. Hence N2,s := max N2,sk < ∞. 1≤k≤N

(4.81)

The auxiliary lemmas below are the adaptations of the ones established in section 4.1 (see also our work [25]) for the studies of the single integro-differential

168

4 Existence of solutions for some non-Fredholm integro-differential equations

equation with mixed diffusion, analogous to system (4.56). We provide them for the convenience of the readers. Lemma 4.3 Let N ≥ 2, 1 ≤ k ≤ N, 0 < sk < 1, 1 ≤ M ≤ N − 1, the functions Gk (x) : R2 → R, so that Gk (x) ∈ L1 (R2 ) and x2 Gk (x) ∈ L1 (R2 ). Let us also assume that (−∆)1−sk Gk (x) ∈ L1 (R2 ). a) Suppose 0 < sk ≤

1 for 1 ≤ k ≤ M . Then N2,sk < ∞ if and only if 2 (Gk (x), 1)L2 (R2 ) = 0,

(4.82)

(Gk (x), x1 )L2 (R2 ) = 0.

(4.83)

1 < sk < 1 for M + 1 ≤ k ≤ N . Then N2,sk < ∞ if and only if 2 orthogonality relations (4.82) and (4.83) along with

b) Let

(Gk (x), x2 )L2 (R2 ) = 0

(4.84)

are valid. Proof. It can be easily verified that in both cases a) and b) of the lemma ck (p) ck (p) G p2 G the boundedness of 2 implies that ∈ L∞ (R2 ) as well. 2 2s k p1 + |p2 | p1 + |p2 |2sk Let us express ck (p) ck (p) ck (p) p2 G p2 G p2 G = χ + χ{|p|>1} . {|p|≤1} p21 + |p2 |2sk p21 + |p2 |2sk p21 + |p2 |2sk

(4.85)

Here and below χA will stand for the characteristic function of a set A ⊆ R2 . Evidently, the first term in the right side of (4.85) can be bounded from above in the absolute value by

G

ck (p) 1} ≤ ≤ C||p|2(1−sk ) G 2 2 2s cos θ+| sin θ|2sk cos θ+| sin θ| k (4.87) Here and further down C will denote a finite, positive constant. By virtue of (4.86), the right side of (4.87) can be bounded from above by |p|2(1−sk )

C ∥(−∆)1−sk Gk (x)∥L1 (R2 ) < ∞. 2π Hence,

ck (p) p2 G ∈ L∞ (R2 ) as well. Clearly, + |p2 |2sk

p21

p21

ck (p) ck (p) ck (p) G G G = χ + χ{|p|>1} . {|p|≤1} 2 2 + |p2 |2sk p1 + |p2 |2sk p1 + |p2 |2sk

(4.88)

The second term in the right side of (4.88) can be easily estimated from above in the absolute value via (4.79) as

|p|2

ck (p)|χ{|p|>1} |G ∥Gk (x)∥L1 (R2 ) ≤ ≤ C∥Gk (x)∥L1 (R2 ) < ∞ 2 2s 2s k k cos θ + |p| | sin θ| 2π(cos2 θ + | sin θ|2sk )

due to the one of our assumptions. We can write c ck (p) = G ck (0) + |p| ∂ Gk (0, θ) + G ∂|p|

|p|

Z 0

s

Z 0

! ck (|q|, θ) ∂2G d|q| ds. ∂|q|2

(4.89)

Equality (4.89) enables us to express the first term in the right side of (4.88) as
R |p| R s ∂ 2 Gck (|q|,θ) ck G ck (0) |p| ∂∂|p| (0, θ) d|q| ds G ∂|q|2 0 0 χ{|p|≤1} + 2 χ{|p|≤1} + χ{|p|≤1} . p21 +|p2 |2sk p1 +|p2 |2sk p21 +|p2 |2sk (4.90) The definition of the standard Fourier transform (4.78) gives us ∂2G 1 2 ck (p) ∥x Gk (x)∥L1 (R2 ) < ∞ (4.91) ≤ 2 ∂|p| 2π via the one of the given conditions. It can be easily checked that p2 χ{|p|≤1} ≤ 1, p21 + |p2 |2sk

p ∈ R2 .

(4.92)

Evidently, the left side of (4.92) can be trivially bounded from above as

170

4 Existence of solutions for some non-Fredholm integro-differential equations

|p|2 χ{|p|≤1} |p|2 cos2 θ + |p|2sk | sin θ|2sk = ≤

|p|2(1−sk )

−

|p|2(1−sk ) χ{|p|≤1} sin2 θ + | sin θ|2sk

|p|2(1−sk )

|p|2(1−sk ) χ{|p|≤1} ≤ 1. |p|2(1−sk ) + | sin θ|2sk − sin2 θ

Hence, (4.92) holds. By virtue of (4.91) along with (4.92), we derive the estimate from above in the absolute value for the third term in (4.90) as ∥x2 Gk (x)∥L1 (R2 ) |p|2 ∥x2 Gk (x)∥L1 (R2 ) χ ≤ 1

≤ ∥Gk (x)∥L1 (R2 ) + ∥x2 Gk (x)∥L1 (R2 ) ,

(4.95)

which is finite. Formulas (4.93) and (4.94) give us ck ∂G i (0, θ) = − {Q1,k cos θ + Q2,k sin θ}. ∂|p| 2π

(4.96)

By virtue of (4.96), the sum of the first two terms in (4.90) can be written as ck (0) G i|p|{Q1,k cos θ + Q2,k sin θ} χ{|p|≤1} − χ{|p|≤1} . θ + |p|2sk | sin θ|2sk 2π(|p|2 cos2 θ + |p|2sk | sin θ|2sk ) (4.97) We fix the polar angle θ = 0 and let |p| → 0. Clearly, (4.97) will be unbounded ck (0) = Q1,k = 0 in both cases a) and b) of the lemma. This is unless G equivalent to orthogonality relations (4.82) and (4.83). Thus, it remains to consider the term |p|2

cos2

4.2 Mixed-diffusion: system case

−

171

i|p|Q2,k sin θ χ{|p|≤1} . 2π(|p|2 cos2 θ + |p|2sk | sin θ|2sk )

(4.98)

1 First we discuss the situation when < sk < 1. Let us fix the polar angle 2 π θ= and let |p| tend to zero. Then (4.98) will be unbounded unless Q2,k 2 vanishes. This is equivalent to orthogonality condition (4.84) and completes the proof of the part b) of the lemma. 1 Finally, we study the case when 0 < sk ≤ . Then (4.98) can be easily 2 bounded from above in the absolute value as |p||Q2,k || sin θ| 1 1−2sk χ{|p|≤1} ≤ |p| | sin θ|1−2sk |Q2,k |χ{|p|≤1} 2π(|p|2 cos2 θ + |p|2sk | sin θ|2sk ) 2π |Q2,k | ≤ , 2π which is finite as discussed above. Therefore, in the case a) of our lemma no any orthogonality conditions other than (4.82) and (4.83) are needed. ⊔ ⊓ In order to study the systems of equations auxiliary expressions

(

G

[ k,m (p) (m) N2,sk := max 2 ,

p1 + |p2 |2sk ∞ 2 L

(R )

(4.66), we introduce the following

p2 G

[ k,m (p)

2

p1 + |p2 |2sk

) (4.99) L∞ (R2 )

with 0 < sk < 1, 1 ≤ k ≤ N, N ≥ 2 and m ∈ N. Under the assumptions of Lemma 4.4 below, all expressions (4.99) will be finite. This will allow us to define (m) (m) N2,s := max N2,sk < ∞, m ∈ N. (4.100) 1≤k≤N

Our final technical statement is as follows. Lemma 4.4 Let m ∈ N, N ≥ 2, 1 ≤ k ≤ N , 0 < sk < 1, 1 ≤ M ≤ N − 1, the functions Gk,m (x) : R2 → R, Gk,m (x) ∈ L1 (R2 ), x2 Gk,m (x) ∈ L1 (R2 ), so that Gk,m (x) → Gk (x), x2 Gk,m (x) → x2 Gk (x) in L1 (R2 ) as m → ∞. Furthermore, (−∆)1−sk Gk,m (x) ∈ L1 (R2 ), such that (−∆)1−sk Gk,m (x) → (−∆)1−sk Gk (x) in L1 (R2 ) as m → ∞. a) Let 0 < sk ≤

1 for 1 ≤ k ≤ M and 2 (Gk,m (x), 1)L2 (R2 ) = 0,

m ∈ N,

(4.101)

(Gk,m (x), x1 )L2 (R2 ) = 0,

m ∈ N.

(4.102)

172

4 Existence of solutions for some non-Fredholm integro-differential equations

1 < sk < 1 for M + 1 ≤ k ≤ N , orthogonality relations (4.101), 2 (4.102) along with

b) Suppose

(Gk,m (x), x2 )L2 (R2 ) = 0,

m∈N

(4.103)

are valid. Let in addition √ (m) 2 2πN2,s L ≤ 1 − ε

(4.104)

for all m ∈ N with some fixed 0 < ε < 1. Then, for all 1 ≤ k ≤ N , we have \ [ G G k,m (p) k (p) → 2 , m → ∞, (4.105) 2 2s k p1 + |p2 | p1 + |p2 |2sk ck (p) [ p2 G p2 G k,m (p) → 2 , 2 2s p1 + |p2 | k p1 + |p2 |2sk

m→∞

in L∞ (R2 ), so that

G

[ k,m (p)

2

p1 + |p2 |2sk

L∞ (R2 )



G

ck (p) → 2

p1 + |p2 |2sk

L∞ (R2 )



p2 G

[ k,m (p)

2

p1 + |p2 |2sk

L∞ (R2 )



p2 G ck (p)

→ 2

p1 + |p2 |2sk

L∞ (R2 )

Moreover,

(4.106)

,

m → ∞,

(4.107)

,

m → ∞.

(4.108)

√ 2 2πN2,s L ≤ 1 − ε

(4.109)

is valid. Proof. By virtue of inequality (4.79) along with the one of the given conditions, we derive for 1 ≤ k ≤ N that 1 c [ ∥G ∥Gk,m (x) − Gk (x)∥L1 (R2 ) → 0, k,m (p) − Gk (p)∥L∞ (R2 ) ≤ 2π

m → ∞.

(4.110) Clearly, under the stated assumptions by means of the result of Lemma 4.3 (m) above we have N2,s < ∞. Using (4.101), we estimate for 1 ≤ k ≤ N that |(Gk (x), 1)L2 (R2 ) | = |(Gk (x) − Gk,m (x), 1)L2 (R2 ) | ≤ ∥Gk,m (x) − Gk (x)∥L1 (R2 ) → 0,

m→∞

as we assume. Hence, orthogonality relations (4.82) are valid in the limit with 1 ≤ k ≤ N . By virtue of (4.102) along with the given conditions, we easily arrive at for 1 ≤ k ≤ N that

4.2 Mixed-diffusion: system case

173

|(Gk (x), x1 )L2 (R2 ) | =|(Gk (x) − Gk,m (x), x1 )L2 (R2 ) | Z ≤ |Gk,m (x) − Gk (x)||x1 |dx 2 ZR Z ≤ |Gk,m (x) − Gk (x)||x|dx + |x|≤1

|Gk,m (x) − Gk (x)||x|dx |x|>1

≤∥Gk,m (x) − Gk (x)∥L1 (R2 ) + ∥x2 Gk,m (x) − x2 Gk (x)∥L1 (R2 ) → 0,

m → ∞.

Thus, orthogonality conditions (4.83) hold in the limit with 1 ≤ k ≤ N . When M + 1 ≤ k ≤ N , by the similar argument we can easily demonstrate that orthogonality relations (4.84) are valid in the limit as well. Using the result of Lemma 4.3 above, we obtain that N2,s < ∞. Let us show that (4.105) yields (4.106). Evidently, we have the equality c c c \ \ \ p 2 [G p 2 [G p 2 [G k,m (p) − Gk (p)] k,m (p) − Gk (p)] k,m (p) − Gk (p)] = χ{|p|≤1} + χ{|p|>1} . 2 2 2 2s 2s 2s k k p1 + |p2 | p1 + |p2 | p1 + |p2 | k (4.111)

Obviously, the second term in the right side of (4.111) can be bounded from above in the absolute value as

≤

c [ |p|2(1−sk ) |G k,m (p) − Gk (p)| χ{|p|>1} 2(1−s ) 2 k |p| cos θ + | sin θ|2sk c [ |p|2(1−sk ) |G k,m (p) − Gk (p)|

cos2 θ + | sin θ|2sk c [ ≤C∥|p|2(1−sk ) [G k,m (p) − Gk (p)]∥L∞ (R2 ) ≤

C ∥(−∆)1−sk [Gk,m (x) − Gk (x)]∥L1 (R2 ) 2π

via the analog of inequality (4.86). Hence,

p 2 [G

c

\

k,m (p) − Gk (p)] χ

{|p|>1}

p21 + |p2 |2sk

L ∞ ( R2 )

≤

C ∥(−∆)1−sk [Gk,m (x) − Gk (x)]∥L1 (R2 ) → 0 2π

as m → ∞ as we assume. Clearly, the first term in the right side of (4.111) can be estimated from above in the norm as

p2 [ G

c [

k,m (p) − Gk (p)] χ

{|p|≤1}

∞ 2 p21 + |p2 |2sk L (R )

G

c [ Gk (p)

k,m (p) ≤ 2 − 2 → 0, m → ∞

p1 + |p2 |2sk p1 + |p2 |2sk ∞ 2 L

(R )

assuming that (4.105) is valid. Thus, (4.106) will hold as well. Evidently,

174

4 Existence of solutions for some non-Fredholm integro-differential equations

c c ck (p) \ \ \ G G G G k,m (p) k,m (p) − Gk (p) k,m (p) − Gk (p) − 2 = χ{|p|≤1} + χ{|p|>1} . p21 + |p2 |2sk p1 + |p2 |2sk p21 + |p2 |2sk p21 + |p2 |2sk (4.112)

The second term in the right side of (4.112) can be bounded from above in the absolute value using (4.110) as c [ |G k,m (p) − Gk (p)| χ{|p|>1} 2 2s cos θ + |p| k | sin θ|2sk c [ |G k,m (p) − Gk (p)| c [ ≤ ≤ C∥G k,m (p) − Gk (p)∥L∞ (R2 ) 2 cos θ + | sin θ|2sk C ≤ ∥Gk,m (x) − Gk (x)∥L1 (R2 ) , 2π |p|2

so that

G

c

\

k,m (p) − Gk (p) χ

{|p|>1}

p21 + |p2 |2sk

≤

L ∞ ( R2 )

C ∥Gk,m (x) − Gk (x)∥L1 (R2 ) → 0, 2π

m→∞

according to our assumption. Analogously to (4.89), we write for 1 ≤ k ≤ N, m ∈ N ! Z |p| Z s 2 [ [ ∂G ∂ Gk,m (|q|, θ) k,m [ [ Gk,m (p) = Gk,m (0) + |p| (0, θ) + d|q| ds. ∂|p| ∂|q|2 0 0 (4.113) Orthogonality relations (4.82) and (4.101) imply that ck (0) = 0, G

[ G k,m (0) = 0,

1 ≤ k ≤ N,

m ∈ N.

(4.114)

By virtue of (4.113) along with (4.89) and (4.114) the first term in the right side of (4.112) is equal to h \ i ck ∂ Gk,m G |p| ∂|p| (0, θ) − ∂∂|p| (0, θ) χ{|p|≤1} + p21 + |p2 |2sk R |p|  R s h ∂ 2 G \ k,m (|q|,θ) +

0

0

∂|q|2

−

ck (|q|,θ) ∂2G ∂|q|2

p21 + |p2 |2sk

i

 d|q| ds χ{|p|≤1} .

(4.115)

Using the definition of the standard Fourier transform (4.78), we easily obtain ∂2G ck (|p|, θ) ∂2G 1 2 [ k,m (|p|, θ) − ∥x Gk,m (x) − x2 Gk (x)∥L1 (R2 ) . (4.116) ≤ 2 2 2π ∂|p| ∂|p| Inequalities (4.116) and (4.92) allow us to derive the estimate from above in the absolute value for the second term in (4.115) as

4.2 Mixed-diffusion: system case

175

|p|2 ∥x2 Gk,m (x) − x2 Gk (x)∥L1 (R2 ) 1 2 χ{|p|≤1} ≤ ∥x Gk,m (x)−x2 Gk (x)∥L1 (R2 ) . 2 2s k 4π(p1 + |p2 | ) 4π Thus, i 

R |p|  R s h ∂ 2 G \ ck (|q|,θ) ∂2G k,m (|q|,θ)

0

− d|q| ds ∂|q|2 ∂|q|2 0

χ

{|p|≤1} 2 2s k

p1 + |p2 |

L∞ (R2 )

1 ≤ ∥x2 Gk,m (x) − x2 Gk (x)∥L1 (R2 ) , 4π which tends to zero as m → ∞ as assumed. An elementary computation gives us [ ∂G i k,m (0, θ) = − [(Gk,m (x), x1 )L2 (R2 ) cos θ + (Gk,m (x), x2 )L2 (R2 ) sin θ], ∂|p| 2π (4.117) c ∂ Gk i (0, θ) = − [(Gk (x), x1 )L2 (R2 ) cos θ + (Gk (x), x2 )L2 (R2 ) sin θ]. (4.118) ∂|p| 2π Let us first discuss the case b) of our lemma. By virtue of orthogonality conditions (4.102) and (4.103) along with (4.117), we have [ ∂G k,m (0, θ) = 0, M + 1 ≤ k ≤ N, m ∈ N. Analogously, (4.83) and (4.84) ∂|p| ck ∂G along with (4.118) yield that (0, θ) = 0, M + 1 ≤ k ≤ N . Thus, in ∂|p| the situation b), the first term in (4.115) is trivial. Let us turn our attention to the case a) of our lemma. We recall inequality (4.95). Since it is assumed that Gk,m (x), x2 Gk,m (x) ∈ L1 (R2 ), we have |x|Gk,m (x) ∈ L1 (R2 ). Similarly, ∥|x|Gk,m (x) − |x|Gk (x)∥L1 (R2 ) Z Z = |x||Gk,m (x) − Gk (x)|dx + |x|≤1

|x||Gk,m (x) − Gk (x)|dx |x|>1

≤∥Gk,m (x) − Gk (x)∥L1 (R2 ) + ∥x2 Gk,m (x) − x2 Gk (x)∥L1 (R2 ) → 0,

m→∞

as we assume. Hence, |x|Gk,m (x) → |x|Gk (x) in L1 (R2 ) as m → ∞. Orthogonality conditions (4.102) along with (4.117) give us [ ∂G i k,m (0, θ) = − sin θ ∂|p| 2π

Z Gk,m (x)x2 dx, R2

By means of (4.83) and (4.118) we obtain

1 ≤ k ≤ M,

m ∈ N.

176

4 Existence of solutions for some non-Fredholm integro-differential equations

ck ∂G i (0, θ) = − sin θ ∂|p| 2π

Z Gk (x)x2 dx,

1 ≤ k ≤ M,

m ∈ N.

R2

This enables us to derive the upper bound on the first term in (4.115) in the absolute value as R |p|| sin θ| R2 |x||Gk,m (x) − Gk (x)|dx χ{|p|≤1} 2π(|p|2 cos2 θ + |p|2sk | sin θ|2sk ) |p|1−2sk | sin θ|1−2sk ≤ ∥|x|Gk,m (x) − |x|Gk (x)∥L1 (R2 ) χ{|p|≤1} 2π 1 ≤ ∥|x|Gk,m (x) − |x|Gk (x)∥L1 (R2 ) , 2π since 0 < sk ≤

1 for 1 ≤ k ≤ M . Therefore, in the situation a) we derive 2

i \

h ∂G

d G k,m k

|p| ∂|p|

(0, θ)− ∂∂|p| (0, θ)

χ

{|p|≤1}

p21 + |p2 |2sk

L ∞ ( R2 )

≤

1 ∥|x|Gk,m (x)−|x|Gk (x)∥L1 (R2 ) → 0 2π

as m → ∞ as discussed above. Thus, by means of the argument above (4.105) is valid in both cases a) and b) of our lemma. Obviously, by virtue of the standard triangle inequality (4.107) and (4.108) follow easily from (4.105) and (4.106) respectively. Inequality (4.109) holds via a trivial limiting argument, which relies on (4.107) and (4.108). ⊔ ⊓

Chapter 5

Non-Fredholm Schr¨ odinger type operators

We study solvability of some linear nonhomogeneous elliptic problems and establish that under reasonable technical conditions the convergence in L2 (Rd ) of their right sides implies the existence and the convergence in H 4 (Rd ) of the solutions. In this chapter in contrast to previous ones (chapters 2-4) the coefficients of the operators are not constant anymore and we cannot use the Fourier transform directly obtain solvability conditions similar to the Fredholm operator: the right-hand side is orthogonal to all solutions of the homogeneous formally adjoint problem. In this chapter we consider two classes of non Fredholm operators (4th and 2nd order operators Schr¨ odinger type) and establish the solvability conditions for the equations involving them. To this end, we use the methods of the spectral and scattering theory for Schr¨ odinger type operators. Here we will mainly follow to our work (see [26], [30] and the references therein). The chapter 5 consists of 4 sections. In section 5.1 we consider the problems which contain the squares of the sums of second order non-Fredholm differential operators Schr¨odinger types. We especially emphasize that here we are dealing with the fourth order operators. Note that in the sum of the two Schr¨odinger type operators has the physical meaning of the resulting Hamiltonian of the two non-interacting quantum particles. The scalar potential functions involved to the operators (see below) are assumed to be shallow and shortrange with a few extra regularity conditions. Sections 5.2 and 5.3 we deal with the solvability in the sense of sequences of the operator equation consisting of the squares of the sums of second order non-Fredholm differential operators of Schr¨odinger type with a single potential bothe in the regular and singular cases. The last section of chapter 5, that is, section 5.4 is devoted to the solvability of generalized Poisson type equations with a scalar potential. For our purposes we start with some preliminaries. Consider the problem −∆u + V (x)u − au = f,

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Efendiev, Linear and Nonlinear Non-Fredholm Operators, https://doi.org/10.1007/978-981-19-9880-5_5

(5.1)

177

178

5 Non-Fredholm Schr¨ odinger type operators

where u ∈ E = H 2 (Rd ) and f ∈ F = L2 (Rd ), d ∈ N, a is a constant and the scalar potential function V (x) tends to 0 at infinity (it is well known that if V (x) → ∞ as |x| → ∞, it leads only to the discreteness of the spectrum). For a ≥ 0, the essential spectrum of the operator A : E → F corresponding to the left side of equation (5.1) contains the origin. Consequently, this operator fails to satisfy the Fredholm property. Its image is not closed, for d > 1 the dimensions of its kernel and the codimension of its image are not finite. As we already indicates in the previous chapters that, one of the important questions about problems with non-Fredholm operators concerns their solvability which we addressed in the following setting. Let fn be a sequence of functions in the image of the operator A, such that fn → f in L2 (Rd ) as n → ∞. Denote by un a sequence of functions from H 2 (Rd ) such that Aun = fn , n ∈ N. Since the operator A does not satisfy the Fredholm property, the sequence un may not be convergent. Let us call a sequence un the solution in the sense of sequences of the equation Au = f if Aun → f (see [57]). If such sequence converges to a function u0 in the norm of the space E, then u0 is a solution of this equation. Solution in the sense of sequences is equivalent in this case to the usual solution. However, in the case of the non-Fredholm operators, this convergence may not hold or it can occur in some weaker sense. In such case, solution in the sense of sequences may not imply the existence of the usual solution. In this chapter we will find sufficient conditions of equivalence of solutions in the sense of sequences and the usual solutions. In the other words, the conditions on sequences fn under which the corresponding sequences un are strongly convergent. Solvability in the sense of sequences for the sums of non-Fredholm Schr¨odingertype operators was studied in [30]. In section 5.1 we consider such operators squared, namely {−∆x + V (x) − ∆y + U (y)}2 u − a2 u = f (x, y),

x, y ∈ R3 ,

(5.2)

HU,V := {−∆x + V (x) − ∆y + U (y)}2 : H 4 (R6 ) → L2 (R6 )

(5.3)

with the constant a > 0. The operator

under the technical conditions on the scalar potential functions V (x) and U (y) stated below. Here and throughout sections 5.1-5.3 the Laplace operators ∆x and ∆y are with respect to the x and y variables respectively, such that cumulatively ∆ = ∆x + ∆y . Similarly for the gradients, ∇x and ∇y are with respect to the x and y variables respectively. In the applications the sum of the two Schr¨odinger type operators has the physical meaning of the resulting Hamiltonian of the two non-interacting quantum particles . Assumption 5.1 The potential functions V (x), U (y) : R3 → R satisfy the estimates

5 Non-Fredholm Schr¨ odinger type operators

C , 1 + |x|3.5+ε

|V (x)| ≤

179

|U (y)| ≤

C 1 + |y|3.5+ε

with some ε > 0 and x, y ∈ R3 a.e. such that

and

√

1 8 1 9 2 4 9 (4π)− 3 ∥V ∥L9 ∞ (R3 ) ∥V ∥ 9 4 3 < 1, L 3 (R ) 8

(5.4)

1 8 1 9 2 4 9 (4π)− 3 ∥U ∥L9 ∞ (R3 ) ∥U ∥ 9 4 3 < 1 L 3 (R ) 8

(5.5)

cHLS ∥V ∥

3

L 2 (R3 )

< 4π,

√

cHLS ∥U ∥

3

L 2 (R3 )

< 4π.

Moreover, |∇x V (x)|, ∆x V (x), |∇y U (y)|, ∆y U (y) ∈ L∞ (R3 ). Here and further down C denotes a finite positive constant and cHLS given on p.98 of [45] is the constant in the Hardy-Littlewood-Sobolev inequality Z Z 3 f1 (x)f1 (y) dxdy ≤ cHLS ∥f1 ∥2 32 3 , f1 ∈ L 2 (R3 ). 2 L (R ) R3 R3 |x − y| The norm of a function f1 ∈ Lp (Rd ), 1 ≤ p ≤ ∞, d ∈ N is designated as ∥f1 ∥Lp (Rd ) . Proposition 5.1 The function V (x) =

C , where C is small enough 1 + |x|4

satisfies Assumption 5.1. Proof. A straightforward computation yields |∇x V (x)| =

4C|x|3 ∈ L∞ (R3 ) (1 + |x|4 )2

and ∆x V (x) = −4C as well.

5|x|2 − 3|x|6 ∈ L∞ (R3 ) (1 + |x|4 )3

⊔ ⊓

Let us denote the inner product of two functions as Z (f (x), g(x))L2 (Rd ) := f (x)¯ g (x)dx,

(5.6)

Rd

We use the spaces H 2 (Rd ) and H 4 (Rd ) equipped with the norms ∥u∥2H 2 (Rd ) := ∥u∥2L2 (Rd ) + ∥∆u∥2L2 (Rd )

(5.7)

180

5 Non-Fredholm Schr¨ odinger type operators

and ∥u∥2H 4 (Rd ) := ∥u∥2L2 (Rd ) + ∥∆2 u∥2L2 (Rd )

(5.8)

respectively. Throughout the chapter, the sphere of radius r > 0 in Rd centered at the origin will be designated by Srd , the unit sphere is denoted by S d and |S d | stands for its Lebesgue measure. Under Assumption 5.1 above on the scalar potentials, operator (5.3) considered as acting in L2 (R6 ) with domain H 4 (R6 ) is self-adjoint and is unitarily equivalent to {−∆x − ∆y }2 on L2 (R6 ) via the product of the wave operators (see [40], [53]) ΩV± := s − lim eit(−∆x +V (x)) eit∆x ,

ΩU± := s − lim eit(−∆y +U (y)) eit∆y ,

t→∓∞

t→∓∞

with the limits here understood in the strong L2 sense (see e.g. [52] p.34, [13] p.90). Hence, operator (5.3) has no nontrivial L2 (R6 ) eigenfunctions . Its essential spectrum fills the nonnegative semi-axis [0, +∞). Therefore, operator (5.3) does not satisfy the Fredholm property. On the contrary, the operator hu,v := −∆x + V (x) − ∆y + U (y) + a considered as acting in L2 (R6 ) with domain H 2 (R6 ) satisfies the Fredholm property, has only the essential spectrum , which fills the interval [a, +∞), 2 6 2 6 such that the inverse h−1 u,v : L (R ) → H (R ) is bounded. The functions of the continuos spectrum of the first operator involved in (5.3) are the solutions of the Schr¨odinger equation [−∆x + V (x)]φk (x) = k 2 φk (x),

k ∈ R3 ,

in the integral form the Lippmann-Schwinger equation (see e.g. section 1.8 and the references therein) φk (x) =

eikx (2π)

3 2

−

1 4π

Z R3

ei|k||x−y| (V φk )(y)dy |x − y|

(5.9)

for the perturbed plane waves and the orthogonality conditions (φk (x), φk1 (x))L2 (R3 ) = δ(k − k1 ), k, k1 ∈ R3 . The integral operator involved in (5.9) (Qφ)(x) := −

1 4π

Z R3

ei|k||x−y| (V φ)(y)dy, |x − y|

φ(x) ∈ L∞ (R3 ).

Let us consider Q : L∞ (R3 ) → L∞ (R3 ). Its norm ∥Q∥∞ < 1 under Assumption 5.1 via Lemma 1.14. In fact, this norm is bounded above by the k-independent quantity, which is the left side of inequality (5.4). Similarly, for the second operator involved in (5.3) the functions of its continuous spectrum solve

5 Non-Fredholm Schr¨ odinger type operators

181

[−∆y + U (y)]ηq (y) = q 2 ηq (y),

q ∈ R3 ,

in the integral formulation ηq (y) =

eiqy (2π)

3 2

−

1 4π

Z R3

ei|q||y−z| (U ηq )(z)dz, |y − z|

(5.10)

such that the orthogonality conditions (ηq (y), ηq1 (y))L2 (R3 ) = δ(q − q1 ), q, q1 ∈ R3 hold. η0 (y) will correspond to the case of q = 0. The integral operator involved in (5.10) is 1 (P η)(y) := − 4π

Z R3

ei|q||y−z| (U η)(z)dz, |y − z|

η(y) ∈ L∞ (R3 ).

For P : L∞ (R3 ) → L∞ (R3 ) its norm ∥P ∥∞ < 1 under Assumption 5.1 by means of Lemma 1.14. As before, this norm can be estimated from above by the q-independent quantity I(U ), which is the left side of inequality (5.5). Let us denote by the double tilde sign the generalized Fourier transform with the product of these functions of the continuous spectrum ˜ f˜(k, q) := (f (x, y), φk (x)ηq (y))L2 (R6 ) ,

k, q ∈ R3 .

(5.11)

(5.11) is a unitary transform on L2 (R6 ). Our first main proposition of section 5.1 is as follows. Theorem 5.1 Let Assumption 5.1 hold, a > 0 and f (x, y) ∈ L2 (R6 ). Assume also that |x|f (x, y), |y|f (x, y) ∈ L1 (R6 ). Then problem (5.2) has a unique solution u(x, y) ∈ H 4 (R6 ) if and only if (f (x, y), φk (x)ηq (y))L2 (R6 ) = 0,

6 (k, q) ∈ S√ a

a.e.

(5.12)

In the very special case when the scalar potential functions V (x) and U (y) vanish identically in R3 , condition (5.12) gives us the orthogonality to the products of the corresponding standard Fourier harmonics. Then we turn our attention to the issue of the solvability in the sense of sequences for our equation. The corresponding sequence of approximate equations with n ∈ N is given by {−∆x + V (x) − ∆y + U (y)}2 un − a2 un = fn (x, y),

x, y ∈ R3 ,

(5.13)

with the constant a > 0 and the right sides converge to the right side of (5.2) in L2 (R6 ) as n → ∞. Theorem 5.2 Let Assumption 5.1 hold, a > 0, n ∈ N and fn (x, y) ∈ L2 (R6 ), such that fn (x, y) → f (x, y) in L2 (R6 ) as n → ∞. Let in addition |x|fn (x, y), |y|fn (x, y) ∈ L1 (R6 ), n ∈ N, such that |x|fn (x, y) → |x|f (x, y), |y|fn (x, y) → |y|f (x, y) in L1 (R6 ) as n → ∞ and the orthogonality

182

5 Non-Fredholm Schr¨ odinger type operators

relations (fn (x, y), φk (x)ηq (y))L2 (R6 ) = 0,

6 (k, q) ∈ S√ a

a.e.

(5.14)

hold for all n ∈ N. Then problems (5.2) and (5.13) admit unique solutions u(x, y) ∈ H 4 (R6 ) and un (x, y) ∈ H 4 (R6 ) respectively, such that un (x, y) → u(x, y) in H 4 (R6 ) as n → ∞. Section 5.2 is devoted to the studies of the equation {−∆x − ∆y + U (y)}2 u − a2 u = ϕ(x, y),

x ∈ Rd ,

y ∈ R3 ,

(5.15)

where d ∈ N, the constant a > 0 and the scalar potential function involved in (5.15) is shallow and short-range under Assumption 5.1 above. The more singular case of a = 0 will be treated in section 5.3 below in higher dimensions. The operator LU := {−∆x − ∆y + U (y)}2 : H 4 (Rd+3 ) → L2 (Rd+3 ).

(5.16)

Similarly to (5.3), under the given assumptions operator (5.16) considered as acting in L2 (Rd+3 ) with domain H 4 (Rd+3 ) is self-adjoint and is unitarily equivalent to {−∆x − ∆y }2 . Thus, operator (5.16) does not have nontrivial L2 (Rd+3 ) eigenfunctions . Its essential spectrum fills the nonnegative semiaxis [0, +∞). Therefore, operator (5.16) is non Fredholm. On the contrary, the operator lU := −∆x − ∆y + U (y) + a considered as acting in L2 (Rd+3 ) with domain H 2 (Rd+3 ) satisfies the Fredholm property, has only the essential spectrum, which fills the interval [a, +∞), such that the inverse lU −1 : L2 (Rd+3 ) → H 2 (Rd+3 ) is bounded. Let us consider another generalized Fourier transform with the standard Fourier harmonics and the perturbed plane waves ! eikx ˜ˆ ϕ(k, q) := ϕ(x, y), , k ∈ Rd , q ∈ R3 . (5.17) d ηq (y) (2π) 2 L2 (Rd+3 ) (5.17) is a unitary transform on L2 (Rd+3 ). We have the following statement. Theorem 5.3 Let the potential function U (y) satisfy Assumption 5.1, a > 0 and additionally ϕ(x, y) ∈ L2 (Rd+3 ), |x|ϕ(x, y), |y|ϕ(x, y) ∈ L1 (Rd+3 ), d ∈ N. Then problem (5.15) possesses a unique solution u(x, y) ∈ H 4 (Rd+3 ) if and only if ! eikx d+3 ϕ(x, y), = 0, (k, q) ∈ S√ a.e. (5.18) d ηq (y) a (2π) 2 L2 (Rd+3 )

5.1 Solvability in the sense of sequences with two potentials

183

Our final main proposition is devoted to the issue of the solvability in the sense of sequences for our problem. The corresponding sequence of approximate equations with n ∈ N is given by {−∆x − ∆y + U (y)}2 un − a2 un = ϕn (x, y),

x ∈ Rd ,

y ∈ R3 , (5.19) where the right sides converge to the right side of (5.15) in L2 (Rd+3 ) as n → ∞. d ∈ N,

Theorem 5.4 Let the potential function U (y) satisfy Assumption 5.1, a > 0, n ∈ N and ϕn (x, y) ∈ L2 (Rd+3 ), d ∈ N, such that ϕn (x, y) → ϕ(x, y) in L2 (Rd+3 ) as n → ∞. Let in addition |x|ϕn (x, y), |y|ϕn (x, y) ∈ L1 (Rd+3 ), such that |x|ϕn (x, y) → |x|ϕ(x, y), |y|ϕn (x, y) → |y|ϕ(x, y) in L1 (Rd+3 ) as n → ∞ and the orthogonality relations ! eikx d+3 ϕn (x, y), = 0, (k, q) ∈ S√ d ηq (y) a (2π) 2 L2 (Rd+3 )

a.e.

(5.20)

hold for all n ∈ N. Then problems (5.15) and (5.19) admit unique solutions u(x, y) ∈ H 4 (Rd+3 ) and un (x, y) ∈ H 4 (Rd+3 ) respectively, such that un (x, y) → u(x, y) in H 4 (Rd+3 ) as n → ∞. Remark 5.1 Let us note that (5.12), (5.14), (5.18), (5.20) are the orthogonality conditions containing the functions of the continuous spectrum of our Schr¨odinger operators, as distinct from the Limiting Absorption Principle in which one orthogonalizes to the standard Fourier harmonics (see e.g. Lemma 2.3 and Proposition 2.4 of [36]). We proceed to the proof of our statements.

5.1 Solvability in the sense of sequences with two potentials Proof of Theorem 5.1. First of all, let us observe that it is sufficient to solve equation (5.2) in H 2 (R6 ), since this solution will belong to H 4 (R6 ) as well. Indeed, it can be easily shown that ∆2 u + [V 2 (x) + U 2 (y)]u − [∆x V (x) + ∆y U (y)]u − 2[V (x) + U (y)]∆u− −2∇x V (x).∇x u − 2∇y U (y).∇y u + 2V (x)U (y)u − a2 u = f (x, y),

(5.21)

184

5 Non-Fredholm Schr¨ odinger type operators

with u(x, y) a solution of (5.2) belonging to H 2 (R6 ). The dot symbol in the fifth and the sixth terms in the left side of (5.21) and throughout this chapter denotes the standard scalar product of two vectors in R3 . Evidently, all the terms in the left side of (5.21) starting from the second one are square integrable since according to Assumption 5.1 our scalar potential functions are bounded along with |∇x V (x)|, |∇y U (y)|, ∆x V (x), ∆y U (y) and u(x, y) ∈ H 2 (R6 ). The right side of (5.21) is square integrable as well as assumed. Therefore, ∆2 u(x, y) ∈ L2 (R6 ), which yields that u(x, y) ∈ H 4 (R6 ). To show the uniqueness of solutions for our equation, we suppose that problem (5.2) admits two solutions u1 (x, y), u2 (x, y) ∈ H 4 (R6 ). Then their difference w(x, y) := u1 (x, y) − u2 (x, y) ∈ H 4 (R6 ) solves the equation HU,V w = a2 w. But the operator HU,V : H 4 (R6 ) → L2 (R6 ) has no nontrivial eigenfunctions as discussed above. Therefore, w(x, y) vanishes in R6 . Let us apply the generalized Fourier transform (5.11) to both sides of problem (5.2). This yields ˜ f˜(k, q) ˜˜(k, q) = u . (k 2 + q 2 )2 − a2 Hence ˜˜(k, q) = g˜˜1 (k, q) + g˜˜2 (k, q), u

(5.22)

where g˜˜1 (k, q) :=

˜ f˜(k, q) , 2a(k 2 + q 2 − a)

g˜˜2 (k, q) := −

˜ f˜(k, q) . 2a(k 2 + q 2 + a)

It is worth noting that in the right side of (5.22) the second term there reflects the presence of the fourth order operator. Evidently, the functions g1 (x, y) and g2 (x, y) satisfy the equations {−∆x + V (x) − ∆y + U (y)}g1 − ag1 =

1 f (x, y) 2a

and

(5.23)

1 f (x, y) (5.24) 2a respectively. The operator involved in the left side of problem (5.24) has a 2 6 2 6 bounded inverse h−1 u,v : L (R ) → H (R ) as discussed above and the right side of (5.24) is square integrable as assumed. Therefore, equation (5.24) admits a unique solution g2 (x, y) ∈ H 2 (R6 ). It is not difficult to see that, under {−∆x + V (x) − ∆y + U (y)}g2 + ag2 = −

5.1 Solvability in the sense of sequences with two potentials

185

the given conditions equation (5.23) has a unique solution g1 (x, y) ∈ H 2 (R6 ) if and only if orthogonality condition (5.12) holds. Note that the solvability of problem (5.23) in L2 (R6 ) is equivalent to its solvability in H 2 (R6 ) since the right side of (5.23) is square integrable and the scalar potentials involved in (5.23) are bounded according to the one of our assumptions. ⊔ ⊓ Let us turn our attention to the solvability in the sense of sequences for our equation in the case of two scalar potentials. Proof of Theorem 5.2. First of all, let us demonstrate that if u(x, y) and un (x, y), n ∈ N are the unique H 4 (R6 ) solutions of (5.2) and (5.13) respectively and un (x, y) → u(x, y) in H 2 (R6 ) as n → ∞, then we have un (x, y) → u(x, y) in H 4 (R6 ) as n → ∞ as well. Indeed, (5.2) and (5.13) yield that for n ∈ N and x, y ∈ R3 {−∆x + V (x) − ∆y + U (y)}2 (un − u) − a2 (un − u) = fn (x, y) − f (x, y). Hence ∆2 (un − u) + [V 2 (x) + U 2 (y)](un − u) − [∆x V (x) + ∆y U (y)](un − u)− −2[V (x) + U (y)]∆(un − u) − 2∇x V (x).∇x (un − u) − 2∇y U (y).∇y (un − u)+ +2V (x)U (y)(un − u) − a2 (un − u) = fn (x, y) − f (x, y).

(5.25)

Since un (x, y) → u(x, y) in H 2 (R6 ) as n → ∞ as assumed, we have here un (x, y) → u(x, y),

∇x un (x, y) → ∇x u(x, y),

∇y un (x, y) → ∇y u(x, y),

∆un (x, y) → ∆u(x, y) in L2 (R6 ) as n → ∞ and V (x),

U (y),

|∇x V (x)|,

∆x V (x),

|∇y U (y)|,

∆y U (y)

are bounded functions due to Assumption 5.1 above. Therefore, all the terms in the left side of identity (5.25) starting from the second one tend to zero in L2 (R6 ) as n → ∞. The right side of (5.25) converges to zero in L2 (R6 ) as n → ∞ as assumed. Hence, ∆2 un → ∆2 u in L2 (R6 ) as n → ∞. By means of norm definition (5.8) we obtain that un (x, y) → u(x, y) in H 4 (R6 ) as n → ∞. By virtue of Theorem 5.1 above, under the given conditions equation (5.13) admits a unique solution un (x, y) ∈ H 4 (R6 ), n ∈ N. Hence, under the stated assumptions we arrive at the limiting orthogonality relation (f (x, y), φk (x)ηq (y))L2 (R6 ) = 0,

6 (k, q) ∈ S√ a

a.e.

Then by means of Theorem 5.1 above problem (5.2) possesses a unique solution u(x, y) ∈ H 4 (R6 ). Let us apply the generalized Fourier transform

186

5 Non-Fredholm Schr¨ odinger type operators

(5.11) to both sides of problems (5.2) and (5.13). This yields the representation (5.22) as in the proof of Theorem 5.1 above, where the functions g1 (x, y), g2 (x, y) ∈ H 2 (R6 ) under the given conditions are the unique solutions of equations (5.23) and (5.24) respectively. Similarly, ˜˜n (k, q) = g˜˜1,n (k, q) + g˜˜2,n (k, q), u

n ∈ N,

(5.26)

where g˜˜1,n (k, q) :=

˜ f˜n (k, q) , 2a(k 2 + q 2 − a)

g˜˜2,n (k, q) := −

˜ f˜n (k, q) . 2a(k 2 + q 2 + a)

Apparently, the functions g1,n (x, y) and g2,n (x, y) solve the equations {−∆x + V (x) − ∆y + U (y)}g1,n − ag1,n =

1 fn (x, y) 2a

(5.27)

and {−∆x + V (x) − ∆y + U (y)}g2,n + ag2,n = −

1 fn (x, y) 2a

(5.28)

respectively. Since the operator involved in the left side of (5.28) has a −1 2 6 2 6 bounded inverse h−1 u,v : L (R ) → H (R ), such that its norm ∥hu,v ∥ < ∞ 2 6 as discussed above and the right side of (5.28) belongs to L (R ) as assumed, (5.28) admits a unique solution g2,n (x, y) ∈ H 2 (R6 ). Because fn (x, y) → f (x, y) in L2 (R6 ) as n → ∞ via the one of our assumptions, we have ∥g2,n − g2 ∥H 2 (R6 ) ≤

1 −1 ∥h ∥∥fn − f ∥L2 (R6 ) → 0, 2a u,v

n → ∞,

such that g2,n (x, y) → g2 (x, y) in H 2 (R6 ) as n → ∞. By virtue of the result [30], we have that equation (5.27) possesses a unique solution g1,n (x, y) ∈ H 2 (R6 ), such that g1,n (x, y) → g1 (x, y) in H 2 (R6 ) as n → ∞. Using formulas (5.26) and (5.22) considered in the x, y space, we easily arrive at ∥un (x, y) − u(x, y)∥H 2 (R6 ) ≤ ≤ ∥g1,n (x, y) − g1 (x, y)∥H 2 (R6 ) + ∥g2,n (x, y) − g2 (x, y)∥H 2 (R6 ) → 0 as n → ∞. Therefore, un (x, y) → u(x, y) in H 4 (R6 ) as n → ∞ as discussed above. ⊔ ⊓ Below we treat the case when the free Laplacian is added to our three dimensional Schr¨odinger operator .

5.2 Solvability in the sense of sequences with Laplacian and a single potential

187

5.2 Solvability in the sense of sequences with Laplacian and a single potential: regular case Proof of Theorem 5.3. First of all, we show that it is sufficient to solve problem (5.15) in H 2 (Rd+3 ), because such solution will belong to H 4 (Rd+3 ) as well. Apparently, ∆2 u+U 2 (y)u−2U (y)∆u−u∆y U (y)−2∇y U (y).∇y u−a2 u = ϕ(x, y), (5.29) where u(x, y) is a solution of (5.15), which belongs to H 2 (Rd+3 ). Clearly, all the terms in the left side of (5.29) starting from the second one are square integrable because by means of Assumption 5.1 our scalar potential function is bounded along with |∇y U (y)| and ∆y U (y) and u(x, y) ∈ H 2 (Rd+3 ). The right side of (5.29) is square integrable as well as assumed. Hence, ∆2 u ∈ L2 (Rd+3 ), which implies that u(x, y) ∈ H 4 (Rd+3 ). To establish the uniqueness of solutions for our equation, we suppose that (5.15) possesses two solutions u1 (x, y), u2 (x, y) ∈ H 4 (Rd+3 ). Then their difference w(x, y) := u1 (x, y) − u2 (x, y) ∈ H 4 (Rd+3 ) satisfies the equation LU w = a2 w. Apparently, the operator LU : H 4 (Rd+3 ) → L2 (Rd+3 ) has no nontrivial eigenfunctions as discussed above. Thus, w(x, y) vanishes in Rd+3 . We apply the generalized Fourier transform (5.17) to both sides of problem (5.15) and obtain ˜ˆ ˜ˆ ˜ˆ(k, q) = G u (5.30) 1 (k, q) + G2 (k, q), where ˜ˆ G 1 (k, q) :=

˜ˆ ϕ(k, q) , 2a(k 2 + q 2 − a)

˜ˆ G 2 (k, q) := −

˜ ˆ q) ϕ(k, . 2a(k 2 + q 2 + a)

Clearly, the functions G1 (x, y) and G2 (x, y) solve the equations {−∆x − ∆y + U (y)}G1 − aG1 =

1 ϕ(x, y) 2a

and

(5.31)

1 ϕ(x, y) (5.32) 2a respectively. The operator involved in the left side of equation (5.32) has a −1 bounded inverse lU : L2 (Rd+3 ) → H 2 (Rd+3 ) as discussed above and the right side of (5.32) is square integrable due to the one of our assumptions. Hence, problem (5.32) possesses a unique solution G2 (x, y) ∈ H 2 (Rd+3 ).It is not difficult to see that( see also [30]) , under the given assumptions equation {−∆x − ∆y + U (y)}G2 + aG2 = −

188

5 Non-Fredholm Schr¨ odinger type operators

(5.31) admits a unique solution G1 (x, y) ∈ H 2 (Rd+3 ) if and only if orthogonality relation (5.18) holds. Evidently, the solvability of equation (5.31) in L2 (Rd+3 ) is equivalent to its solvability in H 2 (Rd+3 ) because the right side of (5.31) is square integrable and the scalar potential involved in (5.31) is bounded due to our assumptions. ⊔ ⊓ We finish this section with establishing the solvability in the sense of sequences for our problem when the free Laplacian is added to a three dimensional Schr¨odinger operator . Proof of Theorem 5.4. First of all we establablish that if u(x, y) and un (x, y), n ∈ N are the unique H 4 (Rd+3 ) solutions of equations (5.15) and (5.19) respectively and un (x, y) → u(x, y) in H 2 (Rd+3 ) as n → ∞, then un (x, y) → u(x, y) in H 4 (Rd+3 ) as n → ∞ as well. Clearly, (5.15) and (5.19) imply that for n ∈ N and x ∈ Rd , y ∈ R3 , d ∈ N {−∆x − ∆y + U (y)}2 (un − u) − a2 (un − u) = ϕn (x, y) − ϕ(x, y). Hence ∆2 (un − u) + U 2 (y)(un − u) − 2U (y)∆(un − u) − (un − u)∆y U (y)− −2∇y U (y).∇y (un − u) − a2 (un − u) = ϕn (x, y) − ϕ(x, y).

(5.33)

The fact that un (x, y) → u(x, y) in H 2 (Rd+3 ) as n → ∞ as assumed implies that un (x, y) → u(x, y),

∇y un (x, y) → ∇y u(x, y),

∆un (x, y) → ∆u(x, y)

in L2 (Rd+3 ) as n → ∞ and U (y),

|∇y U (y)|,

∆y U (y)

are bounded functions via Assumption 5.1. Thus, all the terms in the left side of (5.33) starting from the second one converge to zero in L2 (Rd+3 ) as n → ∞. The right side of (5.33) tends to zero in L2 (Rd+3 ) as n → ∞ via the one of our assumptions. Therefore, ∆2 un (x, y) → ∆2 u(x, y) in L2 (Rd+3 ) as n → ∞. By virtue of norm definition (5.8) we have that un (x, y) → u(x, y) in H 4 (Rd+3 ) as n → ∞. By means of Theorem 5.3 above, under our assumptions problem (5.19) has a unique solution un (x, y) ∈ H 4 (Rd+3 ), n ∈ N. Thus, under the given conditions we obtain the limiting orthogonality relation ! eikx d+3 ϕ(x, y), = 0, (k, q) ∈ S√ a.e. d ηq (y)) a (2π) 2 2 d+3 L (R )

5.2 Solvability in the sense of sequences with Laplacian and a single potential

189

Therefore, by virtue of Theorem 5.3 above equation (5.15) admits a unique solution u(x, y) ∈ H 4 (Rd+3 ). We apply the generalized Fourier transform (5.17) to both sides of equations (5.15) and (5.19). This gives us the representation (5.30) given in the proof of Theorem 5.3, where the functions G1 (x, y), G2 (x, y) ∈ H 2 (Rd+3 ) under our assumptions are the unique solutions of problems (5.31) and (5.32) respectively. Apparently, ˜ ˆ˜ 1,n (k, q) + G ˆ˜ 2,n (k, q), u ˆn (k, q) = G

n ∈ N,

(5.34)

where ˜ˆ G 1,n (k, q) :=

˜ ϕˆn (k, q) , 2a(k 2 + q 2 − a)

˜ˆ G 2,n (k, q) := −

˜ ϕˆn (k, q) . 2a(k 2 + q 2 + a)

Evidently, the functions G1,n (x, y) and G2,n (x, y) satisfy the equations {−∆x − ∆y + U (y)}G1,n − aG1,n =

1 ϕn (x, y) 2a

(5.35)

and

1 ϕn (x, y) (5.36) 2a respectively. Because the operator involved in the left side of (5.36) has a −1 −1 bounded inverse lU : L2 (Rd+3 ) → H 2 (Rd+3 ), such that its norm ∥lU ∥1} . {k +q ≤1} 2 2 2 2 2 2 (k + q ) (k + q ) (k 2 + q 2 )2 (5.39) Here and below χA stands for the characteristic function of a set A ⊆ Rd+3 . Clearly, the second term in the right side of (5.39) can be bounded from ˜ˆ above in the absolute value by |ϕ(k, q)| ∈ L2 (Rd+3 ) due to the one of our assumptions. We apply the generalized Fourier transform (5.17) to both sides of (5.38) and use (5.39) to obtain ˜ˆ(k, q) = u

˜ˆ ˆ˜ q) ϕ(k, q) ϕ(k, 2 +q 2 ≤1} + v˜ˆ(k, q) = 2 χ χ{k2 +q2 >1} . {k k + q2 k2 + q2

(5.40)

Evidently, the second term in the right side of (5.40) can be estimated from ˜ˆ above in the absolute value by |ϕ(k, q)| ∈ L2 (Rd+3 ). The first term in the right side of (5.40) can be trivially bounded from above in the absolute value by virtue of (5.39) by

5.3 Solvability in the sense of sequences with Laplacian and a single potential

191

˜ˆ |ϕ(k, q)| ˜ˆ(k, q)|. = |u 2 (k + q 2 )2 Therefore, if we have u(x, y) ∈ L2 (Rd+3 ) then v(x, y) ∈ L2 (Rd+3 ) as well. Let us first treat the case when the dimension d ≥ 6. By means of Lemma 1.14 under the given assumptions we have ∥ηq (y)∥L∞ (R3 ) ≤

1

1 < ∞, 1 − I(U ) (2π) 3 2

q ∈ R3 ,

(5.41)

such that via (5.17) we obtain ˜ˆ |ϕ(k, q)| ≤

1 (2π)

d+3 2

1 ∥ϕ(x, y)∥L1 (Rd+3 ) . 1 − I(U )

This allows us to estimate the first term in the right side of (5.39) in the norm as s

˜

ϕ(k,

∥ϕ(x, y)∥L1 (Rd+3 ) |S d+3 | 1

ˆ q)

χ 2 2 ≤ 1} . {k +q ≤1} k + q2 k2 + q2 k2 + q2

(5.48)

Evidently, the second term in the right side of (5.48) can be bounded from ˜ above in the absolute value by |ϕˆn (k, q)| ∈ L2 (Rd+3 ) via the one of our assumptions. The first term in the right side of (5.48) can be trivially estimated from above in the absolute value using (5.47) by ˜ |ϕˆn (k, q)| ˜ˆn (k, q)|. = |u (k 2 + q 2 )2 Hence, un (x, y) ∈ H 4 (Rd+3 ) will imply that vn (x, y) ∈ L2 (Rd+3 ). By the similar reasoning via (5.39) and (5.40) we easily deduce that u(x, y) ∈ H 4 (Rd+3 ) yields v(x, y) ∈ L2 (Rd+3 ). IV) Formulas (5.40) and (5.48) give us v˜ˆn (k, q) − v˜ ˆ(k, q) = =

˜ ˜ˆ ˜ ˜ ˆ q) ϕˆn (k, q) − ϕ(k, q) ϕˆn (k, q) − ϕ(k, 2 2 χ + χ{k2 +q2 >1} . {k +q ≤1} 2 2 2 k +q k + q2

(5.49)

Clearly, the second term in the right side of (5.49) can be bounded above in ˜ ˜ˆ the absolute value by |ϕˆn (k, q) − ϕ(k, q)|. Thus

˜ˆ

ϕ˜ˆ (k, q) − ϕ(k,

q)

n

2 2 χ ≤ ∥ϕn (x, y) − ϕ(x, y)∥L2 (Rd+3 ) → 0

{k +q >1}

2 d+3 k2 + q2 L (R

)

as n → ∞ due to the one of our assumptions. The first term in the right side of (5.49) can be easily estimated from above in the absolute value using (5.47) and (5.39) by

194

5 Non-Fredholm Schr¨ odinger type operators

˜ ˜ˆ |ϕˆn (k, q) − ϕ(k, q)| ˜ˆn (k, q) − u ˜ = |u ˆ(k, q)|. 2 2 2 (k + q ) Therefore,

˜ˆ

ϕ˜ˆ (k, q) − ϕ(k,

q)

n

2 +q 2 ≤1} χ

{k

k2 + q2

≤ ∥un (x, y) − u(x, y)∥L2 (Rd+3 ) . L2 (Rd+3 )

Hence, if u(x, y), un (x, y) ∈ H 4 (Rd+3 ), n ∈ N, such that un (x, y) → u(x, y) in L2 (Rd+3 ) as n → ∞, then we arrive at vn (x, y) → v(x, y) in L2 (Rd+3 ) as n → ∞ as well. V) Let us first discuss the situation when d ≥ 6. By means of the part b) of Theorem 5.5 above, under the given conditions equations (5.15) and (5.19) admit unique solutions u(x, y), un (x, y) ∈ H 4 (Rd+3 ), n ∈ N respectively. By ˜ˆn (k, q) − u ˜ virtue of (5.47) and (5.39), we obtain u ˆ(k, q) = =

˜ ˜ ˜ ˆ q) ˆ˜ q) ϕˆn (k, q) − ϕ(k, ϕˆn (k, q) − ϕ(k, 2 +q 2 ≤1} + χ χ{k2 +q2 >1} . {k (k 2 + q 2 )2 (k 2 + q 2 )2

(5.50)

Clearly, the second term in the right side of (5.50) can be bounded above in ˜ ˜ˆ the absolute value by |ϕˆn (k, q) − ϕ(k, q)|, such that

˜ˆ

ϕ˜ˆ (k, q) − ϕ(k,

q)

n

2 2 χ ≤ ∥ϕn (x, y) − ϕ(x, y)∥L2 (Rd+3 ) → 0

{k +q >1}

2 d+3 (k 2 + q 2 )2 L (R

)

as n → ∞ via the one of the given conditions. By means of (5.17) along with (5.41) we easily derive ˜ ˜ˆ |ϕˆn (k, q) − ϕ(k, q)| ≤ Thus

≤

1 (2π)

d+3 2

1 ∥ϕn (x, y) − ϕ(x, y)∥L1 (Rd+3 ) . 1 − I(U )

˜ˆ

ϕ˜ˆ (k, q) − ϕ(k,

q)

n

χ{k2 +q2 ≤1}

2 2 2

(k + q ) 1

(2π)

d+3 2

1 1 − I(U )

s

≤ L2 (Rd+3 )

|S d+3 | ∥ϕn (x, y) − ϕ(x, y)∥L1 (Rd+3 ) → 0, d−5

n→∞

due to our assumptions. Therefore, by virtue of (5.50) we arrive at un (x, y) → u(x, y) in L2 (Rd+3 ) as n → ∞. Steps III) and IV) above give us v(x, y), vn (x, y) ∈ L2 (Rd+3 ), n ∈ N and vn (x, y) → v(x, y) in L2 (Rd+3 ) as n → ∞. Then by means of step II) we have un (x, y) → u(x, y) in H 2 (Rd+3 )

5.3 Solvability in the sense of sequences with Laplacian and a single potential

195

as n → ∞. Finally, step I) above yields un (x, y) → u(x, y) in H 4 (Rd+3 ) as n → ∞, which completes the proof of the part b) of our theorem. VI) Let us proceed to establishing the result of the theorem in the situation when d = 4, 5. We use orthogonality relations (5.44) along with inequality (5.41) to obtain |(ϕ(x, y), η0 (y))L2 (Rd+3 ) | = |(ϕ(x, y) − ϕn (x, y), η0 (y))L2 (Rd+3 ) | ≤ ≤

1

1 ∥ϕn (x, y) − ϕ(x, y)∥L1 (Rd+3 ) → 0, 1 − I(U ) (2π) 3 2

n → ∞.

Note that under the stated assumptions ϕn (x, y) ∈ L1 (Rd+3 ) and ϕn (x, y) → ϕ(x, y) in L1 (Rd+3 ) as n → ∞ . Therefore, the limiting orthogonality condition (ϕ(x, y), η0 (y))L2 (Rd+3 ) = 0 (5.51) holds. By means of the result of the part a) of Theorem 5.5 above, equations (5.15) and (5.19) possess unique solutions u(x, y), un (x, y) ∈ H 4 (Rd+3 ), n ∈ N. Orthogonality relations (5.44) and (5.51) along with definition (5.17) give us ˜ˆ ˜ ϕ(0) = 0, ϕˆn (0) = 0, n ∈ N. (5.52) (5.52) enables us to express Z √k2 +q2 ˜ˆ ∂ ϕ(s, σ) ˜ ˆ ϕ(k, q) = ds, ∂s 0

˜ ϕˆn (k, q) =

Z √k2 +q2 0

˜ ∂ ϕˆn (s, σ) ds, (5.53) ∂s

where n ∈ N. The second term in the right side of (5.50) tends to zero in L2 (Rd+3 ) as n → ∞ as discussed in step V) above. By virtue of (5.53) we write the first term in the right side of (5.50) as i ˜ R √k2 +q2 h ∂ ϕ˜ˆn (s,σ) ∂ ϕ(s,σ) ˆ − ds ∂s ∂s 0 χ{k2 +q2 ≤1} . (5.54) 2 2 2 (k + q ) Clearly, (5.54) can be bounded above in the absolute value by ˜ ˜ˆ ∥(∇k + ∇q )(ϕˆn (k, q) − ϕ(k, q))∥L∞ (Rd+3 ) 3

(k 2 + q 2 ) 2

χ{k2 +q2 ≤1} .

Note that under the given conditions we have ˜ ˜ˆ ∥(∇k + ∇q )(ϕˆn (k, q) − ϕ(k, q))∥L∞ (Rd+3 ) → 0, (see also [30] and the references therein). Hence

n → ∞,

196

5 Non-Fredholm Schr¨ odinger type operators





R√ 0

k2 +q 2

h

˜ ˆn (s,σ) ∂ϕ ∂s

−

(k 2 + q 2 )2

i ˜ ˆ ∂ ϕ(s,σ) ds ∂s



χ{k2 +q2 ≤1}

s

˜ ˜ˆ ≤ ∥(∇k + ∇q )(ϕˆn (k, q) − ϕ(k, q))∥L∞ (Rd+3 )

≤ L2 (Rd+3 )

|S d+3 | → 0, d−3

n → ∞.

Therefore, un (x, y) → u(x, y) in L2 (Rd+3 ) as n → ∞. By means of steps III), IV), II) and I) above we obtain un (x, y) → u(x, y) in H 4 (Rd+3 ) as n → ∞, which completes the proof of the part a) of the theorem. ⊔ ⊓ Remark 5.2 Note that no orthogonality conditions are needed to solve equation (5.15) with a = 0 in H 4 (Rd+3 ) in higher dimensions d ≥ 6. As distinct from the present case, when dealing with the sum of the free Laplacian and our three dimensional Schr¨odinger operator to the first power for a = 0, the solvability relations are not needed for all d ≥ 2 (see [30] and the references therein. Remark 5.3 Our approach can be extended to the higher, even order elliptic equations. For example, in the case of the sixth order operator {−∆x +V (x)− ∆y + U (y)}3 we can check for the analog of Assumption 5.1 of Theorem 5.2.

5.4 Generalized Poisson type equation with a potential In this section we will present two theorems (see Theorems 5.7 and 5.8 below) dealing with the generalized Poisson type equation with the scalar potential. Indeed, consider the equation n o [−∆ + V (x)]s1 + [−∆ + V (x)]s2 u = f (x), x ∈ R3 (5.55) with a square integrable right side and the powers 0 < s1 < s2 < 1 and ∆ = ∆x . The assumptions on our shallow, short-range scalar potential function V (x) were stated in Assumption 5.1. The problems with the sum of the negative Laplacians without a potential raised to different fractional powers arise in the studies of the double scale anomalous diffusion (see e.g. [37]). The probabilistic realization of the anomalous diffusion was discussed in [49]. The non-Fredholm operator in the left side of (5.55) L := [−∆ + V (x)]s1 + [−∆ + V (x)]s2

(5.56)

on L2 (R3 ) is defined via the spectral calculus. It has only the essential spectrum σess (L) = [0, +∞)

5.4 Generalized Poisson type equation with a potential

197

and no nontrivial L2 (R3 ) eigenfunctions. By virtue of the spectral theorem, we have Lφk (x) = (|k|2s1 + |k|2s2 )φk (x) with the functions of the continuous spectrum of our Schr¨ odinger operator φk (x) discussed in the introduction of chapter 5 above. The function φ0 (x) in the theorem below will correspond to the case of k = 0. In the argument below we will use f˜(k) = (f (x), φk (x))L2 (R3 ) ,

k ∈ R3 .

(5.57)

(5.57) is a unitary transform on L2 (R3 ). It is not difficult to show that ( see also section 1.8) |f˜(k)| ≤

1

1 ∥f (x)∥L1 (R3 ) , 1 − I(V ) (2π) 3 2

(5.58)

where I(V ) < 1 is the left side of inequality (5.4). The first result of the section is as follows. Theorem 5.7 Let V (x) satisfy Assumption 5.1 above, the powers 0 < s1 < s2 < 1 and f (x) ∈ L2 (R3 ). 3 , let in addition f (x) ∈ L1 (R3 ). Then equation (5.55) admits 4 a unique solution u(x) ∈ L2 (R3 ). 3 2) If ≤ s1 < 1, let in addition xf (x) ∈ L1 (R3 ). Then problem (5.55) pos4 sesses a unique solution u(x) ∈ L2 (R3 ) if and only if the orthogonality condition (f (x), φ0 (x))L2 (R3 ) = 0 (5.59) 1) If 0 < s1
1} . {|k|≤1} 2s 2s + |k| 2 |k| 1 + |k|2s2

|k|2s1

(5.60)

198

5 Non-Fredholm Schr¨ odinger type operators

Here and below χA will stand for the characteristic function of a set A ⊆ R3 . Clearly, the second term in the right side of (5.60) can be estimated from above in the absolutely value as |f˜(k)| f˜(k) χ ∈ L2 (R3 ) 2s1 ≤ |k| + |k|2s2 {|k|>1} 2 via the one of our assumptions. Let us first consider the situation when 3 0 < s1 < . Then the first term in the right side of (5.60) can be bounded 4 from above in the absolutely value using inequality (5.58) as χ{|k|≤1} f˜(k) 1 1 χ ∥f (x)∥L1 (R3 ) ∈ L2 (R3 ), 2s1 ≤ {|k|≤1} 2s |k| + |k| 2 (2π) 32 1 − I(V ) |k|2s1 which completes the proof of part 1) of our theorem. Let us conclude the 3 argument by treating the situation when the power ≤ s1 < 1. We will use 4 the representation formula f˜(k) = f˜(0) +

|k|

Z 0

∂ f˜(q, σ) dq. ∂q

Here and below σ denotes the angle variables on the sphere and f˜(0) = (f (x), φ0 (x))L2 (R3 ) . This enables us to express the first term in the right side of (5.60) as R |k| ∂ f˜(q,σ) dq f˜(0) ∂q 0 χ + χ{|k|≤1} . {|k|≤1} 2s 2s 2s 2s |k| 1 + |k| 2 |k| 1 + |k| 2 Evidently, the second term in the sum above can be estimated from above in the absolute value as R |k| ∂ f˜(q,σ) dq 0 ∂q χ 2s1 ≤ ∥∇q f˜(q)∥L∞ (R3 ) |k|1−2s1 χ{|k|≤1} ∈ L2 (R3 ). {|k|≤1} |k| + |k|2s2 Note that under the given conditions ∇q f˜(q) ∈ L∞ (R3 ) by means of section 1.8. Therefore, it remains to analyze the term f˜(0) χ{|k|≤1} . + |k|2s2

|k|2s1

(5.61)

It can be easily verified that (5.61) belongs to L2 (R3 ) if and only if orthogonality condition(5.59) holds. ⊔ ⊓

5.4 Generalized Poisson type equation with a potential

199

Note that in the first case of our theorem we do not need an orthogonality condition to solve equation (5.55) in L2 (R3 ). Let us introduce the approximate equations n o [−∆ + V (x)]s1 + [−∆ + V (x)]s2 un = fn (x), x ∈ R3 (5.62) with n ∈ N and 0 < s1 < s2 < 1 and establish the solvability in the sense of sequences for our problem (5.55). The final result of this section is as follows. Theorem 5.8 Let V (x) satisfy Assumption 5.1 above, n ∈ N, the powers 0 < s1 < s2 < 1 and fn (x) ∈ L2 (R3 ), so that fn (x) → f (x) in L2 (R3 ) as n → ∞. 3 , let in addition fn (x) ∈ L1 (R3 ), n ∈ N, so that fn (x) → 4 f (x) in L1 (R3 ) as n → ∞. Then equations (5.55) and (5.62) possess unique solutions u(x) ∈ L2 (R3 ) and un (x) ∈ L2 (R3 ) respectively, so that un (x) → u(x) in L2 (R3 ) as n → ∞. 3 2) If ≤ s1 < 1, let in addition xfn (x) ∈ L1 (R3 ), n ∈ N, so that xfn (x) → 4 xf (x) in L1 (R3 ) as n → ∞ and 1) If 0 < s1
1} . {|k|≤1} |k|2s1 + |k|2s2 |k|2s1 + |k|2s2

(5.66)

Evidently, the second term in the right side of (5.66) can be estimated from above in the absolute value as f˜ (k) − f˜(k) |f˜n (k) − f˜(k)| n χ ≤ . 2s1 {|k|>1} |k| + |k|2s2 2 Hence,

f˜ (k) − f˜(k)

n

χ

2s1

{|k|>1}

|k| + |k|2s2

≤ L2 (R3 )

1 ∥fn (x) − f (x)∥L2 (R3 ) → 0, 2

n→∞

3 as assumed. Let us first discuss the case when 0 < s1 < . By means of (5.58), 4 we have |f˜n (k) − f˜(k)| ≤

1

1 ∥fn (x) − f (x)∥L1 (R3 ) . (2π) 1 − I(V ) 3 2

Then the first term in the right side of (5.66) can be bounded from above in the absolute value as f˜ (k) − f˜(k) 1 1 1 n χ ∥f (x)−f (x)∥L1 (R3 ) 2s1 χ{|k|≤1} , 2s1 ≤ {|k|≤1} 2s |k| + |k| 2 (2π) 32 1 − I(V ) n |k| so that

f˜ (k) − f˜(k)

n

χ{|k|≤1}

2s 2s 1 2

|k|

+ |k|

L 2 ( R3 )

1 1 1 ≤ √ ∥fn (x) − f (x)∥L1 (R3 ) √ →0 3 − 4s1 2π 1 − I(V )

as n → ∞ as assumed, which completes the proof of the first part of the theorem. 3 Finally, we proceed to treating the case when ≤ s1 < 1. Orthogonality re4 lations (5.64) and (5.63) imply that f˜(0) = 0, Therefore,

f˜n (0) = 0,

n ∈ N.

5.4 Generalized Poisson type equation with a potential

f˜(k) =

|k|

Z 0

∂ f˜(q, σ) dq, ∂q

f˜n (k) =

|k|

Z 0

201

∂ f˜n (q, σ) dq, ∂q

n ∈ N.

This enables us to write the first term in the right side of (5.66) as R |k| h ∂ f˜n (q,σ) ∂ f˜(q,σ) i − ∂q dq ∂q 0 χ{|k|≤1} , |k|2s1 + |k|2s2 which can be easily estimated from above in the absolute value by ∥∇q [f˜n (q) − f˜(q)]∥L∞ (R3 ) |k|1−2s1 χ{|k|≤1} . Thus,

f˜ (k) − f˜(k)

n

χ

2s1

|k| + |k|2s2 {|k|≤1}

≤ ∥∇q [f˜n (q) − f˜(q)]∥L∞ (R3 ) √ L2 (R3 )

√ 2 π . 5 − 4s1

Thus (see also [30] and the references therein) under the given conditions we have ∥∇q [f˜n (q) − f˜(q)]∥L∞ (R3 ) → 0, n → ∞, which completes the proof of the theorem.

⊔ ⊓

References

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Index

a priori estimates, 8, 9 algebra, 28 angle variable, 51, 56, 151, 168, 191, 198 anisotropy, 144, 159 anomalous diffusion, 45, 47, 143, 159, 196 appendix, 86, 132, 167 Banach algebra, 13 boundary operator, 5, 7 boundary value problem, 6 boundedness, 10, 37, 87, 168 Carath´ eodory conditions, 9, 61, 96, 144 characteristic function, 51, 53, 55, 56, 150, 168, 190, 198 codimension, 8, 31, 59, 178 cokernel, 8 compact operator, 32, 34 continuous map, 12 contractibility, 32, 33 convergence, 63, 72, 75, 98, 120, 145, 162, 178 convergence in the sense of sequences, 95, 117, 143, 159, 177 convolution, 53 differential operator, 4, 5, 7, 12, 17, 18, 21, 62, 72, 74, 95, 119, 143 diffusion, 44, 45, 49, 95, 143 drift, 62, 66, 69, 72, 74, 78, 82, 86, 96, 97 eigenfunction, 56, 69, 82, 104, 128, 180, 182, 184, 187, 197 eigenvalue, 69, 82, 104, 127, 128 elliptic BVP, 7, 8 embedding, 1, 4, 11

essential spectrum, 60, 62, 97, 144, 161, 178, 180, 182, 196 fixed point, 63, 95, 97, 117, 143 fixed point theorem, 67, 79, 84, 102 Fourier coefficients, 77, 100, 122 Fourier harmonics, 52, 128, 182 Fourier images, 66, 67, 80, 82 Fourier transform, 36, 113, 120, 121 Fr´ echet differentiable, 12 fractional Laplacian, 30, 31 fractional Sobolev spaces, 16 Fredholm index, 8, 31 free Laplacian, 186, 188, 196 generalized Fourier transform, 52, 55, 197 global solvability, 31, 32, 34 H¨ older spaces, 12 Hamiltonian, 178 Hardy-Littlewood-Sobolev inequality, 53 Hilbert space, 2, 29, 64, 99 Hilbert transform, 36 homogeneous equation, 50, 60, 104, 146 integral kernel, 65, 95 invertible, 32, 60 involution, 28 iterated kernel, 63, 75, 145 Laplace operator, 18, 59, 61, 178 Laplace-Beltrami, 6 Lebesgue measure, 63, 64, 76, 79, 82 limiting equations, 65 limiting operators, 59, 60 linear Riemann-Hilbert problem, 34, 35 Lippmann-Schwinger equation, 52, 54, 180

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Efendiev, Linear and Nonlinear Non-Fredholm Operators, https://doi.org/10.1007/978-981-19-9880-5

207

208 necessary condition, 44, 45, 47 Nemytskii operator, 10, 11, 15 non-Fredholm operator, 59, 61, 196, 203 non-interacting quantum particles, 178 nonlinear Riemann-Hilbert problems, 15 normal diffusion, 45, 66, 69, 78 orthogonality condition, 39, 46, 63, 99, 124 parametrix, 32 partition of unity, 7 periodic boundary condition, 73, 95, 117 positivity, 45 potential function, 52, 57, 178 principal symbol, 6 projection, 32 proper, 34 pseudo-local property, 25 pseudodifferential operators, 15, 17, 19

Index Schr¨ odinger operator, 52, 186, 188, 197 self-adjoint operator, 60 Shapiro-Lopatinski condition, 5, 6 short-range potential, ix singular case, 182, 189, 190 singular integral operator, 21, 34 sixth order operator, 196 Sobolev embedding theorem, 11, 25 Sobolev space, 2, 16, 62, 117 solvability condition, vii, 120, 162, 177 spherical layer, 55 superdiffusion, 95, 108, 117 superposition operator, 15 surjective operator, 7 symbol, 5, 17, 132, 184 unbounded domains, 59, 205 unitarily equivalent, 54, 180, 182

reaction-diffusion equation, 45, 205 regular case, 187 Riemann-Hilbert problem, 34, 35 Riesz-Schauder theorem, 32

vector, 1, 72, 79, 117, 128, 152 viral infection, x

scalar potential, 177, 178, 182, 184

zero mode, 50

well-posedness, 46