Free boundary problems. Regularity properties near the fixed boundary 9783319970783, 9783319970790

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Lecture Notes in Mathematics  2218

Darya Apushkinskaya

Free Boundary Problems Regularity Properties Near the Fixed Boundary

Lecture Notes in Mathematics Editors-in-Chief: Jean-Michel Morel, Cachan Bernard Teissier, Paris Advisory Board: Michel Brion, Grenoble Camillo De Lellis, Princeton Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gábor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, New York Anna Wienhard, Heidelberg

2218

More information about this series at http://www.springer.com/series/304

Darya Apushkinskaya

Free Boundary Problems Regularity Properties Near the Fixed Boundary

123

Darya Apushkinskaya Department of Mathematics Saarland University Saarbr¨ucken Germany

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-319-97078-3 ISBN 978-3-319-97079-0 (eBook) https://doi.org/10.1007/978-3-319-97079-0 Library of Congress Control Number: 2018952878 Mathematics Subject Classification (2010): Primary: 35R35; Secondary: 35R37, 35B65, 35K10, 35J15 © Springer Nature Switzerland AG 2018 Translation from the English language edition: Free Boundary Problems: Regularity Properties Near the Fixed Boundary by Darya Apushkinskaya, Copyright © Springer International Publishing AG, part of Springer Nature 2018. All Rights Reserved. This work is subject to copyright. All rights are reserved 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, express 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Free boundary problems (FBPs) belong to the most striking component of the modern theory of partial differential equations (PDEs). The expression FBP refers to a problem in which one or several variables are governed in different subdomains of the space, or space-time, by the different state laws. These subdomains are a priori unknown and have to be determined as a part of the problem. The boundaries of these unknown subdomains are called the free boundaries. FBPs are the typical example of nonlinear problems where singularities arise. Therefore, a particular direction in FBPs has been to study the regularity properties of solutions and those of the free boundaries. Such questions are important for experiments and numerics and are usually considered extremely hard. Since the free boundary is not known a priori, the classical techniques of elliptic/parabolic PDEs do not apply. In the last two decades, many new approaches, combining the ideas from PDEs with ones from geometric measure theory, calculus of variations, harmonic analysis, and so on, have been developed and provided interesting results. This book treats some parts of this subject and its recent development. To be precise, this book is devoted to the so-called obstacle-type problems, a class of FBPs which may be characterized by the following property: gradient of a solution is continuous across the interface. For several elliptic and parabolic oneand two-phase obstacle-type problems, the qualitative properties of solutions and free boundaries near the fixed boundary of a domain are studied. It is supposed that the Dirichlet data are prescribed on the fixed boundary. The book is divided in to three chapters and three appendices. Figure 1 illustrates the main dependencies among the book’s components. Chapter 1 is introductory in nature. It contains a narrative introduction into the field of FBPs as well as a brief observation of other parts of the book and the outlines of the main steps. Chapters 2 and 3 form the core of the book. They can be treated independently. The largest part of the text contains results on parabolic problems. It includes the whole Chap. 2 and a half of Chap. 3. In Chap. 2, the complete study of regularity issues up to the fixed boundary is carried out for the case of the no-sign parabolic obstacle-type problem with the v

vi

Preface

Fig. 1 The main dependencies among chapters

Chapter 1

Appendix A

Chapter 2

Appendix B

Chapter 3

Appendix C

homogeneous Dirichlet boundary data. As the first step the optimal regularity of solutions is established; after that the analysis of global solutions is given. Based on these results, the fine properties of the free boundary are obtained, which boil down to proving a parabolic-tangential touch between the free and fixed boundaries. The latter in turn can be used to show C 1 properties of the free boundary. In the case of the nonhomogeneous Dirichlet boundary data, the situation is much more complicated, especially for the two-phase problems. Even to prove the optimal regularity of solutions is not easy; it requires a special control for dependence of all the estimates on the distance to the fixed boundary. These questions are discussed in Chap. 3. Some of the results obtained there are stronger than ones known for the classical obstacle problem. In Appendix A, one can find all necessary information about various monotonicity formulas which are the most important technical tools in studying the free boundary problems. In Appendix B, we recall and explain several general facts. Most of these auxiliary results are known, but probably not well known in the context used in this book. For the reader’s convenience, we collect in Appendix C some facts concerning various problems with free boundaries. These facts are involved essentially in our arguments. All these appendices are very much in regular use in Chaps. 2 and 3. The intended audience of this book includes graduate students and young researchers entering this field of mathematics. The reader is assumed to be familiar with classical calculus and the standard elliptic/parabolic theory (maximum/comparison principle, interior/boundary estimates, compactness arguments, etc.). This text is based on the revised version of my habilitation thesis at Saarland University. Parts of the book have been used as material for graduate courses on FBPs that I taught at Saarland University (2010, 2011, and 2017) and at Peoples’ Friendship University of Russia in Moscow (2016). I would like to mention just a few names standing for the long list of persons who contributed to this book in one way or the other. First and foremost, I am

Preface

vii

deeply indebpted to Nina Uraltseva. She led me to the Diploma and PhD. We have been collaborating for more than 20 years. I am very grateful to her for support, numerous discussions and valuable advice. My special thanks go to Martin Fuchs. Without his kind support and sometimes also his pressure, this book would never have been completed. Furthermore, I wish to thank Michael Bildhauer who carefully read the manuscript and gave me useful suggestions. I would like to express my sincere thanks to my coauthors Henrik Shahgholian and Norayr Matevosyan for the contributions they have made to our joint publications. Finally, I am very thankful to my family for their efforts to not disturb me too much and sharing with me the pressure associated with a project like this. I thank the Mathematical Sciences Research Institute (MSRI), Berkeley, USA, for hosting a program on Free Boundary Problems, Theory and Applications in Spring 2011, where I was in residence and wrote portions of this book. It remains only to note that this work was partially supported by the Russian Foundation of Basic Research (RFBR) through the grant 17-01-00099 and by the German-Russian Interdisciplinary Science Center (G-RISC) through the grant M-2016b-3. Saarbrücken, Germany July 2018

Darya Apushkinskaya

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 No-Sign Parabolic Obstacle-Type Problems . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.2 Main Results for No-Sign Parabolic Obstacle-Type Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.3 Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.4 Main Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Boundary Estimates for Solutions of Free Boundary Problems .. . . . . 1.2.1 Estimates for Solutions to the Elliptic Obstacle Problem . . . . 1.2.2 Estimates for Solutions to the Two-Phase Elliptic Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.3 Estimates for Solutions to the Two-Phase Parabolic Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.4 Estimates Near the Initial State for Solutions to the Two-Phase Parabolic Problem .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.1 Appendix A: Monotonicity Formulas .. . . .. . . . . . . . . . . . . . . . . . . . 1.3.2 Appendix B: Auxiliary Results . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.3 Appendix C: Additional Facts . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2 No-Sign Parabolic Obstacle-Type Problems . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Statement of the Problem and Main Results . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Optimal Regularity of Solutions .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Useful Properties of Solutions . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Nondegeneracy .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 Measure of Γ (u) . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.3 Convergence .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.4 Blow-Up and Blow-Down . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.5 Balanced Energy . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 4 4 7 8 9 12 12 15 20 22 25 25 25 26 26 26 29 29 33 44 44 47 48 49 51 ix

x

Contents

2.4 2.5 2.6 2.7 2.8

Classification of the Nonnegative Global Solutions . . . . . . . . . . . . . . . . . . Geometric Classification of the Global Solutions with No Sign Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Characterization of the Free Boundary Points Near Π . . . . . . . . . . . . . . . Regularity Properties of Solutions .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Regularity Properties of the Free Boundary .. . . . . .. . . . . . . . . . . . . . . . . . . .

53 57 60 63 70

3 Boundary Estimates for Solutions of Free Boundary Problems . . . . . . . . 73 3.1 One-Sided Estimates up to the Boundary for Solutions to the Elliptic Obstacle Problem . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 73 3.1.1 Statement of the Problem and Main Results . . . . . . . . . . . . . . . . . . 73 3.1.2 Estimates for Mixed Derivatives on the Boundary . . . . . . . . . . . 75 3.1.3 Estimates for Pure Second Derivatives .. . .. . . . . . . . . . . . . . . . . . . . 80 3.2 Boundary Estimates for Solutions to the Two-Phase Elliptic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 85 3.2.1 Statement of the Problem and Main Result . . . . . . . . . . . . . . . . . . . 85 3.2.2 Estimates of the Tangential Gradient Near the Boundary .. . . 86 3.2.3 Boundary Estimates of the Second Derivatives .. . . . . . . . . . . . . . 89 3.3 Estimates Near the Given Boundary of the Second-Order Derivatives for Solutions to the Two-Phase Parabolic Problem .. . . . . 92 3.3.1 Statement of the Problem and Main Result . . . . . . . . . . . . . . . . . . . 92 3.3.2 Lipschitz Estimate of the Normal Derivative at the Boundary Points .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 93 3.3.3 Boundary Estimates of the Second Derivatives .. . . . . . . . . . . . . . 95 3.4 Uniform Estimates Near the Initial State for Solutions to the Two-Phase Parabolic Problem . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 98 3.4.1 Statement of the Problem and Main Result . . . . . . . . . . . . . . . . . . . 98 3.4.2 Estimate of the Time Derivative .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 100 3.4.3 Estimates of the Second Derivatives . . . . . .. . . . . . . . . . . . . . . . . . . . 102 A Monotonicity Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.1 C-monotonicity Formula .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2 ACF-monotonicity Formula . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.3 W-monotonicity Formula . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

107 107 113 115

B Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 123 C Additional Facts .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 131 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 137 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 143

List of Figures

Fig. 1

The main dependencies among chapters . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

vi

Ice melting .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Classical obstacle problem . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Contact points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Caloric continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Evolution of nonparametric surface with speed depending on curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 1.6 Elliptic obstacle problem in 1-dim . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 1.7 Two-phase membrane problem . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 1.8 Temperature control problem . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3 3 5 5 6 13 16 20

Fig. 2.1 Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

33

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4 Fig. 1.5

xi

List of Tables

Table 1.1 Some results on one-phase parabolic obstacle-type problems . . . . . Table 1.2 Some results on two-phase elliptic problems . . .. . . . . . . . . . . . . . . . . . . . Table 1.3 Some results on two-phase parabolic problems .. . . . . . . . . . . . . . . . . . . .

10 19 24

xiii

Basic Notation and Conventions

z = (x, t) x = (x1 , x  ) = (x1 , x2 , . . . , xn ) |x| x ·y e1 , . . . , en e0 Rn+ Rn+1 + Rn+1 − Π Πr Πr (t 0 ) Br (x) Br+ (x) Br , Br+ Sr (x 0 ) Sr Qr (z0 ) = Qr (x 0 , t 0 ) 0 Q+ r (z ) Qr , Q+ r Qr ∂  Qr (z0 ), ∂  Qr Zr,s (z0 ) + (z0 ) Zr,s Kr (z0 )

Generic points in Rn+1 , where x ∈ Rn and t ∈ R1 Points in Rn , (if n ≥ 2) Euclidean norm of x in Rn Inner product in Rn Standard basis in the x-space Rnx Standard basis vector in the t-space R1t {x ∈ Rn : x1 > 0}   (x, t) ∈ Rn+1 : x1 > 0   n+1 (x, t) ∈ Rn+1 : x1 < 0 : x1 = 0 (x, t) ∈ R   (x, t) ∈ Π : |x|  r, −r 2 < t  0   Πr ∩ t = t 0 Open ball in Rn with center x and radius r Br (x) ∩ Rn+ + r (0) Br (0), B  n x ∈ R : |x − x 0 | = r Sr (0) Br (x 0 )×]t 0 − r 2 , t 0 ] Qr (z0 ) ∩ Rn+1 + Qr (0, 0), Q+ r (0, 0) Br ×]0, 1] Parabolic boundary, i.e., the topological boundary minus the top of the cylinder Br (x 0 )×]t 0 − s, t 0 + s[ Z (z0 ) ∩ Rn+1 +  r,s   |x1 − x10 | < r × |x  − (x 0 ) | < r ×]t 0 − r 2 , t 0 [

xv

xvi

Basic Notation and Conventions

Kr+ (z0 ) Kδ (x 0 )

Kr (z0 ) ∩ Rn+1 +     x ∈ Rn : x1 − x10 > |x  − x 0 | tan (δ)

Kδ ∂t Di D  = (D2 , . . . , Dn ) D = (D1 , D  ) ∇u = (Du, ∂t u) D2 u D3 u H = Δ − ∂t ν, e e⊥ν Dν Dνν Hm Lm v+ , v−

Kδ (0) Differential operator with respect to t Differential operator with respect to xi Tangential gradient Spatial gradient The complete gradient of u in the space Rnx × R1t D(Du) D(D 2 u) The heat operator Arbitrary unit vectors e is orthogonal to ν Operator of differentiation along the direction ν Dν (Dν ) m-dimensional Hausdorff measure m-dimensional Lebesgue measure max {v, 0} , max {−v, 0}.

With a slight abuse of notation, we will consider the sets E ⊂ Rnx as well as E ⊂ Rn+1 = Rnx × R1t . In both cases, χE denotes the characteristic function of the set E, while ∂E stands for the topological boundary of E. We emphasize that the precise meaning of E will always be evident by the context. ffl . . . stands for the average integral over the set E, i.e., E

··· = E

1 meas {E}

ˆ ...; E

The reader is assumed to be familiar with the classical • Hölder spaces C k,α (E) • Lebesgue spaces Lp (E) with the norm · p,E , 1 < p  ∞ as well as with the notions of the • Isotropic and anisotropic Sobolev spaces W 1,p (E), W 2,p (E), and Wp2,1 (E) with the norms u W 1,p (E) = Du p,E + u p,E , u W 2,p (E) = D(Du) p,E + u p,E , u W 2,1 (E) = ∂t u p,E + D(Du) p,E + u p,E , p

respectively.

Basic Notation and Conventions

xvii

Here, for Hölder, Lebesgue, and isotropic Sobolev spaces, we follow the definitions and notations as introduced in book [AF03]. In particular, local variants are denoted p 1,2 (E), etc. For definitions of anisotropic Sobolev spaces, by Lloc (E), Cloc (E), Wloc we refer the reader to [Kry08]. By ζ = ζ (|x|) is denoted a time-independent cut-off function belonging to C 2 (B1/2 ), having support in B1/4 , and satisfying ζ ≡ 1 in B1/8 . ξ = ξ (|x|) stands for a time-independent cut-off function belonging to C 2 (B2 ), having support in satisfying ξ ≡ 1 in B1 .  B2 , and |x − x 0 | ξr,x 0 (x) = ξ . r It remains to mention four conventions which are not restated each time: • The indices i, j always vary from 1 to n, while the indices τ, μ vary from 2 to n. n

a ij xi xj . Repeated indices indicate summation, for example, a ij xi xj = i,j =1

• If necessary (and possible), we usually pass to subsequence without relabeling. • We use letters M, N, A, L, and C (with or without indices) to denote various constants. To indicate that, say, N depends on some parameters, we list them in parenthesis: N(. . . ). • Positive constants are not relabeled. Moreover, they are not necessarily being the same in any two occurrences.

Chapter 1

Introduction

Go I know not whither and fetch I know not what Russian Fairy Tales

In my childhood, I read oftentimes in various Russian fairy tales the expression from the epigraph. It was a standard task for a fairy tale hero, to be accomplished by any means. Nowadays, a similar task is faced by mathematicians studying free boundary problems. You may ask, why is this so? I will try to explain. Free boundary problems play one of the most important roles in the modern analysis of partial differential equations and have numerous applications in various sciences. Among these applications we just mention phase transitions, problems in surface science, filtration through porous media, fluids dynamics, as well as mathematical finance and biology. The expression free boundary problem means that we deal with a problem with two a priori unknown objects: an unknown set (= ‘go I know not whither’) coming up in a solution of a partial differential equation (= ‘fetch I know not what’). The boundary of this unknown set is called the free boundary. Observe that the free boundary is not completely free. For a well posed problem the free boundary is determined as a set of points where some conditions on a solution are satisfied. The only problem is that this set of points is not a-priori known (it is free in this sense). The history of problems with free boundaries has more than 180 years. The first detailed study of free boundary problems begins with the works of Lamé and Clapeyron [LC31] and Stefan [Ste89], and concerns the ice formation in the polar seas. Later major contributions regarded free surface flows, shock waves and water waves. As a separate mathematical discipline free boundary problems manifested in the mid 1970s. We just refer to several excellent books on free boundaries such

© Springer Nature Switzerland AG 2018 D. Apushkinskaya, Free Boundary Problems, Lecture Notes in Mathematics 2218, https://doi.org/10.1007/978-3-319-97079-0_1

1

2

1 Introduction

as [DL76, KS80, Cra87, Rod87, Fri82], where the reader will find an intensive discussion of various problems and references. In the last 20 years problems with free boundaries have gained renewed attention. A lot of new methods have been developed and many challenging problems have arisen. We refer the reader to the two modern books [CS05] and [PSU12], where some of the most recent developments and corresponding references can be found. In these notes we consider free boundary problems of the so-called obstacle-type. More precisely, we study the following problems: to find a function u and a set (u) satisfying H [u] = f (x, t, u)

in

Ω(u) ∩ A ⊂ Rn+1 x,t

B(u, Du) = 0

on

∂Ω(u) ∩ A

u=φ

on

∂ A

Δu = f (x, u)

in

Ω(u) ∩ A ⊂ Rnx

B(u, Du) = 0

on

∂Ω(u) ∩ A

u=φ

on

∂A

(1.1)

or

(1.2)

(for definitions of H [u], Δu, Rn+1 x,t etc. see Notation and Conventions). We treat both problems (1.1) and (1.2) under the assumption that the function f is bounded. It is allowed that f has a jump discontinuity across the surface where u vanishes. The discontinuity set is a-priori unknown and therefore it is called free. The condition that the function B imposes on u is usually called the free boundary condition. A typical example of the problem (1.1) is the one-phase Stefan problem describing the melting of an ice cube in a glass of water. If ice begins to melt then the region occupied by water will grow and the interface—surface between the ice and the water (the free boundary)—will move and change its shape (see Fig. 1.1). As a typical example for the free boundary problem (1.2), we mention the classical obstacle problem describing the position of a membrane attached to a fixed wire, which is forced to stay above the given obstacle. The membrane can actually touch the obstacle. The boundary of the set where our membrane coincides with the obstacle (it is called the coincidence set) is a-priori unknown and, therefore, it is the free boundary (see Fig. 1.2). Our studies are concerned with the regularity properties of the solutions of (1.1) and (1.2) as well as of the free boundaries. Roughly speaking, we look for the answers to the following two questions: 1. How regular is a solution u (if such a solution exists)? 2. How regular is the free boundary? We emphasize that for problems like (1.1) and (1.2) the most part of the classical parabolic (elliptic) techniques is not applicable. The reason for this is an absence

1 Introduction

Fig. 1.1 Ice melting

Fig. 1.2 Classical obstacle problem

3

4

1 Introduction

of any a-priori information about regularity of the free boundary. To find such an information is a part of the problem. So, to get the answers to our two main questions, we have to combine ideas from the calculus of variations with geometrical observations, rescaling and blow-up techniques together with an analysis of caloric and harmonic functions.

1.1 No-Sign Parabolic Obstacle-Type Problems In Chap. 2, a one-phase problem of type (1.1) is considered. The exact mathematical formulation is as follows. Let a function u and an open set Ω ⊂ Rn+1 + satisfy the problem:  H (u) = χΩ =

1, 0,

if (x, t) ∈ Ω otherwise

in

Q+ 1 ,

u = |Du| = 0 in Q+ 1 \Ω ,   u = 0 on (x, t) ∈ Rn+1 : x1 = 0 ∩ Q1 ,

(1.3)

where the first equation in (1.3) is understood in the sense of distributions. Remark 1.1 If we add to (1.3) the condition u  0, then Ω = {u > 0} and u becomes a solution of the classical parabolic obstacle problem. It is obvious that the right-hand side of the first equation in (1.3) has a jump across the surface where u vanishes together with its gradient. This implies that (1.3) is a nonlinear problem. Moreover, this nonlinearity has a non-monotone character, since χΩ is equal 1 for positive and negative values of u as well. We will study the regularity properties of the free boundary in a neighborhood of the fixed boundary {x1 = 0}. Observe that due to the condition u = 0 on {x1 = 0} ∩ Q1 , one cannot exclude that the free boundary in (1.3) touches somewhere the fixed boundary {x1 = 0}. Therefore, the contact points between free and fixed boundaries may exist (see Fig. 1.3).

1.1.1 Motivation The motivation to study a problem without any sign assumption on a solution originates in parabolic potential theory. The potential theoretic interpretation is as follows.

1.1 No-Sign Parabolic Obstacle-Type Problems Fig. 1.3 Contact points

free boundary

5

x1

free boundary

t contact points

Fig. 1.4 Caloric continuation

Ω the heat potential of a density Let Ω be a domain in Rn+1 x,t . Denote by U function χΩ :

U Ω (x, t) =

1 cn

ˆ χΩ (y, s)G(x − y, t − s)dyds, Rn ×R

where cn is a normalization constant, while G(x, t) stands for the heat kernel defined by formula (A.1). Then U Ω satisfies H [U Ω ] = χΩ . Observe that U Ω is caloric in Ω c , i.e., in the compliment of Ω. Next, we fix some point z0 = (x 0 , t 0 ) ∈ ∂Ω and consider for small r > 0 the cylinder Zr (z0 ) := Br (x 0 )×]t 0 − r 2 , t 0 + r 2 [. Suppose now that there exists a caloric function v in Zr (z0 ) such that v = U Ω in Zr (z0 ) ∩ Ω c (see Fig. 1.4). This property is called caloric continuation.

6

1 Introduction

Considering the difference u := U Ω − v we observe that u satisfies H [u] = χΩ u = |Du| = 0

in

Zr (z0 ) ,

in Zr (z0 ) \ Ω .

So, we ask how the existence of the caloric continuation for Ω will result in the regularity of ∂Ω. The motivation for studying the one-phase problem near contact points is related to geometrical and physical problems, where certain data produce contact points between the free (unknown) and the given boundary. Such problems appear, for instance, in evolution of nonparametric surfaces with speed depending on curvature. To clarify this, consider an evolution of the nonparametric surface S above the given domain A ⊂ Rn . It is assumed that ∂S remains fixed, while S propagates in the normal direction with speed controlled by the curvature of S (see Fig. 1.5). In the series of publications [OU93a, OU93b, OU95, OU97], Oliker and Uraltseva gave a weak formulation of this evolution problem, where a generalized solution may develop for finite time in such manner that it will not satisfy the boundary condition on some parts of the boundary. The authors of [OU93a, OU93b, OU95, OU97] proved the existence of such weak solutions converging as t → ∞ to weak solutions of the corresponding stationary problem. Since in [OU93a, OU93b, OU95, OU97] generalized solutions were considered, the question about uniqueness arises naturally. In order to prove the uniqueness Fig. 1.5 Evolution of nonparametric surface with speed depending on curvature

1.1 No-Sign Parabolic Obstacle-Type Problems

7

result for weak solutions we have to get the following information about • the behaviour of a solution near the “bad” parts of the boundary. (Here, under the “bad” parts we understand the parts of the boundary where the boundary condition may be violated); • the behaviour of ∂S above ∂A ; • the regularity properties of ∂S near the points of possible contact with ∂A . It remains only to observe that, after simple transformations, ∂S can be locally considered as a free boundary in a parabolic obstacle problem with contact points. Among other free boundary problems with contact points we mention problems in filtration (see [Bai72, Fri82]), and one- or two-phase problems in flame combustion describing the propagation of curved premixed flame when the fixed and the free boundary touch (see [BL89, BLN88, BCN90, Gur99]).

1.1.2 Main Results for No-Sign Parabolic Obstacle-Type Problems Assuming only u ∞,Q+  M, we have found the following answers to our two 1 main questions. 1.

Regularity of solutions: u ∈ Cx1,1 ∩ Ct0,1 .

(1.4)

It is well known from numerous counterexamples that D 2 u as well as ∂t u may have jumps across the free boundary ∂Ω. Thus, the obtained regularity result (1.4) is the optimal one. Speaking about regularity of the free boundary we emphasize that, in general, the free boundary contains both regular and singular points. We say, that z0 ∈ ∂Ω is a regular free boundary point, if in a neighborhood of z0 the free boundary ∂Ω can be locally represented by an at least Lipschitz-regular surface. Otherwise, z0 is a singular free boundary point. Examples of singularities which may occur even in the classical elliptic obstacle problem can be found in [Sch77]. In a neighborhood of the given boundary the situation is different. We prove that near the fixed boundary the free boundary is always regular. More precisely, 2.

Regularity of the free boundary near the fixed boundary: 1. There are no singular points of the free boundary. 2. ∂Ω is Lipschitz-regular up to the contact points. 3. At interior points the free boundary is C 1,α -regular with α ∈ (0, 1). By using standard techniques, this result can be improved up to C ∞ -regularity in space and time.

8

1 Introduction 1/2

4. At contact points we have ∂Ω ∈ Cx1 ∩ Ct . 5. Even in the elliptic case (see [SUW07]) C 1,α -regularity w.r.t. the x-variables is, in general, not true. A counterexample, showing that in t-direction the free boundary in the one-phase parabolic obstacle problem may intersect the fixed boundary transversally, is constructed in Sect. 2.1.

1.1.3 Historical Review The C 1,1 -regularity up to the fixed boundary for solutions to the classical elliptic obstacle problem was proved by Jensen [Jen80]. The properties of the free boundary for the classical elliptic obstacle problem were studied by Caffarelli in the papers [Caf77, Caf80, Caf98]. He showed that the free boundary is regular if certain local density conditions on the set {u = 0} are satisfied. He also got estimates on the set where the free boundary is singular and gave the description of this set. The behavior of the free boundary near the contact points in the classical elliptic obstacle problem was studied in [AU95], where the tangential contact between free and fixed boundaries was proved. The improvement of the latter result up to C 1 -contact between free and fixed boundaries was made by Uraltseva in [Ura96]. The C 1,1 -regularity for solutions to the no-sign elliptic obstacle problem without contact points was proved in the two-dimensional case by Sakai [Sak91]. In higher dimensions the elliptic obstacle-type problem without sign restriction on the solution and without contact points was treated by Caffarelli et al. in [CKS00]. They established the optimal regularity for solutions and proved the analyticity of the free boundary. The latter was proved under the assumption that the complement of Ω does not develop cusp singularities in a neigborhood of a free boundary point. The structure of the singular set for this problem was described by Caffarelli and Shahgholian in [CS04]. The results for an elliptic problem with contact points related to (1.3) were obtained by Shahgholian and Uraltseva in [SU03]. Here the whole program was realized. Namely, the authors obtained the optimal regularity for solutions, established the tangential behaviour of the free boundary near contact points as well as C 1 -regularity of the free boundary, and described the singular set. These results were also extended in [SU03] to general subdomains of the unit ball with C 3 -boundary. Considering parabolic problems connected with (1.3), we begin our exposition with the one-phase Stefan problem. It is well-known that the one-phase Stefan problem can be transformed in a standard way (see [Duv73] for details) to the classical obstacle problem, where certain natural boundary and initial conditions guarantee the additional information ut  0 and u  0. Below we will use the notion “Stefan problem” for the parabolic obstacle problem with restrictions ut  0 and u  0.

1.1 No-Sign Parabolic Obstacle-Type Problems

9

The Stefan problem was studied by Caffarelli in [Caf77]. He proved the optimal regularity for solutions and examined the regularity of the free boundary under local density condition on the set {u = 0}. Such a density condition is called “thickness condition”. A sufficiently complete characterization of the singular set for the parabolic obstacle problem (i.e., the parabolic problem with an additional assumption u  0) was made by Blanchet et al. in the one-dimensional case (see [BDM06]), and by Blanchet in higher dimensions (see [Bla06b]). The results of [Bla06b] were strengthened by Lindgren and Monneau, who studied more general problem H [u] = f χ{u>0} under the assumptions that u  0 and f is a Dini continuous function (see [LM15]). In addition, the pointwise decay estimates for a certain Lp -average of u at singular points were established in [LM15]. We observe that the Stefan problem and the parabolic obstacle problem were considered in [Caf77, BDM06, Bla06b] far away from the given boundary. So, in both problems contact points were not allowed. The parabolic problem without sign restriction on the solution was studied by Caffarelli et al. in [CPS04] where only the interior case (case with no contact points) was considered. The authors of [CPS04] proved the optimal regularity of solutions and showed that the free boundary is regular if a certain “thickness condition” is satisfied locally. We note that even in the case without contact points there is no information about the singular set. For the parabolic no-sign obstacle-type problem the characterization of the singular set is an open problem. For the readers convenience we summarized partially this brief overview in the Table 1.1 on the next page. It contains only the results on parabolic obstacle-type problems.

1.1.4 Main Strategy The main strategy used in Chap. 2 is as follows. 1.

The usual first step in studying free boundary problems is the proof of the optimal regularity of the solution u. Observe that Wq2,1 -regularity of u with arbitrary q < ∞ comes for “free” from the standard parabolic theory, since the right-hand side of the first 2,1 equation in (1.3) is bounded. However, the optimal (i.e., W∞ ) regularity is not obvious. The main difficulty here is to prove a growth result of the type sup |u|  Cr 2

Q+ r (z)

for Qr (z) ⊂ Q1 ,

with z belonging to the free boundary and r being small enough. To show this quadratic bound we argue by contradiction and combine this with a local rescaled version of Caffarelli’s monotonicity formula.

At contact points C 1,α -regularity is, in general, impossible!

∂Ω ∈ C 1,α at interior points only

contact points may exist!

There are no singular points of ∂Ω near the given boundary

u ∈ Cx1,1 ∩ Ct0,1

Structure of the set ∂Ω ∩ {singular points}

∂Ω is locally

C 1,α -surface

u ∈ Cx1,1 ∩ Ct0,1

Results

∂Ω ∈ Lip

u0

“thickness condition”

u  0, ut  0 “thickness condition”

Additional assumptions

H [u] = χΩ in Q+ u = |Du| = 0 in Q+ \ Ω u = 0 on {x1 = 0}

no contact points !

H [u] = χΩ in Q u = |Du| = 0 in Q \ Ω

Problem

Table 1.1 Some results on one-phase parabolic obstacle-type problems

[Caf77]

Chapter 2

n > 1 Blanchet ’06 [Bla06a] ; Lindgren, Monneau ’15 [LM15]

[BDM06]

[CPS04]

n = 1 Blanchet, Dolbeaut, Monneau ’06

Caffarelli, Petrosyan, Shahgholian ’04

Caffarelli ’77

Author’s

10 1 Introduction

1.1 No-Sign Parabolic Obstacle-Type Problems

11

The crucial importance of the optimal regularity lies in its relation with the parabolic scaling of the problem (1.3). We note that if u(x, t) is a solution to (1.3) in Q+ 1 then the scaling ur (x, t) =

u(rx, r 2 t) r2

2,1 will be a solution in Q+ 1/r . If u ∈ W∞ , then the sequence {ur } is uniformly bounded and, therefore, the limit lim urj exists for some subsequence rj . Such rj →0

a limit is called a blow-up of u. It is clear that the blow-up limit is a solution of the problem similar to (1.3) in the entire half-space Rn+1 + and has near infinity at most quadratic growth in x and linear growth in t. 2. The next step is to investigate global solutions u defined in the whole halfspace Rn+1 + and satisfying to zero Dirichlet condition on the boundary of that half-space. It is also assumed that, near infinity, global solutions may have quadratic growth in the space variables and linear growth in time. Remark 1.2 To find a complete description of general, nonhomogeneous, global solutions defined in Rn ×{t  0} is an open and extremely difficult problem. Actually, a complete classification of global solutions is known only for the two-dimensional classical elliptic obstacle problem as it was analyzed by Sakai in [Sak81]. Several attempts (see [Sha92, FS86, Kar94, KM96, KM12]) have been made to fully classify solutions of elliptic problems in higher dimensions, but the theory (even in the elliptic case) is still far away from being complete. Thus, we are unable to classify all possible global solutions. But we can provide a complete classification of all homogeneous global solutions. This global analysis is based essentially on the monotonicity formulas due to Caffarelli and Weiss and on a Liouville-type theorem. As a result of this step we get representation formulas for all blow-up limits of local solutions at a fixed point of the free boundary. Such a global analysis is of fundamental importance for the study of local properties of a free boundary, since the homogeneous global solutions appear naturally as blow-up limits of local solutions at a fixed point of the free boundary. The idea to combine information about global solutions with the blow-up technique to study local properties of surfaces originated in the 1970s in investigations of minimal surfaces. For free boundary problems, this approach was used in the papers [Caf77, Caf80, CKS00, SU03]. 3. As a last step we study the local properties of the free boundary. We essentially use the closeness of the solution to the corresponding blow-up limit at a point z0 of the free boundary. We prove a certain directional monotonicity of the solution which implies the Lipschitz continuity of the free boundary. Namely, we get that in a neighborhood of the fixed boundary the free boundary is a Lipschitz graph x1 = f (x  , t), where the Lipschitz constant of f with

12

1 Introduction

respect to the x-variables is an absolute one while the Lipschitz constant of f with respect to t depends on the distance to the fixed boundary Π. Finally, we prove the higher regularity of the free boundary.

1.2 Boundary Estimates for Solutions of Free Boundary Problems In Chap. 3, we concentrate only on our first main question about regularity properties of a solution. As a result, we prove in Chap. 3 the explicit boundary estimates for higher derivatives of solutions to several elliptic and parabolic free boundary problems. Such estimates require the additional investigation of the behavior of solutions near the given boundaries.

1.2.1 Estimates for Solutions to the Elliptic Obstacle Problem We begin with an obstacle problem for the Laplace equation. In the simplest case, it can be formulated as the minimization problem for the functional J (v) =

ˆ

|Dv|2 + 2vg dx D

on a convex constrained set like  v ∈ W 1,2 (D) : v  0

a.e. in D;

 v − φ ∈ W01,2 (D) .

Here, D ⊂ Rn denotes a bounded domain, and g and φ are given functions. The function φ is assumed to be nonnegative on ∂D in the sense that φ − ∈ W01,2 (D). It is well-known (see, for instance, [Caf77] and the book [Fri82]) that for g ∈ L2 (D) and φ ∈ W 1,2 (D) an unique minimizer u exists if the continuous functional J is convex on a convex subset of the Hilbert space W 1,2 (D). We emphasize also that in the case when φ vanishes on some part of ∂D, one cannot exclude that arbitrary neighborhoods of some points x ∗ ∈ ∂D contain both points of the coincidence set, {x ∈ D : u(x) = 0}, and points of ∂ {x ∈ D : u(x) > 0}. We call the set of such points x ∗ the contact set.

1.2 Boundary Estimates for Solutions of Free Boundary Problems

13

Fig. 1.6 Elliptic obstacle problem in 1-dim

free boundary

Obstacle

1.2.1.1 Example of the Elliptic Obstacle Problem An example of an one-dimensional obstacle problem is shown in Fig. 1.6. Such onedimensional classical problem arises when an elastic string is held fixed at two ends and pass over the smooth obstacle which stays between these two ends. Then at some points the string and the obstacle will touch. Observe that the region where our string and the obstacle coincide is a-priori unknown. The free boundary here is the set of points where the string leaves the obstacle.

1.2.1.2 Main Result It is known that the minimizer u has second derivatives D 2 u ∈ L∞ (D). This regularity result is optimal, i.e., it cannot be improved. Thus, without loss of generality we may assume that D 2 u ∞,D  M,

M = const > 0.

(1.5)

Observe that directly from (1.5) we are unable to get information about the behavior of the free boundary. For this reason we refine the classical result (1.5) by obtaining more detailed estimates of tangential-normal derivatives of u near the contact set and the explicit one-sided estimates of the pure second derivatives of u from below. Such one-sided estimates play a special role in the analysis of the obstacle problem. In the 1970s some of these estimates permitted to start the investigation of the properties of the free boundary. Later, different methods of the analysis of the free boundary were developed (see, for example, [CKS00] and [SU03]). Nevertheless, the refined estimates of the second derivatives are interesting by themselves. In particular, they show that for a free boundary point x 0 the corresponding blow-up u(rx + x 0 ) are convex. limits of scalings ur (x) = r2

14

1 Introduction

The estimates of the second derivatives D 2 u near the boundary ∂D for solution of the obstacle problem satisfying the homogeneous Dirichlet condition on the part of the fixed boundary ∂D are the main subject of Sect. 3.1. The obtained results can be summarized in the following statements: The first statement asserts the existence of a modulus of continuity ϕ |Dτ u(x)| characterizing the behaviour of , τ = 2, . . . , n, near the set of contact x1 points. This modulus of continuity is common for all weak bounded solutions of the elliptic obstacle problem with zero Dirichlet condition on the part of the given boundary. ii The second statement concerns one-sided estimates of pure second directional derivatives Dξ ξ u from below. In other words, we prove the estimate i

inf Dξ ξ u  −C| log r|−1/(2n−2).

Br+ (x 0 )

Here x 0 is either a contact point or a free boundary point lying near a contact point, r is an arbitrary small radius, and ξ is a direction lying in the cone with the vertex at the contact point. Observe that the positive constant C is completely defined by the angle of the cone and the data of the obstacle problem.

1.2.1.3 Historical Review The interior W 2,∞ -regularity for solution of the simplest obstacle problem was obtained by Frehse in [Fre72], while the case of more general elliptic operator was treated by Brezis and Kinderlehrer in [BK74]. The corresponding W 2,∞ -regularity of solutions up to the boundary was established by Jensen in [Jen80]. The interior one-sided estimates of D 2 u from below were established by Caffarelli in [Caf76] for solutions of the three-dimensional filtration problem and in [Caf77] for the multidimensional obstacle problem, respectively. It is clear that the estimates from [Caf76] and [Caf77] depend on the distance to ∂D.

1.2.1.4 Outline of the Proof If inf φ > 0, it is evident that the distance between the boundary of the coincidence ∂D

set {u = 0} and ∂D is positive. Consequently, the results of [Caf77] remain valid for this case. Otherwise, the behaviour of D 2 u near the contact set needs a more detailed analysis.

1.2 Boundary Estimates for Solutions of Free Boundary Problems

15

We begin with estimates for mixed derivatives on the boundary. Here we use the approach close to the method developed by Krylov and Safonov (see [Kry84, Saf85]) for obtaining similar estimates for solutions of the Dirichlet problem for fully nonlinear equations. These preliminary boundary estimates of mixed derivatives permit us to generalize the result of Caffarelli about one-sided estimates of pure second derivatives to the case of contact points. The main difficulties here are related to the lack of any information about the structure of the contact set.

1.2.2 Estimates for Solutions to the Two-Phase Elliptic Problem The second problem considered in Chap. 3 is the following two-phase obstacle problem: Δu = λ+ χ{u>0} − λ− χ{u 0, and Eq. (1.6) is understood in the weak (distributional) sense. The solution of this problem minimizes the energy functional ˆ J (u) = B1+

1 ( |Du|2 + λ+ max {u, 0} + λ− max {−u, 0})dx 2

(1.8)

  over the set of functions belonging to W 1,2 B1+ and having fixed data on ∂B1+ . From the elliptic theory (see, for instance, [Wei01]) it follows that u ∈ + W 2,q (B1−ε ) for any small ε > 0 and any 1 < q < ∞ if we impose the additional assumption ϕ ∈ W 2,∞ (π1 ). Observe also that if the boundary data are non-negative (non-positive) then the solution u is so too, and we arrive at the classical one-phase obstacle problem. If, in addition, ϕ ∈ C 2,α (π1 ) then it is well-known (see [Jen80]) that the solution of the one-phase obstacle problem is a W 2,∞ -function up to π1 , and this regularity is optimal. In this subsection we wish to prove the L∞ -estimates of the second derivatives 2 D u near the given boundary π1 for solutions of (1.6)–(1.7) without any assumption on the sign of u on ∂B1+ and, in particular, on the sign of ϕ.

16

1 Introduction

Fig. 1.7 Two-phase membrane problem

1.2.2.1 Examples of Two-Phase Elliptic Problems To give a physical interpretation of the problem (1.6)–(1.7), consider a tank filled up with two liquids with different densities and a thin film (membrane) with density in between the densities of the liquids. Suppose that this thin film is fixed on the boundary of the tank, and some part of the boundary data is below the heavy liquid while the rest is above the lighter liquid. Due to gravity, the film will be pushed up with a force λ− in the heavier phase u < 0 and pushed down with a force λ+ in the lighter phase u > 0 (see Fig. 1.7). The equilibrium state of the film is given by a minimization of the energy functional (1.8) which is equivalent to (1.6)–(1.7). This model is described in the book [PSU12] (see Section 1.1.5). Another application of the elliptic two-phase problem arises in the optimal control problem with pointwise restrictions on the functions. So, we minimize the functional ˆ ˆ

|Dvf |2 + |vf | dx − θ vf dH n−1 I (f ) := Ω

∂Ω

over the set of control functions ⎧ ⎫ ˆ ˆ ⎨ ⎬ f dx = θ dH n−1 . f ∈ L∞ (Ω) : ess sup |f |  1, ⎩ ⎭ Ω Ω

∂Ω

1.2 Boundary Estimates for Solutions of Free Boundary Problems

17

Here Ω ⊂ Rn is a bounded domain with boundary ∂Ω, θ is a given function on ∂Ω, and vf is a solution of the Neumann problem Δvf = f

in Ω ,

∂vf =θ ∂n

on ∂Ω.

It is easy to see that if u is the solution of the equation Δu = χ{u>0} − χ{u 0,

we prove, in a neighborhood of the boundary π1 , the optimal regularity (of class W 2,∞ ) of a bounded weak solution to the two-phase elliptic problem (1.6)–(1.7).

1.2.2.3 Historical Review The two-phase elliptic obstacle problem can be traced in the literature since the mid 1970s as a limiting case in the interior temperature control problem (see, for example, Chapter I, §2.3.2 [DL76] or §1.1.6 [PSU12]). The study of the two-phase elliptic obstacle problem from the free boundary point of view was initiated by Weiss [Wei98]. For interior W 2,∞ -estimates for solutions of (1.6) we refer the reader to [Ura01] (see also [Sha03]). Unfortunately, the proofs presented in [Ura01, Sha03] do not work near the given boundary. Notice for the sake of completeness that the regularity properties of the true two-phase part of the free boundary with vanishing gradient (i.e., ∂{u(x) > 0} ∩ ∂{u(x) < 0} ∩ {|Du(x)| = 0}) were studied far away from the fixed boundary by Shahgholian and Weiss in [SW06] for dimension 2 and by Shahgholian et al. in [SUW07] for arbitrary space dimension. They established uniform C 1 -regularity of the free boundary and show that this result is optimal in the sense that the graphs of the free boundary are in general not of class C 1, Dini (the corresponding counterexamples can be found in [SUW07, Ura96]). In addition, it was shown in §4.5 [PSU12] that the part of the free boundary with nonzero gradient (i.e., ∂{u(x) > 0} ∩ ∂{u(x) < 0} ∩ {|Du(x)| = 0}) is locally a graph of a real analytic function.

18

1 Introduction

As the first work in a neighborhood of the fixed boundary, we mention the paper [AMM06] where Eq. (1.6) was considered in the unit half-ball B1+ under the assumptions that the free boundary Γ (u) := ∂{x ∈ B1+ : u(x) = 0} ∩ B1+ touches the fixed boundary at 0 and the boundary value ϕ is a C 2,Dini-function on π1 satisfying ϕ(0) = |D  ϕ(0)| = |D  (D  ϕ(0))| = 0. In [AMM06] it was shown that there exists a constant C depending only on u ∞, B + and on the properties of 1 ϕ such that sup |u − De1 u(0)x1|  Cr 2 , Br+

0 0 and if u vanishes at least at a single point of Br (x 0 ) ∩ D, then u = O(r 2 )

in

Br (x 0 ) ∩ D.

(1.9)

Due to the property (1.9), it is possible to estimate |D 2 u| in a neighborhood of Σ. For the readers convenience we summarize briefly this historical overview in the Table 1.2 on the next page.

1.2.2.4 Outline of the Proof The desired estimates of D 2 u are obtained in the following manner. First of all, for τ = 2, . . . , n, we estimate the difference |Dτ u(x) − Dτ ϕ(x  )| near the given boundary through the distance to π1 . We do this with the help of barrier arguments. Observe that direct application of the Alt-Caffarelli-Friedman (ACF) monotonicity formula is not possible in our case. The difficulty lies in the fact that both integrals involved in the ACF monotonicity formula depend on the distance to the given boundary π1 . The obtained estimates of the tangential gradient allow us to control this dependence and to show that all the integrals grow quadratically w.r.t. x1 . Subsequent application of the ACF monotonicity formula finishes the proof.

Δu = f (u) in B u = ϕ on ∂B

Δu = f (u) in B f (u) := λ+ χ{u>0} − λ− χ{u 0} ∩ ∂{u < 0} ∩ {|Du| = 0} is locally real analytical surface

[Ura07]

Chapter 3, Sect. 3.3

Uraltseva ’07

[SUW07]

[PSU12]

Shahgholian, Uraltseva, Weiss ’07

[SW06]

Petrosyan, Shahgholian, Uraltseva ’12

n3

n = 2 Shahgholian, Weiss ’06

Uraltseva ’01 [Ura01] Shahgholian ’03 [Sha03]

D 2 u ∞  c

∂{u > 0} ∩ ∂{u < 0} ∩ {|Du| = 0} is locally C 1 -surface

Author’s

Results

1.2 Boundary Estimates for Solutions of Free Boundary Problems 19

20

1 Introduction

1.2.3 Estimates for Solutions to the Two-Phase Parabolic Problem As the third problem in Chap. 3 we study the generalization of problem (1.6)–(1.7) to the nonstationary case. In other words, we consider that following two-phase parabolic obstacle-like problem H [u] = λ+ χ{u>0} − λ− χ{u 0, and Eq. (1.10) is understood in the weak (distributional) sense.

1.2.3.1 Example of the Two-Phase Parabolic Problem The problem (1.10)–(1.11) arises in the model of temperature control in the interior, regulated by the temperature itself. Suppose that we have a container where we want to keep the temperature u(x, t) close to zero. Assume also that a temperature distribution ϕ is given on the boundary of our container, while the interior of the container is occupied by several cooling/heating devices (see Fig. 1.8). All these devices together can produce only two heat injections, one negative of size −λ1 and

Fig. 1.8 Temperature control problem

1.2 Boundary Estimates for Solutions of Free Boundary Problems

21

one positive of size λ2 , according the following regulation: −λ1 ,

if u > 0,

λ2 ,

if u < 0.

It is clear that the temperature distribution u(x, t) will satisfy our two-phase parabolic problem (1.10)–(1.11). This model is described in the book [DL76] (see Chapter I, §2.3.2).

1.2.3.2 Main Result Assuming that a given function ϕ depends only on the spatial variables and satisfies the conditions D 3 ϕ ∈ L∞ (Π1 ) , ∃L > 0 : |D  ϕ(x)|  L|ϕ(x)|2/3

(1.12) ∀(x, t) ∈ Π1 ,

(1.13)

we prove, in a neighborhood of Π1 , the uniform explicit L∞ -estimates of ∂t u and D 2 u for a bounded weak solution u of the two-phase problem (1.10)–(1.11). The obtained boundary estimates of higher derivatives remain valid in the case ϕ = ϕ(x, t) with ϕ satisfying (1.13) and the additional assumptions ∂t ϕ, D(∂t ϕ) ∈ L∞ (Π1 ) .

1.2.3.3 Historical Review The local (interior) boundedness of the derivatives ∂t u and D 2 u was proved in [SUW09]. The corresponding estimates up to the fixed boundary Π1 were obtained in [Ura07] for the case ϕ = 0. Notice that the case of general Dirichlet data cannot be reduced to the case of zero Dirichlet data since the right hand side of Eq. (1.10) is nonlinear and discontinuous at u = 0. For the completeness of this overview, we mention also the paper by Shahgholian et al. [SUW09], where true two-phase part of the free boundary ∂{u > 0}∩∂{u < 0} was studied far away from the fixed boundary. It was established in [SUW09] that ∂{u > 0}∩∂{u < 0}∩{|Du| = 0} is locally a union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. This result is optimal in the sense that the true two-phase part of the free boundary with vanishing spatial gradient is in general not better that Lipschitz, as shown by a counterexample. In addition, the authors of [SUW09] proved that the part of the free boundary with nonzero spatial gradient is locally C 1 -surface.

22

1 Introduction

1.2.3.4 Outline of the Proof Thanks to the monotonicity of the right-hand side of Eq. (1.10) we get the estimate of ∂t u in L∞ -norm almost “for free”. Repeating all the arguments from the proof of Lemma 3.1 [SUW09] we see that they are valid up to the fixed boundary Π1 . In order to estimate the second derivative D 2 u, we perform successively the following steps: Similarly to the elliptic case, we estimate for τ = 2, . . . , n the difference |Dτ u(x, t) − Dτ ϕ(x  )| through the distance to Π1 . A combination of this result with mollification arguments permits to get the Lipschitz estimate of the normal derivative at the boundary points. 2. Further, for an arbitrary direction e ∈ Rn , a small parameter δ ∈ (0, 1/4) and a point z0 ∈ Π1−δ , we estimate the oscillation of the directional derivative De u 0 in the cylinder Q+ r (z ) by r. The latter enables us to apply the local version of monotonicity formula developed by Caffarelli and Kenig in [CK98]. This application completes the proof. 1.

1.2.4 Estimates Near the Initial State for Solutions to the Two-Phase Parabolic Problem In this subsection we are interested in uniform L∞ -estimates near the initial state for the derivatives D 2 u and ∂t u of the function u satisfying the following two-phase parabolic obstacle-like problem: H [u] = λ+ χ{u>0} − λ− χ{u 0, and Eq. (1.14) is understood in the weak (distributional) sense.

1.2.4.1 Motivation Our reasons to study the behavior of a solution near the initial state are twofold. First and foremost, such L∞ -estimates are interesting by themselves. Together with the results of Sect. 1.2.3 they provide the necessary information about properties of u near the whole given parabolic boundary. On the other hand, our problem (1.14)–(1.15), at least in a particular case of the parabolic obstacle problem, occurs naturally in studying of the American type options in mathematical finance where the properties near the initial state play a

1.2 Boundary Estimates for Solutions of Free Boundary Problems

23

special role. We refer the reader to the papers [PS07, Sha08, Nys08, NPP10] for more detailed explanations of the corresponding financial models.

1.2.4.2 Main Result Under the only assumption that the initial data ψ belong to the space C 1,1 (B1+δ ) 2,1 we prove optimal regularity (of class W∞ ) near the initial state for bounded weak solutions of the two-phase parabolic obstacle problem (1.14)–(1.15). Observe that the obtained result is also optimal in the sense that the condition ψ ∈ C 1,1 (B1+δ ) cannot be weakened.

1.2.4.3 Historical Review As we already have indicated in Sect. 1.2.3.3, interior estimates of ∂t u and D 2 u were obtained in [SUW09]. The corresponding estimates up to the lateral surface were proved in [Ura07] for zero Dirichlet data, and in Sect. 3.3 for general Dirichlet data satisfying certain structure conditions, respectively. Unfortunately the proofs presented in [SUW09], [Ura07] and in Sect. 3.3 do not work near the initial state. To this end, the additional investigation of the behaviour of the solution u close to the initial state is required. Speaking about regularity up to the initial state, we are only aware of the results of [Ura07, NPP10, Sha08, Nys08]. In the papers [Nys08, Sha08] the authors studied the parabolic obstacle problem near the initial state for quasilinear and fully nonlinear equations, respectively. In both cases, the estimates of the second derivatives D 2 u were not considered and hence only the gradient Du and the time derivative ∂t u were estimated. The results in [NPP10] are most close to those obtained in Sect. 3.4. Indeed, the authors of [NPP10] considered the parabolic obstacle problem with more general differential operator of Kolmogorov type and established the L∞ -estimates of D 2 u and ∂t u under the assumption that the initial data ψ belong to the space C 2,α . It remains only to note that the two-phase parabolic problem with initial data ψ ∈ C 2,α was studied in [Ura07] under the additional structure assumption that ψ(x) = 0 implies |Dψ(x)| = 0. We summarize the results of Sects. 1.2.3.3 and 1.2.4.3 in the Table 1.3 on the next page. Note that in the Table 1.3 we denote the lateral surface of the cylinder Q by ∂L Q and the bottom of Q by ∂I Q, respectively.

1.2.4.4 Outline of the Proof The main strategy used in this subsection is as follows. At first, we prove the estimate of the time derivative. We do this with the help of regularizations. The estimate of ess sup |D 2 u| is based essentially on the famous local monotonicity

H [u] = f (u) in Q u = ψ on ∂I Q

H [u] = f (u) in Q u = ϕ(x) on ∂L Q

H [u] = f (u) in Q f (u) := λ+ χ{u>0} − λ− χ{u 0} ∩ ∂{u < 0} ∩ {|Du| = 0} is locally C 1 -surface

∂{u > 0} ∩ ∂{u < 0} ∩ {|Du| = 0} ∈ Lip ∩ Cx1

D 2 u ∞  c ∂t u ∞  c

Results

Table 1.3 Some results on two-phase parabolic problems

[Ura07]

Chapter 3, Sect. 3.4

Chapter 3, Sect. 3.3

Uraltseva ’07

Shahgholian, Uraltseva, Weiss ’09

Author’s

[SUW09]

24 1 Introduction

1.3 Appendices

25

formula due to Caffarelli. The latter requires some preliminary estimates where we trace the exact dependencies of all terms on t. Remark 1.3 It should be emphasized that the arguments from Sect. 1.2.4 are not applicable to the problem (1.10)–(1.11). The difficulty lies in the fact that the standard parabolic cylinder Qr (z0 ) has the quadratic rate of growth in r in the timedirection and only linear rate of growth in r in the space-directions.

1.3 Appendices We finish our studies with three appendices.

1.3.1 Appendix A: Monotonicity Formulas In Appendix A, we collect all information about various monotonicity formulas which are the most important technical tools in the analysis of the free boundary problems. The monotonicity formulas will appear in almost every section. We will use three different kinds of monotonicity formulas: the parabolic monotonicity formula due to Caffarelli [Caf93, CK98], the elliptic monotonicity formula due to Alt-Caffarelli-Friedman [ACF84] and the parabolic monotonicity formula due to Weiss [Wei99]. These formulas are presented in global and local forms. Since we study the regularity properties of solution and the free boundary near the fixed boundary, we also have to pay special attention to various rescaled versions of monotonicity formulas.

1.3.2 Appendix B: Auxiliary Results In Appendix B, we recall and explain some general facts. Among them are the Liouville-Type Theorem, various properties of caloric and subcaloric functions etc. Most of these auxiliary results are known but probably not well known in the context of free boundary problems. For the readers convenience the proofs of almost all technical lemmas are included.

26

1 Introduction

1.3.3 Appendix C: Additional Facts In Appendix C, we collect several facts concerning various problems with free boundaries. These facts were proved by other authors in the papers [CPS04, AS91, Caf77, Ura96], and [SUW09], respectively. We will essentially use these results in our arguments.

1.4 Notes • The optimal regularity result of Chap. 2 was established in [ASU00]. • The classification of nonnegative global solutions of the parabolic obstacle problem was announced in [AUS02]. We emphasize that in Sect. 2.4 of Chap. 2 this classification is established by using more transparent and elegant arguments in comparison to [AUS02]. • The geometric classification of global solutions for parabolic problems without sign restrictions, the directional monotonicity in a cone and regularity properties of the free boundary in Chap. 2 follow the lines of [AUS03, AMU09]. • One-sided estimates up to the boundary for solutions to the elliptic obstacle problem from Chap. 3 were established in [AU95]. In fact, [AU95] was the first publication, where the behavior of the free boundary near the given boundary has been studied. • Uniform L∞ -estimates for the derivatives of solutions to the two-phase elliptic problem up to the lateral surface in Chap. 3 are due to [AU06]. • Estimates near the fixed boundary for the derivatives of solutions to the two-phase parabolic problem in Chap. 3 are outlined in [AU08]. • Uniform estimates near the initial state for derivatives of solutions to the twophase parabolic problem in Chap. 3 were established in [AU13]. They improve substantially the previously known results.

1.5 Open Problems We finally conclude by listing several open problems/areas of further work. • Characterize the singular set for no-sign parabolic obstacle-type problem (i.e., parabolic problem without any sign assumption on a solution). • For two-phase elliptic and parabolic problems study the limiting regularity of Γ ∗ and establish one-sided optimal regularity for solutions up to Γ ∗ . We observe that for the two-phase problems we do not have the property that the space gradient of a solution vanishes on the free boundary, as it was in the one-phase problems. So, here we denote by Γ ∗ the part of the free boundary where the space-gradient of a solution does not vanish.

1.5 Open Problems

27

• Study the asymptotic behavior of the free boundary as t → ∞ for twophase parabolic problem with non-homogeneous Dirichlet condition on the given boundary. • Study regularity properties of the free boundary for hysteresis-type problems. Such problems can be considered as a generalization of the two-phase parabolic problem to the case when we have a spatially distributed multivalued hysteresis operator instead of λ+ χ{u>0} − λ− χ{u 0} is the free boundary; that is Γ (u) is the part of ∂Ω(u) ∩ Λ(u) which “can be seen from below”. • Γ (u) ∩ Π is the set of contact points. Without loss of generality we may assume throughout this chapter that Γ (u) = ∅.

© Springer Nature Switzerland AG 2018 D. Apushkinskaya, Free Boundary Problems, Lecture Notes in Mathematics 2218, https://doi.org/10.1007/978-3-319-97079-0_2

29

30

2 No-Sign Parabolic Obstacle-Type Problems

Theorem 2.1 (Optimal Regularity of Solutions) Let u satisfy (2.1). There exists a universal constant M = M(n, M0 ) such that, if u ∞,Q+  M0 , then 1

sup

Q+ 1/8 ∩Ω

  |Di Dj u(x, t)| + |∂t u(x, t)|  M.

(2.2)

2,1 + Definition 2.1 We say that a function u ∈ W∞ (Q+ R ) belongs to the class PR (M) if u satisfies:

(a) (b) (c) (d)

+ H [u] = χΩ in Q+ R , for some open set Ω = Ω(u) ⊂ QR ; + u = |Du| = 0 in QR \ Ω; u = 0 on Π ∩ QR ; ess sup(|∂t u| + |D 2 u|)  M Q+ R

and the equation in (a) is understood in the sense of distributions. The elements of PR+ (M) will be called local solutions. + (M) be the class of functions u ∈ W 2,1 (Rn+1 ∩ {t  0}) Definition 2.2 Let P∞ ∞ + such that n+1 (a ) H [u] = χΩ in Rn+1 + for some open set Ω = Ω(u) ⊂ R+ ∩ {t  0}; (b ) u = |Du| = 0 in Rn+1 + \ Ω(u); (c ) u = 0 on Π; (d ) ess sup (|∂t u| + |D 2 u|)  M, Rn+1 + ∩{t 0}

where the equation in (a ) is understood in the sense of distributions. The elements + (M) will be called global solutions. of P∞ We also introduce the class P∞ (M). Here we consider Rn+1 ∩ {t  0} instead of Rn+1 + ∩ {t  0} and omit the condition u|Π = 0. The class PR (M) with R < ∞ is defined similarly. + (M), and let z0 = Theorem 2.2 (Classification of Global Solutions) Let u ∈ P∞ 0 0 (x , t ) ∈ Γ (u). Then u is independent of t and the variables x3 , . . . , xn . More 0 precisely, for (x, t) ∈ Rn+1 + ∩ {t  t } we have

x10 = 0

⇒

u(x, t) =

x12 + ax1 x2 + bx1 2

in some suitable rotated coordinate system in Rn that leaves e1 fixed, and for some real numbers a and b; x10 > 0

⇒

u(x, t) =

((x1 − x10 )+ )2 . 2

2.1 Statement of the Problem and Main Results

31

Theorem 2.3 (Lipschitz Property of the Free Boundary) Let u ∈ P2+ (M), let 0 z0 = (x 0 , t 0 ) ∈ Γ (u) ∩ Q+ 1/8 , and let r := x1 . There exist a positive constant δ = δ(n, M) and a Lipschitz continuous nonnegative function f defined on {|x  − (x 0 ) | < r}×]t 0 − r 2 , t 0 ] such that if r < δ/2 then u0

+ (z0 ) in K3r/2

and Ω(u) ∩ Kr (z0 ) = {(x, t) ∈ Kr (z0 ) : x1 > f (x  , t)}. Here the Lipschitz constant of f with respect to the x-variables depends on n and M only, while the Lipschitz constant of f with respect to t equals C(n, M)r −1 . Let f be the same Theorem 2.4 (C 1,α -regularity of the Free Boundary) function as in Theorem 2.3. Then in a neighborhood of every point (x  , t) satisfying     0 f (x , t), x , t ∈ Kr (z ), the function f belongs to the class C 1, α for some 0 < α < 1. Remark 2.1 In fact, applying the Kinderlehrer-Nirenberg technique (see [KN77], and also [Isa75]), one can show that ∂Ω(u) ∩ Kr+ (z0 ) is not only C 1, α but also space-time C ∞ -regular. It should be noted that Theorem 2.4 and Remark 2.1 ensure the C ∞ -regularity of the free boundary in space and time only at the interior points. Unfortunately, even C 1, α -regularity with 0 < α < 1 may fail to occur at the points of contact between the free boundary and the fixed boundary. The following counterexample shows that in t-direction the free boundary may intersect the fixed boundary transversally.   Counterexample Let n = 1, and let Cr = (x, t) : 0 < x < r, −r 2 < t < r 2 . Suppose that a function u on C1 is a solution of the one-phase Stefan problem , i.e., u is a nonnegative solution of the equation H [u] = χ{u>0}

in C1

and ∂t u  0 a.e. in C1 . Assume that u(0, t) = 0 for −1 < t < 1 and ess sup(|∂t u| + |D 2 u|)  M. C1

Assume also that (0, 0) is a free boundary point, i.e., Du(0, 0) = 0. From Lemma 2.25 and Theorem 2.3 it follows that for some r > 0 the derivative ∂t u is continuous in the closure C r of the rectangle Cr and the set ∂ {u > 0} ∩ C r is a graph x = f (t) of some Lipschitz continuous function f . Under our assumptions it

32

2 No-Sign Parabolic Obstacle-Type Problems

is evident that f is a monotone nonincreasing function, f (t) = 0 for t  0, and u(x, t) = 0

if

0  x  f (t),

u(x, t) > 0

if

x > f (t),

(x, t) ∈ C r ,

(x, t) ∈ Cr .

We exclude the case where f ≡ 0 for −r 2 < t < r 2 from our consideration; there is no loss of generality in assuming that 0 < f (t) < r/2 for −r 2 < t < 0. Now we set v = ∂t u and y = x − f (t). Then, in the rectangle   C = (y, t) : 0 < y < r/2, −r 2 < t < r 2 , the function v is a nonnegative solution of the equation Dy Dy v − ∂t v + f  (t)Dy v = 0. Moreover, Harnack’s inequality guarantees that v is strictly positive inside the set C . So, an application of the barrier arguments similar in spirit to those from the  Hopf lemma, and the boundary condition v {y=0} = 0 provide the estimate v(y, t)  βy

in

  (y, t) : 0 < y < ρ, −ρ 2 < t < ρ 2

with some positive constants β and ρ. Returning to the x-variable, we see that ∂t u  β [x − f (t)]

in

Cρ ∩ {u > 0} .

Since Dx u = 0 for x = f (t) and t  0, on the set Cρ ∩ {t  0} we have the estimate |Dx u|  M [x − f (t)] . Therefore, if e1 and e0 are the standard basis vectors in Rx and Rt , respectively, and if e = a0 e0 + a1 e1 with a02 + a12 = 1, a0 > 0, and a1  0, then for such a direction e in Cρ ∩ {t  0} ∩ {u > 0} we have De u = a0 ∂t u + a1 Dx u  (a0 β + a1 M) [x − f (t)] . It follows that in Cρ ∩ {t  0} ∩ {u > 0} the function u is monotone increasing in the directions e satisfying a0 β > −a1 M. Since u(0, 0) = 0, we obtain u(x, t) = 0

in

  β (x, t) ∈ Cρ : t  0, 0 < x < − t . M

2.2 Optimal Regularity of Solutions

33

x

x = f (t)

C

x=

M

t

u=0 2

2

0

t

Fig. 2.1 Counterexample

Thus, we have shown that the free boundary x = f (t) intersects the t-axis at the point (0, 0) transversally (see Fig. 2.1).

2.2 Optimal Regularity of Solutions Lemma 2.1 Let u be a solution of the problem (2.1). Then for Qρ (z0 ) ⊂ Q1 , we have   (2.3) Du 22,Q+ (z0 )  C(n)ρ n ρ 2 u ∞,Q+ρ (z0 ) + u 2∞,Q+ (z0 ) . ρ

ρ/2

Proof Consider two cut-off functions: ς = ς(x) ∈ C0∞ (B ρ (x 0 )), η = η(t) ∈ C0∞



ς(x) ≡ 1

t 0 − ρ2, t 0 + ρ2

 ,

in

η(t) ≡ 1

|Dς| 

Bρ/2 (x 0 ), in

]t 0 −

ρ2 0 , t ], 4

4 , ρ

∂t η 

4 . ρ2

Multiplying both sides of H [u] = χΩ on uς 2 η2 and integrating by parts, we obtain ˆ ˆ ˆ  1 (Du)2 ς 2 η2 dxdt = − (ut =t 0 )2 ς 2 dx + u2 ς 2 η (∂t η) dxdt 2 0 Q+ ρ (z )

0 Q+ ρ (z )

Bρ (x 0 )

ˆ

−

ˆ

2uη ς Di uDi ς dxdt − 2

0 Q+ ρ (z )

uς 2 η2 χΩ dxdt 0 Q+ ρ (z )

(2.4)

34

2 No-Sign Parabolic Obstacle-Type Problems

Using Young’s inequality in the third integral of the right hand side of (2.4), we get (2.3).   Remark 2.2 It follows from Lemma 2.1 that for any δ ∈]0, 1[ we have

Du 22,Q+  C(n, δ) 1 + u 2∞,Q+ , 1−δ

D  u ∞,Q+  C(n, δ) D  u 2,Q+ 1−δ

(2.5)

1

1−δ/2

 C(n, δ) 1 + u ∞,Q+ , 1

(2.6)

where the first inequality in (2.6) follows from the fact that H [D  u] = div f

in

Q+ 1,

f ∈ L∞ (Q+ 1)

and well-known local estimates (see Lemma B.1). Before stating the next results we need to extend a solution u of the problem (2.1) across the plane Π to the whole unit cylinder Q1 by the odd reflection, i.e., by setting it as −u(−x1 , x2 , . . . , xn , t) for (x, t) ∈ Q1 , x1 < 0. For simplicity of notation, we will use the same letter u for the extended function. From Lemma B.2 it follows that for any unit vector e ∈ Rn orthogonal to e1 (x1 -axis) the extended functions (De u)± are subcaloric in Q1 . Lemma 2.2 Let u satisfy (2.1), let z0 = (x 0 , t 0 ) ∈ Γ (u) ∩ Q+ 1/4 , and let e be an arbitrary unit vector in Rn , orthogonal to e1 (x1 -axis). Suppose that we extend u to the whole cylinder Q1 by the odd reflection. Then for v = De u and for r ∈ (0; 1/2], we have

2 Φ(r, v + ψ, v − ψ, x 0 , t 0 )  C(n) 1 + u 2∞,Q+ ,

(2.7)

1

where the functional Φ is defined by the formula (A.2) (see Appendix A), ψ(x) := ζ(|x − x 0 |), while ζ is a standard time-independent cut-off function (See Notation and Conventions). Proof Applying Lemma A.2 to the functions h1 (x, t) = v + (x + x 0 , t + t 0 )ζ (|x|),

h2 (x, t) = v − (x + x 0 , t + t 0 )ζ(|x|)

and making change of the variables, we get Φ(r, v + ψ, v − ψ, x 0 , t 0 )  C(n) Du 42,Q Now, by inequality (2.5) we immediately arrive at (2.7).

1/2 (z

0)

.  

2.2 Optimal Regularity of Solutions

35

Lemma 2.3 Let u satisfy (2.1), and let u ∞,Q+  M0 . Suppose also that z = 1

(x, t) ∈ Λ(u) ∩ Q+ 1/4 , R(z) = x1 = dist {z, Π} and M (ρ, z, u) = sup |u|

for R(z) < ρ 

Q+ ρ (z)

1 . 2

Then there exists a constant C = C(n, M0 ) such that T (z, u) :=

M (ρ, z, u)  C. ρ2 ρ∈(R(z),1/2] sup

(2.8)

Proof We argue by contradiction. Suppose (2.8) fails. Then for every j ∈ N (natural numbers) there are uj , satisfying (2.1) and the inequalities uj ∞,Q+  M0 , and j zj = (x j , t j ) ∈ Λ(uj ) ∩ Q+ 1/4 and rj ∈ (R(z ), 1/2] such that

M (rj , zj , uj ) rj2



1

1 j T z , uj > j 2

(2.9)

(here and in the sequel we regard uj as functions extended by the odd reflection to   the whole cylinder Q1 . It is evident that the extended functions satisfy H uj = fj in Q1 and sup |fj |  1.) Q1

It is evident that inequality (2.9) gives the estimate  rj
n, to a caloric function u0 , satisfying

|u0 (x, t)|  1 + 2 |x|2 + |t| , u0 (0, 0) = |Du0 (0, 0)| = 0,

(2.16)

sup |u0 | = 1,

(2.17)

Q1

 u 0 x

1 =−d0

= 0,

(2.18)

where R(zj ) , j →0 rj

d0 = lim

d0 ∈ [0, 1].

According to Lemma B.3 (see Appendix B), there exist constants a ij , a i , a0 such that  n   ij ii u0 (x, t) = a xi xj + 2 a t + a i xi + a0 . i=1

2.2 Optimal Regularity of Solutions

37

Obviously, conditions (2.16) and (2.18) imply n 

a jj = 0,

j =1

and a0 = a i = a τ μ = 0,

for all indices i, τ, μ.

(We recall that according our notation the index i varies from 1 to n, while the indices τ and μ vary from 2 to n). Consequently, a 11 = 0 and the function u0 takes the form u0 (x, t) = 2a 1τ x1 xτ .

(2.19)

 ± Applying Lemma B.5 (see Appendix B) to the functions Dτ uj (x, t) we get j for (x, t) ∈ Q+ rj (z ) the inequalities |Dτ uj (x, t)|  C(n)x1 Dτ uj ∞,Q+

15/16

 C(n)x1 (1 + M0 )  C(n)rj (1 + M0 ),

(2.20)

where the second inequality follows from (2.6), while the third one follows from the assumption rj  R(zj ). By (2.20) and (2.9) it is evident that sup |Dτ u˜ j | = Q+ 1

=

supQ+ |Dτ uj (rj x + x j , rj2 t + t j )| 1

M (rj , zj , uj ) rj · sup |Dτ uj (x, ˜ t˜)| M (rj , zj , uj ) (x, j ˜ t˜)∈Q+ r (z ) j



C(n)rj2 (1 + M0 ) M (rj , zj , uj )


2 such that if u ∞,Q+  M0 1

+

and z = (x, t) ∈ Γ (u) ∩ Q1/4 , then sup |u|  Cr 2

Q+ r (z)

for Qr (z) ⊂ Q1 .

(2.25)

Remark 2.4 We observe that Lemma 2.3 guarantees (2.25) for all r > x1 . In particular, for z ∈ Π the statement of Lemma 2.5 follows from Lemma 2.3.

2.2 Optimal Regularity of Solutions

39

Proof Define Mk (z, u) := sup |u|, Q+−k (z)

k ∈ N.

2

To obtain (2.25) it is sufficient to show that for some C > 0   4k+1 Mk+1 (z, u)  max 4M1 (z, u), . . . , 4k Mk (z, u), C

(2.26)

for all k ∈ N. Suppose (2.26) fails. Then for every j ∈ N there exist uj , satisfying (2.1) and the

inequality uj ∞,Q+  M0 , zj = (x j , t j ) ∈ Γ (uj ) ∩ Q+ 1/4 and kj ∈ N such that 1

  4kj +1 Mkj +1 (zj , uj ) > max 4M1 (zj , uj ), . . . , 4kj Mkj (zj , uj ), j .

(2.27)

We regard uj , in (2.27) and below, as functions extended to the whole Q1 by the odd reflection. Obviously, (2.27) and inequalities Mkj (zj , uj )  M0 imply kj → ∞

as j → ∞.

(2.28)

In addition, in view of Remark 2.4 we have 2−kj  x1

j

∀j ∈ N.

(2.29)

Define now u˜ j (x, t) =

uj (2−kj x + x j , 2−2kj t + t j ) Mkj +1 (zj , uj )

in

Q2kj −1 .

Then it follows from the definition of uj and (2.27) that u˜ j (0, 0) = |D u˜ j (0, 0)| = 0,

(2.30)

sup |u˜ j | = 1,

(2.31)

Q1/2

H [u˜ j ] ∞,Q

k −1 2 j



1 4kj Mkj +1 (zj , uj )



4kj +1 → 0 as 4kj j

j → ∞.

(2.32)

40

2 No-Sign Parabolic Obstacle-Type Problems

Observe also that for integers m satisfying 1  m  kj − 1 we have u˜ j ∞,Q2m 

Mkj −m (zj , uj ) Mkj +1 (zj , uj )

= 4m+1

4kj −m Mkj −m (zj uj ) 4kj +1 Mkj +1 (zj , uj )

 4 · 22m ,

(2.33)

where the second inequality follows from (2.27). Now, by (2.28)–(2.33) we infer that a subsequence of u˜ j weakly converges in 2,1 Wq,loc (Rn+1 ∩{t  0}), q > n, to a caloric function u0 . Moreover, the limit function u0 satisfies u0 (0, 0) = |Du0 (0, 0)| = 0,

(2.34)

sup |u0 | = 1,

(2.35)

Q1/2

|u0 (x, t)|  16(1 + |x|2 + |t|). Using Lemma B.3 and equalities (2.34) we deduce that there are constants a ij such that   n  (2.36) u0 (x, t) = a ij xi xj + 2 a ii t. i=1

We now proceed to show that u0 is, in fact, one-space dimensional, i.e.,

u0 (x, t) = u0 (x1 , t) = a 11 x12 + 2t .

(2.37)

For any fixed direction e such that e ⊥ e1 define v = De u0 ,

vj = De uj ,

v˜j = De u˜ j .

Then, for a subsequence, v˜j converges in Cloc (Rn+1 ) to v, where H  [v] = 0.  j (x) := ζ 2−kj |x| , where Consider the functions ψj (x) := ζ (|x − x j |) and ψ ζ is a standard time-independent cut-off function (See Notation and Conventions). According Lemma 2.2, for all j ∈ N and for all r  12 , we have

2 Φ(r, vj+ ψj , vj− ψj , x j , t j )  C(n) 1 + uj 2∞,Q+ . 1

(2.38)

2.2 Optimal Regularity of Solutions

41

Taking into account the inequalities uj ∞,Q+  M0 , making change of the 1

variables in (2.38) and letting r = 2−kj , we will obtain  j , j , 0, 0) Φ(1, vj+ ψ vj− ψ

=

2−2kj Mkj +1 (zj , uj ) 

4 Φ(2−kj , vj+ ψj , vj− ψj , x j , t j )

1 + uj ∞,Q+

 4C(n)

4

1

4kj +1 Mkj +1 (zj , uj )  1 + M0 4  4C(n) for all j, j (2.39) where in the second inequality we have used (2.27). j ≡ 1 in B kj ⊃ B1/4 and, consequently, Observe that ψ 2 /8 j )|2  |D vj± |2 χB1/4 |D( vj± ψ

− 1  t < 0,

for

x ∈ B1/2 .

In addition, for ε > 0 (small and fixed) we have G(x, −t)  N(n, ε) > 0

for

− 1  t < −ε,

x ∈ B1/4 .

Hence, ˆ−ε ˆ

|D vj± |2 dxdt 

N(n, ε) −1 B1/4

ˆ0 ˆ

j )|2 G(x, −t)dxdt. |D( vj± ψ

(2.40)

−1 Rn

Next, using (2.40) and invoking the Poincare inequality we may reduce (2.39) to ˆ−ε ˆ

| vj+

2 − m+ j (t)| dxdt

−1 B1/4

ˆ−ε ˆ

2 | vj− − m− j (t)| dxdt

−1 B1/4

 C(n)N

−2



1 + M0 (n, ε) j

4 for all j,

where m± vj± on t-sections over B1/4 . j (t) denotes the corresponding average of  Letting j tend to infinity (and then ε tend to zero), we obtain ˆ

+

+ 2

ˆ

|v − m | dxdt Q1/4

Q1/4

|v − − m− |2 dxdt = 0,

(2.41)

42

2 No-Sign Parabolic Obstacle-Type Problems

where m± is the corresponding average of v ± over B1/4 . Observe that, due to (2.36), m± do not depend on t. Obviously (2.41) implies that either of v ± is equivalent to m± in Q1/4 , and thus independent of the spatial variables. Assume for the definiteness that v − ≡ m− . In view of (2.34) we have v(0, 0) = 0, and, consequently, v − ≡ 0 in Q1/4 . Note that a derivative Di u0 of a polynomial (2.36) changes its sign in Qρ for any ρ > 0, otherwise Di u0 ≡ 0. Hence we obtain v ≡ 0 in

Rn+1 ∩ {t  0}.

(2.42)

Recall that v = De u0 , and e is arbitrary direction in Rn orthogonal to e1 . In such a manner, (2.42) in particular implies that u0 depends only on x1 and t, which together with (2.36) gives (2.37). Now we can use Lemma 2.4 and prove that D1 u0 ≡ 0 too. For this purposes we now define  |x − x j | j , v = D1 u0 , vj = D1 uj , Rj = x1 , ψRj (x) = ζ Rj where ζ is a standard time-independent cut-off function (See Notation and Conventions). Using Lemma 2.4 and taking into account the inequalities uj ∞,Q+  M0 , we 1 get for r  Rj the inequality Φ(r, vj+ ψRj , vj− ψRj , x j , t j )  C(n, M0 ).

(2.43)

Making change of the variables in the left hand side of (2.43) and repeating all the arguments used to derive (2.42), we end up with D1 u0 = 0 in Q1/4 . The latter, combined with (2.37), gives u0 ≡ 0 and we get the contradiction with (2.35). The proof is completed.   Proof (Proof of Theorem 2.1) Let z0 = (x 0 , t 0 ) ∈ Γ (u) ∩ Q+ 1/4 . By Lemma 2.5   |u(x, t)|  C(n, M0 ) |x − x 0 |2 + (t − t 0 )

  for (x, t) ∈ Q1 ∩ t  t 0 . (2.44)

  To obtain a similar estimate for t > t 0 we consider the barrier function   w(x, t) = C1 (n, M0 ) |x − x 0 |2 + (2n + 1)(t − t 0 ) , where C1 (n, M0 ) = max {C(n, M0 ); 2M0 }.

2.2 Optimal Regularity of Solutions

43

One can easily verify that   in Q1 ∩ t > t 0 .

|u(x, t)|  w(x, t)

(2.45)

Let us introduce now the parabolic distance to Γ (u) dp (x, t) :=

inf

(y,τ )∈Γ (u)



1/2 |x − y|2 + |t − τ | .

Then, combining (2.44) and (2.45) we have |u(x, t)|  N(n, M0 )d 2 (x, t)

for all (x, t) ∈ Q+ 1/8 .

(2.46)

+ Now, we fix (x, t) ∈ Q+ 1/8 ∩ Ω and consider Qd/2 (x, t) with d = dp (x, t). Since H [u] = 1 in Q+ d/2 (x, t) and u = 0 on Π, we can apply inequality (B.7) (see Appendix B) for v = u, R = d/2 and obtain ⎧ ⎫ ⎨ ⎬ |Di Dj u(x, t)| + |∂t u(x, t)|  C(n) 1 + d −2 sup |u| . (2.47) ⎩ ⎭ Q+ (x,t ) d/2

 

Combining (2.46) and (2.47) we get (2.2). Theorem 2.1 has the following obvious implication. Corollary 2.1 Let u satisfy (in the sense of distributions) the equation H [u] = χΩ

in

Rn+1 +

for some open set Ω = Ω(u) ⊂ Rn+1 + ∩ {t  0} ,

let u = |Du| = 0 in Rn+1 + \ Ω(u), and let u = 0 on Π. There exists a universal constant C = C(n, M) such that, if

|u(x, t)|  M 1 + |x|2 + |t| ,

(2.48)

then |Di Dj u(x, t)| + |∂t u(x, t)|  C

for all (x, t) ∈ Rn+1 + ∩ {t  0} .

In particular, if (x 0 , t 0 ) ∈ Γ (u), then  |u(rx + x 0 , r 2 t + t 0 )|  Cr 2

∀(x, t) ∈ Q1 ∩ x1 > −

x10 r

Proof Consider the scaling ur (x, t) =

u(rx, r 2 t) r2

in Q+ 1,

r > 1.

and r > 0.

44

2 No-Sign Parabolic Obstacle-Type Problems

Inequality (2.48) implies ur ∞,Q+  3M. Hence by Theorem 2.1 we will have 1

  sup |Di Dj ur (x, t)| + |∂t ur (x, t)|  C(n, 3M),

Q+ 1/8

i.e.,   sup |Di Dj u(x, t)| + |∂t u(x, t)|  C(n, 3M).

Q+ r/8

Letting r tend to infinity we will have the first statement of the corollary. The second part follows by using the first part and that  |u(rx + x 0 , r 2 t + t 0 )|  r 2

sup



0 0 Q+ r (x ,t )

  |Di Dj u(x, t)| + |∂t u(x, t)|  C(n, 3M)r 2 .

 

2.3 Useful Properties of Solutions In this section we collect all necessary facts about local and global solutions of the no-sign parabolic obstacle-type problem that will be used frequently in what comes later.

2.3.1 Nondegeneracy Lemma 2.6 Let u ∈ PR+ (M) with 0 < R  +∞. Then there exists a constant C(n) > 0 such that (i) for all z0 ∈ {u > 0}, and for all ρ satisfying Qρ (z0 ) ⊂ QR we have sup u  u(z0 ) + C(n)ρ 2 ;

0 Q+ ρ (z )

(ii) for all z0 ∈ Λ(u), and for Qρ (z0 ) as above we have either sup u  C(n)ρ 2

0 Q+ ρ (z )

(2.49)

0 or u ≡ 0 in Q+ ρ/2 (z ); 0 (iii) for all z ∈ Γ (u), and for Qρ (z0 ) as above, inequality (2.49) holds true;

2.3 Useful Properties of Solutions

45

(iv) if u(x, t) < 0 for all (x, t) ∈ Qρ (z0 ) ⊂ Q+ R then we have u(z0 )  −C(n)ρ 2 . 1 Proof The proof of Case (i) with C(n) = 2n+1 is similar to that of [Caf80] (see Lemma 1). Without loss of generality, we may assume that z0 ∈ {u > 0}. Consider the function

w(x, t) = u(x, t) − u(x 0 , t 0 ) −

1 (|x − x 0 |2 − (t − t 0 )). 2n + 1

(2.50)

Then w is caloric in {u > 0}, and w(x 0 , t 0 ) = 0. The maximum principle yields sup

0 {u>0}∩Q+ ρ (z )

w=

sup

  0 ∂  {u>0}∩Q+ ρ (z )

w  0.

Since w is strictly negative on ∂ {u > 0}, the nonnegative supremum of w is 0 attained at some point (x ∗ , t ∗ ) on the parabolic boundary of the cylinder Q+ ρ (z ). In particular, u(x ∗ , t ∗ )  u(x 0 , t 0 ) +

ρ2 . 2n + 1

0 The proof of Case (ii) follows from Case (i), if {u > 0} ∩ Q+ ρ/2 (z ) = ∅. If u  0 + + in Qρ/2 (z0 ) then u ≡ 0 in Qρ/2 (z0 ), since u is subcaloric. In the latter case for z0 ∈ Π we have to use additionally Hopf’s lemma. Case (iii) is a easy consequence of Cases (i) and (ii). It remains only to verify Case (iv). It is clear that H [u] = 1 in Qρ (z0 ). Further, we again consider the function w, defined by the formula (2.50). Then w is caloric in Qρ (z0 ), and w(x 0 , t 0 ) = 0. Moreover, the maximum principle yields

0  sup w = Qρ (z0 )



sup w ∂  Qρ (z0 )

sup u − u(x 0 , t 0 ) −

∂  Qρ (z0 )

 −u(x 0 , t 0 ) − Choosing C(n) =

1 2n+1

(|x − x 0 |2 − (t − t 0 )) 2n + 1 ∂  Qρ (z0 ) inf

ρ2 . 2n + 1

we are done.

 

46

2 No-Sign Parabolic Obstacle-Type Problems

Lemma 2.7 Suppose that u ∈ PR+ (M). If z0 = (x 0 , t 0 ) ∈ (∂Ω(u) ∩ Λ(u)) \ Γ (u), then there exists r0 = r0 (z0 ) > 0 such that 0 Q+ r0 (z ).

(2.51)

0 0 2 Q+ r0 (x , t + r0 ).

(2.52)

u ≡ 0 in If, in addition, t 0 < 0 then u < 0 in

Proof The first statement of this lemma follows from the definition of the set Γ (u) / Γ (u). and our choice of z0 ∈ The second statement is easy consequences of (2.51) and Lemma 2.6. Indeed, consider a point z∗ = (x 0 , t 0 + h2 ), where h is a positive parameter which will be chosen later. In view of (2.51) we have sup u  Mh2 ,

∗ Q+ r (z )

(2.53)

provided h < r/2  r0 /2. ∗ Suppose, towards a contradiction, that there exists zˆ ∈ Q+ r/2 (z ) such that u(ˆz) > 0. Then, item (i) from Lemma 2.6 gives 1 r 2  sup u. 4(2n + 1) Q+ (ˆz)

(2.54)

r/2

It is easy to see that (2.54) contradicts (2.53) if h < (4M(2n + 1))−1/2 r. Thus, for sufficiently small h, the cylinder Qr/2 (z∗ ) does not contain points where u is positive. Moreover, the subcaloricity of u implies that Qh (z∗ ) does not contain points where u equals zero. So, we get (2.52) with r0 = h.   Lemma 2.8 Let u ∈ PR+ (M), let z0 ∈ ∂Ω(u) ∩ Λ(u), and let r0 = r0 (z0 ) be the constant from Lemma 2.7. Then there exists a constant C(n) > 0 such that for all ρ  r0 (z0 ) satisfying Zρ,ρ 2 (z0 ) ⊂ QR we have sup

Z + 2 (z0 )

|u|  C(n)ρ 2 .

(2.55)

ρ,ρ

Proof Inequality (2.55) is an easy consequence of Lemmas 2.6 and 2.7. Indeed, if z0 ∈ Γ (u) then item (iii) from Lemma 2.6 provides sup u  C(n)ρ 2 ,

0 Q+ ρ (z )

2.3 Useful Properties of Solutions

47

and we are done (here C(n) is the same constant as in Lemma 2.6). Otherwise, successive application of Lemma 2.7 and item (iv) from Lemma 2.6 implies u(z∗ )  −C(n)

z∗ = x10 + ρ/2, (x 0 ) , t 0 + ρ 2 .

ρ2 , 16

Taking into account that inf

Z + 2 (z0 )

u  u(z∗ )  −C(n)ρ 2

ρ,ρ

 

we complete the proof of the lemma.

2.3.2 Measure of Γ (u) From nondegeneracy and Cx1,1 ∩ Ct0,1 -regularity of solution one can deduce some information about the free boundary.   Lemma 2.9 Let u ∈ PR+ (M) with 0 < R  +∞, and let z = x, t ∈ Q+ R . For any parabolic cylinder Zρ,ρ 2 (z) ⊂ QR the set Zρ,ρ 2 (z) \ (∂Ω(u) ∩ Λ(u)) contains a subset of proportional volume. Remark 2.5 We emphasize that in Lemma 2.9 the cylinder Zρ,ρ 2 (z) is not necessarily centered at a point from ∂Ω(u) ∩ Λ(u). Proof Consider two mutually excluded cases. + 0 = (x 0 , t 0 ) ∈ ∂Ω(u) ∩ Λ(u), then due to If Zρ/2,ρ 2 /4 (z) contain a point z + 0 Lemma 2.8, there exists another point z∗ = (x ∗ , t ∗ ) ∈ Zρ/2,ρ 2 /4 (z ) such that

|u(z∗ )| 

ρ2 . 8n + 4

On the other hand, Theorem 2.1 and Corollary 2.1 imply that |u(z) − u(z∗ )|  C(n, M)ερ 2

∗ in Q+ ερ (z ).

Choosing ε < [9(2n + 1)C(n, M)]−1 , we obtain inf |u| 

∗ Q+ ερ (z )

ρ2 − C(n, M)ερ 2 > 0. 8n + 4

∗ So, we have proved that the set Ω(u) ∩ Zρ,ρ 2 (z) contains the subset Q+ ερ (z ) of proportional volume.

48

2 No-Sign Parabolic Obstacle-Type Problems + Otherwise, if Zρ/2,ρ 2 /4 (z) does not contain any point from ∂Ω(u) ∩ Λ(u) then

+ we can choose as a subset of proportional volume the cylinder Zρ/2,ρ 2 /4 (z) itself.

 

Lemma 2.10 Let u ∈ PR+ (M) with 0 < R  +∞. Then Γ (u) is a set of (n + 1)dimensional Lebesgue measure zero. Proof This is proved in a standard way. We have to show that for every z0 ∈ ∂Ω ∩ Λ(u) and every sufficiently small hyperbolic cylinder Zρ,ρ (z0 ) the difference + (z0 ) \ (∂Ω(u) ∩ Λ(u)) contains a set E of Lebesgue measure proportional to Zρ,ρ ρ + 0 that of Zρ,ρ (z ). Without loss of generality we may assume that 1/ρ ∈ N. Using the same arguments as in the proof of Theorem 5.1 [Wei99] (see also Subsection 5.3 [CPS04]) + (z0 ) into 1/ρ parabolic cylinders we decompose Zρ,ρ Zk = Zρ,ρ 2 (x 0 , t k ),

t k = t 0 + ρ {1 − (2k − 1) ρ} ,

k = 1, 2, . . . ,

1 . ρ

Applying Lemma 2.9 to Zk , k = 1, . . . , 1/ρ, and taking into account Remark 2.5, !+ of proportional we conclude that every Zk+ \(∂Ω(u) ∩ Λ(u)) contains a cylinder Z k volume. It remains only to define the set Eρ with the volume proportional to + (z0 ) by the formula Zρ,ρ Eρ :=

1/ρ "

!+ . Z k

k=1

Thus, ∂Ω(u) ∩ Λ(u) (and, consequently, Γ (u)) contains no density points, i.e., the free boundary has zero Lebesgue measure.  

2.3.3 Convergence Lemma 2.11 Suppose that vk ∈ Pr (M) and zk ∈ Γ (vk ) for k = 1, 2, . . . . If vk → v in Cloc (Qr ) and zk → z0 , as k → ∞, then v ∈ Pr (M) and z0 ∈ Γ (v). Proof This statement is easy consequences of Lemma 2.6 and results of [CPS04] (see subsection 5.2 in [CPS04]).   + (M) across Lemma 2.12 Suppose we extend a nonnegative global solution u ∈ P∞ n+1 the plane Π to the entire space R ∩ {t  0} so as to obtain an odd function, i.e., we set it to be equal to −u(−x1 , x  , t) for (x, t) ∈ Rn+1 − ∩ {t  0}; we keep the notation u for the extended function.

2.3 Useful Properties of Solutions

49

For a point z0 = (x 0 , t 0 ) ∈ Rn+1 + ∩ {t  0} ∩ Λ(u) and a sequence rm  ∞, we define the scaling um (x, t) =

2 t + t 0) u(rm x + x 0 , rm . 2 rm

Then um converge (for a subsequence) to a nonnegative function u∞ uniformly on the compact subsets of Rn+1 ∩ {t  0}. The derivatives Di Dj um and ∂t um q converge in Lloc (Rn+1 ∩ {t  0}), q < ∞, to the corresponding derivatives of u∞ . Moreover, the restriction of u∞ to the set {x1  0} belongs to the class P + (M). Proof Observe that um ∈ P + 0

−x1 /rm

Di Dj um (z) → Di Dj u∞ (z),

(M). By Lemma 2.10, it suffices to check that

∂t um (z) → ∂t u∞ (z)

for a.e.

z ∈ Rn+1 + ∩ {t  0}.

If z = (x, t) ∈ Ω(u∞ ), then for some ρ > 0 and sufficiently large m in the ρ-neighborhood of z we have H [um − u∞ ] = 0. By the general parabolic theory, it follows that the sequence Di Dj um (z) converges to Di Dj u∞ (z). The convergence of ∂t um (z) is established in the same way. By Lemma 2.6, the same convergence occurs in the case where z is an interior point of Λ(u∞ ). Combined with Lemma 2.10, this completes the proof.  

2.3.4 Blow-Up and Blow-Down For a function u ∈ PR+ (M) with 0 < R  +∞ and for a point z0 = (x 0 , t 0 ) ∈ ∂Ω(u) ∩ Λ(u) we consider the parabolic scaling ur defined as ur (x, t) =

u(rx + x 0 , r 2 t + t 0 ) . r2

(2.56)

By a standard compactness arguments we may pass to the limit along a subsequence rk → 0, obtaining as a result a global solution u0 from P∞ (M) or + (M), respectively, for the cases x 0 > 0 and x 0 = 0. In both cases the point P∞ 1 1 (0, 0) ∈ ∂Ω(u0). Moreover, if z0 ∈ Γ (u) then due to Lemma 2.11 we have (0, 0) ∈ Γ (u0 ). Otherwise, from Lemma 2.7 we get u0 ≡ 0 for t  0. Usually, such a process is referred to as passage to blow-up limit. Any global solution u0 thus obtained is called a blow-up of the function u at the point z0 . + (M) and z0 = (x 0 , t 0 ) ∈ Γ (u) we can Similarly, for a global solution u ∈ P∞ 0 consider the scaled functions ur around z and let r → +∞. Then ur converge (for a subsequence) to a function u∞ uniformly on compact subsets of (Rn+1 + ∪Π)∩{t  + (M) and (0, 0) ∈ Γ (u ). The limit function u 0}. It is easy to see that u∞ ∈ P∞ ∞ ∞ is called blow-down of the global solution u at the point z0 .

50

2 No-Sign Parabolic Obstacle-Type Problems

In general, it is a priori not clear if u0 is unique. Different subsequences rk → 0 may lead to different blow-up limits u0 at the same point z0 . Analogous remark is also true for blow-down limits. Definition 2.3 Let z∗ = (x ∗ , t ∗ ) be a point in Rn+1 . We say that a function v is parabolic homogeneous of degree 2 with respect to z∗ if either of the following statements is satisfied: (i) a function v is defined in Rn+1 ∩ {t  t ∗ } and the identity v(θ x + x ∗ , θ 2 t + t ∗ ) = θ 2 v(x + x ∗ , t + t ∗ )

(2.57)

holds for all x ∈ Rn , θ > 0, and for all t  0. (ii) x1∗  0, a function v is defined in {(x, t) ∈ Rn+1 : x1  0, t  t ∗ }, and the identity (2.57) holds for all θ > 0, t  0, and x ∈ Rn such that θ x1 + x1∗  0. Lemma 2.13 The functions u0 and u∞ are degree 2 parabolic homogeneous with respect to the origin. Proof We prove the statement for the blow-up u0 of u at the point z0 . The case of u∞ is treated in the same way. Observe that for any θ > 0 we have W (θ, 0, 0, u0 ) = lim W1/r (θ, 0, 0, ur ) = lim W1 (θ r, x 0 , t 0 , u) = const, r→0+

r→0+

(2.58) where W , W1/r and W1 are global and local Weiss functionals, respectively. (For the definitions of the Weiss functionals we refer the reader to Appendix A.) The first equality in (2.58) follows from the regularity properties of u, while the scaling property (A.29) for the local Weiss functional (see Appendix A) provides the second equality. On the other hand, from Lemma A.6 and Remark A.4, it follows that dW (θ, 0, 0, u0 ) 1 = dθ θ

ˆ−1ˆ −4 Rn

1 = 5 θ

where (u0 )θ (x, t) =

|L (u0 )θ |2 G(x, −t)dxdt −t

−θ 2 ˆ ˆ −4θ 2 Rn

|L u0 |2 G(x, −t)dxdt, −t

u0 (θ x, θ 2 t) , and θ2

L u0 (x, t) := x · Du0 (x, t) + 2t∂t u0 (x, t) − 2u0 (x, t).

(2.59)

2.3 Useful Properties of Solutions

51

Now, combining (2.58) and (2.59) we get the identity L u0 (x, t) ≡ 0 for (x, t) ∈ Rn ×] − 4θ 2 , −θ 2 [.

(2.60)

It is evident that equality (2.60) gives the degree 2 parabolic homogeneity of the   function u0 with respect to the origin.

2.3.5 Balanced Energy Lemma 2.14 Let u ∈ P2+ (M), and let (x 0 , t 0 ) ∈ Λ(u) ∩ Q1 . Suppose that the function u is extended by zero across the plane Π to the set Q2 ∩ {x1 < 0}. We preserve the notation u for this extension. Then for arbitrary ρ and α satisfying 12  ρ > α > 0 we have W1 (ρ, x 0 , t 0 , u) − W1 (α, x 0 , t 0 , u)  −C(n, M)(ρ − α),

(2.61)

where W1 is the local Weiss functional defined in Appendix A.  

Proof Inequality (2.61) follows immediately from Lemma A.6. has a limit as r → Observe that the function W1 lary A.1 in Appendix A). The corresponding limit (r, x 0 , t 0 , u)

ω(x 0 , t 0 , u) := lim W1 (r, x 0 , t 0 , u) r→0+

0+

(see Corol-

(2.62)

will be called the balanced energy of the function u at the point (x 0 , t 0 ) ∈ Λ(u) ∩ ∂Ω(u). Remark 2.6 It is easy to see that ω(x 0 , t 0 , u) =

lim Wb (r, x 0 , t 0 , u) for an

r→0+

+ (M) the convergence arbitrary fixed b > 0. Moreover, for global solutions from P∞ + of the Weiss functional W to the balanced energy as r → 0 is monotone.

Lemma 2.15 For u ∈ P2+ (M) and any point z0 = (x 0 , t 0 ) ∈ ∂Ω(u) ∩ Λ(u) ∩ Q+ 2 the following statements hold: (i) the balanced energy ω(x 0 , t 0 , u) may equal one of the following values 0, A := 15 0 4 or 2A. Correspondingly, z is called a zero energy point , a low energy point or a high energy point . (ii) If z0 ∈ / Γ (u) then ω(x 0 , t 0 , u) = 0, while Γ (u) contains low and high energy points only. (iii) If z0 is a zero energy point then u ≡ 0 in Qr (z0 ) and u < 0 in Qr (x 0 , t 0 + r 2 ) for some r = r(z0 ) > 0.

52

2 No-Sign Parabolic Obstacle-Type Problems

(iv) If z0 is a low energy point then the sets Λ(u) and {u > 0} have non-empty interiors in Qρ (z0 ) for all ρ > 0. (v) If the set Λ(u) does not have non-empty interior in Qρ (z0 ) for some ρ > 0 then z0 is a high energy point. Remark 2.7 We emphasize that Lemma 2.15 concerns the interior points of Q+ 2 only, that is x10 > 0. Proof All the statements follows from results of Sections 6 and 7, [CPS04]. We explain some details here. Case (i) From definition of ω(x 0 , t 0 , u), scaling property (A.29) and the regularity properties of u it follows that ω(x 0 , t 0 , u) = lim W1 (r, x 0 , t 0 , u) = lim W1/r (1, 0, 0, ur ) r→0+

r→0+

(2.63)

= W (1, 0, 0, u0 ), where W1/r and W are local and global Weiss functionals from Appendix A, respectively, and u0 is a blow-up of u at the point (x 0 , t 0 ). Observe also that for validation (2.63) we use exactly the same arguments as in deriving of (2.58). If u0 ≡ 0 in Rn+1 ∩ {t  0} then relation (2.63) provides W (1, 0, 0, u0 ) = 0 and, consequently, (x 0 , t 0 ) is a zero energy point for u. Otherwise, if u0 does not vanish identically in Rn+1 ∩ {t  0}, we can apply successively Lemmas 2.13, C.2 and A.5 to the function u0 , which yields either ω(x 0 , t 0 , u) = A or ω(x 0 , t 0 , u) = 2A. Case (ii) It follows from Lemma 2.7 that for z0 ∈ / Γ (u) we have u0 ≡ 0 in Rn+1 ∩ 0 0 {t  0}, and therefore ω(x , t , u) = 0 due to (2.63). On the other hand, if z0 ∈ Γ (u) then from item (iii) Lemma 2.6 we have that sup u  C(n)ρ 2

0 Q+ ρ (z )

for all ρ > 0 satisfying Qρ (z0 ) ⊂ Q2 . Therefore the point z0 ∈ Γ (u) is either of low or high energy, since no blow-up u0 at z0 = (x 0 , t 0 ) can vanish identically in Rn+1 ∩ {t  0}. Case (iii) follows immediately from Case (ii) and Lemma 2.7. Case (iv) We begin with the set Λ(u). Suppose, towards a contradiction, that there exists ρ0 > 0 such that Λ(u) has an empty interior in Qρ0 (z0 ). Let u0 be an arbitrary blow-up of u at (x 0 , t 0 ). Then u0 ∈ P∞ (M), Λ(u0 ) has an empty interior in Rn+1 ∩ {t  0} and, consequently, u0 satisfies Δu0 − ∂t u0 = 1

(2.64)

2.4 Classification of the Nonnegative Global Solutions

53

in all of Rn+1 ∩ {t  0}. Differentiation of (2.64) combined with Lemmas B.3 and 2.13 provide the representation u0 (x, t) = mt + P (x) where P is a homogeneous quadratic polynomial such that ΔP = m + 1. But then relation (2.63) and Lemma A.5 give ω(x 0 , t 0 , u) = W (1, 0, 0, u0 ) = W (1, 0, 0, mt + P (x)) = 2A, which contradicts the assumption that z0 is a low energy point.  

Case (v) is proved exactly in the same manner as Case (iv) for the set Λ(u).

2.4 Classification of the Nonnegative Global Solutions We start with considering nonnegative solutions of the obstacle-type problem, since the proofs are simpler in that case. + (M), u  0 in Rn+1 ∩ {t  0} and Γ (u) = ∅. Then Lemma 2.16 Suppose u ∈ P∞ + u(x, t) is a one-space dimensional, i.e.,

u(x, t) = u(x1 , t).

(2.65)

Proof We prove this lemma in two steps. 1.

Let ν be any spatial direction such that ν · e1 > 0. Since u  0, we have Dν u  0 on

Π ∩ {t  0}.

(2.66)

Rn+1 + ∩ {t  0}.

(2.67)

We claim that Dν u  0 in

To prove this, first we extend u across the plane Π to the entire space Rn+1 ∩ {t  0} so as to have an odd function, preserving the notation u for the extension. Also, we define w as follows:  w(x, t) =

Dν u(x, t)

if

(x, t) ∈ Rn+1 + ∩ {t  0},

(Dν u(x, t))+

if

(x, t) ∈ Rn+1 − ∩ {t  0}.

(2.68)

54

2 No-Sign Parabolic Obstacle-Type Problems

The function possesses the following properties: (1) the function w and the derivatives De w for e⊥e1 are continuous across (Π ∩ {t  0}) \ Λ(u); (2) the jump of D1 w on (Π # e1 ) > 0;  ∩ {t  0}) \ Λ(u) equals  2 cos (ν, (3) H [w] = 0 on the set (x, t) ∈ Rn+1 ∩ {t  0} \ Π : w(x, t) = 0 . From these facts it is easy to deduce that w+ is caloric in Rn+1 ∩ {t  0}∩{w(x, t) > 0} and that off this set the function w+ is continuously extended by zero. Therefore, by Lemma B.2, w+ is subcaloric in the entire space Rn+1 ∩ {t  0}. Since, by the definition (2.68), w− (x, t) = 0 for x1  0, we see that w− is also subcaloric in Rn+1 ∩ {t  0}. Let t 0 be a point satisfying     t 0 = max t : Rn+ × {t} ∩ Γ (u) = ∅ . We fix a point z0 = (x 0 , t 0 ) ∈ Λ(u) with x10  0. By Lemma A.1, if 0 < r < rj and rj → ∞, then 0  Φ(r, w+ , w− , z0 )  Φ(rj , w+ , w− , z0 )  lim Φ(rj , w+ , w− , z0 ) =: Cν . rj →∞

(2.69)

Observe that the above limit exists because Dw is bounded and the functional Φ, defined by (A.2), is monotone. Next, we consider the parabolic scaling uk (x, t) =

u(rk x + x 0 , rk2 t + t 0 ) rk2

.

By Lemma 2.12, it converges uniformly on the compact subsets of Rn+1 ∩ {t  0} (for a subsequence) to a limit nonnegative function u∞ , which is odd with respect to the hyperplane Π. Moreover, the derivatives Di Dj uk and ∂t uk q converge in Lloc (Rn+1 ∩ {t  0}), q < ∞, to the corresponding derivatives ∞ of u . We define w∞ by (2.68) with u∞ in place of u, and put

wk (x, t) =

⎧ ⎪ ⎪ ⎪ ⎨ Dν uk (x, t)

if

x1  −

x10 , rk

⎪ ⎪ ⎪ ⎩ (Dν uk (x, t))+

if

x1 < −

x10 . rk

2.4 Classification of the Nonnegative Global Solutions

55

Letting k → ∞ (see Lemma 2.12) and using (2.69), we see that for every s>0 Cν = lim Φ(srk , w+ , w− , z0 ) = lim Φ(s, (wk )+ , (wk )− , 0, 0) rk →∞

rk →∞

= Φ(s, (w∞ )+ , (w∞ )− , 0, 0),

(2.70)

i.e., Φ(s, (w∞ )+ , (w∞ )− , 0, 0) is constant for all s > 0. Now we show that Cν cannot be positive. Suppose Cν > 0. Then, by (2.70) and Lemma A.1, each of the sets   S + (s) = x ∈ Rn : Dν u∞ (x, −s 2 ) > 0 ,   S − (s) = x ∈ Rn : Dν u∞ (x, −s 2 ) < 0 , must be one of the half-spaces Rn± . However, w∞  0 in Rn+1 ∩ {t  0} − by (2.68). Therefore, for any s > 0 the set S − (s) must coincide with the halfn+1 space Rn+1 + . This is equivalent to the fact that w∞ < 0 in R+ ∩ {t  0}, n+1 ∞ which contradicts the condition that u  0 in R+ ∩ {t  0}. Thus, we have Cν = 0; consequently, (2.69) implies that either w+ ≡ 0 or n+1 − 0 w ≡ 0 in Rn+1 + ∩ {t  t }. Since u  0 in R+ ∩ {t  0}, the latter case occurs. We have shown that Dν u  0

in

0 Rn+1 + ∩ {t  t }.

If t 0 = 0 then inequality (2.67) is proved. Otherwise, applying Lemma B.4, we again arrive at (2.67). We proceed to show that, in fact, u depends on only one spatial variable, i.e. 2. to prove (2.65). If for all directions e orthogonal to e1 we have De u ≡ 0, then u depends only on x1 and t, and we arrive at (2.65). Therefore, we may assume (for the definiteness) that there is a point (x ∗ , t ∗ ) ∗ ∗ such that De u(x ∗ , t ∗ ) > 0 for some e orthogonal to e1 . Then D−e u(x √ ,t ) < 0, and there exists a > 0 such that Dν u(x ∗ , t ∗ ) < 0 for ν = ae1 − 1 − a 2 e, which contradicts the result of Step 1. . The proof is complete.   + (M) be a nonnegative, not identically zero, degree 2 Lemma 2.17 Let u ∈ P∞ parabolic homogeneous function with respect to the origin, and let (0, 0) ∈ Γ (u). Then

u(x, t) =

x12 2

in

Rn+1 + ∩ {t  0}.

(2.71)

56

2 No-Sign Parabolic Obstacle-Type Problems

Proof First, we observe that due to Lemma 2.16 our global solution u is a one-space dimensional, i.e., the equality (2.65) takes place. In other words, u is a one-space dimensional homogeneous global solution, which does not vanish identically in R2+ ∩ {t  0} and satisfy u(0, t) = 0 for t  0. Then, in view of Lemma C.1, we get the desired representation (2.71).   + (M) be nonnegative, and let z0 = (x 0 , t 0 ) ∈ Γ (u). Then Lemma 2.18 Let u ∈ P∞ the function u is a parabolic homogeneous function of degree 2 with respect to z0 , and

W (r, x 0 , t 0 , u) =

15 =: A f or any 4

r > 0.

(2.72)

Proof We only need to prove (2.72). Then the first statement of our lemma follows immediately (see Sect. A.3 in Appendix A). Due to the scaling property (A.24) and the monotonicity of the Weiss functional we have W (1, 0, 0, u0 )  W (1, 0, 0, ur ) = W (r, x 0 , t 0 , u)  W (1, 0, 0, u∞ ). (2.73) Here u0 and u∞ are blow-up and blow-down limits of the function u at the point z0 respectively. We recall that according to Lemma 2.13 the functions u0 and u∞ are degree 2 parabolic homogeneous with respect to the origin. If x10 > 0 then u0 ∈ P∞ (M). Hence we can apply successively Lemmas 2.13, C.2 and A.5 to the function u0 , which yields the bound W (1, 0, 0, u0 )  A.

(2.74)

+ (M) and (0, 0) ∈ Otherwise we will have that the function u0 belongs to the class P∞

Γ (u0 ). In this case Lemma 2.17 guarantees the exact representation u0 (x, t) = Item (i) from Lemma A.5 now gives W (1, 0, 0, u0 ) = A.

x12 2 .

(2.75)

+ (M) with Similarly, the application of Lemma 2.17 to the function u∞ ∈ P∞ (0, 0) ∈ Γ (u∞ ) and Lemma A.5 provide the identity

W (1, 0, 0, u∞ ) = A.

(2.76)

Finally, combining together the inequalities (2.73)–(2.76) we get the desired result.   + (M) be nonnegative, and let z0 = (x 0 , t 0 ) ∈ Γ (u). Theorem 2.5 Let u ∈ P∞ Then u is independent of t and the variables x3 , . . . , xn . More precisely, for (x, t) ∈

2.5 Geometric Classification of the Global Solutions with No Sign Restrictions

57

0 Rn+1 + ∩ {t  t } we have

x10 = 0

⇒

u(x, t) =

x12 , 2

x10 > 0

⇒

u(x, t) =

((x1 − x10 )+ )2 . 2

Proof Suppose first that x10 = 0. In this case the successive application of Lemmas 2.18 and 2.17 to the function u gives the desired representation. We now turn to the case x10 > 0. By Lemma 2.18 the function u is parabolic homogeneous of degree 2 with respect to z0 . This together with the assumption z0 ∈ Γ (u) guarantees that u(x, t) ≡ 0 in the infinite set {(x, t) : 0 < x1  x10 , t  u(x, t) = u(x + x 0, t + t 0 ) belongs to the t 0 }. From here it follows that the function  + class P∞ (M) and (0, 0) ∈ Γ ( u). It is obvious that  u is parabolic homogeneous of degree 2 with respect to the origin. Therefore, applying Lemma 2.17 to  u, we get  u(x, t) =

x12 2

as x1  0, t  0.

Consequently, for t  t 0 we have ⎧ 0 2 ⎪ ⎨ (x1 − x1 ) , if 2 u(x, t) = ⎪ ⎩ 0, otherwise.

x1  x10 ,

 

This finishes the proof.

2.5 Geometric Classification of the Global Solutions with No Sign Restrictions In this section we consider the global solutions of the obstacle-type problem and prove results similar to those in the previous section, but with no sign restrictions on solutions. + (M) be a degree 2 parabolic homogeneous function with Lemma 2.19 Let u ∈ P∞ respect to the origin, and let (0, 0) ∈ Γ (u). Then for some a ∈ R we have

u(x, t) =

x12 + ax1 x2 2

in Rn+1 + × {t  0}.

(2.77)

Proof Let e be a unit spatial direction orthogonal to e1 . We claim that the function v := De u does not change its sign. To prove this, first we extend v by zero across

58

2 No-Sign Parabolic Obstacle-Type Problems

the plane Π to the entire space Rn+1 × {t  0} and keep the notation v for the extension. From homogeneity of u with respect to the origin it follows that Φ(λ, v+ , v− , 0, 0) = C(e),

(2.78)

where the functional Φ is defined by the formula (A.2). However, in our case the equality (2.78) is possible only if C(e) = 0, see Sect. A.1 in Appendix A. This implies v  0 or v  0 for all points of Rn+1 + × {t  0}. So, we have proved that De u preserves its sign. Since this is true for all spatial directions e orthogonal to e1 , it follows that u(x, t) is two-space dimensional, i.e., in suitable spatial coordinate u(x, t) = u(x1 , x2 , t).

(2.79)

For definiteness, we assume in the rest of the proof that D2 u  0

(2.80)

(otherwise we replace e2 by −e2 ). Further, we consider the case where the interior of Λ(u) is empty. For (x, t) ∈ Rn+1 + ∩ {t  0} we define the function w by the formula w(x, t) = u(x, t) −

x12 . 2

It is easy to see that w is caloric in Rn+1 ∩ {t  0} and has at most quadratic + growth with respect to x and at most linear growth with respect to t. Then we extend w by the odd reflection to the whole space Rn+1 ∩ {t  0} and preserve the notation w for the extended function. By the Liouville theorem (see Lemma B.3 in Appendix B), the function w, and, consequently, the function u, is a polynomial of degree 2. Taking into account the homogeneity of u, the equality (2.79) and the condition u|Π = 0 we get the exact representation u(x, t) =

x12 + ax1 x2 , 2

(x, t) ∈ Rn+1 + ∩ {t  0}

for some constant a. Now we claim that the interior of Λ(u) is always empty. Suppose, towards a contradiction, that we may fix a cylinder Q2r (z0 ) in the interior of Λ(u). Without loss of generality we may suppose that t 0 < 0. In this part our arguments are similar to that of the proof of Theorem B in [SU03]. Due to (2.80) we must have u  0 in G2r (z0 ) := {(x1 , x2 − s, t) : (x1 , x2 , t) ∈ Q2r (z0 ), s  0}.

2.5 Geometric Classification of the Global Solutions with No Sign Restrictions

59

From here one infers that for the smaller set Gr (z0 ) the following holds ∂Ω(u) ∩ Gr (z0 ) = ∅.

(2.81)

Indeed, if there exists z∗ ∈ ∂Ω(u) ∩ Gr (z0 ), then the maximum principle applied to the subcaloric function u in Qr (z∗ ) gives that u ≡ 0 in Qr (z∗ ). Then, by homogeneity of u, it vanishes also in Bλr (λx ∗ ) × {λ2 t ∗ } for any λ > 0. Hence z∗ ∈ / ∂Ω(u). Combining (2.81) and the fact Qr (z0 ) ⊂ Λ(u), we conclude that Gr (z0 ) ⊂ Λ(u). Hence we can translate u in the x2 -direction by considering the functions um (x, t) = u(x1 , x2 −m, t) in Rn+1 + ∩{t  0}. Since u is homogeneous with respect to the origin, each element of {um } satisfies um (λx, λ2 t) = λ2 u(x1 , x2 − λ−1 m, t) = λ2 um/λ (x1 , x2 , t).

(2.82)

In addition, for all x2  x20 we have |u(x, t)|  M(|x1 − x10 |2 + |t − t 0 |), and, hence |um (x, t)|  C(1 + x12 + |t|).

(2.83)

Due to (2.80) and (2.83), the sequence {um } is non-increasing and bounded for any fixed x1 and t. Therefore, by compactness {um } converges to a limit function  u, which is a global solution independent of x2 . It should be mentioned also that  u is parabolic homogeneous of degree 2 function with respect to the origin, as provided by (2.82). In other words,  u is a one-space dimensional homogeneous global solution with Qr (z0 ) ⊂ Λ( u), satisfying  u(0, t) = 0 for t  0. If  u does not vanish identically in R1+ × R− then, in view of Lemma C.1, we get the representation  u(x, t) = (x1 )2 /2. The latter contradicts to the fact Qr (z0 ) ⊂ Λ( u). If  u ≡ 0 in R1+ × R− then due to the monotone convergence of um we may conclude that u  0 in R1+ × R− . Now, taking into account that (0, 0) ∈ Γ (u), we can apply Theorem 2.5 to the function u. This gives the exact representation u(x, t) = (x1 )2 /2, which contradicts to our assumption about the interior of Λ(u).   + (M) and z0 = (x 0 , t 0 ) ∈ Γ (u). Then the function u is a Lemma 2.20 Let u ∈ P∞ parabolic homogeneous function of degree 2 with respect to z0 , and

W (r, x 0 , t 0 , u) =

15 =: A f or any 4

r > 0.

60

2 No-Sign Parabolic Obstacle-Type Problems

Proof The statement is proved along the same lines as Lemma 2.18. The only difference is that we have to use Lemma 2.19 instead of Lemma 2.17.   Proof (Proof of Theorem 2.2) Let us suppose first that x10 = 0. In this case the successive application of Lemmas 2.20 and 2.19 to the function u gives the desired representation. We now turn to the case x10 > 0. By Lemma 2.20 the function u is parabolic homogeneous of degree 2 with respect to z0 . This together with the assumption z0 ∈ Γ (u) guarantees that u(x, t) ≡ 0 in the infinite set {(x, t) : 0 < x1  x10 , t  t 0 }. From here it follows that the function  u(x, t) = u(x + x 0 , t + t 0 ) belongs to + the class P∞ (M) and (0, 0) ∈ Γ ( u). It is obvious that  u is parabolic homogeneous of degree 2 with respect to the origin. Therefore, applying Lemma 2.19 to  u, we get that in some suitable rotated coordinate system that leaves e1 fixed  u has the following representation  u(x, t) =

x12 + ax1 x2 2

as x1  0, t  0.

Consequently, for t  t 0 we have ⎧ 0 2 ⎪ ⎨ (x1 − x1 ) + a(x − x 0 )(x − x 0 ), if 1 2 1 2 2 u(x, t) = ⎪ ⎩ 0, otherwise.

x1  x10 ,

(2.84)

Since Du is a continuous function we can conclude that in (2.84) the parameter a = 0. This finishes the proof.  

2.6 Characterization of the Free Boundary Points Near Π The information about global solutions that we obtained in Theorem 2.2 can be applied to study the behavior of the free boundary near the fixed boundary Π. First we present a preliminary result concerning contact points z0 ∈ Π ∩ Γ (u). We show that the free boundary touches the fixed one in a parabolically tangential manner. Lemma 2.21 For any ε > 0 there exists ρ = ρε > 0 such that if u ∈ P1+ (M) and z0 = (x 0 , t 0 ) ∈ Π ∩ Γ (u) ∩ Q1/2 then Γ (u) ∩ Kε (z0 ) ∩ {0 < x1 < ρε } = ∅.

(2.85)

Here  Kε (z ) := (x, t) : x1 > ε 0

%

 |x 

− (x 0 ) |2

+ |t

− t0|

.

2.6 Characterization of the Free Boundary Points Near Π

61

Proof Suppose, towards a contradiction, that the statement of Lemma 2.21 is not true for some ε > 0. Then there exist functions uj ∈ P1+ (M), points! zj = (! x j ,! tj ) ∈ j j j j Π ∩ Γ (uj ) ∩ Q1/2 , and points z = (x , t ) ∈ Γ (uj ) ∩ Kε (! z ) such that rj := j x1 → 0 as j → ∞. We consider two cases: t j > ! t j , for a subsequence, and t j  ! t j . In the first case let us consider for t  0 the functions vj (x, t) =  We observe that vj x

1 =0

uj (x j − rj e1 + rj x, t j + rj2 t) rj2

.

= 0, (e1 , 0) ∈ Γ (vj ), and



tj (! x j ) − (x j ) tj −!  z := 0, ,− rj rj2

 ∈ Γ (vj ) ∩ Π.

j

zj ) that It follows from inclusions zj ∈ Kε (! |(! x j ) − (x j ) |  ε−1 , rj

0
0 such that if u ∈ P2+ (M) and z0 = (x 0 , t 0 ) ∈ Γ (u) ∩ Q1 with x10  δ0 , then for the balanced energy at the point z0 (see (2.62)) we have ω(x 0 , t 0 , u) =

15 = A. 4

(2.86)

Proof From (2.62) and (A.29) it follows that ω(x 0 , t 0 , u) = W (1, 0, 0, u0 ), where u0 is an arbitrary blow-up limit of the solution u at the point z0 . As we have + (M) or u ∈ mentioned in Sect. 2.3.4, there are only two possibilities: u0 ∈ P∞ 0 P∞ (M). In the first case z0 ∈ Π and Lemma 2.20 immediately provides (2.86). Whereas for the second case Lemmas 2.11 and 2.15 imply either W (1, 0, 0, u0 ) = A (and we are done), or W (1, 0, 0, u0 ) = 2A. Now we claim that ω(x 0 , t 0 , u) = 2A, if δ0 is small enough. Indeed, suppose, towards a contradiction, that there exist sequences uk ∈ P2+ (M) and zk = (x k , t k ) ∈ k k ∗ ∗ ∗ + Γ (uk ) ∩ Q+ 1 such that uk → u, z → z = (x , t ) ∈ Q1 ∩ Π, rk := x1 → 0 as k k k → ∞ and ω(x , t , uk ) = 2A. For (x, t) ∈ Q+ 1/rk we consider the functions vk (x, t) =

uk (rk x + x k − rk e1 , rk2 t + t k ) rk2

.

We observe that the sequence {vk } converges, for a subsequence, to a global solution + (M). It is evident that (e , 0) = (1, 0, . . . , 0) ∈ Γ (v) (see Lemma 2.11). v ∈ P∞ 1 By Lemma 2.20 we have W (ρ, e1 , 0, v) = A.

(2.87)

On the other hand, for arbitrary ρ > 0 elementary computations combined with estimate (2.61) and scaling property (A.29) give 2A = ω(x k , t k , uk )  W1 (ρrk , x k , t k , uk ) + C0 ρrk = W1/rk (ρ, e1 , 0, vk ) + C0 ρrk = W (ρ, e1 , 0, v) + ϑk (ρ) + C0 ρrk .

(2.88)

2.7 Regularity Properties of Solutions

63

Here 1 ϑk (ρ) := 4 ρ

−ρ ˆ



ˆ

2

v2 − v2 |Dvk | − |Dv| + 2(vk − v) + k t 2

2

 ×

−4ρ 2 B1/rk (e1 )

×G(x − e1 , −t)dxdt 1 − 4 ρ

−ρ ˆ

2

−4ρ 2

(2.89)

ˆ Rn \B1/rk (e1 )

 v2 2 |Dv| + 2v + G(x − e1 , −t)dxdt. t

It is evident that for fixed ρ the sequence {ϑk (ρ)} converges to zero as k → ∞. Indeed, the integrand in the first integral on the right-hand side of (2.89) uniformly + (M) provided the converges to zero as k → ∞. Whereas, the property v ∈ P∞ convergence to zero for the second integral. Thus, (2.88) contradicts (2.87) for large k. The proof is complete.  

2.7 Regularity Properties of Solutions Lemma 2.23 For any ε > 0 there exists δ1 = δ1 (n, ε, M) > 0 such that if 0 u ∈ P2+ (M) and z0 = (x 0 , t 0 ) ∈ Γ (u) ∩ Q+ 1/8 ∩ {x1 < δ1 } then for r := x1   2 and φ(x) = (x1 − x10 )+ /2 we have sup |u(x, t) − φ(x)|  εr 2 ,

0 Q+ 8r (z )

sup |Du(x, t) − Dφ(x)|  εr.

0 Q+ 8r (z )

Proof We begin with considering the first inequality. Suppose, towards a contradiction, that it fails. Then there exists a number ε0 > 0 and sequences uj ∈ P2+ (M) j and zj = (x j , t j ) ∈ Γ (uj ) ∩ Q1/8 such that rj := x1  0 and   sup uj (x, t) − φ(x) > ε0 rj2 .

j Q+ 8r (z ) j

We define vj by the formula vj (x, t) =

uj (rj x + x j , rj2 t + t j ) rj2

(2.90)

64

2 No-Sign Parabolic Obstacle-Type Problems

for (x, t) ∈ Q1/rj ∩ {x1 > −1}. Observe that (0, 0) ∈ Γ (vj ) and vj |{x1 =−1} = 0. Moreover, in an appropriate function space the functions vj converge (along a subsequence) to a continuous function v0 satisfying H [v0 ] = χ{v0 >0} v0 = 0

on

in Rn+1 ∩ {x1 > −1} ∩ {t  0} ,   (x, t) ∈ Rn+1 : x1 = −1 and t  0 ,

( |∂t v0 | + |D 2 v0 |)  M.

ess sup Rn+1 ∩{x1 >−1}∩{t 0}

Taking into account the relations (0, 0) ∈ Γ (v0 ) and v0 |x1 =−1 = 0, we get from Theorem 2.2 that v0 ≡ ((x1 )+ )2 /2. Therefore, for all sufficiently large j we have the inequality 2  vj (x, t) − ((x1 )+ )   ε0 . 2 2 Q8 ∩{x1 >−1}

sup

(2.91)

On the other hand, the inequality (2.90) implies    2    uj (rj x + x j , rj2 t + t j ) ((x1 )+ )2  vj (x, t) − ((x1 )+ )  =   − sup     2 2 rj2 Q8 ∩{x1 >−1} Q8 ∩{x1 >−1} sup

2 j  (y1 − x1 )+   uj (y, τ )  > ε0 . = sup  −  2 2 + r 2r j Q (z ) j j 8rj

This contradiction with (2.91) completes the proof of the first inequality. It remains only to observe that the second inequality is proved in the same way as the first one.   Remark 2.8 It should be mentioned that the additional assumption u  0 in Lemma 2.23 allows to prove a more general statement. Namely, let all the 0 assumptions of Lemma 2.23 be satisfied, and let u  0 in Q+ 2 . Then for ρ ∈ x1 , δ1 and the same function φ as in Lemma 2.23 we have sup |u(x, t) − φ(x)|  ερ 2 ,

0 Q+ ρ (z )

sup |Du(x, t) − Dφ(x)|  ερ.

0 Q+ ρ (z )

Lemma 2.24 Let u ∈ P2+ (M), let 0 < ε < 1, and let z0 = (x 0 , t 0 ) ∈ Γ (u) ∩ Q+ 1/8 ∩ {x1 < δ1 }, where δ1 = δ1 (n, ε, M) is the constant occurring in Lemma 2.23.

2.7 Regularity Properties of Solutions

65

Then the following statements hold: (i) there exists a positive number N = N(n) such that for r := x10 and Σr := √ Q7r (z0 ) ∩ {0 < x1 < r(1 − N ε)} we have u(x, t)  0

in Σr .

(2.92)

z) = 0, then (ii) If there exists a point ! z = (! x ,! t) ∈ Σr such that u(! u≡0

in Σr ∩ {t  ! t }.

(2.93)

(iii) The value t ∗ := sup{t : z = (x, t) ∈ Σr and u(z) = 0} satisfies the inequality t ∗  t 0 − ε(2n + 1)r 2 .

(2.94)

0 Proof Suppose that there is a point z(1) = (x (1) , t (1) ) ∈ Q+ 7r (z ) ∩ {x1 < r} such (1) that u(z ) > 0; otherwise we already have (2.92) with N(n) = 0. (1) Then for ρ := r − x1 we deduce from Lemmas 2.6 and 2.23 the inequalities

ρ2  sup u − u(z(1))  sup |u|  εr 2 , 2n + 1 Q+ρ (z(1) ) + 0 Q (z )∩{x1 r (2n + 1)ε. Therefore, for all z = (x, t) ∈ Q+ 7r (z ) √ √ with x1 < r(1 − (2n + 1)ε) we have u(z)  0. Choosing N(n) = 2n + 1 we arrive at (2.92). Finally, with (2.92) at hand, application of the maximum principle to the subcaloric function u implies (2.93), since u(! z) = 0. Now we can prove (2.94). If t ∗ = t 0 then (2.94) is true. Otherwise, there exists ρ > 0 such that for z(1) := (x10 /2, (x 0 ) , t 0 ) we have u < 0 in Qρ (z(1) ) and Qρ (z(1)) ⊂ Σ. It is easy to see that item (iv) from Lemmas 2.6 and 2.23 imply the inequalities ρ2  |u(z(1))|  εr 2 , 2n + 1 which are only possible if & ρ  r ε(2n + 1) holds. The latter inequality is equivalent to (2.94).

66

2 No-Sign Parabolic Obstacle-Type Problems

Lemma 2.25 Let u ∈ P2+ (M) and z0 = (x 0 , t 0 ) ∈ Γ (u). If for some ρ > 0 the cylinder Qρ (x 0 , t 0 + ρ 2 /2) contains only low energy points, then lim

Ω(u)z→z0

∂t u(z) = 0.

Proof First of all, we observe that this lemma concerns the continuity of ∂t u locally at the interior free boundary point z0 ∈ / Π. It means that we should not worry about the presence of Π and, consequently, can follow the arguments from the proof of Lemma 7.7 [CPS04] (see also Theorem 9 [AUS03]). We sketch the details. It suffices to show that the lower limit m :=

lim

Ω(u)z→z0

inf ∂t u(z)

(2.95)

is nonnegative, whereas the corresponding upper limit is nonpositive. Let m be defined by (2.95), and let zk = (x k , t k ) ∈ Ω(u) be a sequence such that k z → z0 as k → ∞ and lim ∂t u(zk ) = m.

k→∞

For each point zk we define the corresponding distance to the free boundary as follows:   dk = sup r > 0 : Qr (zk ) ∩ Γ (u) = ∅ . Clearly, dk → 0 as k → ∞. Consider the functions uk (x, t) =

u(dk x + x k , dk2 t + t k ) dk2

.

We observe that Q1 ⊂ Ω(uk ), and that ∂t uk (0, 0) = ∂t u(x k , t k ) tends to m as k → ∞. Therefore, the uk converge (along a subsequence) to a global solution u0 ∈ P∞ (M) with the following properties: H [u0] = 1

in Q1 ,

∂t u0 (0, 0) = lim ∂t uk (0, 0) = m, k→∞

∂t u0 (x, t)  m

for any (x, t) ∈ Q1 .

The latter inequality follows from the fact that in Q1 the functions ∂t uk converge pointwise to ∂t u0 , and from assumption (2.95).

2.7 Regularity Properties of Solutions

67

Thus, the function ∂t u0 is caloric in Q1 and has a local minimum at the point (0, 0). Consequently, by the maximum principle, ∂t u0 (x, t) ≡ m

in Q1 .

(2.96)

On the other hand, in accordance with our definition of dk , for each k there exists a point (y k , s k ) ∈ Γ (uk ) ∩ ∂  Q1 and a corresponding point (dk y k + x k , dk2 s k + t k ) ∈ Γ (u) ∩ ∂  Qdk (zk ) such that (dk y k + x k , dk2 s k + t k ) → (x 0 , t 0 )

as k → ∞.

(2.97)

Let (y 0 , s 0 ) denote the limit of a subsequence of points (y k , s k ) as k → ∞. Obviously, (y 0 , s 0 ) ∈ Γ (u0 ) ∩ ∂  Q1 . Observe also that the convergence of W1 (r, x, t, u) to A = 15/4 as r  0 is uniform with respect to z = (x, t). This fact can be proved easily by an argument similar to the proof of the Dini theorem, because for r > 0 the functions W1 (r, x, t, u) are continuous with respect to (x, t), the limits function is a constant, and the convergence in question is monotone up to exponentially small terms. In particular, if rj  0 as j → ∞, z˜ j = (x˜ j , t˜j ) ∈ Γ (u) ∩ Qρ (x 0 , t 0 + ρ 2 /2), and z˜ j → z0 = (x 0 , t 0 ), then lim W1 (rj , x˜ j , t˜j , u) = ω(x 0 , t 0 , u) =

j →∞

15 . 4

(2.98)

Next, using (A.29), (2.97) and (2.98), we see that W (r, y 0 , s 0 , u0 ) = lim W1/dk (r, y k , s k , uk ) k→∞

= lim W1 (rdk , dk y k + x k , dk2 s k + t k , u) k→∞

=A

(2.99)

for every r > 0. In particular, u0 is a homogeneous function of degree 2 with respect  to (y 0 , s 0 ) in Rn+1 ∩ t  s 0 . Now, using Lemmas C.2 and A.5 and relation (2.99), we obtain the representation u0 (x, t) =

  2 x − y0 · e + 2

  in Rn+1 ∩ t  s 0 ,

(2.100)

where e is a direction in the x-space Rn . We want to show that (2.96) and (2.100) contradict each other, unless m = 0. Indeed, if s 0 > −1, then for m = 0 representation (2.100) contradicts formula (2.96)

68

2 No-Sign Parabolic Obstacle-Type Problems

in the cylinder B1 ×] − 1, s 0 [. If s 0 = −1, then by (2.96), we have  u0 (x, t) =

 2 x − y0 · e + 2

+ m(t + 1) in Q1 ,

whence H [u0 ] = 1 − m in Q1 . Assuming that m = 0, we arrive at a contradiction with the equality H [u0] = 1 in Q1 obtained above. This shows that lim inf ∂t u(x, t) = 0 as (x, t) → (x 0 , t 0 ). Similarly one can prove that lim sup ∂t u(x, t) = 0, which will conclude the proof of the lemma.   Remark 2.9 Notice that the result of Lemma 2.25 has also been established in [BDM06] for the solution of the one-dimensional parabolic obstacle problem with variable coefficients depending on x and t. Lemma 2.26 Let N0 and Nτ (with τ = 2, . . . , n) be arbitrary constants satisfying 1 , |N0 |  32(2n + 1)M

n 

|Nτ |  1.

(2.101)

τ =2

There exists δ = δ(n, M) > 0 such that if u ∈ P2+ (M) and z0 = (x 0 , t 0 ) ∈ 0 Γ (u) ∩ Q+ 1/8 ∩ {x1 < δ} then for r = x1 we have v := rD1 u + r

n 

Nτ Dτ u + r 2 N0 ∂t u − u  0

in Kr (z0 ),

(2.102)

τ =2 + (z0 ). u  0 in K3r/2

(2.103)

1 and set δ = min{δ0 , δ1 }, where δ0 = δ0 (n, M) and Proof We take ε := 32(2n+1) δ1 = δ1 (n, ε, M) are the constants defined in Lemmas 2.22 and 2.24, respectively. The proof of (2.102) falls naturally into four parts.

Step 1. Consider the function vs := rD1 u − u that is the special case of v with N0 = Nτ = 0, τ = 2, . . . , n. It is easy to see that vs  0 in

Qr/2 (z0 ),

vs  0 in

+ K3r/2 (z0 )

(2.104) ∩ {t  t ∗ },

(2.105)

where t ∗ is the same as in Lemma 2.24. 2 +) Note that the function ψ(x) = ((x1 −r) satisfies for x1  3r the 2 inequality rD1 ψ − ψ  0.

(2.106)

2.7 Regularity Properties of Solutions

69

+ 0 This together with Lemma 2.23 imply the following estimate in K2r (z )

vs  −2εr 2 .

(2.107)

Suppose, towards a contradiction to (2.104), that there exists a point z(1) = (x (1) , t (1) ) ∈ Qr/2 (z0 ) with vs (z(1)) < 0. Then we consider the auxiliary function w(x, t) = vs (x, t) +

|x − x (1) |2 + t (1) − t . 2n + 1

It is evident that w is caloric in Ω(u), continuous in Ω(u), and w(z(1) ) < 0. Hence, applying the maximum principle to the function w in Ω(u) ∩ Qr/2 (z0 ) and taking into account (2.107) together with the estimate w  0 on ∂Ω(u), we get the inequalities −2εr 2 +

 r2 inf w  w(z(1) ) < 0,  wΩ(u)∩∂  Q (z(1) )  r/2 4(2n + 1) Ω(u)∩Qr/2 (z(1) )

1 which are not possible if ε  8(2n+1) . Suppose now that inequality (2.105) is not true in a point z(1) = + (x (1), t (1) ) ∈ K3r/2 (z0 ) ∩ {t  t ∗ }. In this case we repeat the above arguments and take into account that w  0 on the set Π ∩ {t  t ∗ } as well. Step 2. We claim that t ∗ = t 0 (here t ∗ is the parameter occurring in (2.105)). If

t ∗ < t 0 and   int Λ(u) ∩ {t ∗ < t < t 0 } ∩ Kr/4(z0 ) = ∅,

(2.108)

then, in view of item (v) in Lemma 2.15, z0 is a high energy point, which contradicts to Lemma 2.22. Suppose now that t ∗ < t 0 and (2.108) is false. The latter means that there exist a point zˆ = (x, ˆ tˆ) and some cylinder Qρ (ˆz) satisfying Qρ (ˆz) ⊂ Λ(u) ∩ {t ∗ < t < t 0 } ∩ Kr/4(z0 ). From here, using (2.104) we may conclude that u0

in

Q := {|x  − (! x ) | < ρ} ×



 r x1 × { ! t }.  x1  ! t − ρ2 < t  ! 2

70

2 No-Sign Parabolic Obstacle-Type Problems

Since u is a subcaloric, we say that u ≡ 0 in Q . Combining the last equality with (2.93) we get u≡0

in Σr ∩ {t  ! t },

which contradicts to definition of t ∗ . Thus, we have proved that t ∗ = t 0 and, consequently, u≡0

in Σr .

(2.109)

+ Step 3. We claim that all the points z = (x, t) ∈ ∂Ω(u) ∩ K3r/2 (z0 ) are low energy points, i.e. ω(x, t, u) = A. Indeed, inequalities (2.105) and (2.109) + (z0 ), that is guarantee the nonnegativity of the function u on the set K3r/2 the desired inequality (2.103). Hence we have no zero energy points on this set (see item (iii) in Lemma 2.15). + It remains only to note that, in view of Lemma 2.22, the set K3r/2 (z0 ) does not contain the high energy points as well. Step 4. Using Step 3 and Lemma 2.25 we conclude that ∂t u is continuous in + (z0 ). This allows us to prove the statements of the lemma in the K3r/2 full form. Namely, we consider the function

v = rD1 u + r

n 

Nτ Dτ u + r 2 N0 ∂t u − u

τ =2

with constants N0 and Nτ satisfying conditions (2.101). Inequalities (2.101) and (2.106) guarantee for v the estimate v−

r2 8(2n + 1)

in

+ 0 K2r (z ).

Then, repeating the arguments from Step 1 and taking into account the relation t ∗ = t 0 , we finish the proof of (2.102).  

2.8 Regularity Properties of the Free Boundary Lemma 2.27 Let u ∈ P2+ (M), let z0 = (x 0 , t 0 ) ∈ Γ (u) ∩ Q+ 1/8 ∩ {x1 < δ}, where δ = δ(n, M) is the constant occurring in Lemma 2.26, and let r := x10 . + 0 (z ) implies z ∈ Γ (u). Moreover, there exists Then the inclusion z ∈ ∂Ω(u)∩K2r a positive constant C = C(n, M) such that for a cone   %  |2 + C 2 r −2 t 2 : x > |x K := (x, t) ∈ Rn+1 1 +

2.8 Regularity Properties of the Free Boundary

71

and an arbitrary point z ∈ Γ (u) ∩ Kr (z0 ) we have Kr (z0 ) ∩ {z − K } ⊂ Λ(u).

(2.110)

Proof Let us take C = 32(2n + 1)M. The first statement of the lemma was already proved in Step 3 of the proof of Lemma 2.26. Further, from Lemma 2.26 it follows that for an arbitrary unit vector e ∈ K the inequality De u  0 holds in Kr (z0 ). Then, evidently, D−e u  0 in Kr (z0 ). The latter together with the assumptions z ∈ Γ (u) and (2.103) gives the desired inclusion (2.110).   Proof (Proof of Theorem 2.3) First we observe that the estimate for the function u was proved already in Lemma 2.26. To prove the second statement we set r = x10 and define f as follows: f (x  , t) = sup{x1 ∈ [0, 2r] : u(x1 , x  , t) = 0}. We claim that f is a Lipschitz function. Indeed, Lemma 2.27 guarantees for every 0 point z ∈ Γ (u) ∩ Q+ 1/8 ∩ Kr (z ) the existence of standard cone having top at z and lying in Λ(u). This implies the corresponding space-time Lipschitz regularity of Γ (u) ∩ Kr (z0 ). The theorem is proved.   Proof (Proof of Theorem 2.4) The statement of the theorem is an obvious consequence of Theorem 2.3 and Lemmas 2.27, C.3.  

Chapter 3

Boundary Estimates for Solutions of Free Boundary Problems

In this chapter we concentrate only on our first main question about regularity properties of a solution. So, we study here several elliptic and parabolic free boundary problems of types (1.1) and (1.2) and prove in a neighborhood of the given boundary the explicit estimates for higher derivatives of solutions.

3.1 One-Sided Estimates up to the Boundary for Solutions to the Elliptic Obstacle Problem 3.1.1 Statement of the Problem and Main Results In this section we deal with an obstacle problem for the Laplace equation. In the simplest case, it can be formulated as the minimization problem for the functional J (v) =

ˆ

|Dv|2 + 2vg dx D

on a convex constrained set  v ∈ W 1,2 (D) : v  0

a.e. in D;

 v − φ ∈ W01,2 (D) .

Here, D ⊂ Rn denotes a bounded domain, and g and φ are given functions. The function φ is assumed to be nonnegative on ∂D in the sense that φ − ∈ W01,2 (D). We emphasize also that in the case when φ vanishes on some part of ∂D, one cannot exclude that arbitrary neighborhoods of some points x ∗ ∈ ∂D

© Springer Nature Switzerland AG 2018 D. Apushkinskaya, Free Boundary Problems, Lecture Notes in Mathematics 2218, https://doi.org/10.1007/978-3-319-97079-0_3

73

74

3 Boundary Estimates for Solutions of Free Boundary Problems

contain both points of the coincidence set, {x ∈ D : u(x) = 0}, and points of ∂ {x ∈ D : u(x) > 0}. We call the set of such points x ∗ the contact set. The one-sided estimates of the second derivatives D 2 u near the boundary ∂D for solution of the obstacle problem are the main subject of this section. Further we consider a model case where D is the unit half-ball B1+ , the boundary condition φ = 0 is given on the flat part of ∂B1+ , the point x = 0 belongs to the contact set and g ≡ 1. It is easy to see that from the minimality of J it follows that the solution u satisfies (in distributional sense) the equation Δu = χ{u>0}

in B1+ ,

and u = |Du| = 0 in B1+ \ {u > 0}. Definition 3.1 We say a function u ∈ W 2,∞ (BR+ ) belongs to the class PR+ (M, L) if u satisfies: (a) Δu = χ{u>0} in BR+ ; (b) u  0 in BR+ and u = 0 on BR ∩ {x1 =  0};  (c) the set ∂ {u > 0} ∩ BR is the graph x ∈ BR : x1 = f (x  ) of a nonnegative function satisfying the Lipschitz condition with constant L; (d) ess sup(|D 2 u|)  M, BR+

and the equation in (a) is understood in the sense of distributions. Remark 3.1 We require in Definition 3.1 that ∂ {u > 0}∩BR is Lipschitz continuous for simplicity of further explanation only. In [Ura97] it is shown how to get rid of this Lipschitz continuity assumption. Without loss of generality we can assume that L  1. Throughout this section, for u ∈ PR+ (M, L), we will suppose also that     Γ (u) = x ∈ BR : x = f (x  ), x  , f (x  ) > 0 is the free boundary, 0 ∈ Γc (u), where Γc (u) = Γ (u) ∩ {x1 = 0} is the contact set,   Σint (u) = Σ(u) \ Γc (u), Σ(u) = x  ∈ BR ∩ {x1 = 0} : f (x  ) = 0 ,   Ω(u) = x ∈ BR+ : u(x) > 0 ,   Λ(u) = x ∈ BR+ : u(x) = |Du(x)| = 0 . (3.1) Now we can formulate the main results of this section. The first theorem asserts |Dτ u(x)| that there exists a modulus of continuity ϕ characterizing the behavior of x1 near the contact set Γc (u). We emphasize that this modulus of continuity is common to all functions belonging to the class P1+ (M, L).

3.1 One-Sided Estimates up to the Boundary for Solutions to the Elliptic. . .

Theorem 3.1 (Estimates for Mixed Derivatives) x ∗ ∈ Γc (u) ∩ B1/2 There exists a non-decreasing function ϕ :]0; 1/2[→ R+ ,

75

Let u ∈ P1+ (M, L), and let

ϕ(r) → 0 as r → 0,

such that |Dτ u(x)|  ϕ(ρ), x1 x∈Bρ+ (x ∗ ) sup

τ = 2, . . . , n.

(3.2)

In particular, the estimate |D1 (Dτ u(x)) |  ϕ (dist {x, Γc (u)}) ,

τ = 2, . . . , n

(3.3)

holds true for every x ∈ Σint (u) ∩ B1/2 . Theorem 3.2 (Estimates for Pure Second Derivatives) Let u ∈ P1+ (M, L), and let δ be a constant satisfying 0 < δ < π2 . There exists a positive number Cδ such that the estimate inf Dξ ξ u  −Cδ | log r|−β ,

∀x 0 ∈ Γ (u) ∩ B1/8

Br+ (x 0 )

(3.4)

is valid for any direction ξ lying in the cone Kδ . Here Cδ depends only on δ, M and L, β = (2(n − 1))−1 , while r is an arbitrary radius satisfying 0 < r < 1/2.

3.1.2 Estimates for Mixed Derivatives on the Boundary Lemma 3.1 Let u ∈ P1+ (M, L), let x ∗ ∈ Γc (u) ∩ B1/2 , and let δ ∈]0; π/2[. Assume that there exists x 0 ∈ Kδ (x ∗ ) ∩ Γ (u) satisfying |x 0 | < (10L)−1 . Then osc

x∈Br+

Dτ u(x)  (1 − x1

where r = |x 0 | and a constant

1)

1

osc

+ x∈B5Lr

Dτ u(x) , x1

τ = 2, . . . , n,

(3.5)

∈]0; 1[ is determined by n, L and δ.

Proof Without loss of generality we assume that x ∗ = 0. We introduce the notation 1 γ1 = √ , 2 1 + L2

γ2 =

1 − γ1 , 2

βk = γk sin (δ),

γ1 γ3 = √ , n−1

k = 1, 2, 3,

76

3 Boundary Estimates for Solutions of Free Boundary Problems

and set   0 n 0  0  K− (x ) = x ∈ R : x − x < −L|x − (x ) | , 1 1 L   x10 0 , x , . . . , xn0 . ρ = β1 r, ρ = β3 r, z= 2 2 + 0 It is clear that Bρ (z) ⊂ K− L (x )∩B1 ⊂ Λ(u), where the last inclusion follows from the Lipschitz continuity of the function f . In addition, for each x ∈ Bρ (z), we have

x1 

1 0 x − ρ  β2 r. 2 1

(3.6)

Choose a tangential direction τ (2  τ  n) and put m+ = sup

+ x∈B5Lr

Dτ u(x) , x1

−m− = inf

+ x∈B5Lr

Dτ u(x) , x1

ω = m+ + m− = osc

+ x∈B5Lr

Dτ u(x) . x1

We note that the condition Dτ u(x) = 0 for x ∈ Γ (u) implies m+  0 and m−  0. Therefore, at least one of the numbers m+ and m− is not less than ω/2. Let m+  ω/2 for definiteness. Consider the nonnegative function v(x) = m+ x1 − Dτ u(x)   + . Next, we fix an arbitrary point x = 2Lr, x  with |x  |  r, and consider in B5Lr an orthogonal basis e1 , . . . , en such that e1 is directed along the vector z − x. Let the origin be at the point x ∗ = x − ρe1 . The coordinates of points in this basis will be denoted by y1 , . . . , yn . We are going to apply Lemma B.7 to the function v. Let N = Ω(u). Choose a parameter h from the definition of Q so that hρ = ρ + |x − z|. It is clear that the top {y ∈ Rn : |yτ |  ρ, τ = 2, . . . , n, y1 = hρ} is contained in Bρ (z) ⊂ Λ(u) and h  10Lβ3−1 . Since Dτ u(x) = 0 for x ∈ ∂Ω(u) ∩ Q, we have v = m+ x 1 

ω x1 2

on ∂Ω(u) ∩ Q.

On the other hand, inequalities (3.6) show that   inf x1 : x ∈ Ω(u) ∩ Q  β2 r. Hence, v  (ω/2) β2 β3−1 ρ on ∂Ω(u) ∩ Q. Consequently, all the conditions of Lemma B.7 are satisfied.

3.1 One-Sided Estimates up to the Boundary for Solutions to the Elliptic. . .

77

Taking into account the inclusions Bρ (x) ⊂ KL ⊂ Ω(u) provided the Lipschitz continuity of the function f , we arrive at the estimate min v  ωβ4 ρ,

Bρ/2 (x)

β4 =

β2 C(10Lβ3−1 , 1/2), 2β3

(3.7)

where C is the constant from Lemma B.7 with h = 10Lβ3−1 and = 1/2. With the help of estimate (3.7) it is not difficult to obtain the inequality sup

x∈Br+

Dτ u(x)  m+ − x1

1 ω,

1

=

1 2



β3 4L

n−1 .

(3.8)

For this purposes we compare v with the barrier function |x − x|−s − (2Lr)−s w(x) = εβ4  ρ −s − (2Lr)−s 2  in the domain D = Ω(u) ∩ x :

 < |x − x| < 2Lr . It is easy to see that

ρ 2

Δw  0 in

D

for s  n − 2. Moreover, w = 0 for |x − x| = 2Lr and w  v on the sphere |x − x| = ρ/2 if ε  1. Furthermore, one can check that  w(x)∂D∩∂Ω(u)  εβ5ωxn ,

β5 =

1−

2β4 s −s . 4L β3

On the other hand, v = m+ x1  (ω/2) x1 on ∂Ω(u). Thus, setting ε = (1/2) β5−1 , we obtain v(x)  w(x) on ∂D, and, consequently, v(x)  w(x)  in D. In particular, it follows from the last inequality with x = x1 , x  that 1 v(x1 , x )  2 



β3 4L

s+1 ωx1 .

Since x  is an arbitrary point such that |x  |  r, the last inequality with s = n − 2 implies estimate (3.8) under the assumption m+  ω/2. Otherwise, if m+ < ω/2, a similar argument applied to the function v(x) = m− x1 + Dτ u(x),

m− >

ω , 2

78

3 Boundary Estimates for Solutions of Free Boundary Problems

leads to the inequality inf

x∈Br+

Dτ u(x)  −m− + x1

1 ω.

Thus, in all cases the estimate osc

x∈Br+

Dτ u(x)  (1 − x1

1 ) ω,

P1+ (M, L),

x∗

ω = osc

+ x∈B5Lr

Dτ u(x) x1  

is true.

let ∈ Γc (u) ∩ B1/2 , let k  2, and let Lemma 3.2 Let u ∈ −1 r  (10Lk) . Assume that for τ = 2, . . . , n the following conditions hold: osc

+ x∈B5Lkr (x ∗ )

4M Dτ u(x) .  x1 k−1

Then the estimate osc

x∈Br+ (x ∗ )

holds true with

Dτ u(x) Dτ u(x)  (1 − ) osc , + x1 x1 x∈B5Lkr (x ∗ )

(3.9)

∈]0; 1[ depending only on L, n and k.

Proof Fix k  2 and let r  (10Lk)−1 . We have to consider two different cases. Suppose first that   x ∈ Bkr (x ∗ ) : x1 > r ∩ Γ (u) = ∅. (3.10) Then there exists a point x 0 ∈ Γ (u) such that x10 >r and |x 0 − x ∗ | < kr. The latter means that x 0 ∈ Kδ (x ∗ ) ∩ Γ (u) with δ = arcsin k −1 and, consequently, all the assumptions of Lemma 3.1 are satisfied. Applying Lemma 3.1 we get estimate (3.5) with a positive constant 1 = 1 (L, k). It is evident that r < |x 0 − x ∗ | < kr. Thus, condition (3.10) implies estimate (3.9) with = 1 . Otherwise, condition (C.5) takes place and application of Lemma C.5 provides inequality (3.9) with = 0 (n, k).   Now, choosing = min { 1 , 0 } we complete the proof. Proof (Proof of Theorem 3.1) Consider an arbitrary ε ∈]0; 1[, set kε = 1 + 4Mε−1 and θε = 5Lkε , and define ω(ρ) for small positive radius ρ by the formula ω(ρ) :=

osc

x∈Bρ+ (x ∗ )

Dτ u(x) . x1

3.1 One-Sided Estimates up to the Boundary for Solutions to the Elliptic. . .

79

If ω(θε r) > ε for some r  (2θε )−1 , then Lemma 3.2 yields ω(r)  (1 −

ε )ω (θε r) ,

where ε = (L, n, kε ) while is the constant form Lemma 3.2. Therefore, for all r  (2θε )−1 we have the inequality ω(r)  max {ε; (1 −

ε )ω (θε r)} .

(3.11)

Next, we take ρ < (2θε )−1 and choose a parameter s ∗ such that ∗

(2θε )−1  ρ (θε )s  1/2. Applying successively inequality (3.11) to the sequence ρs := ρ (θε )s with s = 0, . . . , s ∗ − 1, we obtain     ∗ ∗ ω(ρ)  max ε; (1 − ε )s ω(ρ (θε )s )  max ε; 2M (2θε ρ)γε , (3.12) where γε = − logθε {1 − ε }. Since Dτ u(x) = 0 on Γ (u) and Γ (u) ∩ Bρ+ (x ∗ ) = ∅, inequality (3.12) yields osc

x∈Bρ+ (x ∗ )

|Dτ u(x)|  ω(ρ). x1

Thus, we have proved inequality (3.2) with the function ϕ(ρ) defined as follows: ϕ(ρ) =

Dτ u(x) . + ∗ ∗ x1 u∈P + (M,L) τ,x x∈Bρ (x ) sup

sup

osc

(3.13)

1

Indeed, it is evident that the function ϕ from (3.13) is non-decreasing and satisfies ϕ(ρ)  ω(ρ) for every ρ > 0. Moreover, inequality (3.12) implies the estimate  ε 1+1/γε 1 ω(ρ)  ε provided ρ  Rε := 25L . The latter means that for arbitrary 2M ε > 0 we have ϕ(ρ)  ε,

∀ρ  Rε ,

i.e., ϕ(ρ) → 0 as ρ → 0. This completes the proof of the first statement of the theorem. It remains only to observe that inequality (3.3) follows immediately from (3.2), since Dτ u are continuous in Ω(u) ∪ Σint (u) and Dτ u(x) = 0 for x ∈ Σint (u).  

80

3 Boundary Estimates for Solutions of Free Boundary Problems

3.1.3 Estimates for Pure Second Derivatives In this section, we extend Caffarelli’s results concerning estimates for the derivatives of De u to the case u ∈ P1+ (M, L).   Lemma 3.3 If u ∈ P1+ (M, L), x 0 ∈ Γ (u), and ρ0 = min x10 , 1 − |x 0 | , then for any direction e ∈ Rn the following estimate holds:  inf Dee u  −2α M

Bρ (x 0 )

ρ ρ0

α ∀ρ  ρ0 ,

,

(3.14)

where α = − log2 (1 − γ0 ) and the constant γ0 ∈]0; 1[ from Lemma B.8 depends only on M and L. Proof Indeed, the function v = Dee u satisfies the equation Δv = 0

in

E := Ω(u)

and v  0 on Γ (u). We apply Lemma B.8 sequentially to v for R = ρm := 2−m ρ0 , kˆ = 0,

Eρm := E ∩ Bρm (x 0 ),

k(R) = km := inf v,

m = 1, 2, . . . ,

Eρm

where we assume that km < 0. As a result, we arrive at the estimate  km  − (1 − γ0 ) M = −M m

ρm ρ0

α .

The last inequality is also true for km  0. This implies (3.14).

 

Remark 3.2 The continuity of Dee u in Ω(u) ∪ Γ (u) is not essential to derive (3.14) provided that estimate (C.3) from Lemma C.4 is known. Indeed, inequality (C.3) implies that, for any k < 0, R < ρ0 , the distance from Γ (u) ∩ BR (x 0 ) to supp (v − k)− is positive and therefore (v−k)− ∈ W 1,2 (ER ). Thus, one can apply Lemma B.8 with the parameters kˆ = 0 and R < ρ0 to the function v = Dee u. Now we intend to obtain an estimate analogous to (C.3) for x ∗ ∈ Γc (u). So, we prove the following generalization of Lemma C.4. Lemma 3.4 Let u ∈ P1 (M, L), and let x ∗ ∈ Γc (u). Assume that for some direction e ∈ Rn and R0 < 1, the following condition holds: Dee u  0

on Σint (u) ∩ BR0 (x ∗ ).

3.1 One-Sided Estimates up to the Boundary for Solutions to the Elliptic. . .

81

Then inf Dee u  −C| log ρ|−β

for ρ  R0 ,

Bρ+ (x ∗ )

where C = C(M, R0 ) and β = 1/ (2(n − 1)). Proof Without loss of generality we assume that x ∗ = 0. Introduce the notation rm = 2−m R0 ,

−Mm = inf Dee u Br+m

for m = 1, 2, . . . ,

M0 = M.

We are going to estimate Mm successively for all m such that Mm > 0. Choose one of such m, take a point x ∈ Brm+1 ∩ Ω(u), and denote the distance d(x) =   dist {x; Γ (u)} by s. It is clear that s  |x|  rm+1 , so that Bs+ (x) ⊂ Ω(u) ∩ Br+m . We claim that for arbitrary θ satisfying 0 < θ < 1/2 the following estimates are valid: √ (3.15) Dee u(x) + Mm  γ1 Mm − γ2 θ θ n−1 , where the positive constants γi , i = 1, 2 depend only on M. If x1 > s, i.e., if the ball Bs (x) does not intersect the hyperplane {x1 = 0}, inequality (3.15) was proved by Caffarelli (see, for example, estimate (3.13) from Chapter 2 [Fri82]). Moreover, if one replaces in the corresponding arguments of [Fri82] the ball Bs (x) by B(1−θ)s (x), then estimate (3.15) can be obtained as well for x1 > (1 − θ ) s. It remains to consider the case x1  (1 − θ ) s. Let us define R=

% s2

− x12

√

 s θ,

 y=

R  ,x , 2

y 0 = (0, x  ),

and note that BR+ (y 0 ) ⊂ Bs+ (x) ⊂ Ω. The function v = Dee u is harmonic in BR+ (y 0 ) and

v ∈ C BR+ (y 0 ) ,

v  0 on BR (y 0 ) ∩ {x1 = 0} .

Applying Lemma B.8 with the parameters kˆ = 0, F = 0, ν = 1, and L0 = 0 to the function v in the domain ER = BR+ (y 0 ) we obtain inf

+ BR/2 (y 0 )

v  (1 − γ0 ) inf v. BR+ (y 0 )

Here γ0 is the absolute constant from Lemma B.8.

(3.16)

82

3 Boundary Estimates for Solutions of Free Boundary Problems

Since inf v  −Mm , inequality (3.16) takes the form BR+ (y 0 )

inf

+ BR/2 (y 0 )

v  −(1 − γ0 )Mm .

(3.17)

It is obvious that, for x1  R/2, inequality (3.17) implies estimate (3.15) with any γ1  γ0 and γ2  0. If R/2  x1  (1 − θ ) s, in view of v(y)  − (1 − γ0 ) Mm , one can apply the Harnack inequality to the nonnegative harmonic function v +Mm in the ball Bx1 (x). As a result, one obtains the chain of inequalities v(x) + Mm 

1 2



R/2 x1

n−1

[v(y) + Mm ]  2−n θ

n−1 2

[v(y) + Mm ]  2−n γ0 θ

n−1 2

Mm .

Hence, estimate (3.15) is also valid with any γ1  2−n γ0 and γ2  0. Thus, (3.15) is proved for all possible cases. Taking into account that x is an arbitrary point of Brm+1 ∩ Ω(u), we conclude from (3.15) that √ −Mm+1  −Mm + γ1 Mm − γ2 θ θ n−1 .   2 , we obtain the recursion inequalities Choosing θ = (1/4) min M −2 ; γ2−2 Mm 2n−1 Mm+1  Mm − γ3 Mm , m = 0, 1, . . . ,   1 −2(n−1) . γ3 = γ1 min M −2(n−1); γ2 2

Hence, one can show by induction that   −β  rm  Dee u(x)  −C log R  0

for any x ∈ Br+m ∩ Ω(u), where C depends only on M and β = 1/(2(n − 1)).

 

Proof (Proof of Theorem 3.2) Fix δ > 0. For an arbitrary unit vector ξ ∈ Kδ we have Dξ ξ u = D1 (D1 u) (ξ1 )2 + 2

n  τ =2

D1 (Dτ u) ξ1 ξτ +

n 

  Dτ Dμ u ξτ ξμ .

τ,μ=2

Taking into account that Δu = 1 in Ω(u) ∪ Σint (u) and that u(0, x  ) = 0 for |x  | < 1 we get   Dτ Dμ u(x) = 0, τ, μ = 2, . . . , n, D1 (D1 u(x)) = 1

3.1 One-Sided Estimates up to the Boundary for Solutions to the Elliptic. . .

83

for x ∈ Σint (u). Consequently, ' Dξ ξ u =

ξ12

1+2

n  τ =2

ξτ D1 (Dτ u) ξ1

( on Σint (u).

(3.18)

Next, consider Rδ  1/4 satisfying ϕ (Rδ )  β tan (δ), where ϕ is the function from Lemma 3.1. Combining (3.3) and (3.18) we see that the inequality Dξ ξ u(x) 

ξ12

) * 2(n − 1) ϕ (Rδ )  0. 1− tan (δ)

(3.19)

holds true for all x ∈ Σint ∩ B1/2 lying in Rδ -neighborhood of Γc (u). Thanks to estimate (3.19) we may apply Lemma 3.4 and get inf Dξ ξ u  −Cδ | log r|−β ,

Br+ (x ∗ )

∀x ∗ ∈ Γc (u) ∩ B1/4 .

(3.20)

Note that (3.20) is exactly the desired estimate (3.4) for the case when x 0 = x ∗ ∈ Γc (u) ∩ B1/4 . It remains only to prove (3.4) for x 0 ∈ Γ (u) ∩ B1/8 . With (3.20) at hands we see that all the assumptions of Lemma C.6 are satisfied. So, application of Lemma C.6 completes the proof.   Theorem 3.2 has the following implication Corollary 3.1 Let u ∈ P1+ (M, L) and let x ∗ ∈ Γc (u). For any δ ∈]0; π/2[ there exists R > 0 such that Kδ (x ∗ ) ∩ BR (x ∗ ) ⊂ Ω(u). Remark 3.3 The assertion of the corollary means that the function f is differentiable at x ∗ ∈ Γc (u) and |Df (x ∗ )| = 0. Proof Without loss of generality, assume that x ∗ = 0. Assume also that the assertion of the corollary fails for some δ ∈]0; π/2[. This means that there exists a sequence x m ∈ Kδ ∩ Γ (u) such that rm := |x m | → 0 as m → ∞. Consider the sequence of functions um (x) =

u(rm x) , 2 rm

84

3 Boundary Estimates for Solutions of Free Boundary Problems

+ (M, L) and choose a subsequence converging uniformly to some function u0 ∈ P∞ n on compact subsets of R ∩ {x1  0}. It is easy to see that 0 ∈ Γc (u0 ). Since um (x m /|x m |) = 0, there exists a point

x 0 ∈ Kδ ∩ Γ (u0 ),

|x 0 | = 1.

(3.21)

By Theorem 3.2, for any direction ξ lying in Kδ we have the estimate inf Dξ ξ um (y)  −Cδ | log (rm R)|−β ,

BR+ (x 0 )

(3.22)

where the positive constants Cδ and β are the same as in Theorem 3.2, while R is any radius not exceeding (8rm )−1 . Moreover, we observe that sup |D 2 um |  M,

sup |um | 

BR+ (x 0 )

BR+ (x 0 )

MR 2 , 2

(3.23)

where R is the same as in (3.22). If x ∈ Ω(u0 ) then for sufficiently large m we claim that a ball Bε (x) ⊂ Ω(um ) and Δum (x) = 1 in Bε (x). Therefore, D 2 um ⇒ D 2 u0

in

Bε (x),

∀ε < ε,

and, consequently, Δu0 = 1 in Bε (x). Estimates (3.22) and (3.23) show that sup |D 2 u0 |  M, Rn+

inf Dξ ξ u0  0,

BR+ (x 0 )

∀R > 0,

∀ξ ∈ Kδ .

Taking into account that δ is an arbitrary parameter satisfying δ ∈]0; π/2[, we conclude that the function u0 is convex in Rn+ . Since u0 (0, x  ) = 0 for all x  ∈ Rn−1 and u0 (x 0 ) = 0, the convex function u0 also vanishes in the whole strip {0 < x1  1}. Thus,   Λ(u0 ) ⊃ x ∈ Rn : 0 < x1  1 .

(3.24)

On the other hand, conditions (a) and (c) from Definition 3.1 combined with the first assumption in (3.1) imply that Karctan L ∩ B1+ in contained in Ω(um ) for any m. As in [Fri82] (see Lemma 4.1, Chapter 2), one can prove that Ω(u0) contains the set of all limit points of Ω(um ) and, consequently, Karctan L ∩ B1+ ⊂ Ω(u0 ).

(3.25)

3.2 Boundary Estimates for Solutions to the Two-Phase Elliptic Problem

85

Put E = Karctan L ∩ {x ∈ Rn : 0 < x1  1}. It is easy to see that the interior of E is nonempty. It follows from (3.24) and (3.25) that E is contained in the set Λ(u0 ) ∩ Ω(u0 ) that has no interior points. The contradiction completes the proof.  

3.2 Boundary Estimates for Solutions to the Two-Phase Elliptic Problem 3.2.1 Statement of the Problem and Main Result In this section we study the regularity properties of solutions of the following twophase obstacle problem: Δu = λ+ χ{u>0} − λ− χ{u 0, and Eq. (3.26) is understood in the weak (distributional) sense. The Dirichlet data ϕ is supposed to satisfy the following conditions: ϕ is a W 3,∞ -function, ∃ L > 0 such that |D  ϕ(x)|  L|ϕ(x)|2/3

(3.28) ∀x ∈ π1 .

(3.29)

Remark 3.4 Condition (3.29) means that the function ϕ has zeros of sufficiently high order. More precisely, the function |ϕ(x)|1/3 must satisfy the Lipschitz condition. Remark 3.5 Throughout this section we will write C(ϕ) to indicate that C is defined by the Sobolev-norms of ϕ. For a C 1 -function u defined in B1+ , we introduce the following sets: Λ(u) = {x ∈ B1+ : u(x) = |Du(x)| = 0}; Γ (u) = ∂{x ∈ B1+ : u(x) = 0} ∩ B1+ is the free boundary. We emphasize that in the two-phase case we do not have the property that the gradient vanishes on the free boundary, as it was in the classical one-phase case; this causes difficulties. Therefore, we will distinguish the following parts of Γ : Γ 0 (u) = Γ (u) ∩ Λ(u);

Γ ∗ (u) = Γ (u) \ Γ 0 (u).

We observe that Γ ∗ (u) is C 1,α -surface for any α < 1.

86

3 Boundary Estimates for Solutions of Free Boundary Problems

From now on we suppose that sup |u|  M. Together with the assumptions (3.28) B1+

it provides for any δ ∈ (0, 1) the following estimates for u: D 2 u q,B +  N1 (q, M, δ, ϕ) 1−δ

∀q < ∞,

sup |Du|  N2 (M, δ, ϕ),

(3.31)

+ B1−δ

|Du(x) − Du(y)|  N3 (α, M, δ, ϕ) |x − y|α

(3.30)

∀α ∈ (0, 1).

(3.32)

Observe that the constants N1 − N3 depend on W 2,∞ -norm of ϕ. The L∞ -estimates of the second derivatives D 2 u near the boundary π1 for solutions of the two-phase problem (3.26)–(3.27) are of main interest of this section. Now we can state the result. Theorem 3.3 Let u be a solution of the problem (3.26)–(3.27), with a function ϕ satisfying the assumptions (3.28) and (3.29). Suppose also that sup |u|  M. B1+

Then for any δ ∈ (0, 1) there exists a positive constant C completely defined by n, M, λ± , δ, L and by the norm of ϕ in the Sobolev space W 3,∞ (π1 ) such that ess sup |D 2 u|  C. + B1−δ

3.2.2 Estimates of the Tangential Gradient Near the Boundary Lemma 3.5 Let u be a solution of Eq. (3.26), and let e be a direction in Rn . Then for x ∈ B1+ \ Λ(u) we have (i) Δ(De u(x)) = (λ+ + λ− )

De u(x) n−1 ∗ H Γ (u), |Du(x)|

(ii) Δ|u(x)| = λ+ χ{u>0} + λ− χ{u0}∪{u0}

{u>0}

ˆ

{u w(x)}. According to the above arguments A ± have no intersections with Λ(u). If we show that A ± are empty then the proof of (3.34) is complete. Suppose, towards a contradiction, that at least one of the sets A ± is non-empty. It is obvious that an appropriate choice of the constants N4 and N5 guarantees the inequality v±  w

on ∂Qδ .

(3.36)

We emphasize also that assumption (3.28) provides the estimate sup |Δ(Dτ ϕ)| Qδ

 N6 , whereas the assumptions (3.28) and (3.29) guarantee sup Δ|ϕ|  N7 , where Qδ

the constants N6 and N7 are defined by the W 3,∞ -norm and W 2,∞ -norm of ϕ, respectively. Next, the direct computation in combination with the above estimates for Δ(Dτ ϕ) and Δ|ϕ|, and the equalities from Lemma 3.5 yield  N4 Δ(v ± − w)D ±  −N6 − N7 + − 2nN5 + σ ± H n−1 Γ ∗ (u) ∩ A ± , δ where the measure densities σ ± are defined by the formula σ ± (x) = 2|Du(x)| ± λ

Dτ u(x) , |Du(x)|

λ := λ+ + λ− .

We claim that σ ±  0 on Γ ∗ (u) ∩ A ± , respectively. Indeed, it is suffices to show that for x ∈ Γ ∗ (u) ∩ A ± we have  N4 2|Du(x)|2 + λ ±Dτ ϕ(x  ) + |ϕ(x  )| + √ x1  0. 2 δ

(3.37)

Suppose that 2|D1 u(x)|2 < λ|Dτ ϕ(x  )|;

(3.38)

otherwise (3.37) is proved. Arguing in the same way as in deriving (3.35) we get the estimate 

ˆx1

|ϕ(x )| 



ˆx1

|D1 u(t, x )|dt  0

 N3 (x1 )

|D1 u(x1 , x  ) − D1 u(t, x  )|dt + |D1 u(x)|x1

0 3/2

+ |D1 u(x)|x1 .

(3.39)

3.2 Boundary Estimates for Solutions to the Two-Phase Elliptic Problem

89

If N3 (x1 )3/2 < |D1 u(x)|x1 then inequalities (3.29), (3.38) and (3.39) imply & √ |ϕ(x  )|  2|D1 u(x)| x1 < 2 λ|Dτ ϕ(x  )| x1  2 λL|ϕ(x  )|1/3 x1 , √ and, consequently, |Dτ ϕ(x  )|  L|ϕ(x  )|2/3  2L λL x1 . From here, increasing N4 if it is necessary, we arrive at (3.37). In the other case, i.e., if |D1 u(x)|x1  N3 (x1 )3/2 , inequalities (3.29) and (3.39) guarantee that |Dτ ϕ(x  )|  L|ϕ(x  )|2/3  (2N3 )2/3 L x1 . Again, increasing N4 if it is necessary, we arrive at (3.37). Now we are able to conclude that  N4 − 2nN5  0, Δ(v ± − w)A ±  −N6 − N7 + δ

(3.40)

provided by the choice of N4 large enough. Thanks to (3.36) and (3.40) we can apply the comparison principle to the functions v ± and w on the sets A ± , respectively, and deduce the inequalities v ± (x)  w(x)

A ± = Qδ ∩ {x : v ± (x) > w(x)},

in

which give the desired contradiction with our assumption that A ± = ∅ and complete the proof.  

3.2.3 Boundary Estimates of the Second Derivatives Lemma 3.7 Let the assumptions of Theorem 3.3 hold, let an arbitrary δ ∈ (0, 1) + be fixed, and let x 0 be an arbitrary point in B1−δ . Then ˆ

1 R2

BR

(x 0 )

|D 2 u(x)|2 dx  Cδ , |x − x 0 |n−2

(3.41)

where R is defined by the formula  R :=

x10 > δ/2

δ/2,

if

x10 /2,

otherwise,

(3.42)

and Cδ depends on the same arguments as the constant Nδ/2 from Lemma 3.6.

90

3 Boundary Estimates for Solutions of Free Boundary Problems

Proof First of all, we observe that it is enough to show that ˆ

1 R2

BR

(x 0 )

|D(Dτ u)|2 dx  Cδ , |x − x 0 |n−2

(3.43)

for any tangential direction τ , since we can find the derivative D1 D1 u from Eq. (3.26). Each of the derivatives Dτ u, τ = 2, . . . , n, satisfies the integral identity ˆ

ˆ D(Dτ u(x))Dη(x)dx = B1+

∀η ∈ W01,2 (B1+ ),

f Dτ η(x)dx,

(3.44)

B1+

+ where f := λ+ χ{u>0} −λ− χ{u δ/2 we note that similar to (3.41) estimate 4 δ2

ˆ Bδ/2 (x 0 )

|D 2 u(x)|2 dx  Cδ |x − x 0 |n−2  

follows easily from the Hölder inequality and (3.30).

∈ {u > 0} ∪ {u < 0} Proof (Proof of Theorem 3.3) Let δ ∈ (0, 1) be fixed, let Du(x 0 ) 0 n with |x | < 1 − δ, let ν = |Du(x 0 )| , and let a direction e ∈ R be orthogonal to ν. x0

Since De u(x 0 ) = 0, it follows that

C(n)|D(De u)(x 0 )|4  lim F (r, x 0 , (De u)+ , (De u)− ), r→0

92

3 Boundary Estimates for Solutions of Free Boundary Problems

where the functional F is defined by the formula (A.20) (see Appendix A). On the other hand, according to Lemma A.4, we have the inequality F (r, x 0 , (De u)+ , (De u)− )  F (R, x 0 , (De u)+ , (De u)− ), where R is defined by formula (3.42). Application of Lemma 3.7 enable us to estimate the right-hand side of the last relation by the constant Cδ2 . This means that we obtained the estimate of all the derivatives D(De u)(x 0 ) with e ⊥ ν. It is evident that the derivative Dν Dν u(x 0 ) can be now estimated from Eq. (3.26). Since the Lebesgue measure of Γ (u) is zero (see [Wei01]), it remains only to note that the obtained estimate of the second derivatives at the point x 0 does not depend on dist(x 0 , Γ (u)), as well as on x10 . This finishes the proof.  

3.3 Estimates Near the Given Boundary of the Second-Order Derivatives for Solutions to the Two-Phase Parabolic Problem 3.3.1 Statement of the Problem and Main Result We consider a weak solution of the two-phase parabolic obstacle problem H [u] = λ+ χ{u>0} − λ− χ{u 0. We suppose that a given function ϕ depends only on space variables and satisfies the following conditions: D 3 ϕ ∈ L∞ (Π1 ), ∃L > 0 such that |D  ϕ(x)|  L|ϕ(x)|2/3

(3.47) ∀(x, t) ∈ Π1 .

(3.48)

We suppose also that sup |u|  M with M  1. Together with (3.47) it provides for Q+ 1

any δ ∈ (0, 1) the following estimates for u: ∂t u q,Q+ + D 2 u q,Q+  N1 (q, M, δ, ϕ) 1−δ

1−δ

∀q < ∞,

sup |Du|  N2 (M, δ, ϕ).

(3.50)

Q+ 1−δ

|Du(x, t) − Du(y, t ∗ )|  N3 (α, M, δ, ϕ) |x − y|α + |t − t ∗ |α/2

(3.49)

∀α ∈ (0, 1).

(3.51)

3.3 Estimates Near the Given Boundary of the Second-Order Derivatives

93

For a Cx1 ∩ Ct0 -function u defined in Q+ 1 we introduce the set:   Λ(u) = (x, t) ∈ Q+ 1 : u(x, t) = |Du(x, t)| = 0 . Here we extend the results of the Sect. 3.2 to the parabolic case. Theorem 3.4 Let u be a solution of the problem (3.45)–(3.46) with a function ϕ satisfying the assumptions (3.47) and (3.48). Suppose also that sup |u|  M. Q+ 1

Then for any δ ∈ (0, 1/4) there exists a positive constant C completely defined by n, M, λ± , δ, L, and by the Sobolev’s norm of ϕ such that ess sup |D 2 u|  C. Q+ 1−δ

3.3.2 Lipschitz Estimate of the Normal Derivative at the Boundary Points Lemma 3.8 Let u be a solution of Eq. (3.45), and let e be a direction in Rn . Then, in Q+ 1 \ Λ(u) we have   De u n H {u = 0, |Du| = 0}, (i) H De u = (λ+ + λ− ) |∇u|   |Du|2 n H {u = 0, |Du| = 0}. (ii) H |u| = λ+ χ{u>0} + λ− χ{u0}

−

ˆ

De ηdz + λ

{u>0}

η cos (# γ , e) dH n + λ−

ˆ

∂{u 0 and each t ∈ (−(1 − δ)2 , 0] we have the estimate |D1 u(0, x  , t) − D1 u(0, y  , t)|  Nδ |x  − y  |

∀x  , y  ∈ Π1−δ (t),

(3.54)

with the same constant Nδ as in Lemma 3.9. Proof If we have the existence of the second derivatives D  (D1 u) on the surface Π1−δ , then Lemma 3.9 immediately guarantees the boundness of them. However, the derivatives D  (D1 u) are not defined on Π1−δ . By this reason we have to consider instead of u its mollifier uε with respect to x  -variables. It is easy to see that inequality (3.52) preserves with the same constant Nδ , if we replace in (3.52) the derivative Dτ u by Dτ uε and Dτ ϕ by Dτ ϕε , respectively. In other words, from (3.52) it follows that |D  (D1 uε )|  Nδ

in

Q+ 1−δ .

The latter inequality means that for t ∈ (−(1 − δ)2 , 0] and x  , y  ∈ Π1−δ (t) we have, in fact, the estimate |D1 uε (0, x  , t) − D1 uε (0, y  , t)|  Nδ |x  − y  |.

(3.55)

Now, letting ε → 0, we get from (3.55) the desired estimate (3.54).

 

3.3.3 Boundary Estimates of the Second Derivatives Lemma 3.11 Let the assumptions of Theorem 3.4 hold, let an arbitrary δ ∈ (0, 1/4) be fixed, and let z0 = (x 0 , t 0 ) be an arbitrary point on Π1−δ . Then for any direction e ∈ Rn and a cylinder Qr (z0 ) ⊂ Q1−δ we have osc De u  Cδ r,

0 Q+ r (z )

(3.56)

where Cδ depends on the same arguments as the constant Nδ from Lemma 3.9. Proof The proof will be divided into three steps. Step 1. For almost all t ∈ (−(1−2δ)2, 0) the function u(·, t) can be regarded as a solution of an elliptic equation Δu(x, t) = F (x) ≡ λ+ χ{u>0} − λ− χ{u0} − λ− χ{u 0. Observe that

3.4 Uniform Estimates Near the Initial State

99

Eq. (3.68) is understood in distributional sense. We suppose that a given function ψ satisfies ψ ∈ C 1,1 (B10 ) .

(3.70)

We suppose also that sup |u|  M with M  1. Q10

Remark 3.7 Throughout this section we will write C(ψ) to indicate C is defined by the sum D 2 ψ ∞,B10 + ψ ∞,B10 . For a Cx1 ∩ Ct0 -function u defined in Q10 we introduce the following sets: Λ(u) = {(x, t) ∈ Q10 : u(x, t) = |Du(x, t)| = 0} ; Γ (u) = ∂ {(x, t) ∈ Q10 : u(x, t) = 0} ∩ Q10

is the free boundary.

Similarly to Sect. 3.2 we note that in the two-phase case we do not have the property that the gradient vanishes on the free boundary, as it was in the classical one-phase case; this causes difficulties. Therefore, we will distinguish the following parts of Γ : Γ 0 (u) = Γ (u) ∩ Λ(u),

Γ ∗ (u) = Γ (u) \ Γ 0 (u).

From [SUW09] it follows that the set Γ ∗ (u) can be locally described as x1 = f (x2 , . . . , xn , t) in some suitable rotated coordinate system in Rn with f ∈ C 1,α for any α < 1. The main result of this section is Theorem 3.5 Let u be a weak solution of (3.68)–(3.69) with a function ψ satisfying the assumption (3.70). Suppose also that sup |u|  M. Q10

Then there exists a positive constant c completely defined by n, M, λ± , and ψ such that   ess sup |D 2 u| + |∂t u|  C. Q1

Remark 3.8 The cylinder Q10 is chosen only for simplicity. In fact, the problem (3.68)–(3.69) can be treated in Q1+δ for arbitrary δ > 0. In this case, the constant C in Theorem 3.5 will also depend on δ. Remark 3.9 Let the assumptions of Theorem 3.5 hold. From the general parabolic theory (see, for example, [LSU67]) it follows that u and Du are continuous in Q 9 . Remark 3.10 From Theorem 3.5 and the well-known interpolation theory it follows 1,1/2 that Du ∈ Cx,t (Q1 ).

100

3 Boundary Estimates for Solutions of Free Boundary Problems

3.4.2 Estimate of the Time Derivative For ε > 0 we consider the regularized problem H [uε ] = f ε (uε )

Q9 ,

in

(3.71)

uε = u

on ∂B9 ×]0, 1],

(3.72)

u = ψε

on B9 × {0} .

(3.73)

ε

Here u satisfies (3.68)–(3.69), f ε is a smooth non-decreasing function such that f ε (s) = λ+ as s  ε and f ε (s) = −λ− as s  ε; while ψε is a mollifier of ψ with the radius depending on the distance to ∂B9 such that sup |ψ − ψε |  ε. B9

By the parabolic theory, for each ε > 0, the regularized problem (3.71)–(3.73) has a solution uε with Duε and ∂t uε belonging to L2 (Q9 ). Moreover, the functions uε are continuous in Q 9 and smooth in the closure of the cylinder Q9−δ for arbitrary small δ > 0. Lemma 3.12 Let ε > 0, let u satisfy (3.68)–(3.69), and let uε be a solution of (3.71)–(3.73). Then sup |uε − u|  ε.

(3.74)

Q9

 Proof Setting w = uε − u we observe that (w − ε)+ ∂  Q = 0. Then Eqs. (3.71) 9 and (3.68) together with integration by parts provide for arbitrary t ∈]0, 1] the following identity ¨



¨



f (u) − f (u ) (w − ε)+ dz = ε

ε

B9 ×]0,t ]

−H [w] (w − ε)+ dz B9 ×]0,t ]

1 = 2

ˆ

t ¨  |Dw|2 dz. [(w − ε)+ ] dx  + 2

B9

{w>ε}

(3.75) Taking into account the relations f (u)−f ε (uε )  0 on the set {uε > ε}∪{u < 0} and (w − ε)+ = 0 on the set {uε  ε}∩{u  0}, we conclude that the left-hand side of identity (3.75) is nonpositive. The latter means that sup w  ε. Q9

(3.76)

3.4 Uniform Estimates Near the Initial State

101

Replacing in identity (3.75) the term (w − ε)+ by (w + ε)− and repeating the above arguments we end up with inf w  −ε.

(3.77)

Q9

Combination inequalities (3.76) and (3.77) finishes the proof. We observe also that   ∂t uε t =0 = −f ε (uε )t =0 + Δψε ,

x ∈ B9−δ .

 Thanks to condition (3.70) we may conclude that ∂t uε t =0 are bounded in B9−δ uniformly with respect to ε. With the estimate (3.74) at hands it is easy to check that ∂t uε 2,Q8  C, and the latter inequality is also uniform with respect to ε. Hence we can easily deduce the following result. Lemma 3.13 For each small δ > 0 the uniform estimate sup |∂t uε |  N1 (n, M, λ± , ψ, δ)

Q8−δ

(3.78)

holds true for solutions uε of the regularized problem (3.71)–(3.73). Proof We set v = ∂t uε . It is easy to see that v ± are subcaloric in Q9 . Now we may apply the well-known parabolic estimate (see Lemma B.10 from Appendix B) and get  ±

sup v  N

Q8−δ

Q8

 2 v ± (z)dz + sup v ± , B8 ×{0}

 

which implies the desired inequality (3.78).

Remark 3.11 By virtue of Lemma 3.12, solutions uε of the regularized problem (3.71)–(3.73) converge to u as ε → 0 uniformly in Q9 . Remark 3.12 It follows from Lemma 3.13 and Remark 3.11 that the estimate sup |∂t u|  N1 (n, M, λ± , ψ, 1) Q7

(3.79)

holds true for a function u satisfying (3.68)–(3.69). Here N1 is the same constant as in Lemma 3.13.

102

3 Boundary Estimates for Solutions of Free Boundary Problems

3.4.3 Estimates of the Second Derivatives 0 0 0 Lemma 3.14 Let the √ assumptions of Theorem 3.5 hold, let z = (x , t ) be a point 0 in Q1 , and let R = t . Then there exists a positive constant N2 completely defined by the values of n, M, λ± , and the norms of ψ such that

sup |Du(·, t) − Dψ(·)|  N2 R

for t ∈]0, t 0 ].

(3.80)

B2R (x 0 )

Remark 3.13 It is evident that B6R (x 0 ) ⊂ B7 . Proof First, we observe that the assumption (3.70) implies Δψ ∈ L∞ (B10 ). Therefore, for almost all t ∈]0, 1] the difference u(·, t) − ψ(·) can be regarded in B7 as a solution of an elliptic equation Δ (u(x, t) − ψ(x)) = F (x) ≡ f (u(x, t)) + ∂t u(x, t) − Δψ(x).

(3.81)

Due to estimate (3.79), ess sup |F | is bounded by the known constant. Q7

Thus, for a test function η ∈ W01,2 (B6R (x 0 )) we have the integral identity ˆ ˆ D(u − ψ)Dηdx = − F ηdx. (3.82) B6R (x 0 )

B6R (x 0 )

2 , where ς (x) := ξ We set in (3.82) η = (u − ψ) ς3R 3R 3R,x 0 (x) is a standard timeindependent cut-off function (see Notation). Then, after integrating by parts and subsequent application of Young’s inequality, identity (3.82) takes the form ˆ ˆ 2 2 2 |D(u − ψ)| ς3R dx  C1 (u − ψ)ς3R dx B6R (x 0 )

B6R (x 0 )

ˆ

+C2

(u − ψ)2 |Dς3R |2 dx.

(3.83)

B6R (x 0 )

It is easy to see that inequality (3.79) implies the estimate sup B6R

(x 0 )×]0,t 0 ]

|u − ψ|  CR 2 .

Putting together (3.83) and the above inequality we arrive at ˆ |D(u(x, t) − ψ(x))|2 dx  N2 R n+2 , t ∈ [0, t 0 ]. B3R (x 0 )

(3.84)

3.4 Uniform Estimates Near the Initial State

103

Now we observe that for almost all t ∈]0, 1] and for any direction e ∈ Rn the difference De u − De ψ may be considered as a weak solution of the equation Δ (De u(x, t) − De ψ(x)) = −De F (x) in B7 . The well known results (see Lemma B.11 from Appendix B) applied to the difference De u − De ψ yield the inequality + , sup |De u − De ψ|  C , -

B2R (x 0 )

|De u − De ψ|2 dx + cR sup |F |. B3R (x 0 )

B3R (x 0 )

Combining the last inequality with the estimate (3.84), we get (3.80) and finish the proof.   Lemma 3.15 Let u be a solution of Eq. (3.68), and let e be a direction in Rn . Then   H (De u)±  0 in

Q10 ,

(3.85)

where the inequalities (3.85) are understood in the sense of distributions. Proof Since De u is continuous, the result follows immediately from Lemma 3.8 (see Item (i)).   Lemma 3.16 Let the assumptions of Lemma 3.14 hold. Then ˆ 0

R2

ˆ Rn

|D 2 u(x, t)|2 ςR2 (x)G(x − x 0 , t 0 − t)dxdt  N3 (n, M, λ± , ψ)R 2 , (3.86)

where ςR (x) := ξR/2,x 0 (x) is a standard time-independent cut-off function (see Notation and Conventions) and the heat kernel G is defined by the formula (A.1) from Appendix A. Proof Suppose that e is an arbitrary direction in Rn if Du(z0 ) = 0 and e ⊥ ν, where ν = Du(z0 )/|Du(z0 )|, otherwise. From (3.70), (3.80) and our choice of e it follows that sup |De ψ|  CR. B2R (x 0 )

(3.87)

104

3 Boundary Estimates for Solutions of Free Boundary Problems

According to Lemma 3.15, the functions v = (De u)± are subcaloric in Q2 , i.e., H [v]  0 in the sense of distributions. Since |Dv|2 + vH [v] = 12 H [v 2 ] we have ˆR 2 ˆ | Dv(x, t)|2 ςR2 (x)G(x − x 0 , t 0 − t)dxdt 0 BR (x 0 )

1  2

ˆR 2 ˆ H [v 2 (x, t)]ςR2 (x)G(x − x 0 , t 0 − t)dxdt.

(3.88)

0 BR (x 0 )

After successive integration the right-hand side of (3.88) by parts we get ˆ

ˆR2 ˆ



|Dv|2 ςR2 Gdxdt  − 0 BR (x 0 )

BR (x 0 )

ˆR2 ˆ +

R 2  v2 2 ςR G dx  2 0 v2 2 ς [∂t G + ΔG] dxdt 2 R

0 BR (x 0 )

ˆR2 ˆ +

  v 2 2ςR DςR DG + G|DςR |2 + GςR ΔςR dxdt

0 BR (x 0 )

=: I1 + I2 + I3 .

It is evident that due to (3.69) we have ˆ I1 

|De ψ(x)|2 2 ςR (x)G(x − x 0 , t 0 )dx  CR 2 , 2

BR (x 0 )

where the last inequality provided by (3.87). Taking into account the relation ∂t G + ΔG = ∂t G(x − x 0 , t 0 − t) + ΔG(x − x 0 , t 0 − t) = 0 for t < t 0 , we conclude that I2 = 0. 2 we observe  Finally,  that the integral in I3 is really taken over the set E =]0, R ]× 0 0 BR (x ) \ BR/2 (x ) . Therefore, in E we have the following estimates for functions

3.4 Uniform Estimates Near the Initial State

105

involved into I3 −

R2

e 16(R2 −t) |G(x − x 0 , t 0 − t)|  C 2  CR −n ; (R − t)n/2 |DG(x − x 0 , t 0 − t)DςR (x)|  C|G(x − x 0 , t 0 − t)| −

|x − x 0 | R(R 2 − t)

R2

e 16(R2 −t) C 2  CR −n−2 . (R − t)1+n/2 Consequently, I3  CR −n−2

¨ v 2 dxdt  C sup v 2  CR 2 , E

E

where the last inequality follows from (3.80) and (3.87). Thus, collecting all inequalities we get ˆR 2 ˆ |Dv|2 ςR2 Gdz  CR 2 0 BR

(3.89)

(x 0 )

Inequalities (3.89) mean that we obtained the desired integral estimate for all the derivatives D (De u) with e ⊥ ν. Similar estimate for the derivative Dν (Dν u) follows from (3.89), (3.79) and Eq. (3.68).   Corollary 3.2 Let the assumptions of Lemma 3.14 hold, and let e be an arbitrary direction in Rn . Then Ψe (R/2) = Ψ (R/2, (De u)+ , (De u)− , ςR , z0 )  N4 (n, M, λ± , ψ), where the functional Ψ is defined by the formula (A.16), while ςR (x) := ξR/2,x 0 (x) is a standard time-independent cut-off function (see Notation and Conventions). Proof The desired inequality follows immediately from the definition (A.16) combined with (3.86).   Lemma 3.17 Let the assumptions of Lemma 3.14 hold, let e be an arbitrary direction in Rn if |Du(z0 )| = 0 and e ⊥ ν = Du(z0 )/|Du(z0 )| otherwise. Then (De u)± 22,Q

R (z

0)

 N5 (n, M, λ± , ψ)R n+4 .

(3.90)

Proof It is evident that inequalities (3.80) and (3.87) imply the estimate (3.90).

106

3 Boundary Estimates for Solutions of Free Boundary Problems

Proof (Proof of Theorem 3.5) Let z0 = (x 0 , t 0 ) be an arbitrary point from Q1 such that |u(z0 )| > 0, and let e ∈ Rn be the same direction as in Lemma 3.17. Since De u(z0 ) = 0, it follows that C(n)|D(De u)(z0 )|4  lim Ψe (r), r→0

where Ψe (r) = Ψ (r, (De u)+ , (De u)− , ςR , z0 ) with the functional Ψ defined by the formula (A.16), and ςR (x) = ξR/2,x 0 (x). On the other hand, according to Lemma A.2 and Remark A.1 after that, we have for 0 < r  inequality Ψe (r)  Ψe (R/2) +

R 2

:=

√ t0 2

the

N(n) (De u)+ 22,Q (z0 ) (De u)− 22,Q (z0 ) . R R R 2n+8

Application of Corollary 3.2 and Lemma 3.17 enable us to estimate the right-hand side of the last inequality by the known constant. This means that we proved the L∞ -estimate for all the derivatives D (De u) (z0 ) with e ⊥ ν. It is clear that the corresponding estimate of Dν (Dν u) (z0 ) can be now deduced from (3.79) and Eq. (3.68). So, we establish the L∞ -estimates for D 2 u(z0 ) for all points z0 ∈ {±u > 0} ∩ Q1 , and these estimates do not depend on the distance of z0 from the free boundary Γ (u). Since |D 2 u| = 0 almost everywhere on Λ(u) and the (n + 1)-dimensional Lebesgue measure of the set Γ ∗ (u) equals zero, we get the uniform estimate of the Lipschitz constant for Du(·, t) for each t ∈ [0, 1]. It remains only to observe that the uniform L∞ -estimate of ∂t u were established in (3.79). This finishes the proof.  

Appendix A

Monotonicity Formulas

Here we collect all informations about various monotonicity formulas which are the most important technical tools in studying the free boundary problems. The monotonicity formulas appear in almost every section.

A.1 C-monotonicity Formula We denote ˆt 0 ˆ I (r, v, z0 ) =

|Dv(x, t)|2 G(x − x 0 , t 0 − t)dxdt, t 0 −r 2 Rn

where r ∈]0, R], z0 = (x 0 , t 0 ) is a point in Rn+1 , a function v is defined in the strip Rn × [t 0 − R 2 , t 0 ], and the heat kernel G(x, t) is defined by G(x, t) =

exp (−|x|2/4t) for t > 0 and G(x, t) = 0 for t  0, (4πt)n/2

(A.1)

Our main tool is the following version of the monotonicity formula due to Caffarelli (see [Caf93, CK98]). Lemma A.1 Suppose that two nonnegative, subcaloric and continuous functions h1 , h2 on Rn × [t 0 − R 2 , t 0 ] satisfy h1 (x, t) · h2 (x, t) ≡ 0

in Rn × [t 0 − R 2 , t 0 ],

|Dhi | ∈ L2loc (Rn × [t 0 − R 2 , t 0 ]),

© Springer Nature Switzerland AG 2018 D. Apushkinskaya, Free Boundary Problems, Lecture Notes in Mathematics 2218, https://doi.org/10.1007/978-3-319-97079-0

i = 1, 2.

107

108

A Monotonicity Formulas

Also, suppose that the functions Dh1 and Dh2 have at most polynomial growth in x at infinity. Then the functional 1 I (r, h1 , z0 )I (r, h2 , z0 ) r4

Φ(r) = Φ(r, h1 , h2 , z0 ) =

(A.2)

is monotone nondecreasing in r. More precisely, if at least one of the sets   Si (r) = x ∈ Rn : hi (x, t 0 − r 2 ) = 0 ,

i = 1, 2,

differs from the half-space containing x 0 on its boundary by the set of nonzero measure, then either Φ  (r) > 0, or Φ(ς ) ≡ 0 for ς ∈ (0, r]. Proof In essence, the proof of this statement was given in [CK98]. For the completeness, we present some details here. We may assume that z0 = (0, 0) and restrict ourselves to the case where all the functions involved in Φ(r) (and also supports of the hi , i = 1, 2) are sufficiently smooth. In the general case the result can be obtained by approximation. We denote ˆ0 ˆ 2

|Dhi (x, t)|2 G(x, −t)dxdt,

Ji (r ) = I (r, hi , 0, 0) = −r 2

i = 1, 2.

Rn

Differentiation of Φ yields 

J  (r 2 ) J  (r 2 ) 2 Φ (r) = 2rΦ(r) 1 2 + 2 2 − 2 J1 (r ) J2 (r ) r 

 .

(A.3)

  Since H h2i /2  |Dhi |2 , for # = r 2 we get '

ˆ0 ˆ Ji (#) 

H −# Rn

( ˆ 2 hi h2i (x, t) G(x, −t)dxdt  (x, −#)G(x, #)dx 2 2 Rn

(the latter identity follows from the assumptions hi (0, 0) = 0 by integration by parts). Also, we have ˆ  Ji (#) = |Dhi (x, −#)|2 G(x, #)dx. Rn

A.1 C-monotonicity Formula

109

Therefore, ´ ´ 2 Ji (#) 1 Rn |D h˜ i (y, −1/2)|2G(y, 1/2)dy Rn |Dhi (x, −#)| G(x, #)dx 2 ´ = , ´ 2 ˜2 Ji (#) # Rn hi (x, −#)G(x, #)dx Rn hi (y, −1/2)G(y, 1/2)dy (A.4) √ where h˜ i (y, τ ) := hi ( 2#y, 2#τ ). Observe that h˜ i (y, −1/2) = hi (x, −#). We introduce the quantity ´ φ(S˜i ) = inf

S˜i

´

|Dw(y)|2 dμ(y)

S˜ i

w2 (y)dμ(y)

,

where   S˜i = y ∈ Rn : h˜ i (y, −1/2) = 0 , dμ(y) = G(y, 1/2)dy is Gaussian measure, and the infimum is taken over all nonzero functions on Rn with compact support in S˜i . Then from (A.4) it follows that Ji (#) 1  φ(S˜i ). Ji (#) #

(A.5)

Recalling that # = r 2 , from (A.3) and (A.5) we deduce the inequality Φ  (r) 



2 Φ(r) φ(S˜1 ) + φ(S˜2 ) − 2 . r

(A.6)

By the results of Beckner-Kenig-Pipher [BKP] (cf. Corollary 2.4.6 [CK98]), we have φ(S˜1 ) + φ(S˜2 ) − 2  0.

(A.7)

Precisely this inequality requires that the supports of the hi be disjoint. In Remark 2.4.8 [CK98] it was shown that equality occurs in (A.7) if and only if, up to rotation, we have S˜1 = {y ∈ Rn : y1 > 0} and S˜2 = {y ∈ Rn : y1 < 0}. Combining this with (A.6) and the observation that S˜i = Si (r), we complete the proof.   In some situations we need to use a local version of the Cafferelli monotonicity formula: Lemma A.2 Let ζ = ζ (|x|) be a standard time-independent cut-off function (see Notation and Conventions).

110

A Monotonicity Formulas

Let h1 , h2 be nonnegative, subcaloric and continuous functions in Q1/2 , satisfying h1 (0, 0) = h2 (0, 0) = 0,

h1 (x, t) · h2 (x, t) = 0 in

Q1/2 .

Then, there exists N = N(n) > 0 such that, for the functional Φ defined in (A.2) and for 0 < r  14 , we have:   Φ(r, h1 ζ, h2 ζ, 0, 0)  Φ(1/4, h1 ζ, h2 ζ, 0, 0) + N h1 42,Q1/2 + h2 42,Q1/2 . (A.8) Proof For the proof of this lemma we refer the reader to (the proof of) Theorem 2.3.1 in [CK98] (see also Theorem 12.12 in [CS05]). As usual we provide some details here. First, we introduce the additional notation F (#) :=

1 J1 (#) · J2 (#), #2

where ˆ0 ˆ Ji (#) =

|D[hi (x, t)ζ (x)]|2 G(x, −t)dxdt,

i = 1, 2,

−# Rn

and compute F  (#) = −

2 1 1 J1 (#)J2 (#) + 2 J1 (#)J2 (#) + 2 J1 (#)J2 (#). 3 # # #

(A.9)

In view of the presence of ζ , the estimates of Ji and Ji differ a little from the analogous bounds from the proof of Lemma A.1. It is evident that  1  H (hi ζ )2 − (hi ζ ) H [hi ζ ] 2  1  = H (hi ζ )2 − ζ 2 hi H [hi ] − (hi ζ ) [hi Δζ + 2Dhi Dζ ] 2  1   H (hi ζ )2 − (hi ζ ) [hi Δζ + 2Dhi Dζ ] (A.10) 2

|D(hi ζ )|2 =

(the latter inequality in (A.10) follows from the assumptions hi  0 and H [hi ]  0).

A.1 C-monotonicity Formula

111

Taking into account the properties of ζ , the inequality |G(x, −t)|  Ce−C/# for −# < t < 0, |x|  18 , and the “energy estimate” ˆ

ˆ |Dhi | dxdt  C 2

Q1/4

h2i dxdt,

Q1/2

we get from (A.10) the estimate for Ji (#) with 0 < #  1/16 ˆ0 ˆ

 1  H (hi (x, t)ζ (x))2 G(x, −t)dxdt + Ce−C/# hi 22,Q1/2 2

Ji (#)  −# Rn

=

1 2

ˆ

Rn

h2i (x, −#)ζ 2 (x)G(x, #)dx + Ce−C/# hi 22,Q1/2

 C#−n/2 hi (·, −#) 22,B1/4 + Ce−C/# hi 22,Q1/2

(A.11)

(here in the third inequality we use simply that |G(x, #)|  C#−n/2 for |x|  1/4). Also, we have ˆ Ji (#) = |D [hi (x, −#)ζ (x)] |2 G(x, #)dx Rn

1 = 2#

ˆ

  |D h˜ i (y, −#)ζ˜ (y) |2 dμ(y),

(A.12)

Rn

√ √ where h˜ i (y) := hi ( 2#x, −#), ζ˜ (y) = ζ ( 2#x) and dμ(y) = G(y, 1/2)dy is Gaussian measure. Applying the Poincare inequality to the right-hand side of (A.12) and going back to the x-variables we can conclude that Ji (#)

ϑi  2#

ˆ Rn

ϑi |h˜ i (y, −#)ζ˜ (y)|2dμ(y) = 2#

 ϑi   Ji (#) − Ce−C/# hi 22,Q1/2 , #

ˆ |hi (x, −#)ζ(x)|2 G(x, #)dx Rn

(A.13)

where the last inequality follows from the second line of (A.11). We note also that the value of ϑi depends only on the support of the product h˜ i (·, −#)ζ˜ (·).

112

A Monotonicity Formulas

Substituting (A.13) into (A.9) we arrive at F  (#) 

J1 (#)J2 (#) [−2 + ϑ1 + ϑ2 ] #3

C − 3 e−C/# ϑ1 J2 (#) h1 22,Q1/2 + ϑ2 J1 (#) h2 22,Q1/2 . (A.14) #

We observe that ϑ1 + ϑ2  2 provided that the supports of the hi are disjoint. Therefore, substituting (A.11) into (A.14) and changing the constant in e−C/# in order to absorb all negative powers of # we get   F  (#)  − Ce−C/# h1 (·, −#) 22,B1/4 h2 22,Q1/2 + h2 (·, −#) 22,B1/4 h1 22,Q1/2 − Ce−C/# h1 22,Q1/2 h2 22,Q1/2 . Hence, for 0 < #
0} ∩ ∂B1 . Moreover, if Dθ hi denotes the tangential derivative of hi along ∂B1 , we have J˜i (1) =

ˆ |Dhi |2 dσ = S˜i

ˆ 

 (Dr hi )2 + (Dθ hi )2 dσ.

S˜i

Therefore, ´

 (Dr hi )2 + (Dθ hi )2 dσ

J˜i (1) S˜ i  ´ ˜ Ji (1) hi Dr hi + S˜ i

´

 2εi ´  S˜ i

S˜i

n−2 2 2 hi

dσ

 (Dr hi )2 + (Dθ hi )2 dσ

 , (Dr hi )2 + εi (εi + n − 2)h2i dσ

(A.22)

where εi > 0 and the second inequality in (A.22) follows from the splitting of the term hi Dr hi . If we denote by λi = λi (S˜i ) the first eigenvalue of the spherical Laplacian in S˜i , then we can write ˆ ˆ 2 (A.23) (Dθ hi ) dσ  λi h2i dσ. S˜i

S˜i

Choosing εi = εi (S˜i , n) such that εi (εi + n − 2) = λi and substituting (A.22) and (A.23) into (A.21) we obtain F  (1)  2 (ε1 + ε2 − 2) . F (1) Observe now that the set function εi as a function of S˜i and n was studied by Friedland and Hayman in [FH76]. Since the functions h1 and h2 have disjoint supports, it follows from results of [FH76] that ε1 + ε2  2. This completes the proof.  

A.3 W-monotonicity Formula

115

A.3 W-monotonicity Formula Here we use the functional introduced by Weiss (cf. [Wei99]) for the study of some free boundary problems in the entire space Rn+1 . Changing Weiss’s notation somewhat, we shall write this functional as follows: 1 W (r, x 0 , t 0 , v) := 4 r

0 −r 2 tˆ ˆ

 |Dv|2 + 2v +

t 0 −4r 2 Rn

v2 t − t0

G(x − x 0 , t 0 − t)dxdt,

where (x 0 , t 0 ) is a point in Rn+1 , r is a positive constant, the function G(x, t) is defined by (A.1), and v is a continuous function defined on Q := Rn × [t 0 − 4R 2 , t 0 ], R  r. We also suppose that Di v ∈ L2loc (Q) and Dv have at most polynomial growth with respect to x, as |x| → ∞. It is easy to check that for any ∈]0, R/r] the functional W has the following scaling property: W ( r, x 0 , t 0 , v) = W ( , 0, 0, vr ),

(A.24)

where vr (x, t) =

v(rx + x 0 , r 2 t + t 0 ) r2

(A.25)

is the parabolic scaling of v around z0 = (x 0 , t 0 ). In [Wei99] it was shown that the functional W is monotone nondecreasing with dW = 0 for all r > 0 is equivalent to the degree 2 respect to r and that the identity dr parabolic homogeneity of the function v. Lemma A.5 (i) For every spatial direction e ∈ Rn   1 15 1 =: A. W r, 0, 0, (x · e)2+ = W r, 0, 0, (x1 )2+ = 2 2 4 (ii) For every constant m and homogeneous quadratic polynomial P (x) satisfying ΔP = m + 1  1 2 W (r, 0, 0, mt + P (x)) = W r, 0, 0, (x1 ) = 2A. 2 Proof Both statements were proved in [CPS04] (see Lemma 6.2). They follow from the direct computations. We provide some details here.

116

A Monotonicity Formulas

In Part (i) for v(x, t) = (x · e)2+ /2 = (y1 )2+ /2 we have ˆ

ˆ−r 2

1 W (r, 0, 0, v) = 4 r

dt −4r 2

( ˆ∞ ' y14 2 2y1 + G(y, −t)dy1 dy2 . . . dyn 4t

Rn−1

ˆ−r 2

1 = 4 2r

(−t)dt =

0

15 . 4

(A.26)

−4r 2

Observe that the second equality in (A.26) we get from the definition of the heat kernel (see (A.1)) combined with an elementary integration by parts: ˆ∞ '

ˆ dy2 . . . dyn Rn−1

y12 ( ( ˆ∞ ' − 4(−t) 4 4 y y e 2y12 + 1 G(y, −t)dy1 = 2y12 + 1 √ dy1 4t 4t 4π(−t)

0

0

4t = √ π

ˆ∞ 

 2 −2ξ 2 + ξ 4 e−ξ dξ

0

t =− . 2 The case v(x, t) = 12 (x1 )2 from Part (ii) is treated completely analogous (A.26). It remains only to consider v(x, t) = mt + P (x). Observe that for given t 0 and M  1 be given constants, let u ∈ P2b (M), and let

z0 = (x 0 , t 0 ) ∈ Γ (u) ∩ Qb . Suppose that the function u is extended by zero across the plane Π to the whole cylinder set Qb (z0 ); we preserve the notation u for this extension.

118

A Monotonicity Formulas

Then for 0 < r  b we have 1 dWb (r, x 0 , t 0 , u) = dr r

ˆ−1ˆ Bb/r

−4

x0 + 12 r

|L ur |2 G(x, −t)dxdt + Jb (r, u) −t

ˆ−1ˆ −4

 |D1 ur |2 G(x, −t)dx  dt,  −x 0 Bb/r ∩ x1 = r 1

(A.30)

where ur is the same parabolic scaling around z0 as in (A.25), L ur (x, t) := x · Dur (x, t) + 2t∂t ur (x, t) − 2ur (x, t),

(A.31)

and Jb (r, u) satisfies the estimate  n+4 

b 1 exp −b 2 /16r 2 |Jb (r, u)|  C0 1 + r r

(A.32)

with the universal constant C0 = C0 (n, M). Remark A.3 If u ∈ P2b (M) then the same statement is true without the third term on the right-hand side of (A.30). Besides, in the case of b = ∞ we have J∞ (r, u) = 0. Proof First, we assume that the function u has all derivatives with respect to x and t up to the second order. Using (A.29) and the relation d (Di ur ) = Di dr



dur dr

,

we obtain d d Wb (r, x 0 , t 0 , u) = Wb/r (1, 0, 0, ur ) = I1 + I2 , dr dr

(A.33)

where ˆ−1ˆ I1 = 2 −4

I2 = −

b r2

x0 − 12 r

 ) * ur dur dur dur   Dur D + + G(x, −t)dxdt, −x 0 dr dr t dr Bb/r ∩ x1 > r 1

ˆ−1ˆ −4

  −x 0 Sb/r ∩ x1 > r 1

 (ur )2 2  |Dur | + 2ur +  G(x, −t)dx  dt. −x 0 t Bb/r ∩ x1 = r 1

ˆ−1ˆ −4

 (ur )2 G(x, −t)dSb/r dt |Dur |2 + 2ur + t

A.3 W-monotonicity Formula

119

Then, integrating the term 2Dur D identity



d dr ur

Di G(x, −t) =



G(x, −t) in I1 by parts and using the

xi G(x, −t), 2t

we get ˆ−1ˆ I1 = 2

  −x 0 Bb/r ∩ x1 > r 1

−4

ˆ−1ˆ +2 −4

 0 −x Sb/r ∩ r 1

dur  xi ur  −Δur − Di ur + 1 + G(x, −t)dxdt dr 2t t

dur (γ · Dur )G(x, −t)dSb/r dt dr

ˆ−1ˆ −2 −4

  −x 0 Bb/r ∩ x1 = r 1

dur D1 ur G(x, −t)dx  dt. dr

(A.34)

 The assumption u(0, x  , t)Π∩Q = 0 implies that 2b

 



ur = |D ur | = ∂t ur = 0 on E := Bb/r ∩ x1 =

−x10 r

×] − 4, −1[, (A.35)

whence   (ur )2  2 2 |Dur | + 2ur +  = |D1 ur | . t E

(A.36)

L ur dur = , dr r

(A.37)

Moreover, since

from (A.35) and (A.31) it follows that    x0 dur D1 ur  = 12 |D1 ur |2 . − dr r E

(A.38)

120

A Monotonicity Formulas

Substituting (A.36) and (A.38) in (A.34), and using (A.37), (A.31), and (A.33), we obtain the representation d Wb (r, x 0 , t 0 , u) dr ˆ−1ˆ =2

  −x 0 Bb/r ∩ x1 > r 1

−4

x0 +Jb (r, u) + 12 r

) * L ur L ur 1 − H [ur ] − G(x, −t)dxdt r 2t

ˆ−1ˆ −4

 |D1 ur |2 G(x, −t)dxdt,  −x 0 Bb/r ∩ x1 = r 1

(A.39)

where ˆ−1ˆ

L ur (γ · Dur ) G(x, −t)dH n−1 dt r

Jb (r, u) := 2 −4

b − 2 r

Sb/r

ˆ−1ˆ −4

Sb/r

 (ur )2 G(x, −t)dH n−1 dt, |Dur |2 + 2ur + t (A.40)

and γ is the unit vector of the outward normal to Sb/r . It is evident that estimate (A.32) can be easily deduced from formula (A.40). The formal calculations given above are correct if the function u has all derivatives with respect to x and t up to second order. Therefore, in the case of + an arbitrary function u ∈ P2b (M), we must regularize the function with respect to t-variable. For instance, this can be done by using the Steklov average. For the smoothed function u, the representation (A.39) is proved as above. Now, letting the parameter of the averaging tend to zero, we easily show that (A.39) is true for the initial u. + From the assumption u ∈ P2b (M) it follows that Γ (ur ) has zero Lebesgue measure, and L ur = 0 H [ur ] = 1   where Q := Bb/r ∩ x1 >

−x10 r

a.e. in in

{ur = 0} ,

Q ∩ {ur = 0} ,

 ×] − 4, −1[.

A.3 W-monotonicity Formula

121

Therefore, for a.e. (x, t) ∈ Q we have ) * |L ur |2 L ur L ur 1 − H [ur ] − =− . r 2t 2rt

(A.41)  

Combining (A.39) and (A.41), we complete the proof. (r, x 0 , t 0 , u)

Remark A.4 Under the conditions of Lemma A.6, the functional Wb is uniformly bounded for 0 < r  b. Moreover, estimate (A.32) provides the equality lim |Jb (r, u)| = 0,

r→0+

b>0

(A.42)

+ for all functions of class P2b (M).

Corollary A.1 Let all the assumptions of Lemma A.6 be satisfied. Then the function Wb (r, x 0 , t 0 , u) has a limit as r → 0+ . Moreover, this limit does not depend on the parameter b.

Appendix B

Auxiliary Results

For the future references, we recall and explain some general facts. Most of these auxiliary results are known, but probably not well known in the context used in this book. For the readers convenience we have included the proofs of almost all technical lemmas. Lemma B.1 Let v be a generalized solution of the problem H [v] = div f v=0

in

Q+ 1,

f ∈ L∞ (Q+ 1 ),

on Π ∩ Q1 .

Then, for any δ ∈ (0, 1) there exists a positive constant C = C(n, δ) such that v ∞,Q+  C(n, δ) v 2,Q+ 1−δ

1−δ/2

+ f ∞,Q+

1−δ/2

.

Proof Successive application of Theorem 8.1 Ch. III and Theorem 6.2 Ch. II from [LSU67] provides the desired estimate.   Let E be a domain in Rn+1 , and let f ∈ L1loc (E). We say that f is subcaloric (supercaloric) in E if ˆ f (Δϕ + ∂t ϕ)dxdt  0

( 0)

E

for every nonnegative C ∞ -function ϕ with compact support in E. We say that f is caloric if it is subcaloric and supercaloric in E. Lemma B.2 Let v be a continuous function in a domain E ⊂ Rn+1 , let Dv ∈ L1loc (E), and let v be caloric in the set {v > 0}. Then v + is subcaloric in E.

© Springer Nature Switzerland AG 2018 D. Apushkinskaya, Free Boundary Problems, Lecture Notes in Mathematics 2218, https://doi.org/10.1007/978-3-319-97079-0

123

124

B Auxiliary Results

Proof Since v is caloric in {v > 0}, it is evident that v is subcaloric in {v > 0}. The latter means that for any nonnegative test-function η ∈ C ∞ (E) with supp η ⊂ {v > 0} we have the inequality ˆ (B.1) (η∂t v + DηDv) dxdt  0. E

Next for arbitrary ε > 0 we set η = ζ ψε (v), where ζ ∈ C0∞ (E) is a nonnegative function, while ψε ∈ C ∞ (R) satisfies  1, s > ε, ψε (s) = and 0  ψε (s)  1 ∀s ∈ R. 0, s < ε/2, Plugging η into (B.1) we get ˆ

ζ ψε (v)∂t v + ζ ψε (v)(Dv)2 + ψε (v)Dζ Dv dxdt  0. E

Taking into account that ζ ψε (v)(Dv)2  0 we arrive at ˆ (ζ ψε (v)∂t v + ψε (v)Dζ Dv) dxdt  0.

(B.2)

E

Defining Ψε (v) =

´v

ψε (s)ds we can easily claim that Ψε (v) ≡ 0 in the set

0

{v < ε/2} and Ψε (v) converges uniformly to v + as ε tending to zero. Furthermore, inequality (B.2) together with the identity ∂t Ψε (v) = ψε (v)∂t v, and integration by parts provide ˆ (−Ψε (v)∂t ζ + ψε (v)Dζ Dv) dxdt  0.

(B.3)

E

Letting in (B.3) ε → 0 we deduce ˆ

  + −v ∂t ζ + χ{v>0} Dζ Dv dxdt  0.

E

It remains only to observe that D(v + ) = 0 in the set {v  0}, and, consequently, ˆ

 +  −v ∂t ζ + Dζ D(v + ) dxdt  0.

E

This finishes the proof.

 

B Auxiliary Results

125

Lemma B.3 (Liouville Type Theorem) Let v be a caloric function in the whole space Rn+1 ∩ {t  0} and satisfy the inequality

|v(x, t)|  C 1 + |x|2 + |t| ,

(x, t) ∈ Rn+1 ∩ {t  0} .

Then there exist constants a ij , a i , a0 such that v(x, t) ≡ a xi xj + 2 ij

 n 

 a

ii

t + a i xi + a0 .

i=1

Proof This result can be easily deduced from well-known estimates for derivatives of caloric functions.   ∩ Lemma B.4 Let h be a caloric function in the whole strip E = Rn+1 + 0  t < t  0 . Moreover, suppose that h is continuous in E and has at most polynomial growth in x at infinity. If, in addition, h(x, t)  0

  ∀(x, t) ∈ Π ∩ t 0 < t  0

and h(x, t 0 )  0

∀x ∈ Rn+ ,

then h  0 in E . Proof The proof of this statement is elementary, but perhaps not obvious. We sketch some details. Since h is caloric in E , we have the identity ˆ 0=−

H [h](x, t)h− (x, t)G(x, ε − t)dxdt,

(B.4)

E

where the heat kernel G is defined by (A.1) and ε is a positive constant. Integration of (B.4) by parts combined with the given boundary inequalities for h yields ˆ 0=

  Di hDi h− Gdxdt +

E

ˆ

− 2

ˆ

|Dh | Gdxdt −

= E

ˆ = E

E

|Dh− |2 Gdxdt +

ˆ Rn+



ˆ Di E

(h− )2  2

(h− )2 2



)

ˆ Di Gdxdt +

 ΔG + ∂ G dxdt + . /0 t 1 =0

t=0  (h− )2 G(x, ε − t) dx. 2

ˆ Rn+

∂t E

* (h− )2 Gdxdt 2

t=0  (h− )2 G(x, ε − t) dx 2

126

B Auxiliary Results

It is evident that the both integrals in the right-hand side of the above identity are nonnegative, and sum of them equals zero. This implies h− ≡ 0 in E and finishes the proof.   Remark B.1 The statement of Lemma B.4 holds true if we replace the condition about at most polynomial growth in x at infinity by the exponential growth 2

|h(x, t)|  A ea|x| , with some positive constants A and a. In this case we repeatedly apply the arguments from the proof of Lemma B.4 on the time intervals [t 0 ; t 0 + T ], [t 0 + T ; t 0 + 2T ] etc., for sufficiently small value of T . +

Lemma B.5 Let v be a continuous function in Q1 . Suppose also that v is smooth at points where it is positive and satisfies H [v] = f v=0

in Q+ 1 ∩ {v > 0} , on Π ∩ Q1 ,

with f ∈ L∞ (Q+ 1 ). There exists a positive constant C, depending on n only, such that

v(x, t) ,  C v ∞,Q+ + f − ∞,Q+ 15/16 15/16 x1 Proof Let M0 = v ∞,Q+

15/16

(x, t) ∈ Q+ 1/2 .

(B.5)

.

For arbitrary (y  , t 0 ) ∈ Π ∩ Q1/2 we introduce the set   1 1 1 K (y  , t 0 ) = (x, t) : 0 < x1 < , |x  − y  | < , t 0 − < t < t 0 , v(x, t) > 0 , 2 4 16

and consider in K (y  , t 0 ) the barrier function w(x, t) = M0 (16|x  − y  |2 + N1 x1 − N2 x12 ) + 16M0 (t 0 − t),

where N1 and N2 are positive constants which will be chosen later. Next, an elementary calculation gives that H [v − w]  0 in  v − w  

∂ K (y ,t 0 )

if N2 = 16(n + 1) +

f − ∞,Q+ 1

2M0

K (y  , t 0 ),  0,

, N1 = 12 N2 + 2.

B Auxiliary Results

127

Applying the maximum principle to the difference v − w in the set K (y  , t 0 ) we get the estimate ∀(x, t) ∈ K (y  , t 0 ).

v(x, t)  w(x, t) In particular, for 0 < x1
0, H [v] = 1 in QR (z) and v = 0 on QR (z) ∩ Π (if the latter set is not empty). Then ⎧ ⎫ ⎨ ⎬   sup |Di Dj v| + |∂t v|  C(n) 1 + R −2 sup |v| . (B.7) ⎩ ⎭ Q+ (z) Q+ (z) R/2

R

Proof Inequality (B.7) can be derived easily by rescaling from the standard parabolic estimates in the unit cylinder (see Exercise 10.2.6 [Kry96]).   Lemma B.7 Let N ⊂ Rn be an open set, let ρ and h be positive constants, and let  Q = y ∈ Rn : |yτ | < ρ

for τ = 2, . . . , n;

 0 < y1 < h · ρ .

  We assume that ∂ N ∩ Q ∩ {y1 = hρ} = ∅. Let us also assume that a function v ∈ W 2,∞ (Q) satisfies the conditions Δv  0

in Q ∩ N ,

v0

in Q,

v  βρ

on Γ = ∂N ∩ Q,

where β is a positive constant. Then, for any ∈]0, 1[, the following inequality holds: v  βC(h, )ρ where Q(

)

= {y : |yτ | < (1 − ) ρ 1

√

e− 2 πh n−1 C(h, ) = 2 √ πh n − 1

in

N ∩ Q( ) ,

for τ = 2, . . . , n;

ρ < y1 < hρ} and

  n−1  1 (1 − )π e 2π cos 2

√

n−1

 −1 .

128

B Auxiliary Results

Proof Consider the barrier function n 2  cos w(y) = ε eμy1 − 1 τ =2



π yτ 2ρ



in Q = Q ∩ N . The parameters ε and μ will be chosen later. An elementary computation gives   n n  2 2 π π yτ + θ yτ  0 ε−1 Δw = eμy1 μ2 − θ cos cos 2ρ 2ρ τ =2

in Q,

τ =2

2 √ π π provided μ  2ρ n − 1 and θ = (n − 1) 2ρ . Moreover, setting   S1 = y ∈ ∂Q : y1 = 0 ,   S2 = y ∈ ∂Q : |yτ | = ρ for some τ = 2, . . . , n , we observe that ∂Q = S1 ∪ S2 ∪ Γ . Therefore, for ε  β (μh)−1 e−μhρ , we have  w S

= 0  v; 1 ∪S2   wΓ  εμy1 eμy1  εμhρeμhρ  βρ  v Γ . √ √ 1 2ρ Finally, choosing ρμ = π2 n − 1 and ε = πh√ βe− 2 πh n−1 , and applying n−1 the maximum principle to the difference w − v in Q , we obtain

vw

in Q ,

and consequently, 

vβ e

1 2π

√

√

n−1

= βC(h, )ρ

   (1 − )π n−1 2e− 21 πh n−1 − 1 cos ρ √ 2 πh n − 1 in

N ∩ Q( ) .  

Lemma B.8 Let a non-negative function g : B1 ∩ {x1 = 0} → R satisfy the Lipschitz condition with the constant L0 , and let g(0) = 0.     Let E = x ∈ B1+ : x1 > g(x  ) , x 0 = g(x 0 ), x 0 , and ER = E ∩ BR (x 0 ) with R satisfying 0 < R < 1 − |x 0 |.

B Auxiliary Results

129

Assume that a function v such that 1,2 v ∈ C ER ∩ Wloc (ER )

(B.8)

or v ∈ W 1,2 (ER )

(B.9)

is a generalized solution to the elliptic equation

f = f 1, . . . , f n , Di a ij Dj v = div f + f 0 , in the domain ER , and let a ij ∈ L∞ (ER ) ; ν|η|2  a ij (x)ηi ηj  ν −1 |η|2 f 0,

∀x ∈ ER , ∀η ∈ Rn ,

n  2 f i ∈ Lq/2 (ER ) ,

ν = const > 0;

q > n.

i=1

Assume also that   v g(x  ), x   kˆ > k(R) := inf v ER

for |x − x 0 | < R.

Then

inf v  k(R) + γ0 kˆ − k(R) − CF R 1−n/q ,

ER/2

where F =

n

 + f i q,ER + f 0 q/2,ER , and the constants C > 0 and γ0 ∈]0; 1[

i=1

are determined by the quantities ν and L0 . Proof The validity of Lemma B.8 follows from Lemmas 6.1 and 7.1 Ch. II [LU68]. The corresponding proof is given in [LU68] for the case when the function v satisfies condition (B.9). But it suffices to require only the truncations (v − k)− := ˆ to belong to the class W 1,2 (E ˜ ) ∀R˜ < R. The last min {v − k, 0}, k(R) < k < k, R fact is also valid if v satisfies (B.8).   Lemma B.9 Let a continuous function v in the cylinder QR (z0 ) satisfies the following conditions: v (z0 ) = 0; v is differentiable at z0 ; v

+

and v − are subcaloric in QR (z0 ).

130

B Auxiliary Results

Then + ˆ , −n−4 |Dv(z )|  C(n), R 0

v 2 dxdt.

QR (z0 )

Proof The above inequality follows immediately from Corollary 1.1.9 [CK98] after elementary rescaling.   0 Lemma B.10 Let a Lipschitz function v be subcaloric in Qλ,θ R (z ) 0 0 2 0 BλR (x )×]t − θ R , t ] with λ > 1 and θ > 0. Then there exists a positive constant N = N(n) such that

:=

 sup v  N 0 Q1,θ R (z )

0 Qλ,θ R (z )

v 2 (z)dz +

sup

BλR (x 0 )×{t 0 −θR 2 }

v.

Proof The above inequality is a particular case of Lemma 3.1 [NU11]. It is necessary also to take into account Remark 6.1 from [NU11].   Lemma B.11 Let E be a domain in Rn , and let f i ∈ L∞ (E ), i = 1, . . . , n. Then if v ∈ W 1,2 (Ω) is a solution of the equation Δv = div f,

f = f 1, . . . , f n

in E , we have, for any ball B2R (x 0 ) ⊂ E ,  sup |u|  C BR (x 0 )

B2R (x 0 )

v 2 dx + cR f ∞,B2R (x 0 ) .

Proof The validity of Lemma B.11 follows from Theorem 8.17 [GT01].

 

Appendix C

Additional Facts

Here we collect some facts concerning various free boundary problems which were proved by other authors. + (M) be a not identically zero, degree 2 Lemma C.1 Let n = 1, and let u ∈ P∞ parabolic homogeneous function with respect to the origin. Then

u(x, t) =

x2 2

in R2+ ∩ {t  0} .

Proof This statement is proved along the same lines as the rest of the proof of Lemma 6.3 from [CPS04]. We sketch the main details. From homogeneity it follows that for all t < 0 we have the representation 

 x x u(x, t) = −t u √ , −1 =: −tf √ . −t −t It is evident that the function f = f (ξ ) is defined on [0, ∞), has at most quadratic growth at infinity and satisfies ξ f  (ξ ) − f  (ξ ) + f (ξ ) = 1 2

in

Ω(u) ∩ {t = −1} ,

(C.1)

f (ξ ) = f  (ξ ) = 0

on

Λ(u) ∩ {t = −1} .

(C.2)

The general solution to the ordinary differential equation (C.1) is given by ⎛ f (ξ ) = 1 + C1 (ξ 2 − 2) + C2 ⎝−2ξ e

ξ 2 /4

⎞

ˆξ + (ξ 2 − 2)

e

s 2 /4

ds ⎠ .

0

© Springer Nature Switzerland AG 2018 D. Apushkinskaya, Free Boundary Problems, Lecture Notes in Mathematics 2218, https://doi.org/10.1007/978-3-319-97079-0

131

132

C Additional Facts

If a is a finite endpoint of a connected component of Ω(u) ∩ {t = −1}, then the boundary conditions (C.1) take the form ⎧ ˆa ⎪ 2 ⎪ 2 a 2 /4 2 ⎪ ⎪ C1 (a − 2) − 2aC2 e + C2 (a − 2) es /4 ds = −1 ⎪ ⎪ ⎪ ⎨ 0

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2aC1 − 4C2 e

a 2 /4

ˆa + 2aC2

es

2 /4

ds = 0.

0

We see that the above linear system has the unique solution 1 1 2 C1 = − ae−a /4 2 4

ˆa es

2 /4

C2 =

ds,

1 −a 2 /4 ae . 4

0

In particular, the uniqueness means that there could not be two different values of a; hence the connected component of Ω(u) ∩ {t = −1} is unbounded. Next, on the unbounded interval we must have C2 = 0, since f has at most quadratic growth at infinity. This implies C1 = 12 and the only possible value of a is 0. Thus, Ω(u) ∩ {t = −1} is (0, ∞) and f (ξ ) = 12 ξ 2 .   Lemma C.2 Let u ∈ P∞ (M) be a not identically zero, degree 2 parabolic homogeneous function with respect to the origin. Then either of the following representations takes place: (i) u(x, t) = 12 (x · e)2+ for a certain spatial vector e; (ii) u(x, t) = mt + P (x), where m is a constant and P (x) is a homogeneous quadratic polynomial satisfying ΔP = m + 1. Proof For a proof of this statement we refer the reader to (the proof of) Lemma 6.3 from [CPS04]. Lemma C.3 Let u(x)  0 in CR := BR (x 0 )×]t 0 − R 2 , t 0 + R 2 [, where (x 0 , t 0 ) ∈ ∂ {u > 0}. Suppose that u is a solution of the equation H [u] = χ{u>0}

in

CR ,

and that D 2 u ∈ L∞ (CR ) and the derivative ∂t u is continuous in CR . Suppose also that in some spatial direction, say e1 , the function u is monotone (i.e., De1 u  0) and that ∂ {u > 0} is a x1 -graph of a Lipschitz function f . Then f is a C 1,α -function for some 0 < α < 1. Proof An elegant proof of this statement is given in [AS91]. Now we recall well-known estimates that follow from [Caf77].

 

C Additional Facts

133

Lemma C.4 For an arbitrary function u ∈ P1+ (M, L) and any direction e ∈ Rn the following estimate holds: Dee u(x)  −C| log (d(x))|−β

∀x ∈ Ω(u),

(C.3)

where d(x) = dist {x, Γ (u)}, β = (2(n − 1))−1 , and C depends on M and on the distance of x to ∂B1+ . Moreover, the second derivatives D 2 u are continuous in Ω(u) ∪ Γ (u) and lim

Ω(u)x→x 0 ∈Γ (u)

Dee u(x)  0.

(C.4)

Proof This result follows directly from Theorems 1 and 3 of [Caf77]. P1+ (M, L),

Lemma C.5 Let u ∈ let −1 r  (10Lk) . Suppose also that

x∗

 

∈ Γc (u) ∩ B1/2 , let k  2, and let

  x ∈ Bkr (x ∗ ) : x1 > r ∩ Γ (u) = ∅.

(C.5)

If, in addition, the inequality osc

+ x∈B5Lkr (x ∗ )

4M Dτ u(x) ,  x1 k−1

τ = 2, . . . , n,

takes place, then the estimate osc

x∈Br+ (x ∗ )

holds true with

0

Dτ u(x)  (1 − x1

0)

osc

+ x∈B5Lkr (x ∗ )

Dτ u(x) x1

(C.6)

∈]0; 1[ depending only on n and k.

Proof The proof of inequality (C.6) can be found in [Ura96] (see Lemma 3 and Remark after that).   Lemma C.6 Let u ∈ P1+ (M, L), and let δ be a constant satisfying 0 < δ < π/2. Suppose also that for any direction ξ lying in the cone Kδ and any r ∈]0; 1/8[ the estimate inf Dξ ξ u  −Cδ | log r|−β

Br+ (x ∗ )

∀x ∗ ∈ Γc (u) ∩ B1/4

(C.7)

takes place with β = (2(n − 1))−1 and with the constant Cδ completely defined by δ, M and L. Then 

δ | log r|−β inf Dξ ξ u  −C

Br+ (x 0 )

∀x 0 ∈ Γ (u) ∩ B1/8 ,

(C.8)

134

C Additional Facts

δ = C δ (δ, M, L) and β = β (M, L, β), whereas the radius r is the same as where C in inequality (C.7). Proof The proof of this statement can be traced in the rest of the proof of Lemma 5 in [Ura96]. We sketch the details. Consider a point x 0 ∈ Γ (u) ∩ B1/8 . From Lemma C.4 it follows that 0 estimate (C.8) holds  0 true for  such x . However, the constant Cδ depends also on ∗ 0 d (x ) := dist x , Γc (u) . We show that due to (C.7) this dependence can be eliminated. Indeed, using (C.4) and repeating the arguments from [Fri82] we deduce the inequality  inf Dξ ξ u  −2α M

Br+ (x 0 )

r ∗ d (x 0 )

α ,

∀x 0 ∈ Γ (u),

r  d ∗ (x 0 ),

(C.9)

where α = α(M, L) ∈]0; 1[, and the dependence on d ∗ (x 0 ) is given in the explicit form. Further, we denote by x ∗ the contact point closest to x 0 , i.e., we assume that ∗ x ∈ Γc (u) and d ∗ (x 0 ) = |x 0 − x ∗ |. If d ∗  r  1/8 then Br (x 0 ) ⊂ B2r (x ∗ ), and condition (C.7) implies inf Dξ ξ u  inf Dξ ξ u  −Cδ | log (2r)|−β ,

Br+ (x 0 )

+ ∗ B2r (x )

Cδ = Cδ (δ, M, L).

Otherwise, if (d ∗ )2  r  d ∗ then Br (x 0 ) ⊂ Bd ∗ (x 0 ) ⊂ B2d ∗ (x ∗ ), and again estimate (C.7) provides inf Dξ ξ u 

Br+ (x 0 )

inf

+ ∗ B2d ∗ (x )

√ Dξ ξ u  −Cδ | log (2d ∗ )|−β  −Cδ | log (2 r)|−β ,

with Cδ = Cδ (δ, M, L). Finally, if 0 < r < (d ∗ )2 then (C.9) guarantees that inf Dξ ξ u  −2α Mr α/2 .

Br+ (x 0 )

δ = C δ (δ, M, L) and β  = min {β, α/2} is proved for Thus, estimate (C.8) with C any x 0 ∈ Γ (u) ∩ B1/8 and r  1/8.   Lemma C.7 Let λ± be non-negative constants such that λ+ + λ− > 0, and let u be a weak solution of the equation H [u] = λ+ χ{u>0} − λ− χ{u 0, and let u be a weak solution of Eq. (C.10). Then the set {u = 0} ∩ {|Du| = 0} is locally in 1 Q+ 1 a C -surface and ∂t u is continuous on that surface. In addition, the unit normal vector to {u = 0} ∩ {|Du| = 0} directed into {u > 0} has the form γ (x, t) =

∇u(x, t) . |∇u(x, t)|

Proof For a proof of this statement we refer the reader to (the proof of) Lemma 7.1 from [SUW09].  

References

[ACF84] H.W. Alt, L.A. Caffarelli, A. Friedman, Variational problems with two phases and their free boundaries. Trans. Am. Math. Soc. 282(2), 431–461 (1984) [AF03] R.A. Adams, J.J.F. Fournier, Sobolev Spaces. Pure and Applied Mathematics (Amsterdam), vol. 140, 2nd edn. (Elsevier/Academic Press, Amsterdam, 2003) [AMM06] J. Andersson, N. Matevosyan, H. Mikayelyan, On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem. Ark. Mat. 44(1), 1–15 (2006) [AMU09] D.E. Apushkinskaya, N. Matevosyan, N.N. Uraltseva, The behavior of the free boundary close to a fixed boundary in a parabolic problem. Indiana Univ. Math. J. 58(2), 583–604 (2009) [AS91] I. Athanasopoulos, S. Salsa, An application of a parabolic comparison principle to free boundary problems. Indiana Univ. Math. J. 40(1), 29–32 (1991) [ASU00] D.E. Apushkinskaya, H. Shahgholian, N.N. Uraltseva, Boundary estimates for solutions of a parabolic free boundary problem. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271, 39–55 (2000) [AU95] D.E. Apushkinskaya, N.N. Uraltseva, On the behavior of the free boundary near the boundary of the domain. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 221(Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsii. 26), 5–19, 253 (1995). English transl. in J. Math. Sci. (N.Y.) 87(2), 3267–3276 (1997) [AU06] D.E. Apushkinskaya, N.N. Uraltseva, Boundary estimates for solutions of two-phase obstacle problems. Probl. Math. Anal. (34), 3–11 (2006). English transl. in J. Math. Sci. (N.Y.) 142(1), 1723–1732 (2007) [AU08] D.E. Apushkinskaya, N.N. Uraltseva, Boundary estimates for solutions to the twophase parabolic obstacle problem. Probl. Math. Anal. (38), 3–10 (2008). English transl. in J. Math. Sci. (N.Y.) 156(4), 569–576 (2009) [AU13] D. E. Apushkinskaya and N. N. Uraltseva. Uniform estimates near the initial state for solutions of the two-phase parabolic problem. Algebra i Analiz 25(2), 63–74 (2013) [AU15] D.E. Apushkinskaya, N.N. Uraltseva, Free boundaries in problems with hysteresis. Phil. Trans. Roy. Soc. A 373(2050), 20140271 (2015) [AUS02] D.E. Apushkinskaya, N.N. Ural’tseva, Kh. Shakhgolyan, On global solutions of a parabolic problem with an obstacle. Algebra i Analiz 14(1), 3–25 (2002) [AUS03] D.E. Apushkinskaya, N.N. Ural’tseva, Kh. Shakhgolyan, On the Lipschitz property of the free boundary in a parabolic problem with an obstacle. Algebra i Analiz 15(3), 78–103 (2003)

© Springer Nature Switzerland AG 2018 D. Apushkinskaya, Free Boundary Problems, Lecture Notes in Mathematics 2218, https://doi.org/10.1007/978-3-319-97079-0

137

138

References

[Bai72] C. Baiocchi, Su un problema di frontiera libera connesso a questioni di idraulica. Ann. Mat. Pura Appl. (4) 92, 107–127 (1972) [BCN90] H. Berestycki, L.A. Caffarelli, L. Nirenberg, Uniform estimates for regularization of free boundary problems, in Analysis and Partial Differential Equations. Lecture Notes in Pure and Applied Mathematics, vol. 122 (Dekker, New York, 1990), pp. 567–619 [BDM06] A. Blanchet, J. Dolbeault, R. Monneau, On the continuity of the time derivative of the solution to the parabolic obstacle problem with variable coefficients. J. Math. Pures Appl. (9) 85(3), 371–414 (2006) [BK74] H. Brézis, D. Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J. 23(9), 831–844 (1974) [BKP] W. Beckner, C. Kenig, J. Pipher, A convexity property of eigenvalues, with applications. Unpublished. [BL89] H. Berestycki, B. Larrouturou, A semi-linear elliptic equation in a strip arising in a two-dimensional flame propagation model. J. Reine Angew. Math. 396, 14–40 (1989) [Bla06a] A. Blanchet, On the regularity of the free boundary in the parabolic obstacle problem. application to American options. Nonlinear Anal. 65(7), 1362–1378 (2006) [Bla06b] A. Blanchet, On the singular set of the parabolic obstacle problem. J. Diff. Equ. 231(2), 656–672 (2006) [BLN88] H. Berestycki, B. Larrouturou, L. Nirenberg, A nonlinear elliptic problem describing the propagation of a curved premixed flame, in Mathematical Modeling in Combustion and Related Topics (Lyon, 1987). NATO Advanced Science Institutes Series E: Applied Sciences, vol. 140 (Nijhoff, Dordrecht, 1988), pp. 11–28 [Caf76] L.A. Caffarelli, The smoothness of the free surface in a filtration problem. Arch. Rational Mech. Anal. 63(1), 77–86 (1976) [Caf77] L.A. Caffarelli, The regularity of free boundaries in higher dimensions. Acta Math. 139(3–4), 155–184 (1977) [Caf80] L.A. Caffarelli, Compactness methods in free boundary problems. Commun. Partial Diff. Equ. 5(4), 427–448 (1980) [Caf93] L.A. Caffarelli, A monotonicity formula for heat functions in disjoint domains, in Boundary Value Problems for Partial Differential Equations and Applications. RMA Research Notes in Applied Mathematics, vol. 29 (Masson, Paris, 1993), pp. 53–60 [Caf98] L.A. Caffarelli, The obstacle problem revisited. J. Fourier Anal. Appl. 4(4–5), 383–402 (1998) [CK98] L.A. Caffarelli, C.E. Kenig, Gradient estimates for variable coefficient parabolic equations and singular perturbation problems. Am. J. Math. 120(2), 391–439 (1998) [CKS00] L.A. Caffarelli, L. Karp, H. Shahgholian, Regularity of a free boundary with application to the Pompeiu problem. Ann. of Math. (2) 151(1), 269–292 (2000) [CPS04] L. Caffarelli, A. Petrosyan, H. Shahgholian, Regularity of a free boundary in parabolic potential theory. J. Am. Math. Soc. 17(4), 827–869 (electronic) (2004) [Cra87] J. Crank, Free and Moving Boundary Problems (The Clarendon Press Oxford University Press, New York, 1987) [CS04] L.A. Caffarelli, H. Shahgholian, The structure of the singular set of a free boundary in potential theory. Izv. Nats. Akad. Nauk Armenii Mat. 39(2), 43–58 (2004) [CS05] L. Caffarelli, S. Salsa, A Geometric Approach to Free Boundary Problems. Graduate Studies in Mathematics, vol. 68 (American Mathematical Society, Providence, RI, 2005) [DL76] G. Duvaut, J.-L. Lions, Inequalities in Mechanics and Physics (Springer, Berlin, 1976). Translated from the French by C. W. John, Grundlehren der Mathematischen Wissenschaften, 219 [Duv73] G. Duvaut, Résolution d’un problème de Stefan (fusion d’un bloc de glace à zéro degré). C. R. Acad. Sci. Paris Sér. A-B 276, A1461–A1463 (1973) [FH76] S. Friedland, W.K. Hayman, Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions. Comment. Math. Helv. 51(2), 133– 161 (1976)

References

139

[Fre72] J. Frehse, On the regularity of the solution of a second order variational inequality. Boll. Un. Mat. Ital. (4) 6, 312–315 (1972) [Fri82] A. Friedman, Variational Principles and Free-Boundary Problems. Pure and Applied Mathematics (Wiley, New York, 1982). A Wiley-Interscience Publication [FS86] A. Friedman, M. Sakai, A characterization of null quadrature domains in RN . Indiana Univ. Math. J. 35(3), 607–610 (1986) [GT01] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics (Springer, Berlin, 2001). Reprint of the 1998 edition. [Gur99] A. Gurevich, Boundary regularity for free boundary problems. Commun. Pure Appl. Math. 52(3), 363–403 (1999) [Isa75] V.M. Isakov, Inverse theorems on the smoothness of potentials. Differ. Uravn. 11, 64–74 (1975) [Jen80] R. Jensen, Boundary regularity for variational inequalities. Indiana Univ. Math. J. 29(4), 495–504 (1980) [Kar94] L. Karp, On the Newtonian potential of ellipsoids. Complex Variables Theory Appl. 25(4), 367–371 (1994) [KM96] L. Karp, A.S. Margulis, Newtonian potential theory for unbounded sources and applications to free boundary problems. J. Anal. Math. 70, 1–63 (1996) [KM12] L. Kapr, A.S. Margulis, Null quadrature domains and a free boundary problem for the Laplacian. Indiana Univ. Math. J. 61(2), 859–882 (2012) [KN77] D. Kinderlehrer, L. Nirenberg, Regularity in free boundary problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4(2), 373–391 (1977) [Kry84] N.V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain. Math. USSR Izv. 22(1), 67–98 (1984) [Kry96] N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, vol. 12 (American Mathematical Society, Providence, RI, 1996) [Kry08] N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. Graduate Studies in Mathematics, vol. 96 (American Mathematical Society, Providence, RI, 2008) [KS80] D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Pure and Applied Mathematics, vol. 88 (Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980) [LC31] G. Lamé, B.P. Clapeyron, Mémoire sur la solidification par refroidissement d’um globe solide. Ann. Chem. Phys. 47, 250–256 (1831) [LM15] E. Lindgren, R. Monneau, Pointwise regularity of the free boundary for the parabolic obstacle problem. Calc. Var. 54(1), 299–347 (2015) [LSU67] O.A. Ladyženskaja, V.A. Solonnikov, N.N. Ural ceva, Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23 (American Mathematical Society, Providence, RI, 1967). Translated from the Russian by S. Smith [LU68] O.A. Ladyzhenskaya, N.N. Ural tseva, Linear and Quasilinear Elliptic Equations (Academic Press, New York, 1968). Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis [NPP10] K. Nyström, A. Pascucci, S. Polidoro, Regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolic operators. J. Diff. Equ. 249(8), 2044– 2060 (2010) [NU11] A.I. Nazarov, N.N. Uraltseva, The Harnack inequality and related properties of solutions of elliptic and parabolic equations with divergence-free lower-order coefficients. Algebra i Analiz 23(1), 136–168 (2011) [Nys08] K. Nyström, On the behaviour near expiry for multi-dimensional American options. J. Math. Anal. Appl. 339(1), 644–654 (2008) [OU93a] V.I. Oliker, N.N. Uraltseva, Evolution of nonparametric surfaces with speed depending on curvature. II. The mean curvature case. Commun. Pure Appl. Math. 46(1), 97–135 (1993)

140

References

[OU93b] V.I. Oliker, N.N. Uraltseva, Evolution of nonparametric surfaces with speed depending on curvature. III. Some remarks on mean curvature and anisotropic flows, in Degenerate Diffusions (Minneapolis, MN, 1991). The IMA Volumes in Mathematics and Its Applications, vol. 47, pp. 141–156 (Springer, New York, 1993) [OU95] V.I. Oliker, N.N. Ural’tseva, Long time behavior of flows moving by mean curvature, in Nonlinear Evolution Equations. American Mathematical Society Translations: Series 2, vol. 164, pp. 163–170 (American Mathematical Society, Providence, RI, 1995) [OU97] V.I. Oliker, N.N. Ural’tseva, Long time behavior of flows moving by mean curvature. II. Topol. Methods Nonlinear Anal. 9(1), 17–28 (1997) [PS07] A. Petrosyan, H. Shahgholian, Parabolic obstacle problems applied to finance, in Recent Developments in Nonlinear Partial Differential Equations. Contemporary Mathematics, vol. 439, pp. 117–133 (American Mathematical Society, Providence, RI, 2007) [PSU12] A. Petrosyan, H. Shahgholian, N. Uraltseva, Regularity of Free Boundaries in ObstacleType Problems. Graduate Studies in Mathematics, vol. 136 (American Mathematical Society, Providence, RI, 2012) [Rod87] J.-F. Rodrigues, Obstacle Problems in Mathematical Physics. North-Holland Mathematics Studies, vol. 134 (North-Holland Publishing Co., Amsterdam, 1987) [Saf85] M.V. Safonov, Smoothness near the boundary of solutions of elliptic Bellman equations. Boundary value problems of mathematical physics and related problems in the theory of functions, no. 17 [Russian. English summary]. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 147, 150–154 (1985) [Sak81] M. Sakai, Null quadrature domains. J. Anal. Math. 40, 144–154 (1982), 1981 [Sak91] M. Sakai, Regularity of a boundary having a Schwarz function. Acta Math. 166(3–4), 263–297 (1991) [Sch77] D.G. Schaeffer, Some examples of singularities in a free boundary. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 4(1), 133–144 (1977) [Sha92] H. Shahgholian, On quadrature domains and the Schwarz potential. J. Math. Anal. Appl. 171(1), 61–78 (1992) [Sha03] H. Shahgholian, C 1,1 regularity in semilinear elliptic problems. Commun. Pure Appl. Math. 56(2), 278–281 (2003) [Sha08] H. Shahgholian, Free boundary regularity close to initial state for parabolic obstacle problem. Trans. Am. Math. Soc. 360(4), 2077–2087 (2008) [Ste89] J. Stefan, Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere. Sitzungsber. Österreich. Akad. Wiss. Math. Naturwiss. Kl. Abt. 2, Math. Astron. Phys. Meteorol. Tech. 98, 965–983 (1889) [SU03] H. Shahgholian, N. Uraltseva, Regularity properties of a free boundary near contact points with the fixed boundary. Duke Math. J. 116(1), 1–34 (2003) [SUW07] H. Shahgholian, N. Uraltseva, G.S. Weiss, The two-phase membrane problem— regularity of the free boundaries in higher dimensions. Int. Math. Res. Not. IRMN (8), 16 (2007) [SUW09] H. Shahgholian, N. Uraltseva, G.S. Weiss, A parabolic two-phase obstacle-like equation. Adv. Math. 221(3), 861–881 (2009) [SW06] H. Shahgholian, G.S. Weiss, The two-phase membrane problem—an intersectioncomparison approach to the regularity at branch points. Adv. Math. 205(2), 487–503 (2006) [Ura96] N.N. Uraltseva, C 1 regularity of the boundary of a noncoincident set in a problem with an obstacle. Algebra i Analiz 8(2), 205–221 (1996) [Ura97] N.N. Uraltseva, On some properties of the free boundary in a neighborhood of the points of contact with a given boundary. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 249(Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 29), 303–312 (1997) [Ura01] N.N. Uraltseva, Two-phase obstacle problem. J. Math. Sci. (New York) 106(3), 3073– 3077 (2001). Function theory and phase transitions

References

141

[Ura07] N.N. Uraltseva, Boundary estimates for solutions of elliptic and parabolic equations with discontinuous nonlinearities, in Nonlinear Equations and Spectral Theory. American Mathematical Society Translations Series 2, vol. 220, pp. 235–246 (American Mathematical Society, Providence, RI, 2007) [Wei98] G.S. Weiss, Partial regularity for weak solutions of an elliptic free boundary problem. Commun. Partial Diff. Equ. 23(3–4), 439–455 (1998) [Wei99] G.S. Weiss, Self-similar blow-up and Hausdorff dimension estimates for a class of parabolic free boundary problems. SIAM J. Math. Anal. 30(3), 623–644 (electronic) (1999) [Wei01] G.S. Weiss, An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary. Interfaces Free Bound. 3(2), 121–128 (2001)

Index

ACF-monotonicity formula, 113

balanced energy, 51 blow-down, 49 blow-up, 49

C-monotonicity formula, 107 local version, 109, 112 scaling property, 113 classical obstacle problem, 2, 4

low energy point, 51 zero energy point, 51 free boundary problem, 1 function caloric, 123 subcaloric, 123 supercaloric, 123

heat kernel, 107

Liouville type Theorem, 125 elliptic obstacle problem, 12, 73 estimates for mixed derivatives, 75 estimates for pure derivatives, 75 example, 13 free boundary tangential touch, 83 PR+ (M, L), 74 Γ (u), 74 Γc (u), 74 Σint (u), 74

free boundary, 1 1/2 Cx1 ∩ Ct regularity, 8 1,α C regularity, 7, 31 C ∞ regularity, 7, 31 (n + 1)-Lebesgue measure, 48 Lipschitz regularity, 7, 31 free boundary points contact points, 4, 29 high energy point, 51

monotonicity formula Alt-Caffarelli-Friedman (see ACFmonotonicity formula) Caffarelli (see C-monotonicity formula) Weiss (see W-monotonicity formula)

no-sign parabolic obstacle problem. see no-sign parabolic problem no-sign parabolic problem, 4, 29 classification of global solutions, 30 continuity of ut , 66 convergence, 48 counterexample, 31 directional monotonicity, 71 free boundary Lipschitz implies C 1,α , 132 Lipschitz regularity, 7, 31 (n + 1)-Lebesgue measure, 48 1/2 Cx1 ∩ Ct regularity, 8

© Springer Nature Switzerland AG 2018 D. Apushkinskaya, Free Boundary Problems, Lecture Notes in Mathematics 2218, https://doi.org/10.1007/978-3-319-97079-0

143

144 C 1,α regularity, 7, 31 C ∞ regularity, 7, 31 Γ (u), 29 global solutions, 30 local solutions, 30 nondegeneracy, 44 optimal regularity, 7, 30 PR+ (M), 30 + (M), 30 P∞ PR (M), 30 P∞ (M), 30 obstacle-type problem, 2 one-phase Stefan problem, 2, 31 parabolic homogeneous of degree 2, 50 two-phase elliptic problem, 15, 85 boundary estimates of derivatives, 86 example, 16 optimal regularity, 17

Index Γ (u), 85 Γ ∗ (u), 85 Γ 0 (u), 85 two-phase parabolic problem estimates of derivatives near {t = 0}, 99 estimates of derivatives near Π1 , 93 example, 20 near {t = 0}, 22, 98 optimal regularity, 23 near Π1 , 20, 92 regularity, 21 regularized problem, 100 Γ (u), 99 Γ ∗ (u), 99 Γ 0 (u), 99

W-monotonicity formula, 115 Weiss functional derivative, 118 global, 115 local, 117 scaling property, 115, 117

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