Nonlocal diffusion and applications 9783319287386, 9783319287393

Table of contents :
Preface......Page 6
Acknowledgments......Page 8
Contents......Page 10
Introduction......Page 12
1 A Probabilistic Motivation......Page 14
1.1 The Random Walk with Arbitrarily Long Jumps......Page 15
1.2 A Payoff Model......Page 17
2.1 Preliminary Notions......Page 19
2.2 Fractional Sobolev Inequality and Generalized Coarea Formula......Page 28
2.3 Maximum Principle and Harnack Inequality......Page 31
2.4 An s-Harmonic Function......Page 36
2.5 All Functions Are Locally s-Harmonic Up to a Small Error......Page 41
2.6 A Function with Constant Fractional Laplacian on the Ball......Page 45
3 Extension Problems......Page 50
3.1 Water Wave Model......Page 51
3.1.1 Application to the Water Waves......Page 53
3.2 Crystal Dislocation......Page 54
3.3 An Approach to the Extension Problem via the Fourier Transform......Page 67
4 Nonlocal Phase Transitions......Page 77
4.1 The Fractional Allen-Cahn Equation......Page 80
4.2 A Nonlocal Version of a Conjecture by De Giorgi......Page 95
5 Nonlocal Minimal Surfaces......Page 106
5.1 Graphs and s-Minimal Surfaces......Page 110
5.2 Non-existence of Singular Cones in Dimension 2......Page 120
5.3 Boundary Regularity......Page 128
6 A Nonlocal Nonlinear Stationary Schrödinger Type Equation......Page 136
6.1 From the Nonlocal Uncertainty Principle to a Fractional Weighted Inequality......Page 145
A.1 Another Proof of Theorem 2.4.1......Page 148
A.2 Another Proof of Lemma 2.3......Page 152
References......Page 158

Citation preview

Lecture Notes of the Unione Matematica Italiana

Claudia Bucur Enrico Valdinoci

Nonlocal Diffusion and Applications

Lecture Notes of the Unione Matematica Italiana

More information about this series at http://www.springer.com/series/7172

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Editorial Board Ciro Ciliberto (Editor in Chief) Dipartimento di Matematica Università di Roma Tor Vergata Via della Ricerca Scientifica  Roma, Italy e-mail: [email protected].it

Franco Flandoli Dipartimento di Matematica Applicata Università di Pisa Via Buonarroti c  Pisa, Italy e-mail: fl[email protected]

Susanna Terracini (Co-editor in Chief) Università degli Studi di Torino Dipartimento di Matematica “Giuseppe Peano” Via Carlo Alberto   Torino, Italy e-mail: [email protected]

Angus MacIntyre Queen Mary University of London School of Mathematical Sciences Mile End Road London E NS, United Kingdom e-mail: [email protected]

Adolfo Ballester-Bollinches Department d’Àlgebra Facultat de Matemàtiques Universitat de València Dr. Moliner,   Burjassot (València), Spain e-mail: [email protected] Annalisa Buffa IMATI – C.N.R. Pavia Via Ferrata   Pavia, Italy e-mail: [email protected] Lucia Caporaso Dipartimento di Matematica Università Roma Tre Largo San Leonardo Murialdo I- Roma, Italy e-mail: [email protected].it Fabrizio Catanese Mathematisches Institut Universitätstraÿe   Bayreuth, Germany e-mail: [email protected] Corrado De Concini Dipartimento di Matematica Università di Roma “La Sapienza” Piazzale Aldo Moro   Roma, Italy e-mail: [email protected].it Camillo De Lellis Institut für Mathematik Universität Zürich Winterthurerstrasse  CH- Zürich, Switzerland e-mail: [email protected]

Giuseppe Mingione Dipartimento di Matematica e Informatica Università degli Studi di Parma Parco Area delle Scienze, /a (Campus)  Parma, Italy e-mail: [email protected] Mario Pulvirenti Dipartimento di Matematica Università di Roma “La Sapienza” P.le A. Moro   Roma, Italy e-mail: [email protected].it Fulvio Ricci Scuola Normale Superiore di Pisa Piazza dei Cavalieri   Pisa, Italy e-mail: [email protected] Valentino Tosatti Northwestern University Department of Mathematics  Sheridan Road Evanston, IL , USA e-mail: [email protected] Corinna Ulcigrai Forschungsinstitut für Mathematik HG G . Rämistrasse   Zürich, Switzerland e-mail: [email protected]

The Editorial Policy can be found at the back of the volume.

Claudia Bucur • Enrico Valdinoci

Nonlocal Diffusion and Applications

123

Claudia Bucur Dipartimento di Matematica Federigo Enriques UniversitJa degli Studi di Milano Milano, Italy

Enrico Valdinoci Dipartimento di Matematica Federigo Enriques UniversitJa degli Studi di Milano Milano, Italy Consiglio Nazionale delle Ricerche Istituto di Matematica Applicata e Tecnologie Informatiche Enrico Magene Pavia, Italy Weierstraß Institut für Angewandte Analysis und Stochasitk Berlin, Germany University of Melbourne School of Mathematics and Statistics Victoria, Australia

ISSN 1862-9113 ISSN 1862-9121 (electronic) Lecture Notes of the Unione Matematica Italiana ISBN 978-3-319-28738-6 ISBN 978-3-319-28739-3 (eBook) DOI 10.1007/978-3-319-28739-3 Library of Congress Control Number: 2016934714 © Springer International Publishing Switzerland 2016 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. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

Preface

The purpose of these pages is to collect a set of notes that are a result of several talks and minicourses delivered here and there in the world (Milan, Cortona, Pisa, Roma, Santiago del Chile, Madrid, Bologna, Porquerolles, and Catania to name a few). We will present here some mathematical models related to nonlocal equations, providing some introductory material and examples. Of course, these notes and the results presented do not aim to be comprehensive and cannot take into account all the material that would deserve to be included. Even a thorough introduction to nonlocal (or even just fractional) equations goes way beyond the purpose of this book. Using a metaphor with fine arts, we could say that the picture that we painted here is not even impressionistic, it is just naïf. Nevertheless, we hope that these pages may be of some help to the young researchers of all ages who are willing to have a look at the exciting nonlocal scenario (and who are willing to tolerate the partial and incomplete point of view offered by this modest observation point). Milano, Italy Milano, Italy November 2015

Claudia Bucur Enrico Valdinoci

v

Acknowledgments

It is a pleasure to thank Serena Dipierro, Rupert Frank, Richard Mathar, Alexander Nazarov, Joaquim Serra, and Fernando Soria for very interesting and pleasant discussions. We are also indebted with all the participants of the seminars and minicourses from which this set of notes generated for the nice feedback received, and we hope that this work, though somehow sketchy and informal, can be useful to stimulate new discussions and further develop this rich and interesting subject.

vii

Contents

1

A Probabilistic Motivation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 The Random Walk with Arbitrarily Long Jumps .. . . . . . . . . . . . . . . . . . . . 1.2 A Payoff Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 2 4

2 An Introduction to the Fractional Laplacian . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Preliminary Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Fractional Sobolev Inequality and Generalized Coarea Formula . . . . 2.3 Maximum Principle and Harnack Inequality .. . . . .. . . . . . . . . . . . . . . . . . . . 2.4 An s-Harmonic Function .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 All Functions Are Locally s-Harmonic Up to a Small Error . . . . . . . . . 2.6 A Function with Constant Fractional Laplacian on the Ball . . . . . . . . .

7 7 16 19 24 29 33

3 Extension Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Water Wave Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 Application to the Water Waves . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Crystal Dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 An Approach to the Extension Problem via the Fourier Transform .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

39 40 42 43

4 Nonlocal Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 The Fractional Allen-Cahn Equation . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 A Nonlocal Version of a Conjecture by De Giorgi . . . . . . . . . . . . . . . . . . .

67 70 85

56

5 Nonlocal Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 97 5.1 Graphs and s-Minimal Surfaces . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101 5.2 Non-existence of Singular Cones in Dimension 2 . . . . . . . . . . . . . . . . . . . . 111 5.3 Boundary Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 119 6 A Nonlocal Nonlinear Stationary Schrödinger Type Equation . . . . . . . . . 127 6.1 From the Nonlocal Uncertainty Principle to a Fractional Weighted Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 136

ix

x

Contents

A Alternative Proofs of Some Results . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 139 A.1 Another Proof of Theorem 2.4.1 .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 139 A.2 Another Proof of Lemma 2.3 . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 143 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 149

Introduction

In the recent years the fractional Laplace operator has received much attention both in pure and in applied mathematics. Starting from the basics of the nonlocal equations, in this set of notes we will discuss in detail some recent developments in four topics of research on which we focused our attention, namely: • A problem arising in crystal dislocation (which is related to a classical model introduced by Peierls and Nabarro) • A problem arising in phase transitions (which is related to a nonlocal version of the classical Allen–Cahn equation) • The limit interfaces arising in the above nonlocal phase transitions (which turn out to be nonlocal minimal surfaces, as introduced by Caffarelli, Roquejoffre, and Savin) • A nonlocal version of the Schrödinger equation for standing waves (as introduced by Laskin) Many fundamental topics slipped completely out of these notes: just to name a few, the topological methods and the fine regularity theory in the fractional cases are not presented here; the fully nonlinear or singular/degenerate equations are not taken into account; only very few applications are discussed briefly; important models such as the quasi-geostrophic equation and the fractional porous media equation are not covered in these notes; we will not consider models arising in game theory such as the nonlocal tug-of-war; the parabolic equations are not taken into account in detail; unique continuation and overdetermined problems will not be studied here, and the link to probability theory that we consider here is not rigorous and only superficial (the reader interested in these important topics may look, for instance, at [8, 10, 14, 15, 17–20, 31, 36–38, 41, 53, 65, 66, 70, 71, 88, 95, 97–100, 109, 113, 127, 133]). Also, a complete discussion of the nonlocal equations in bounded domains is not available here (for this, we refer to the recent survey [119]). In terms

xi

xii

Introduction

of surveys, collections of results, and open problems, we also mention the very nice website [2], which gets1 constantly updated. This set of notes is organized as follows. To start with, in Chap. 1, we will give a motivation for the fractional Laplacian (which is the typical nonlocal operator for our framework) that originates from probabilistic considerations. As a matter of fact, no advanced knowledge of probability theory is assumed from the reader, and the topic is dealt with at an elementary level. In Chap. 2, we will recall some basic properties of the fractional Laplacian, discuss some explicit examples in detail, and point out some structural inequalities that are due to a fractional comparison principle. This part continues with a quite surprising result, which states that every function can be locally approximated by functions with vanishing fractional Laplacian (in sharp contrast with the rigidity of the classical harmonic functions). We also give an example of a function with constant fractional Laplacian on the ball. In Chap. 3 we deal with extended problems. It is indeed a quite remarkable fact that in many occasions nonlocal operators can be equivalently represented as local (though possibly degenerate or singular) operators in one dimension more. Moreover, as a counterpart, several models arising in a local framework give rise to nonlocal equations, due to boundary effects. So, to introduce the extension problem and give a concrete intuition of it, we will present some models in physics that are naturally set on an extended space to start with and will show their relation with the fractional Laplacian on a trace space. We will also give a detailed justification of this extension procedure by means of the Fourier transform. As a special example of problems arising in physics that produce a nonlocal equation, we consider a problem related to crystal dislocation, present some mathematical results that have been recently obtained on this topic, and discuss the relation between these results and the observable phenomena. Chapters 4, 5, and 6 present topics of contemporary research. We will discuss in particular: some phase transition equations of nonlocal type; their limit interfaces, which (below a critical threshold of the fractional parameter) are surfaces that minimize a nonlocal perimeter functional; and some nonlocal equations arising in quantum mechanics. We remark that the introductory part of these notes is not intended to be separated from the one which is more research oriented, namely, even the chapters whose main goal is to develop the basics of the theory contain some parts related to contemporary research trends.

1

It seems to be known that Plato did not like books because they cannot respond to questions. He might have liked websites.

Chapter 1

A Probabilistic Motivation

The fractional Laplacian will be the main operator studied in this book. We consider a function uW Rn ! R (which is supposed1 to be regular enough) and a fractional parameter s 2 .0; 1/. Then, the fractional Laplacian of u is given by C.n; s/ ./ u.x/ D 2

Z

s

Rn

2u.x/  u.x C y/  u.x  y/ dy; jyjnC2s

(1.1)

where C.n; s/ is a dimensional2 constant. One sees from (1.1) that ./s is an operator of order 2s, namely, it arises from a differential quotient of order 2s weighted in the whole space. Different fractional operators have been considered in literature (see e.g. [39, 111, 128]), and all of them come from interesting problems in pure or/and applied mathematics. We will focus here on the operator in (1.1) and we will motivate it by probabilistic considerations (as a matter of fact, many other motivations are possible). The probabilistic model under consideration is a random process that allows long jumps (in further generality, it is known that the fractional Laplacian is an infinitesimal generator of Lèvy processes, see e.g. [7, 13] for further details). A more detailed mathematical introduction to the fractional Laplacian is then presented in the subsequent Sect. 2.1.

To write (1.1) it is sufficient, for simplicity, to take here u in the Schwartz space S .Rn / of smooth and rapidly decaying functions, or in C2 .Rn / \ L1 .Rn /. We refer to [131] for a refinement of the space of definition.

1

2

The explicit value of C.n; s/ is usually unimportant. Nevertheless, we will compute its value explicitly in formulas (2.10) and (2.15). The reason for which it is convenient to divide C.n; s/ by a factor 2 in (1.1) will be clear later on, in formula (2.5). © Springer International Publishing Switzerland 2016 C. Bucur, E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana 20, DOI 10.1007/978-3-319-28739-3_1

1

2

1 A Probabilistic Motivation

1.1 The Random Walk with Arbitrarily Long Jumps We will show here that the fractional heat equation (i.e. the “typical” equation that drives the fractional diffusion and that can be written, up to dimensional constants, as @t uC./s u D 0) naturally arises from a probabilistic process in which a particle moves randomly in the space subject to a probability that allows long jumps with a polynomial tail. For this scope, we introduce a probability distribution on the natural numbers N WD f1; 2; 3;    g as follows. If I  N , then the probability of I is defined to be P.I/ WD cs

X k2I

1 jkj1C2s

:

The constant cs is taken in order to normalize P to be a probability measure. Namely, we take cs WD

X k2N

1 jkj1C2s

!1 ;

so that we have P.N / D 1. Now we consider a particle that moves in Rn according to a probabilistic process. The process will be discrete both in time and space (in the end, we will formally take the limit when these time and space steps are small). We denote by  the discrete time step, and by h the discrete space step. We will take the scaling  D h2s and we denote by u.x; t/ the probability of finding the particle at the point x at time t. The particle in Rn is supposed to move according to the following probabilistic law: at each time step , the particle selects randomly both a direction v 2 @B1 , according to the uniform distribution on @B1 , and a natural number k 2 N , according to the probability law P, and it moves by a discrete space step khv. Notice that long jumps are allowed with small probability. Then, if the particle is at time t at the point x0 and, following the probability law, it picks up a direction v 2 @B1 and a natural number k 2 N , then the particle at time t C  will lie at x0 C khv. Now, the probability u.x; t C / of finding the particle at x at time t C  is the sum of the probabilities of finding the particle somewhere else, say at x C khv, for some direction v 2 @B1 and some natural number k 2 N , times the probability of having selected such a direction and such a natural number. This translates into Z cs X u.x C khv; t/ dH n1 .v/: u.x; t C / D j@B1 j jkj1C2s  @B1 k2N

1.1 The Random Walk with Arbitrarily Long Jumps

3

Notice that the factor cs =j@B1 j is a normalizing probability constant, hence we subtract u.x; t/ and we obtain Z cs X u.x C khv; t/ dH n1 .v/  u.x; t/ 1C2s j@B1 j jkj @B 1 k2N Z X cs u.x C khv; t/  u.x; t/ D dH n1 .v/: j@B1 j jkj1C2s  @B1

u.x; t C /  u.x; t/ D

k2N

As a matter of fact, by symmetry, we can change v to v in the integral above, so we find that Z cs X u.x  khv; t/  u.x; t/ u.x; t C /  u.x; t/ D dH n1 .v/: 1C2s j@B1 j jkj @B 1  k2N

Then we can sum up these two expressions (and divide by 2) and obtain that u.x; t C /  u.x; t/ X Z u.x C khv; t/ C u.x  khv; t/  2u.x; t/ cs dH n1 .v/: D 1C2s 2 j@B1 j jkj  @B1 k2N

Now we divide by  D h2s , we recognize a Riemann sum, we take a formal limit and we use polar coordinates, thus obtaining: u.x; t C /  u.x; t/  Z X cs h u.x C khv; t/ C u.x  khv; t/  2u.x; t/ D dH n1 .v/ 1C2s 2 j@B1 j jhkj  @B1

@t u.x; t/ '

k2N

'

cs 2 j@B1 j

cs D 2 j@B1 j

Z

C1

0

Z

Rn

Z @B1

u.x C rv; t/ C u.x  rv; t/  2u.x; t/ dH n1 .v/ dr jrj1C2s

u.x C y; t/ C u.x  y; t/  2u.x; t/ dy jyjnC2s

D  cn;s ./s u.x; t/ for a suitable cn;s > 0. This shows that, at least formally, for small time and space steps, the above probabilistic process approaches a fractional heat equation. We observe that processes of this type occur in nature quite often, see in particular the biological observations in [90, 140], other interesting observations in [118, 126, 142] and the mathematical discussions in [84, 93, 104, 107, 110].

4

1 A Probabilistic Motivation

Fig. 1.1 The random walk with jumps

Roughly speaking, let us say that it is not unreasonable that a predator may decide to use a nonlocal dispersive strategy to hunt its preys more efficiently (or, equivalently, that the natural selection may favor some kind of nonlocal diffusion): small fishes will not wait to be eaten by a big fish once they have seen it, so it may be more convenient for the big fish just to pick up a random direction, move rapidly in that direction, stop quickly and eat the small fishes there (if any) and then go on with the hunt. And this “hit-and-run” hunting procedure seems quite related to that described in Fig. 1.1.

1.2 A Payoff Model Another probabilistic motivation for the fractional Laplacian arises from a payoff approach. Suppose to move in a domain ˝ according to a random walk with jumps as discussed in Sect. 1.1. Suppose also that exiting the domain ˝ for the first time by jumping to an outside point y 2 Rn n ˝, means earning u0 .y/ sestertii. A relevant question is, of course, how rich we expect to become in this way. That is, if we start at a given point x 2 ˝ and we denote by u.x/ the amount of sestertii that we expect to gain, is there a way to obtain information on u? The answer is that (in the right scale limit of the random walk with jumps presented in Sect. 1.1) the expected payoff u is determined by the equation (

./s u D 0

in ˝;

u D u0

in Rn n ˝:

(1.2)

To better explain this, let us fix a point x 2 ˝. The expected value of the payoff at x is the average of all the payoffs at the points xQ from which one can reach x, weighted by the probability of the jumps. That is, by writing xQ D x C khv, with v 2 @B1 ,

1.2 A Payoff Model

5

k 2 N and h > 0, as in the previous Sect. 1.1, we have that the probability of jump cs is . This leads to the formula j@B1 j jkj1C2s Z cs X u.x C khv/ u.x/ D dH n1 .v/: j@B1 j jkj1C2s  @B1 k2N

By changing v into v we obtain u.x/ D

Z cs X u.x  khv/ dH n1 .v/ 1C2s j@B1 j jkj @B 1  k2N

and so, by summing up, Z cs X u.x C khv/ C u.x  khv/ 2u.x/ D dH n1 .v/: j@B1 j jkj1C2s  @B1 k2N

Since the total probability is 1, we can subtract 2u.x/ to both sides and obtain that 0D

Z cs X u.x C khv/ C u.x  khv/  2u.x/ dH n1 .v/: 1C2s j@B1 j jkj  @B1 k2N

We can now divide by h1C2s and recognize a Riemann sum, which, after passing to the limit as h & 0, gives 0 D ./s u.x/, that is (1.2).

Chapter 2

An Introduction to the Fractional Laplacian

We introduce here some preliminary notions on the fractional Laplacian and on fractional Sobolev spaces. Moreover, we present an explicit example of an sharmonic function on the positive half-line RC , an example of a function with constant Laplacian on the ball, discuss some maximum principles and a Harnack inequality, and present a quite surprising local density property of s-harmonic functions into the space of smooth functions.

2.1 Preliminary Notions We introduce here the fractional Laplace operator, the fractional Sobolev spaces and give some useful pieces of notation. We also refer to [57] for further details related to the topic. We consider the Schwartz space of rapidly decaying functions defined as 

 ˇ ˇ n ˛ S .R / WD f 2 C .R / ˇ 8˛; ˇ 2 N0 ; sup jx @ˇ f .x/j < 1 : n

1

n

x2Rn

For any f 2 S .Rn /, denoting the space variable x 2 Rn and the frequency variable  2 Rn , the Fourier transform and the inverse Fourier transform are defined, respectively, as fO ./ WD F f ./ WD

Z Rn

f .x/e2ix dx

(2.1)

fO ./e2ix d:

(2.2)

and f .x/ D F 1 fO .x/ D

Z Rn

© Springer International Publishing Switzerland 2016 C. Bucur, E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana 20, DOI 10.1007/978-3-319-28739-3_2

7

8

2 An Introduction to the Fractional Laplacian

Another useful notion is the one of principal value, namely we consider the definition Z Z u.x/  u.y/ u.x/  u.y/ dy WD lim dy: (2.3) P.V. nC2s nC2s "!0 n n jx  yj R R nB" .x/ jx  yj Notice indeed that the integrand above is singular when y is in a neighborhood of x, and this singularity is, in general, not integrable (in the sense of Lebesgue): indeed notice that, near x, we have that u.x/  u.y/ behaves at the first order like ru.x/  .x  y/, hence the integral above behaves at the first order like ru.x/  .x  y/ jx  yjnC2s

(2.4)

whose absolute value gives an infinite integral near x (unless either ru.x/ D 0 or s < 1=2). The idea of the definition in (2.3) is that the term in (2.4) averages out in a neighborhood of x by symmetry, since the term is odd with respect to x, and so it does not contribute to the integral if we perform it in a symmetric way. In a sense, the principal value in (2.3) kills the first order of the function at the numerator, which produces a linear growth, and focuses on the second order remainders. The notation in (2.3) allows us to write (1.1) in the following more compact form: ./s u.x/ D D D D D

Z

2u.x/  u.x C y/  u.x  y/ dy jyjnC2s Rn Z 2u.x/  u.x C y/  u.x  y/ C.n; s/ dy lim 2 "!0 Rn nB" jyjnC2s  Z Z u.x/  u.x C y/ u.x/  u.x  y/ C.n; s/ dy C dy lim 2 "!0 Rn nB" jyjnC2s jyjnC2s Rn nB"  Z Z u.x/  u./ u.x/  u./ C.n; s/ d C d lim nC2s 2 "!0 Rn nB" .x/ jx  jnC2s Rn nB" .x/ jx  j Z u.x/  u./ d; C.n; s/ lim "!0 Rn nB" .x/ jx  jnC2s C.n; s/ 2

where the changes of variable  WD x C y and  WD x  y were used, i.e. Z ./ u.x/ D C.n; s/ P.V. s

Rn

u.x/  u.y/ dy: jx  yjnC2s

(2.5)

The simplification above also explains why it was convenient to write (1.1) with the factor 2 dividing C.n; s/. Notice that the expression in (1.1) does not require the P.V. formulation since, for instance, taking u 2 L1 .Rn / and locally C2 , using a Taylor

2.1 Preliminary Notions

9

expansion of u in B1 , one observes that Z

j2u.x/  u.x C y/  u.x  y/j dy jyjnC2s Rn Z Z jD2 u.x/jjyj2 jyjn2s dy C dy kukL1 .Rn / jyjnC2s Rn nB1 B1 Z Z n2s 2 jyj dy C kD ukL1 .Rn / jyjn2sC2 dy; kukL1 .Rn / Rn nB1

B1

and the integrals above provide a finite quantity. Formula (2.5) has also a stimulating analogy with the classical Laplacian. Namely, the classical Laplacian (up to normalizing constants) is the measure of the infinitesimal displacement of a function in average (this is the “elastic” property of harmonic functions, whose value at a given point tends to revert to the average in a ball). Indeed, by canceling the odd contributions, and using that Z

jx  yj2 dy D Br .x/

n Z X kD1

Br .x/

.xk  yk /2 dy D n

Z Br .x/

.xi  yi /2 dy;

for any i 2 f1; : : : ; ng; we see that 1 lim r!0 r2 D lim  r!0

D lim  r!0

 u.x/ 

1 r2 jBr .x/j 1 rnC2 jB1 j

 Z 1 u.y/ dy jBr .x/j Br .x/ Z  u.y/  u.x/ dy Br .x/

Z

1 ru.x/  .x  y/ C D2 u.x/.x  y/  .x  y/ 2 Br .x/

C O.jx  yj3 / dy n Z X 1 @2 u.x/ .xi  yi /.xj  yj / dy D lim  nC2 r!0 2r jB1 j i;jD1 Br .x/ i;j D lim  r!0

D lim  r!0

1 2rnC2 jB1 j

n Z X iD1

1

n X

2n rnC2 jB1 j

iD1

D  Cn u.x/;

Br .x/

@2i;i u.x/ .xi  yi /2 dy

@2i;i u.x/

Z

jx  yj2 dy Br .x/

(2.6)

10

2 An Introduction to the Fractional Laplacian

for some Cn > 0. In this spirit, when we compare the latter formula with (2.5), we can think that the fractional Laplacian corresponds to a weighted average of the function’s oscillation. While the average in (2.6) is localized in the vicinity of a point x, the one in (2.5) occurs in the whole space (though it decays at infinity). Also, the spacial homogeneity of the average in (2.6) has an extra factor that is proportional to the space variables to the power 2, while the corresponding power in the average in (2.5) is 2s (and this is consistent for s ! 1). Furthermore, for u 2 S .Rn / the fractional Laplace operator can be expressed in Fourier frequency variables multiplied by .2jj/2s , as stated in the following lemma. Lemma 2.1 We have that  ./s u.x/ D F 1 .2jj/2s uO ./ :

(2.7)

Roughly speaking, formula (2.7) characterizes the fractional Laplace operator in the Fourier space, by taking the s-power of the multiplier associated to the classical Laplacian operator. Indeed, by using the inverse Fourier transform, one has that u.x/ D .F 1 .Ou//.x/ D  Z D Rn

Z Rn

uO ./e2ix d

 .2jj/2 uO ./e2ix d D F 1 .2jj/2 uO ./ ;

which gives that the classical Laplacian acts in a Fourier space as a multiplier of .2jj/2 . From this and Lemma 2.1 it also follows that the classical Laplacian is the limit case of the fractional one, namely for any u 2 S .Rn / lim ./s u D u

s!1

and also

lim ./s u D u:

s!0

Let us now prove that indeed the two formulations (1.1) and (2.7) are equivalent. Proof (Proof of Lemma 2.1) Consider identity (1.1) and apply the Fourier transform to obtain

 Z F 2u.x/  u.x C y/  u.x  y/ 

C.n; s/ dy F ./s u.x/ D 2 jyjnC2s Rn Z 2  e2iy  e2iy C.n; s/ (2.8) uO ./ dy D 2 jyjnC2s Rn Z 1  cos.2  y/ dy: D C.n; s/ uO ./ n jyjnC2s R

2.1 Preliminary Notions

11

Now, we use the change of variable z D jjy and obtain that Z

1  cos.2  y/ dy jyjnC2s Rn Z 1  cos 2  z jj 2s D jj dz: jzjnC2s Rn

J./ WD

Now we use that J is rotationally invariant. More precisely, we consider a rotation R that sends e1 D .1; 0; : : : ; 0/ into =jj, that is Re1 D =jj, and we call RT its transpose. Then, by using the change of variables ! D RT z we have that J./ D jj2s D jj2s D jj2s

Z Z Z

Rn

1  cos.2Re1  z/ dz jzjnC2s

Rn

1  cos.2RT z  e1 / dz jRT zjnC2s

Rn

1  cos.2!1 / d!: j!jnC2s

Changing variables !Q D 2! (we still write ! as a variable of integration), we obtain that Z 1  cos !1 2s J./ D .2jj/ d!: (2.9) nC2s Rn j!j Notice that this latter integral is finite. Indeed, integrating outside the ball B1 we have that Z Z j1  cos !1 j 2 d!  < 1; nC2s nC2s j!j Rn nB1 Rn nB1 j!j while inside the ball we can use the Taylor expansion of the cosine function and observe that Z Z Z j1  cos !1 j j!j2 d! d!  d!  < 1: nC2s nC2s nC2s2 j!j j!j j!j B1 B1 B1 Therefore, by taking Z C.n; s/ WD Rn

1  cos !1 d! j!jnC2s

it follows from (2.9) that J./ D

.2jj/2s : C.n; s/

1 (2.10)

12

2 An Introduction to the Fractional Laplacian

By inserting this into (2.8), we obtain that 

F ./s u.x/ D C.n; s/ uO ./ J./ D .2jj/2s uO ./; which concludes the proof. Notice that the renormalization constant C.n; s/ introduced in (1.1) is now computed in (2.10). Another approach to the fractional Laplacian comes from the theory of semigroups (or, equivalently from the fractional calculus arising in subordination identities). This technique is classical (see [143]), but it has also been efficiently used in recent research papers (see for instance [33, 51, 134]). Roughly speaking, the main idea underneath the semigroup approach comes from the following explicit formulas for the Euler’s function: for any  > 0, one uses an integration by parts and the substitution  D t to see that s .s/ D .1  s/ Z C1  s e d D 0

Z

C1

 s

D 0

Z

C1

Ds 0

D  ss

Z

d  .e  1/ d d

 s1 .e  1/ d C1

0

ts1 .et  1/ dt;

that is s D

1 .s/

Z

C1

0

ts1 .et  1/ dt:

(2.11)

Then one applies formally this identity to  WD . Of course, this formal step needs to be justified, but if things go well one obtains ./s D

1 .s/

Z

C1 0

ts1 .et  1/ dt;

that is (interpreting 1 as the identity operator) ./s u.x/ D

1 .s/

Z

C1 0

ts1 .et u.x/  u.x// dt:

(2.12)

2.1 Preliminary Notions

13

Formally, if U.x; t/ WD et u.x/, we have that U.x; 0/ D u.x/ and @t U D

@ t .e u.x// D et u.x/ D U; @t

that is U.x; t/ D et u.x/ can be interpreted as the solution of the heat equation with initial datum u. We indeed point out that these formal computations can be justified: Lemma 2.2 Formula (2.12) holds true. That is, if u 2 S .Rn / and U D U.x; t/ is the solution of the heat equation  @t U ˇ D U in t > 0; U ˇtD0 D u; then ./s u.x/ D

1 .s/

Z

C1 0

ts1 .U.x; t/  u.x// dt:

(2.13)

Proof From Theorem 1 on page 47 in [69] we know that U is obtained by Gaussian convolution with unit mass, i.e. Z Z U.x; t/ D G.x  y; t/ u.y/ dy D G.y; t/ u.x  y/ dy; Rn Rn (2.14) 2 =.4t/

where G.x; t/ WD .4t/n=2 ejxj

:

As a consequence, using the substitution  WD jyj2 =.4t/, Z Z

C1 0 C1

ts1 .U.x; t/  u.x// dt Z

D

t Z

0 C1

Z

Z

C1

Z

Rn

D Rn

0

D2

2s1









G.y; t/ u.x  y/  u.x/ dy dt

Rn

D 0

s1

n=2

.4t/ 

n=2

Z

n=2 s1 jyj2 =.4t/

t

e

2 n=2

.jyj /

C1

Z Rn

0



jyj

2s

n 2 Cs1

e



sC1 

e

 u.x  y/  u.x/ dy

C1 0

n



d 4 2

 u.x C y/ C u.x  y/  2u.x/ dy d: jyjnC2s

Now we notice that Z



u.x  y/  u.x/ dy dt

.4/





 2 Cs1 e d D

n 2

 Cs ;

14

2 An Introduction to the Fractional Laplacian

so we obtain that Z

C1 0

ts1 .U.x; t/  u.x// dt

D 22s1  n=2

n 2 n

Z Cs Rn

u.x C y/ C u.x  y/  2u.x/ dy jyjnC2s

22s  n=2 2 C s ./s u.x/: D C.n; s/

This proves (2.13), by choosing C.n; s/ appropriately. And, as a matter of fact, gives the explicit value of the constant C.n; s/ as   22s n2 C s 22s s n2 C s C.n; s/ D  n=2 D n=2 ;  .s/  .1  s/

(2.15)

where we have used again that .1  s/ D s .s/, for any s 2 .0; 1/. It is worth pointing out that the renormalization constant C.n; s/ introduced in (1.1) has now been explicitly computed in (2.15). Notice that the choices of C.n; s/ in (2.10) and (2.15) must agree (since we have computed the fractional Laplacian in two different ways): for a computation that shows that the quantity in (2.10) coincides with the one in (2.15), see Theorem 3.9 in [22]. For completeness, we give below a direct proof that the settings in (2.10) and (2.15) are the same, by using Fourier methods and (2.11). Lemma 2.3 For any n 2 N, n  1, and s 2 .0; 1/, we have that Z

1  cos.2!1 /  2 C2s .1  s/  : d! D j!jnC2s s n2 C s n

Rn

(2.16)

Equivalently, we have that Z Rn

n

1  cos !1  2 .1  s/  : d! D 2s nC2s j!j 2 s n2 C s

(2.17)

Proof Of course, formula (2.16) is equivalent to (2.17) (after the substitution !Q WD 2!). Strictly speaking, in Lemma 2.1 (compare (1.1), (2.7), and (2.10)) we have proved that Z 2

Rn

1 1  cos !1 d! j!jnC2s

Z Rn

 2u.x/  u.x C y/  u.x  y/ dy D F 1 .2jj/2s uO ./ : nC2s jyj (2.18)

2.1 Preliminary Notions

15

Similarly, by means of Lemma 2.2 (compare (1.1), (2.13) and (2.15)) we know that  Z 22s1 s n2 C s 2u.x/  u.x C y/  u.x  y/ dy n  n=2 .1  s/ jyjnC2s R Z C1 1 D ts1 .U.x; t/  u.x// dt: .s/ 0

(2.19)

Moreover (see (2.14)), we have that U.x; t/ WD t  u.x/, where 2 =.4t/

t .x/ WD G.x; t/ D .4t/n=2 ejxj

:

We recall that the Fourier transform of a Gaussian is a Gaussian itself, namely 2

2

F .ejxj / D ejj ; p therefore, for any fixed t > 0, using the substitution y WD x= 4t, Z 1 2 F t ./ D ejxj =.4t/ e2ix dx n=2 .4t/ Rn Z p 2 ejyj e2iy. 4t/ dy D Rn

D e4

2 tjj2

:

As a consequence   F U.x; t/  u.x/ D F t  u.x/  u.x/  D F . t  u/./  uO ./ D F t ./  1 uO ./ D .e4

2 tjj2

 1/Ou./:

We multiply by ts1 and integrate over t > 0, and we obtain Z F

C1 0

 ts1 U.x; t/  u.x/ dt D

Z

C1 0

ts1 .e4

2 tjj2

 1/ dt uO ./

D .s/ .4 2 jj2 /s uO ./; thanks to (2.11) (used here with  WD 4 2 jj2 ). By taking the inverse Fourier transform, we have Z

C1 0

  ts1 U.x; t/  u.x/ dt D .s/ .2/2s F 1 jj2s uO ./ :

16

2 An Introduction to the Fractional Laplacian

We insert this information into (2.19) and we get  Z  22s1 s n2 C s 2u.x/  u.x C y/  u.x  y/ dy D .2/2s F 1 jj2s uO ./ : nC2s n  n=2 .1  s/ jyj R Hence, recalling (2.18),  Z 22s1 s n2 C s 2u.x/  u.x C y/  u.x  y/ dy n  n=2 .1  s/ jyjnC2s R Z 2u.x/  u.x C y/  u.x  y/ 1 dy; D Z 1  cos !1 jyjnC2s Rn 2 d! nC2s Rn j!j which gives the desired result. For the sake of completeness, a different proof of Lemma 2.3 will be given in Sect. A.2. There, to prove Lemma 2.3, we will use the theory of special functions rather than the fractional Laplacian. For other approaches to the proof of Lemma 2.3 see also the recent PhD dissertations [75] (and related [76]) and [92].

2.2 Fractional Sobolev Inequality and Generalized Coarea Formula Fractional Sobolev spaces enjoy quite a number of important functional inequalities. It is almost impossible to list here all the results and the possible applications, therefore we will only present two important inequalities which have a simple and nice proof, namely the fractional Sobolev Inequality and the Generalized Coarea Formula. The fractional Sobolev Inequality can be written as follows:  Theorem 2.2.1 For any s 2 .0; 1/, p 2 1; ns and u 2 C01 .Rn /, Z kuk

np L nsp

.Rn /

Z

C Rn

Rn

ju.x/  u.y/jp dx dy jx  yjnCsp

 1p

;

(2.20)

for some C > 0, depending only on n and p. Proof We follow here the very nice proof given in [117] (where, in turn, the proof is attributed to Haïm Brezis). We fix r > 0, a > 0, P > 0 and x 2 Rn . Then, for any y 2 Rn , ju.x/j  ju.x/  u.y/j C ju.y/j;

2.2 Fractional Sobolev Inequality and Generalized Coarea Formula

17

and so, integrating over Br .x/, we obtain Z jBr j ju.x/j 

Z ju.x/  u.y/j dy C

ju.y/j dy

Br .x/

Br .x/

Z ju.x/  u.y/j a  jx  yj dy C ju.y/j dy jx  yja Br .x/ Br .x/ Z Z ju.x/  u.y/j a r dy C ju.y/j dy: jx  yja Br .x/ Br .x/ Z

D

Now we choose a WD nCsp and we make use of the Hölder Inequality (with p p np np exponents p and p1 and with exponents nsp and n.p1/Csp ), to obtain jBr j ju.x/j  r

r

nCsp p

Z

ju.x/  u.y/j Br .x/

nCsp p

Z Br .x/

jx  yj

nCsp p

Z dy C

ju.x/  u.y/jp dy jx  yjnCsp

Z C

ju.y/j

np nsp

ju.y/j dy Br .x/

 1p Z

 p1 p dy Br .x/

Z  nsp np dy

Br .x/

 n.p1/Csp np dy

Br .x/

Z

 1p ju.x/  u.y/jp  Cr dy nCsp Br .x/ jx  yj Z  nsp np n.p1/Csp np p nsp CCr ju.y/j dy ; nCs

Br .x/

for some C > 0. So, we divide by rn and we possibly rename C. In this way, we obtain " Z # Z  1p  nsp p np np n ju.x/  u.y/j  ju.x/j  Crs dy C r p ju.y/j nsp dy : nCsp Br .x/ jx  yj Br .x/ That is, using the short notation Z ˛ WD

Rn

Z and ˇ WD we have that

ju.x/  u.y/jp dy jx  yjnCsp np

ju.y/j nsp dy Rn

1 nsp  n ju.x/j  Crs ˛ p C r p ˇ np ;

18

2 An Introduction to the Fractional Laplacian

hence, raising both terms at the appropriate power

np nsp

and renaming C

np

1  nsp nsp nsp n Cr nsp ˛ p C r p ˇ np :

np

ju.x/j nsp 

(2.21)

We take now r WD n

With this setting, we have that r p ˇ renaming C, we infer from (2.21) that np

ˇ

nsp n2 1

˛n

nsp np

: 1

is equal to ˛ p . Accordingly, possibly

sp

ju.x/j nsp  C˛ ˇ n Z  spn Z np ju.x/  u.y/jp nsp DC dy ju.y/j dy ; nCsp Rn jx  yj Rn for some C > 0, and so, integrating over x 2 Rn , Z

Z

np

Z

ju.x/j nsp dx  C Rn

Rn

Rn

ju.x/  u.y/jp dx dy jx  yjnCsp

 Z

np

ju.y/j nsp dy

 spn

Rn

:

This, after a simplification, gives (2.20). What follows is the Generalized Co-area Formula of [139] (the link with the classical Co-area Formula will be indeed more evident in terms of the fractional perimeter functional discussed in Chap. 5). Theorem 2.2.2 For any s 2 .0; 1/ and any measurable function u W ˝ ! Œ0; 1 , 1 2

Z Z ˝

˝

ju.x/  u.y/j dx dy D jx  yjnCs

Z

1 0

Z

Z fx2˝; u.x/>tg

fy2˝; u.y/tg

dx dy jx  yjnCs

 dt:

Proof We claim that for any x, y 2 ˝ Z

1

ju.x/  u.y/j D 0

 fu>tg .x/ futg .y/ C futg .x/ fu>tg .y/ dt:

(2.22)

To prove this, we fix x and y in ˝, and by possibly exchanging them, we can suppose that u.x/  u.y/. Then, we define '.t/ WD fu>tg .x/ futg .y/ C futg .x/ fu>tg .y/:

2.3 Maximum Principle and Harnack Inequality

19

By construction  '.t/ D

0 if t < u.y/ and t  u.x/; 1 if u.y/  t < u.x/;

therefore Z

1 0

Z

u.x/

'.t/ dt D

dt D u.x/  u.y/; u.y/

which proves (2.22). So, multiplying by the singular kernel and integrating (2.22) over ˝  ˝, we obtain that Z Z ju.x/  u.y/j dx dy nCs ˝ ˝ jx  yj  Z 1 Z Z fu>tg .x/ futg .y/ C futg .x/ fu>tg .y/ D dx dy dt jx  yjnCs ˝ ˝ 0  Z Z Z Z 1 Z dx dy dx dy dt C D nCs nCs fu>tg futg jx  yj futg fu>tg jx  yj 0  Z 1 Z Z dx dy dt; D2 nCs fu>tg futg jx  yj 0 as desired.

2.3 Maximum Principle and Harnack Inequality The Harnack Inequality and the Maximum Principle for harmonic functions are classical topics in elliptic regularity theory. Namely, in the classical case, if a nonnegative function is harmonic in B1 and r 2 .0; 1/, then its minimum and maximum in Br must always be comparable (in particular, the function cannot touch the level zero in Br ). It is worth pointing out that the fractional counterpart of these facts is, in general, false, as this next simple result shows (see [94]): Theorem 2.3.1 There exists a bounded function u which is s-harmonic in B1 , nonnegative in B1 , but such that inf u D 0. B1

Proof (Sketch of the proof) The main idea is that we are able to take the datum of u outside B1 in a suitable way as to “bend down” the function inside B1 until it reaches

20

2 An Introduction to the Fractional Laplacian

the level zero. Namely, let M  0 and we take uM to be the function satisfying 8 s ˆ ˆ 0, we expect uM to bend down, since the fact that the fractional Laplacian vanishes in B1 forces the second order quotient to vanish in average (recall (1.1), or the equivalent formulation in (2.5)). Indeed, we claim that there exists M? > 0 such that uM?  0 in B1 with inf uM? D 0. Then, the result of Theorem 2.3.1 would be reached by B1

taking u WD uM? . To check the existence of such M? , we show that inf uM ! 1 as M ! C1. B1

Indeed, we argue by contradiction and suppose this cannot happen. Then, for any M  0, we would have that inf uM  a; B1

for some fixed a 2 R. We set uM C M  1 : M

vM WD Then, by (2.23), 8 s ˆ ˆ 0 in B1 , unless u vanishes identically. Proof We observe that we already know that u  0 in the whole of Rn , thanks to Theorem 2.3.2. Hence, if u is not strictly positive, there exists x0 2 B1 such

22

2 An Introduction to the Fractional Laplacian

that u.x0 / D 0. This gives that Z Z 2u.x0 /  u.x0 C y/  u.x0  y/ u.x0 C y/ C u.x0  y/ 0 dy D  dy: nC2s n n jyj jyjnC2s R R Now both u.x0 C y/ and u.x0  y/ are non-negative, hence the latter integral is less than or equal to zero, and so it must vanish identically, proving that u also vanishes identically. A simple version of a Harnack-type inequality in the fractional setting can be also obtained as follows: Proposition 2.3.4 Assume that ./s u  0 in B2 , with u  0 in the whole of Rn . Then Z u.0/  c u.x/ dx; B1

for a suitable c > 0. Proof Let 2 C01 .B1=2 /, with .x/ 2 Œ0; 1 for any x 2 Rn , and .0/ D 1. We fix > 0, to be taken arbitrarily small at the end of this proof and set  WD u.0/ C > 0:

(2.25)

We define a .x/ WD 2 .x/  a. Notice that if a > 2, then a .x/  2  a < 0  u.x/ in the whole of Rn , hence the set f a < u in Rn g is not empty, and we can define a WD inf f a < u in Rn g: a2R

By construction a  2:

(2.26)

If a <  then a .0/ D 2  a >  > u.0/, therefore a  :

(2.27)

a  u in the whole of Rn .

(2.28)

there exists x0 2 B1=2 such that a .x0 / D u.x0 /.

(2.29)

Notice that

We claim that

To prove this, we suppose by contradiction that u > a in B1=2 , i.e.

WD min.u  a / > 0: B1=2

2.3 Maximum Principle and Harnack Inequality

23

Also, if x 2 Rn n B1=2 , we have that u.x/  a .x/ D u.x/  2 .x/ C a D u.x/ C a  a  ; thanks to (2.27). As a consequence, for any x 2 Rn , u.x/  a .x/  minf ; g DW  > 0: So, if we define a] WD a  .  =2/, we have that a] < a and u.x/  a] .x/ D u.x/  a .x/ 



  > 0: 2 2

This is in contradiction with the definition of a and so it proves (2.29). From (2.29) we have that x0 2 B1=2 , hence ./s u.x0 /  0. Also j./s a .x/j D 2 j./s .x/j  C, for any x 2 Rn , therefore, recalling (2.28) and (2.29), C  ./s a .x0 /  ./s u.x0 /



Z a .x0 /  a .x0 C y/  u.x0 /  u.x0 C y/ D C.n; s/ P.V. dy jyjnC2s Rn Z u.x0 C y/  a .x0 C y/ D C.n; s/ P.V. dy n jyjnC2s R Z u.x0 C y/  a .x0 C y/ dy:  C.n; s/ P.V. jyjnC2s B1 .x0 / Notice now that if y 2 B1 .x0 /, then jyj  jx0 j C 1 < 2, thus we obtain C 

C.n; s/ 2nC2s

Z B1 .x0 /

u.x0 C y/  a .x0 C y/ dy:

Notice now that a .x/ D 2 .x/  a  , due to (2.27), therefore we conclude that Z  C.n; s/ u.x0 C y/ dy  jB1 j : C  nC2s 2 B1 .x0 / That is, using the change of variable x WD x0 C y, recalling (2.25) and renaming the constants, we have Z  C u.0/ C D C  u.x/ dx; B1

hence the desired result follows by sending ! 0.

24

2 An Introduction to the Fractional Laplacian

2.4 An s-Harmonic Function We provide here an explicit example of a function that is s-harmonic on the positive line RC WD .0; C1/. Namely, we prove the following result: Theorem 2.4.1 For any x 2 R, let ws .x/ WD xsC D maxfx; 0gs . Then  ./ ws .x/ D s

cs jxjs if x < 0; 0 if x > 0;

for a suitable constant cs > 0. At a first glance, it may be quite surprising that the function xsC is s-harmonic in .0; C1/, since such function is not smooth (but only continuous) uniformly up to the boundary, so let us try to give some heuristic explanations for it (Fig. 2.1). We try to understand why the function xsC is s-harmonic in, say, the interval .0; 1/ when s 2 .0; 1 . From the discussion in Sect. 1.2, we know that the s-harmonic function in .0; 1/ that agrees with xsC outside .0; 1/ coincides with the expected value of a payoff, subject to a random walk (the random walk is classical when s D 1 and it presents jumps when s 2 .0; 1/). If s D 1 and we start from the middle of the interval, we have the same probability of being moved randomly to the left and to the right. This means that we have the same probability of exiting the interval .0; 1/ to the right (and so ending the process at x D 1, which gives 1 as payoff) or to the left (and so ending the process at x D 0, which gives 0 as payoff). Therefore the expected value starting at x D 1=2 is exactly the average between 0 and 1, which is 1=2. Similarly, if we start the process at the point x D 1=4, we have the same probability of reaching the point 0 on the left and the point 1=2 to the right. Since

Fig. 2.1 An s-harmonic function

2.4 An s-Harmonic Function

25

we know that the payoff at x D 0 is 0 and the expected value of the payoff at x D 1=2 is 1=2, we deduce in this case that the expected value for the process starting at 1=4 is the average between 0 and 1=2, that is exactly 1=4. We can repeat this argument over and over, and obtain the (rather obvious) fact that the linear function is indeed harmonic in the classical sense. The argument above, which seems either trivial or unnecessarily complicated in the classical case, can be adapted when s 2 .0; 1/ and it can give a qualitative picture of the corresponding s-harmonic function. Let us start again the random walk, this time with jumps, at x D 1=2: in presence of jumps, we have the same probability of reaching the left interval .1; 0 and the right interval Œ1; C1/. Now, the payoff at .1; 0 is 0, while the payoff at Œ1; C1/ is bigger than 1. This implies that the expected value at x D 1=2 is the average between 0 and something bigger than 1, which produces a value larger than 1=2. When repeating this argument over and over, we obtain a concavity property enjoyed by the s-harmonic functions in this case (the exact values prescribed in Œ1; C1/ are not essential here, it would be enough that these values were monotone increasing and larger than 1) (Fig. 2.2). In a sense, therefore, this concavity properties and loss of Lipschitz regularity near the minimal boundary values is typical of nonlocal diffusion and it is due to the possibility of “reaching far away points”, which may increase the expected payoff. Now we present a complete, elementary proof of Theorem 2.4.1. The proof originated from some pleasant discussions with Fernando Soria and it is based on some rather surprising integral cancellations. The reader who wishes to skip this proof can go directly to Sect. 2.3 on page 19. Moreover, a shorter, but technically more advanced proof, is presented in Sect. A.1. Here, we start with some preliminary computations. Lemma 2.4 For any s 2 .0; 1/ Z

1 0

.1 C t/s C .1  t/s  2 dt C t1C2s

Fig. 2.2 A payoff model: case s D 1 and s 2 .0; 1/

Z

C1 1

.1 C t/s 1 dt D : 1C2s t s

26

2 An Introduction to the Fractional Laplacian

Proof Fixed " > 0, we integrate by parts: Z

1

.1 C t/s C .1  t/s  2 dt t1C2s " Z id 1 1h .1 C t/s C .1  t/s  2 t2s dt D 2s " dt   Z 1 .1 C "/s C .1  "/s  2 1 1 .1 C t/s1  .1  t/s1 s D  2 C 2 C dt 2s "2s 2 " t2s  Z 1 Z 1 1 1 D .1 C t/s1 t2s dt  .1  t/s1 t2s dt ; Œo.1/  2s C 2 C 2s 2 " " (2.30) with o.1/ infinitesimal as " & 0. Moreover, by changing variable Qt WD t=.1  t/, that is t WD Qt=.1 C Qt/, we have that Z

1

"

.1  t/s1 t2s dt D

Z

C1 "=.1"/

.1 C Qt/s1 Qt2s dQt:

Inserting this into (2.30) (and writing t instead of Qt as variable of integration), we obtain Z

1

.1 C t/s C .1  t/s  2 dt t1C2s " Z 1  Z C1

1 1 s s1 2s s1 2s o.1/  2 C 2 C D .1 C t/ t dt  .1 C t/ t dt 2s 2 " "=.1"/ Z "=.1"/  Z C1

1 1 s o.1/  2 C 2 C .1 C t/s1 t2s dt  .1 C t/s1 t2s dt : D 2s 2 " 1 (2.31)

Now we remark that Z

"=.1"/ "

.1 C t/s1 t2s dt 

Z

"=.1"/ "

.1 C "/s1 "2s dt D "22s .1  "/1 .1 C "/s1 ;

therefore Z lim

"&0 "

"=.1"/

.1 C t/s1 t2s dt D 0:

So, by passing to the limit in (2.31), we get Z

1 0

.1 C t/s C .1  t/s  2 2s C 2 1  dt D t1C2s 2s 2

Z

C1 1

.1 C t/s1 t2s dt:

(2.32)

2.4 An s-Harmonic Function

27

Now, integrating by parts we see that 1 2

Z

C1

1

.1 C t/s1 t2s dt D

1 2s

D

Z

C1 1

2 C 2s s

t2s

Z

C1 1

d .1 C t/s dt dt t12s .1 C t/s dt:

By plugging this into (2.32) we obtain that Z

1 0

2s .1 C t/s C .1  t/s  2 2s C 2 C  dt D 1C2s t 2s 2s

Z

C1 1

t12s .1 C t/s dt;

which gives the desired result. From Lemma 2.4 we deduce the following (somehow unexpected) cancellation property: Corollary 2.4.1 Let ws be as in the statement of Theorem 2.4.1. Then ./s ws .1/ D 0: Proof The function t 7! .1 C t/s C .1  t/s  2 is even, therefore Z

1 1

.1 C t/s C .1  t/s  2 dt D 2 jtj1C2s

Z 0

1

.1 C t/s C .1  t/s  2 dt: t1C2s

Moreover, by changing variable Qt WD t, we have that Z

1 1

.1  t/s  2 dt D jtj1C2s

Z

C1 1

.1 C Qt/s  2 dQt: Qt1C2s

Therefore Z

C1

ws .1 C t/ C ws .1  t/  2ws .1/ dt jtj1C2s 1 Z 1 Z 1 Z C1 .1  t/s  2 .1 C t/s C .1  t/s  2 .1 C t/s  2 D dt C dt C dt 1C2s 1C2s jtj jtj jtj1C2s 1 1 1 Z C1 Z 1 .1 C t/s C .1  t/s  2 .1 C t/s  2 dt C 2 dt D2 1C2s t t1C2s 0 1 Z 1  Z C1 Z C1 .1 C t/s C .1  t/s  2 .1 C t/s dt D2 dt C dt  2 t1C2s t1C2s t1C2s 0 1 1   Z C1 1 dt D2 2 ; s t1C2s 1

28

2 An Introduction to the Fractional Laplacian

where Lemma 2.4 was used in the last line. Since Z

C1 1

dt t1C2s

D

1 ; 2s

we obtain that Z

C1 1

ws .1 C t/ C ws .1  t/  2ws .1/ dt D 0; jtj1C2s

that proves the desired claim. The counterpart of Corollary 2.4.1 is given by the following simple observation: Lemma 2.5 Let ws be as in the statement of Theorem 2.4.1. Then ./s ws .1/ > 0: Proof We have that ws .1 C t/ C ws .1  t/  2ws .1/ D .1 C t/sC C .1  t/sC  0 and not identically zero, which implies the desired result. We have now all the elements to proceed to the proof of Theorem 2.4.1. Proof (Proof of Theorem 2.4.1) We let  2 fC1; 1g denote the sign of a fixed x 2 R n f0g. We claim that Z Z

C1 1 C1

D 1

ws ..1 C t// C ws ..1  t//  2ws ./ dt jtj1C2s ws . C t/ C ws .  t/  2ws ./ dt: jtj1C2s

(2.33)

Indeed, the formula above is obvious when x > 0 (i.e.  D 1), so we suppose x < 0 (i.e.  D 1) and we change variable  WD t, to see that, in this case, Z Z

C1 1 C1

D Z

1 C1

D Z

1 C1

D 1

ws ..1 C t// C ws ..1  t//  2ws ./ dt jtj1C2s ws .1  t/ C ws .1 C t/  2ws ./ dt jtj1C2s ws .1 C / C ws .1  /  2ws ./ d jj1C2s ws . C / C ws .  /  2ws ./ d; jj1C2s

2.5 All Functions Are Locally s-Harmonic Up to a Small Error

29

thus checking (2.33). Now we observe that, for any r 2 R, s ws .jxjr/ D .jxjr/sC D jxjs rC D jxjs ws .r/:

That is ws .xr/ D ws .jxjr/ D jxjs ws .r/: So we change variable y D tx and we obtain that Z Z

C1

ws .x C y/ C ws .x  y/  2ws .x/ dy jyj1C2s

1 C1

D 1

D jxjs

Z

ws .x.1 C t// C ws .x.1  t//  2ws .x/ dt jxj2s jtj1C2s C1

1

D jxjs

Z

C1

1

ws ..1 C t// C ws ..1  t//  2ws ./ dt jtj1C2s ws . C t/ C ws .  t/  2ws ./ dt; jtj1C2s

where (2.33) was used in the last line. This says that  ./s ws .x/ D

jxjs ./s ws .1/ if x < 0, jxjs ./s ws .1/ if x > 0,

hence the result in Theorem 2.4.1 follows from Corollary 2.4.1 and Lemma 2.5.

2.5 All Functions Are Locally s-Harmonic Up to a Small Error Here we will show that s-harmonic functions can locally approximate any given function, without any geometric constraints. This fact is rather surprising and it is a purely nonlocal feature, in the sense that it has no classical counterpart. Indeed, in the classical setting, harmonic functions are quite rigid, for instance they cannot have a strict local maximum, and therefore cannot approximate a function with a strict local maximum. The nonlocal picture is, conversely, completely different, as the oscillation of a function “from far” can make the function locally harmonic, almost independently from its local behavior.

30

2 An Introduction to the Fractional Laplacian

We want to give here some hints on the proof of this approximation result: Theorem 2.5.1 Let k 2 N be fixed. Then for any f 2 Ck .B1 / and any " > 0 there exists R > 0 and u 2 H s .Rn / \ Cs .Rn / such that (

./s u.x/ D 0

in B1

uD0

in Rn n BR

(2.34)

and kf  ukCk .B1 /  ": Proof (Sketch of the proof) For the sake of convenience, we divide the proof into three steps. Also, for simplicity, we give the sketch of the proof in the onedimensional case. See [62] for the entire and more general proof. Step 1. Reducing to monomials Let k 2 N be fixed. We use first of all the Stone-Weierstrass Theorem and we have that for any " > 0 and any f 2 Ck Œ0; 1 there exists a polynomial P such that kf  PkCk .B1 /  ": Hence it is enough to prove Theorem 2.5.1 for polynomials. Then, by linearity, it N X cm xm and one finds an is enough to prove it for monomials. Indeed, if P.x/ D mD0

s-harmonic function um such that kum  xm kCk .B1 / 

then by taking u WD

N X

" ; jcm j

cm um we have that

mD1

ku  PkCk .B1 / 

N X

jcm jkum  xm kCk .B1 /  ":

mD1

Notice that the function u is still s-harmonic, since the fractional Laplacian is a linear operator. Step 2. Spanning the derivatives We prove the existence of an s-harmonic function in B1 , vanishing outside a compact set and with arbitrarily large number of derivatives prescribed. That is, we show that for any m 2 N there exist R > r > 0, a point x 2 R and a function u such that ./s u D 0 in .x  r; x C r/; u D 0 outside .x  R; x C R/;

(2.35)

2.5 All Functions Are Locally s-Harmonic Up to a Small Error

31

and Dj u.x/ D 0 for any j 2 f0; : : : ; m  1g; Dm u.x/ D 1:

(2.36)

To prove this, we argue by contradiction. We consider Z to be the set of all pairs .u; x/ of s-harmonic functions in a neighborhood of x, and points x 2 R satisfying (2.35). To any pair, we associate the vector 

u.x/; Du.x/; : : : ; Dm u.x/ 2 RmC1

and take V to be the vector space spanned by this construction, i.e. V WD

n

o u.x/; Du.x/; : : : ; Dm u.x/ ; for .u; x/ 2 Z :

Notice indeed that V is a linear space.

(2.37)

Indeed, let V1 , V2 2 V and a1 , a2 2 R. Then, for any i 2 f1; 2g, we have that Vi D  ui .xi /; Dui .xi /; : : : ; Dm ui .xi / , for some .ui ; xi / 2 Z , i.e. ui is s-harmonic in .xi  ri ; xi C ri / and vanishes outside .xi  Ri ; xi C Ri /, for some Ri  ri > 0. We set u3 .x/ WD a1 u1 .x C x1 / C a2 u2 .x C x2 /: By construction, u3 is s-harmonic in .r3 ; r3 /, and it vanishes outside .R3 ; R3 /, with r3 WD minfr1 ; r2 g and R3 WD maxfR1 ; R2 g, therefore .u3 ; 0/ 2 Z . Moreover Dj u3 .x/ D a1 Dj u1 .x C x1 / C a2 Dj u2 .x C x2 / and thus a 1 V1 C a 2 V2   D a1 u1 .x1 /; Du1 .x1 /; : : : ; Dm u1 .x1 / C a2 u2 .x2 /; Du2 .x2 /; : : : ; Dm u2 .x2 /  D u3 .0/; Du3 .0/; : : : ; Dm u3 .0/ : This establishes (2.37). Now, to complete the proof of Step 2, it is enough to show that V D RmC1 :

(2.38)

32

2 An Introduction to the Fractional Laplacian

Indeed, if (2.38) holds true, then in particular .0; : : : ; 0; 1/ 2 V, which is the desired claim in Step 2. To prove (2.38), we argue by contradiction: if not, by (2.37), we have that V is a proper subspace of RmC1 and so it lies in a hyperplane. Hence there exists a vector c D .c0 ; : : : ; cm / 2 RmC1 n f0g such that ˚  V   2 RmC1 s.t. c   D 0 : That is, taking a pair .u; x/ 2 Z , the vector c D .c0 ; : : : ; cm / is orthogonal to any vector u.x/; Du.x/; : : : ; Dm u.x/ , namely X

cj Dj u.x/ D 0:

jm

To find a contradiction, we now choose an appropriate s-harmonic function u and we evaluate it at an appropriate point x. As a matter of fact, a good candidate for the s-harmonic function is xsC , as we know from Theorem 2.4.1: strictly speaking, this function is not allowed here, since it is not compactly supported, but let us say that one can construct a compactly supported s-harmonic function with the same behavior near the origin. With this slight caveat set aside, we compute for a (possibly small) x in .0; 1/: Dj xs D s.s  1/ : : : .s  j C 1/xsj and multiplying the sum with xms (for x ¤ 0) we have that X

cj s.s  1/ : : : .s  j C 1/xmj D 0:

jm

But since s 2 .0; 1/ the product s.s  1/ : : : .s  j C 1/ never vanishes. Hence the polynomial is identically null if and only if cj D 0 for any j, and we reach a contradiction. This completes the proof of the existence of a function u that satisfies (2.35) and (2.36). Step 3. Rescaling argument and completion of the proof By Step 2, for any m 2 N we are able to construct a locally s-harmonic function u such that u.x/ D xm C O.xmC1 / near the origin (up to a translation). By considering the blow-up u .x/ D

u.x/ D xm C O.xmC1 / m

we have that for  small, u is arbitrarily close to the monomial xm . As stated in Step 1, this concludes the proof of Theorem 2.5.1.

2.6 A Function with Constant Fractional Laplacian on the Ball

33

It is worth pointing out that the flexibility of s-harmonic functions given by Theorem 2.5.1 may have concrete consequences. For instance, as a byproduct of Theorem 2.5.1, one has that a biological population with nonlocal dispersive attitudes can better locally adapt to a given distribution of resources (see e.g. Theorem 1.2 in [104]). Namely, nonlocal biological species may efficiently use distant resources and they can fit to the resources available nearby by consuming them (almost) completely, thus making more difficult for a different competing species to come into place.

2.6 A Function with Constant Fractional Laplacian on the Ball We complete this chapter with the explicit computation of the fractional Laplacian of the function U .x/ D .1  jxj2 /sC . In B1 we have that ./s U .x/ D C.n; s/

!n B.s; 1  s/; 2

where C.n; s/ is introduced in (2.10) and B is the special Beta function (see section 6.2 in [4]). Just to give an idea of how such computation can be obtained, with small modifications respect to [67, 68] we go through the proof of this result. p The reader can find the more general result, i.e. for U .x/ D .1  jxj2 /C for p > 1, in the above mentioned [67, 68]. Let us take uW R ! R as u.x/ D .1  jxj2 /sC . We consider the regional fractional Laplacian restricted to .1; 1/ Z L u.x/ WD P:V:

1 1

u.x/  u.y/ dy: jx  yj1C2s

and we compute its value at zero. By symmetry we have that Z L u.0/ D 2 lim

1

"!0 "

1  .1  y2 /s dy: y1C2s

Changing the variable ! D y2 and integrating by parts we get that Z L u.0/ D 2 lim

"!0

1 "

y12s dy  

Z

1 "

.1  y2 /s y12s dy



 " 1 C lim  .1  !/s ! s1 d! s "!0 s "2  2s  Z 1 "  "2s .1  "2 /s 1 C ! s .1  !/s1 d! : D  C lim s "!0 s "2 D 

2s

Z

1

34

2 An Introduction to the Fractional Laplacian

Using the integral representation of the Gamma function (see [4], formula 6.2.1), i.e. Z B.˛; ˇ/ D

1 0

t˛1 .1  t/ˇ1 dt;

it yields that 1 L u.0/ D B.1  s; s/  : s For x 2 B1 we use the change of variables ! D Z

xy 1xy .

We obtain that

1

.1  x2 /s  .1  y2 /s dy jx  yj1C2s 1 Z 1 .1  !x/2s1  .1  ! 2 /s .1  !x/1 2 s D .1  x / P:V: d! j!j2sC1 1 Z 1 Z 1 1  .1  ! 2 /s .1  !x/2s1  1 D .1  x2 /s P:V: d! C d! j!j2sC1 j!j2sC1 1 1

 ! Z 1 .1  ! 2 /s 1  .1  !x/1 C d! j!j2sC1 1 !

L u.x/ D P:V:

D .1  x2 /s L u.0/ C J.x/ C I.x/ ; (2.39) where we have recognized the regional fractional Laplacian and denoted Z J.x/ WD P:V: Z I.x/ WD P.V.

1 1 1

1

.1  !x/2s1  1 d! j!j2sC1

and

1  .1  !x/1 .1  ! 2 /s d!: j!j2sC1

In J.x/ we have that Z

1

J.x/ D P:V: Z

1 1

D lim

"!0

"

.1  !x/2s1 d!  j!j2sC1

Z

1

j!j

12s

 d!

1

.1 C !x/2s1 C .1  !x/2s1 d!  2 j!j2sC1

Z "

1

!

12s

 d! :

2.6 A Function with Constant Fractional Laplacian on the Ball

1 !

With the change of variable t D J.x/ D

1 C lim s "!0

Z

1="

h

1

35

 i "2s .t C x/2s1 C .t  x/2s1 dt  s

2s

D

1 1 .1 C x/ C .1  x/2s .1 C "x/2s C .1  "x/2s  2  C lim s 2s 2s "!0 "2s

D

1 .1 C x/2s C .1  x/2s  : s 2s

To compute I.x/, with a Taylor expansion of .1  !x/1 at 0 we have that Z

1

I.x/ D P.V.



P1

1

kD1 .x!/ j!j2sC1

k

(2.40)

.1  ! 2 /s d!:

The odd part of the sum vanishes by symmetry, and so Z I.x/ D  2 lim

1

P1

kD1 .x!/ ! 2sC1

"!0 " 1 X

D  2 lim

"!0

x2k

kD1

Z

1 "

2k

.1  ! 2 /s d!

! 2k2s1 .1  ! 2 /s d!:

We change the variable t D ! 2 and integrate by parts to obtain I.x/ D  lim

"!0

D

1 X kD1

1 X

x2k

Z

kD1

x2k lim



"!0

1 "2

tks1 .1  t/s dt;

s "2k2s .1  "2 /s  ks ks

Z

1 "2

 tks .1  t/s1 dt :

For k  1, the limit for " that goes to zero is null, and using the integral representation of the Beta function, we have that I.x/ D

1 X kD1

x2k

s B.k C 1  s; s/: ks

We use the Pochhammer symbol defined as ( .q/k D

1

for k D 0;

q.q C 1/    .q C k  1/

for k > 0

(2.41)

36

2 An Introduction to the Fractional Laplacian

and with some manipulations, we get .s/ .k C 1  s/ .s/ s B.k C 1  s; s/ D ks .k  s/ .k C 1/ .s/ .k  s/ .s/ kŠ .s/k : D B.1  s; s/ kŠ

D

And so I.x/ D B.1  s; s/

1 X kD1

x2k

.s/k : kŠ

By the definition of the hypergeometric function (see e.g. page 211 in [112]) we obtain I.x/ D  B.1  s; s/ C B.1  s; s/

1 X

.s/k

kD0

x2k kŠ

 

1 1  D B.1  s; s/ F  s; ; ; x2  1 : 2 2 Now, by 15.1.8 in [4] we have that

1 1  F  s; ; ; x2 D .1  x2 /s 2 2 and therefore

 I.x/ D B.1  s; s/ .1  x2 /s  1 : Inserting this and (2.40) into (2.39) we obtain L u.x/ D B.1  s; s/  .1  x2 /s

.1 C x/2s C .1  x/2s : 2s

(2.42)

Now we write the fractional Laplacian of u as Z

Z 1 u.x/ u.x/ dy C dy 1C2s jx  yj1C2s 1 jx  yj 1  Z 1  Z 1 2 s 12s 12s jx  yj dy C jx  yj dy D L u.x/ C .1  x /

./s u.x/ D L u.x/ C C.1; s/

1

1

D L u.x/ C .1  x2 /s

.1 C x/

1

2s

C .1  x/ 2s

2s

:

2.6 A Function with Constant Fractional Laplacian on the Ball

37

Inserting (2.42) into the computation, we obtain ./s u.x/ D C.1; s/B.1  s; s/:

(2.43)

To pass to the n-dimensional case, without loss of generality and up to rotations, we consider x D .0; 0; : : : ; xn / with xn  0. We change into polar coordinates xy D th, with h 2 @B1 and t  0. We have that Z ./s U .x/ .1  jxj2 /s  .1  jyj2 /s D P:V: dy C.n; s/ jx  yjnC2s Rn  Z  Z 1 .1  jxj2 /s  .1  jx C htj2 /s D P:V: dt dH n1 .h/: 2 @B1 jtj1C2s R (2.44) p Changing the variable t D jxjhn C  jhn xj2  jxj2 C 1, we notice that 1  jx C htj2 D .1   2 /.1  jxj2 C jhn xj2 / and so Z P:V:

.1  jxj2 /s  .1  jx C htj2 /s dt jtj1C2s R Z .1  jxj2 /s  .1   2 /s .jhn xj2  jxj2 C 1/s p DP:V: jhn xj2  jxj2 C 1 d ˇ ˇ1C2s p ˇ ˇ R ˇ  jxjhn C  jhn xj2  jxj2 C 1ˇ s  jxj2 h2n Z 1  .1   2 /s jhn xj2  jxj2 C 1 d DP:V: ˇ ˇ1C2s ˇ ˇ jxjhn R ˇ  p ˇ ˇ ˇ jhn xj2  jxj2 C 1   jxjhn ./s u p jhn xj2  jxj2 C 1 D C.1; s/

DB.1  s; s/; where the last equality follows from identity (2.43). Hence from (2.44) we have that ./s U .x/ D C.n; s/B.1  s; s/ This concludes the proof of the result.

!n : 2

Chapter 3

Extension Problems

We dedicate this part of the book to the harmonic extension of the fractional Laplacian. We present at first two applications, the water wave model and the Peierls-Nabarro model related to crystal dislocations, making clear how the extension problem appears in these models. We conclude this part by discussing1 in detail the extension problem via the Fourier transform. The harmonic extension of the fractional Laplacian in the framework considered here is due to Luis Caffarelli and Luis Silvestre (we refer to [30] for details). We also recall that this extension procedure was obtained by S. A. Molˇcanov and E. Ostrovski˘ı in [108] by probabilistic methods (roughly speaking “embedding” a long jump random walk in Rn into a classical random walk in one dimension more, see Fig. 3.1). The idea of this extension procedure is that the nonlocal operator ./s acting on functions defined on Rn may be reduced to a local operator, acting on functions defined in the higher-dimensional half-space RnC1 WD Rn  .0; C1/: C n Indeed, take UW RnC1 C ! R such that U.x; 0/ D u.x/ in R , solution to the equation

 div y12s rU.x; y/ D 0

in RnC1 C :

Then up to constants one has that 

 lim y12s @y U.x; y/ D ./s u.x/: y!0

1

Though we do not develop this approach here, it is worth mentioning that extended problems arise naturally also from the probabilistic interpretation described in Sect. 1. Roughly speaking, a stochastic process with jumps in Rn can often be seen as the “trace” of a classical stochastic process in Rn  Œ0; C1/ (i.e., each time that the classical stochastic process in Rn  Œ0; C1/ hits Rn  f0g it induces a jump process over Rn ). Similarly, stochastic process with jumps may also be seen as classical processes at discrete, random, time steps. © Springer International Publishing Switzerland 2016 C. Bucur, E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana 20, DOI 10.1007/978-3-319-28739-3_3

39

40

3 Extension Problems

Fig. 3.1 The random walk with jumps in Rn can be seen as a classical random walk in RnC1

3.1 Water Wave Model n Let us consider the half space RnC1 C D R  .0; C1/ endowed with the coordinates n x 2 R and y 2 .0; C1/. We show that the half-Laplacian (namely when s D 1=2/ arises when looking for a harmonic function in RnC1 with given data C on Rn  fy D 0g. Thus, let us consider the following local Dirichlet-to-Neumann problem:

(

U D 0

in RnC1 C ;

U.x; 0/ D u.x/

for x 2 Rn :

The function U is the harmonic extension of u, we write U D Eu, and define the operator L as L u.x/ WD @y U.x; 0/:

(3.1)

We claim that L D

p

x ;

in other words L 2 D x :

(3.2)

3.1 Water Wave Model

41

Indeed, by using the fact that E.L u/ D @y U (that can be proved, for instance, by using the Poisson kernel representation for the solution), we obtain that  L 2 u.x/ D L L u .x/  D  @y E L u .x; 0/  D  @y  @y U .x; 0/  D @yy U C x U  x U .x; 0/ D U.x; 0/  u.x/ D  u.x/; which concludes the proof of (3.2). One remark in the above calculation lies in the choice of the sign of the square root of the operator. Namely, if we set LQ u.x/ WD @y U.x; 0/, the same computation as above would give that LQ 2 D . In a sense, there is no surprise that a quadratic equation offers indeed two possible solutions. But a natural question is how to choose the “right” one. There are several reasons to justify the sign convention in (3.1). One reason is given by spectral theory, that makes the (fractional) Laplacian a negative definite operator. Let us discuss a purely geometric justification, in the simpler n D 1dimensional case. We wonder how the solution of problem (

./s u D 1

in .1; 1/;

uD 0

in R n .1; 1/:

(3.3)

should look like in the extended variable y. First of all, by Maximum Principle (recall Theorems 2.3.2 and 2.3.3), we have that u is positive2 when x 2 .1; 1/ (since this is an s-superharmonic function, with zero data outside). Then the harmonic extension U in y > 0 of a function u which is positive in .1; 1/ and vanishes outside .1; 1/ should have the shape of an elastic membrane over the halfplane R2C that is constrained to the graph of u on the trace fy D 0g. We give a picture of this function U in Fig. 3.2. Notice from the picture that @y U.x; 0/ is negative, for any x 2 .1; 1/. Since ./s u.x/ is positive, we deduce that, to make our picture consistent with the maximum principle, we need to take the sign of L opposite to that of @y U.x; 0/. This gives a geometric justification of (3.1), which is only based on maximum principles (and on “how classical harmonic functions look like”).

As a matter of fact, the solution of (3.3) is explicit and it is given by .1  x2 /s , up to dimensional constants. See [68] for a list of functions whose fractional Laplacian can be explicitly computed (unfortunately, differently from the classical cases, explicit computations in the fractional setting are available only for very few functions).

2

42

3 Extension Problems

Fig. 3.2 The harmonic extension

Fig. 3.3 The water waves model

3.1.1 Application to the Water Waves We show now that the operator L arises in the theory of water waves of irrotational, incompressible, inviscid fluids in the small amplitude, long wave regime. Consider a particle moving in the sea, which is, for us, the space Rn  .0; 1/, where the bottom of the sea is set at level 1 and the surface at level 0 (see Fig. 3.3). The velocity of the particle is vW Rn  .0; 1/ ! RnC1 and we write v.x; y/ D  vx .x; y/; vy .x; y/ , where vx W Rn  .0; 1/ ! Rn is the horizontal component and vy W Rn  .0; 1/ ! R is the vertical component. We are interested in the vertical velocity of the water at the surface of the sea which we call u.x/, namely u.x/ WD vy .x; 0/. In our model, the water is incompressible, thus div v D 0 in Rn  .0; 1/. Furthermore, on the bottom of sea (since water cannot penetrate into the sand), the velocity has only a non-null horizontal component, hence vy .x; 1/ D 0. Also, in our model we assume that there are no vortices: at a mathematical level, this gives that v is irrotational, thus we may write it as the gradient of a function UW RnC1 ! R. We are led to the problem 8 ˆ ˆ 0 and u.1/ D 0; u.0/ D 1=2; u.C1/ D 1:

(3.9)

For the existence of such solution and its main properties see [114] and [25]. Furthermore, the solution decays polynomially at ˙1 (see [60] and [59]), namely ˇ ˇ ˇ x ˇˇ C 1 ˇu.x/  H.x/ C  # ˇ ˇ 00 1C2s 2sW .0/ jxj jxj

for any x 2 Rn ;

(3.10)

where # > 2s and H is the Heaviside step function ( H.x/ D

1; x  0 0; x < 0:

3 As a matter of fact, the solution of (3.9) coincides with the one of a one-dimensional fractional Allen-Cahn equation, that will be discussed in further detail in the forthcoming Sect. 4.1.

48

3 Extension Problems

Fig. 3.5 The initial datum when " ! 0

We take the initial condition of the solution of (3.8) to be the superposition of transitions all occurring with the same orientation, i.e. we set v .x; 0/ WD

  N X 2s x  x0i ; .0; x/ C u W 00 .0/ iD1

(3.11)

where x01 ; : : : ; x0N are N fixed points. The main result in this setting is that the solution v approaches, as ! 0, the superposition of step functions  (Fig. 3.5). The discontinuities of the limit function occur at some points xi .t/ iD1;:::;N which move accordingly to the following4 dynamical system  8 X ˆ , and [59] for 2 2 1 the case s < (in these papers, it is also carefully stated in which sense the limit 2 in (3.14) holds true). We would like to give now a formal (not rigorous) justification of the ODE system in (3.12) that drives the motion of the transition layers. Proof (Justification of ODE system (3.12)) We assume for simplicity that the external stress  is null. We use the notation ' to denote the equality up to negligible terms in . Also, we denote   x  xi .t/ ui .t; x/ WD u and, with a slight abuse of notation u0i .t; x/

WD u

0



 x  xi .t/ :

By (3.10) we have that the layer solution is approximated by 

x  xi .t/ ui .t; x/ ' H



 2s x  xi .t/  ˇ ˇ1C2s : 2sW 00 .0/ˇx  xi .t/ˇ

(3.15)

We use the assumption that the solution v is well approximated by the sum of N transitions and write v .t; x/ '

N X iD1

ui .t; x/ D

N

x  x .t/  X i : u iD1

50

3 Extension Problems

For that 1X 0 u .t; x/xPi .t/ iD1 i N

@t v .t; x/ D 

and, since the basic layer solution u is the solution of (3.9), we have that ./s v ' 

N X

./s ui .t; x/

iD1

D 

N

x  x .t/  1 X i s ./ u 2s iD1

D

  N 1 X 0 x  xi .t/  u W 2s iD1

D

N 1 X 0 W ui .t; x/ : 2s iD1

Now, returning to the parabolic equation (3.8) we have that 1 1X 0 u .t; x/xPi .t/ D 2sC1  iD1 i N

X N

N

X   0 W ui .t; x/  W ui .t; x/ : 0

iD1

(3.16)

iD1

Fix an integer k between 1 and N, multiply (3.16) by u0k .t; x/ and integrate over R. We obtain 1X  xPi .t/ iD1 N

D

Z R

u0i .t; x/u0k .t; x/ dx

1

X N Z

2sC1

iD1

R

 W 0 ui .t; x/ u0k .t; x/ dx 

Z

W0 R

N

X

  ui .t; x/ u0k .t; x/ dx :

iD1

(3.17) We compute the left hand side of (3.17). First, we take the kth term of the sum (i.e. we consider the case i D k). By using the change of variables y WD

x  xk .t/

(3.18)

3.2 Crystal Dislocation

51

we have that 1  xPk .t/

Z R

.u0k /2 .t; x/ dx

  Z 1 0 2 x  xk .t/ dx D  xPk .t/ .u / R Z D  xPk .t/ .u0 /2 .y/ dy

(3.19)

R

D 

xPk .t/ ; 

where  is defined by (3.13). Then, we consider the ith term of the sum on the left hand side of (3.17). By performing again the substitution (3.18), we see that this term is    Z Z  1 1 0 0 0 x  xi .t/ 0 x  xk .t/  xPi .t/ ui .t; x/uk .t; x/ dx D  xPi .t/ u u dx R R  Z  xk .t/  xi .t/ 0 D  xPi .t/ u0 y C u .y/ dy R ' 0;   xk .t/  xi .t/ where, for the last equivalence we have used that for small, u0 y C is asymptotic to u0 .˙1/ D 0. We consider the first member on the right hand side of the identity (3.17), and, as before, take the kth term of the sum. We do the substitution (3.18) and have that Z Z   1 W 0 uk .t; x/ u0k .t; x/ dx D W 0 u.y/ u0 .y/ dy R R  ˇˇC1 D W u.y/ ˇ 1

D W.1/  W.0/ D 0 by the periodicity of W. Now we use (3.15), the periodicity of W 0 and we perform a Taylor expansion, noticing that W 0 .0/ D 0. We see that !   2s x  xi .t/ x  xi .t/  W ui .t; x/ ' W H ˇ ˇ1C2s 2sW 00 .0/ˇx  xi .t/ˇ    2s x  xi .t/ 0 'W  ˇ ˇ1C2s 2sW 00 .0/ˇx  xi .t/ˇ   2s x  xi .t/ ' ˇ ˇ1C2s : 2sˇx  xi .t/ˇ 0





0



52

3 Extension Problems

Therefore, the ith term of the sum on the right hand side of the identity (3.17) for i ¤ k, by using the above approximation and doing one more time the substitution (3.18), for small becomes    2s x  xi .t/ 0 x  xk .t/ dx ˇ ˇ1C2s u R 2sˇx  xi .t/ˇ  Z 2s y C xk .t/  xi .t/ 0 D  ˇ ˇ1C2s u .y/ dy R 2sˇ y C xk .t/  xi .t/ˇ  Z 2s xk .t/  xi .t/ '  ˇ u0 .y/ dy ˇ1C2s R 2sˇxk .t/  xi .t/ˇ  2s xk .t/  xi .t/ D  ˇ ˇ1C2s : 2sˇxk .t/  xi .t/ˇ (3.20) We also observe that, for small, the second member on the right hand side of the identity (3.17), by using the change of variables (3.18), reads 1

1

Z

 1 W ui .t; x/ u0k .t; x/ dx D  R

Z

0

W0 R

N

X

Z

 ui .t; x/ u0k .t; x/ dx

iD1

1 D

Z R

 X W 0 uk .t; x/ C ui .t; x/ u0k .t; x/ dx i¤k

  X

xk .t/  xi .t/  0 0 D u .y/ dy: W u.y/ C u yC R Z

i¤k

  xk .t/  xi .t/ is asymptotic either to u.C1/ D 1 for xk > xi , For small, u y C or to u.1/ D 0 for xk < xi . By using the periodicity of W, it follows that 1

Z

W0 R

N

X

Z

  ui .t; x/ u0k .t; x/ dx D W 0 u.y/ u0 .y/ dy D W.1/  W.0/ D 0; R

iD1

again by the asymptotic behavior of u. Concluding, by inserting the results (3.19) and (3.20) into (3.17) we get that X xPk .t/ xk .t/  xi .t/ D ˇ ˇ1C2s ;  ˇ 2s xk .t/  xi .t/ˇ i¤k

which ends the justification of the system (3.12).

3.2 Crystal Dislocation

53

We recall that, till now, in Theorem 3.2.1 we considered the initial data as a superposition of transitions all occurring with the same orientation (see (3.11)), i.e. the initial dislocation is a monotone function (all the atoms are initially moved to the right). Of course, for concrete applications, it is interesting to consider also the case in which the atoms may dislocate in both directions, i.e. the transitions can occur with different orientations (the atoms may be initially displaced to the left or to the right of their equilibrium position). To model the different orientations of the dislocations, we introduce a parameter i 2 f1; 1g (roughly speaking i D 1 corresponds to a dislocation to the right and i D 1 to a dislocation to the left). The main result in this case is the following (see [115]): Theorem 3.2.2 There exists a viscosity solution of 8  1

1 ˆ  ./s v  2s W 0 .v / C  ˆ@t v D <   N 2s X x  x0i ˆ ˆv .0; x/ D .0; x/ C u i : W 00 .0/ iD1

in .0; C1/  R; for x 2 R

such that lim v .t; x/ D

!0



where xi .t/

iD1;:::;N

iD1

is solution to

 8 X ˆ 0, c > 0, T > Tc and  > 0 satisfying lim T D Tc

!0

and lim % D 0 !0

3.2 Crystal Dislocation

55

such that for any < 0 we have jv .t; x/j  % e

c

T t 2sC1

;

for any x 2 R and t  T :

(3.24)

The estimate in (3.24) states, roughly speaking, that at a suitable time T (only slightly bigger than the collision time Tc ) the dislocation function gets below a small threshold  , and later it decays exponentially fast (the constant of this exponential becomes large when is small). The reader may compare Theorems 3.2.3 and 3.2.4 and notice that different asymptotics are considered by the two results. A result similar to Theorem 3.2.4 holds for a larger number of dislocated atoms. For instance, in the case of three atoms with alternate dislocations, one has that, slightly after collision, the dislocation function decays exponentially fast to the basic layer solution. More precisely (see again [116]), we have that: Theorem 3.2.5 Let N D 3, 1 D 3 D 1, 2 D 1, and let v be the solution given by Theorem 3.2.2, with  0. Then there exist 0 > 0, c > 0, T 1 ; T 2 > Tc and  > 0 satisfying lim T 1 D lim T 2 D Tc ;

!0

!0

and lim % D 0 !0

and points yN and zN satisfying lim jNz  yN j D 0

!0

such that for any < 0 we have  v .t; x/  u

x  yN





c.tT 1 / 2sC1



c.tT 2 / 2sC1

C % e

;

for any x 2 R and t  T 1 ;

(3.25)

;

for any x 2 R and t  T 2 ;

(3.26)

and 

x  zN v .t; x/  u

  % e

where u is the basic layer solution introduced in (3.9). Roughly speaking, formulas (3.25) and (3.26) say that for times T 1 , T 2 just slightly bigger than the collision time Tc , the dislocation function v gets trapped between two basic layer solutions (centered at points yN and zN ), up to a small error. The error gets eventually to zero, exponentially fast in time, and the two basic layer solutions which trap v get closer and closer to each other as goes to zero (that is, the distance between yN and zN goes to zero with ).

56

3 Extension Problems

We refer once more to [116] for a series of figures describing in details the results of Theorems 3.2.4 and 3.2.5. We observe that the results presented in Theorems 3.2.1, 3.2.2, 3.2.3, 3.2.4 and 3.2.5 describe the crystal at different space and time scale. As a matter of fact, the mathematical study of a crystal typically goes from an atomic description (such as a classical discrete model presented by Frenkel-Kontorova and Prandtl-Tomlinson) to a macroscopic scale in which a plastic deformation occurs. In the theory discussed here, we join this atomic and macroscopic scales by a series of intermediate scales, such as a microscopic scale, in which the PeierlsNabarro model is introduced, a mesoscopic scale, in which we studied the dynamics of the dislocations (in particular, Theorems 3.2.1 and 3.2.2), in order to obtain at the end a macroscopic theory leading to the relaxation of the model to a permanent deformation (as given in Theorems 3.2.4 and 3.2.5 , while Theorem3.2.3 somehow describes the further intermediate features between these schematic scalings).

3.3 An Approach to the Extension Problem via the Fourier Transform We will discuss here the extension operator of the fractional Laplacian via the Fourier transform approach (see [30] and [134] for other approaches and further results and also [81], in which a different extension formula is obtained in the framework of the Heisenberg groups). Some readers may find the details of this part rather technical: if so, she or he can jump directly to Chap. 4 on page 67, without affecting the subsequent reading. We fix at first a few pieces of notation. We denote points in RnC1 WD Rn  C .0; C1/ as X D .x; y/, with x 2 Rn and y > 0. When taking gradients in RnC1 C , we write rX D .rx ; @y /, where rx is the gradient in Rn . Also, in RnC1 , we will often C take the Fourier transform in the variable x only, for fixed y > 0. We also set a WD 1  2s 2 .1; 1/: bs .Rn / defined as the set of We will consider the fractional Sobolev space H functions u that satisfy kukL2 .Rn / C ŒOu G < C1; where sZ Œv G WD

jj2s jv./j2 d: Rn

3.3 An Approach to the Extension Problem via the Fourier Transform

57

1;1 For any u 2 Wloc ..0; C1//, we consider the functional

Z

ˇ ˇ2 ˇ ˇ2  ta ˇu.t/ˇ C ˇu0 .t/ˇ dt:

C1

G.u/ WD 0

(3.27)

By Theorem 4 of [128], we know that the functional G attains its minimum among 1;1 all the functions u 2 Wloc ..0; C1// \ C0 .Œ0; C1// with u.0/ D 1. We call g such minimizer and C] WD G.g/ D

min

1;1 u2Wloc ..0;C1//\C0 .Œ0;C1// u.0/D1

G.u/:

(3.28)

The main result of this section is the following. Theorem 3.3.1 Let u 2 S .Rn / and let

 U.x; y/ WD F 1 uO ./ g.jjy/ :

(3.29)

div .ya rU/ D 0

(3.30)

Then

for any X D .x; y/ 2 RnC1 C . In addition, ˇ ˇ  ya @y U ˇ

fyD0g

D C] ./s u

(3.31)

in Rn , both in the sense of distributions and as a pointwise limit. In order to prove Theorem 3.3.1, we need to make some preliminary computations. At first, let us recall a few useful properties of the minimizer function g of the operator G introduced in (3.27). We know from formula (4.5) in [128] that 0  g  1;

(3.32)

g0  0:

(3.33)

and from formula (2.6) in [128] that

We also cite formula (4.3) in [128], according to which g is a solution of g00 .t/ C at1 g0 .t/ D g.t/

(3.34)

58

3 Extension Problems

for any t > 0, and formula (4.4) in [128], according to which lim ta g0 .t/ D C] :

t!0C

(3.35)

1;1 Now, for any V 2 Wloc .RnC1 C / we set

sZ ŒV a WD

ya jrX V.X/j2 dX:

RnC1 C

Notice that ŒV a is well defined (possibly infinite) on such space. Also, one can compute ŒV a explicitly in the following interesting case: Lemma 3.1 Let

2 S .Rn / and U.x; y/ WD F 1



 ./ g.jjy/ :

(3.36)

Then ŒU 2a D C] Œ 2G :

(3.37)

Proof By (3.32), for any fixed y > 0, the function  7! ./ g.jjy/ belongs to L2 .Rn /, and so we may consider its (inverse) Fourier transform. This says that the definition of U is well posed. By the inverse Fourier transform definition (2.2), we have that Z rx U.x; y/ D rx Z D Rn

Rn

./ g.jjy/ eix d

i ./ g.jjy/ eix d

 D F 1 i ./g.jjy/ .x/: Thus, by Plancherel Theorem, Z Rn

jrx U.x; y/j2 dx D

Z Rn

ˇ ˇ ˇ ./g.jjy/ˇ2 d:

3.3 An Approach to the Extension Problem via the Fourier Transform

59

Integrating over y > 0, we obtain that Z RnC1 C

ya jrx U.X/j2 dX D

Z Rn

Z D Z

Rn

ˇ2 ˇ jj2 ˇ ./ˇ

Z

ˇ2 ˇ jj1a ˇ ./ˇ

C1

D 0

D Π2G

 ˇ2 ˇ ya ˇg.jjy/ˇ dy d

0

Z

ˇ2 t g.t/ˇ dt  ˇ

aˇ

C1

C1

 ˇ2 ˇ ta ˇg.t/ˇ dt d

0

Z

jj

ˇ

2s ˇ

Rn

Z

C1

ˇ2 ./ˇ d

(3.38)

ˇ2 ˇ ta ˇg.t/ˇ dt:

0

Let us now prove that the following identity is well posed

 @y U.x; y/ D F 1 jj ./ g0 .jjy/ :

(3.39)

For this, we observe that jg0 .t/j  C] ta :

(3.40)

To check this, we define .t/ WD ta jg0 .t/j. From (3.33) and (3.34), we obtain that  0 .t/ D 

 da 0 t g .t/ D ta g00 .t/ C at1 g0 .t/ D ta g.t/  0: dt

Hence .t/  lim ./ D lim  a jg0 ./j D C] ;  !0C

 !0C

where formula (3.35) was used in the last identity, and this establishes (3.40). From (3.40) we have that jj j ./j jg0 .jjy/j  C] ya jj1a j ./j 2 L2 .Rn /, and so (3.39) follows. Therefore, by (3.39) and the Plancherel Theorem, Z

2

Rn

j@y U.x; y/j dx D

Z Rn

ˇ2 ˇ ˇ2 ˇ jj2 ˇ ./ˇ ˇg0 .jjy/ˇ d:

60

3 Extension Problems

Integrating over y > 0 we obtain Z RnC1 C

ya j@y U.x; y/j2 dx D

Z Rn

Z D Z

Rn

ˇ2 ˇ jj2 ˇ ./ˇ

Z

ˇ2 ˇ jj1a ˇ ./ˇ

C1

D 0

D Π2G

ˇ2 ˇ ta ˇg0 .t/ˇ dt 

Z

C1 0

C1 0

Z Z

C1

0

Rn

 ˇ2 ˇ ya ˇg0 .jjy/ˇ dy d  ˇ ˇ2 ta ˇg0 .t/ˇ dt d

ˇ2 ˇ jj2s ˇ ./ˇ d

ˇ ˇ2 ta ˇg0 .t/ˇ dt:

By summing this with (3.38), and recalling (3.28), we obtain the desired result ŒU 2a D C] Œ 2G . This concludes the proof of the Lemma. Now, given u 2 L1loc .Rn /, we consider the space Xu of all the functions V 2  such that, for any x 2 Rn , the map y 7! V.x; y/ is in C0 Œ0; C1/ , with V.x; 0/ D u.x/ for any x 2 Rn . Then the problem of minimizing Œ  a over Xu has a somehow explicit solution. 1;1 Wloc .RnC1 C /

Lemma 3.2 Assume that u 2 S .Rn /. Then min ŒV 2a D ŒU 2a D C] ŒOu 2G ;

(3.41)

 U.x; y/ WD F 1 uO ./ g.jjy/ :

(3.42)

V2Xu

Proof We remark that (3.42) is simply (3.36) with have that

WD uO , and by Lemma 3.1 we

ŒU 2a D C] ŒOu 2G : Furthermore, we claim that U 2 Xu :

(3.43)

In order to prove this, we first observe that jg.T/  g.t/j 

C] jT 2s  t2s j : 2s

(3.44)

3.3 An Approach to the Extension Problem via the Fourier Transform

61

To check this, without loss of generality, we may suppose that T  t  0. Hence, by (3.33) and (3.40), Z

T

jg.T/  g.t/j  t

jg0 .r/j dr

Z

T

 C]

ra dr

t

C] .T 1a  t1a / D ; 1a that is (3.44). Then, by (3.44), for any y, yQ 2 .0; C1/, we see that ˇ C jj2s jy2s  yQ 2s j ˇ ] ˇ ˇ : ˇg.jj y/  g.jj yQ /ˇ  2s Accordingly, ˇ 

ˇˇ ˇ ˇ ˇ ˇU.x; y/  U.x; yQ /ˇ D ˇF 1 uO ./ g.jj y/  g.jj yQ / ˇ ˇ ˇ Z ˇ

ˇ ˇ ˇ  ˇOu./ g.jj y/  g.jj yQ / ˇ d Rn



C] jy2s  yQ 2s j 2s

Z Rn

jj2s jOu./j d;

and this implies (3.43). Thanks to (3.43) and (3.37), in order to complete the proof of (3.41), it suffices to show that, for any V 2 Xu , we have that ŒV 2a  ŒU 2a :

(3.45)

To prove this, let us take V 2 Xu . Without loss of generality, since ŒU a < C1 thanks to (3.37), we may suppose that ŒV a < C1. Hence, fixed a.e. y > 0, we have that Z Z a 2 a y jrx V.x; y/j dx  y jrX V.x; y/j2 dx < C1; Rn

Rn

62

3 Extension Problems

hence the map x 2 jrx V.x; y/j belongs to L2 .Rn /. Therefore, by Plancherel Theorem, Z Z ˇ ˇˇ2 ˇ  jrx V.x; y/j2 dx D (3.46) ˇF rx V.x; y/ ./ˇ d: Rn

Rn

Now by the Fourier transform definition (2.1)  F rx V.x; y/ ./ D

Z Z

Rn

rx V.x; y/ eix dx i V.x; y/ eix dx

D Rn

 D i F V.x; y/ ./; hence (3.46) becomes Z Z jrx V.x; y/j2 dx D Rn

Rn

 jj2 jF V.x; y/ ./j2 d:

(3.47)

On the other hand   F @y V.x; y/ ./ D @y F V.x; y/ ./ and thus, by Plancherel Theorem, Z Rn

j@y V.x; y/j2 dx D

Z Rn

ˇ  ˇ ˇF @y V.x; y/ ./ˇ2 d D

Z Rn

 j@y F V.x; y/ ./j2 d:

We sum up this latter result with identity (3.47) and we use the notation .; y/ WD F V.x; y/ ./ to conclude that Z Rn

jrX V.x; y/j2 dx D

Z Rn

jj2 j.; y/j2 C j@y .; y/j2 d:

(3.48)

Accordingly, integrating over y > 0, we deduce that ŒV 2a D

Z RnC1 C

 ya jj2 j.; y/j2 C j@y .; y/j2 d dy:

(3.49)

Let us first consider the integration over y, for any fixed  2 Rn n f0g, that we now omit from the notation when this does not generate any confusion. We set h.y/ WD .; jj1 y/. We have that h0 .y/ D jj1 @y .; jj1 y/ and therefore,

3.3 An Approach to the Extension Problem via the Fourier Transform

63

using the substitution t D jj y, we obtain Z

C1 0

D jj1a D jj1a

ˇ ˇ2  ya jj2 j.; y/j2 C ˇ@y .; y/ˇ dy Z Z

C1 0 C1 0

ˇ2  ˇ ta j.; jj1 t/j2 C jj2 ˇ@y .; jj1 t/ˇ dt

(3.50)



ta jh.t/j2 C jh0 .t/j2 dt

D jj2s G.h/: Now, for any  2 R, we show that min

1;1 w2Wloc ..0;C1//\C0 .Œ0;C1//

w.0/ D G.w/ D 2 C] :

(3.51)

Indeed, when  D 0, the trivial function is an allowed competitor and G.0/ D 0, which gives (3.51) in this case. If, on the other hand,  ¤ 0, given w as above with w.0/ D  we set w .x/ WD 1 w.x/. Hence we see that w .0/ D 1 and thus G.w/ D 2 G.w /  2 G.g/ D 2 C] , due to the minimality of g. This proves (3.51). From (3.51) and the fact that  h.0/ D .; 0/ D F V.x; 0/ ./ D uO ./; we obtain that ˇ2 ˇ G.h/  C] ˇuO ./ˇ : As a consequence, we get from (3.50) that Z

C1 0

ˇ ˇ ˇ2  ˇ2 ya jj2 j.; y/j2 C ˇ@y .; y/ˇ dy  C] jj2s ˇuO ./ˇ :

Integrating over  2 Rn n f0g we obtain that Z RnC1 C

ˇ ˇ2  ya jj2 j.; y/j2 C ˇ@y .; y/ˇ d dy  C] ŒOu 2G :

Hence, by (3.49), ŒV 2a  C] ŒOu 2G ; which proves (3.45), and so (3.41). We can now prove the main result of this section.

64

3 Extension Problems

Proof (Proof of Theorem 3.3.1) Formula (3.30) follows from the minimality property in (3.41), by writing that ŒU 2a  ŒUC ' 2a for any ' smooth and compactly supported inside RnC1 C and any 2 R. Now we take ' 2 C01 .Rn / (notice that its support may now hit fy D 0g). We define u WD u C ', and U as in (3.29), with uO replaced by uO (notice that (3.29) is nothing but (3.42)), hence we will be able to exploit Lemma 3.2. We also set

 O g.jjy/ : ' .x; y/ WD F 1 './ We observe that



 O g.0/ D F 1 './ O D '.x/ ' .x; 0/ D F 1 './

(3.52)

and that

 U D U C F 1 './ O g.jjy/ D U C ' : As a consequence ŒU 2a

D

ŒU 2a

Z C 2

RnC1 C

ya rX U  rX ' dX C o. /:

Hence, using (3.30), (3.52) and the Divergence Theorem, ŒU 2a

D D

ŒU 2a ŒU 2a

Z C 2 Z  2

RnC1 C

 div ' ya rX U dX C o. / (3.53) ' y @y U dx C o. /: a

Rn f0g

Moreover, from Plancherel Theorem, and the fact that the image of ' is in the reals, uO t2G

Z D ŒOu G C 2 Z D ŒOu G C 2 Z D ŒOu G C 2

Rn

Rn

Rn

jj2s uO ./ './ O d C o. / 

F 1 jj2s uO ./ .x/ '.x/ dx C o. / ./s u.x/ '.x/ dx C o. /:

3.3 An Approach to the Extension Problem via the Fourier Transform

65

By comparing this with (3.53) and recalling (3.41) we obtain that ŒU 2a  2

Z Rn f0g

' ya @y U dx C o. /

D ŒU 2a D C] Œu 2G

Z

D C] ŒOu G C 2C] D

ŒU 2a

Z C 2C]

Rn

Rn

./s u.x/ '.x/ dx C o. /

./s u ' dx C o. /

and so Z

Z



' y @y U dx D C] a

Rn f0g

Rn

./s u ' dx;

for any ' 2 C01 .Rn /, that is the distributional formulation of (3.31). Furthermore, by (3.29), we have that



  ya @y U.x; y/ D F 1 jj uO ./ ya g.jjy/ D F 1 jj1a uO ./ .jjy/a g.jjy/ : Hence, by (3.35), we obtain

 lim ya @y U.x; y/ D  C] F 1 jj1a uO ./

y!0C

 D  C] F 1 jj2s uO ./ D  ./s u.x/;

that is the pointwise limit formulation of (3.31). This concludes the proof of Theorem 3.3.1.

Chapter 4

Nonlocal Phase Transitions

We consider a nonlocal phase transition model, in particular described by the AllenCahn equation. A fractional analogue of a conjecture of De Giorgi, that deals with possible one-dimensional symmetry of entire solutions, naturally arises from treating this model, and will be consequently presented. There is a very interesting connection with nonlocal minimal surfaces, that will be studied in Chap. 5. We introduce briefly the classical case.1 The Allen-Cahn equation has various applications, for instance, in the study of interfaces (both in gases and solids), in the theory of superconductors and superfluids or in cosmology. We deal here with a two-phase transition model, in which a fluid can reach two pure phases (say 1 and 1) forming an interface of separation. The aim is to describe the pattern and the separation of the two phases. The formation of the interface is driven by a variational principle. Let u.x/ be the function describing the state of the fluid at position x in a bounded region ˝. As a first guess, the phase separation can be modeled via the minimization of the energy Z E0 .u/ D

 W u.x/ dx; ˝

where W is a double-well potential. More precisely, WW Œ1; 1 ! Œ0; C1/ such that W 2 C2 .Œ1; 1 / ; W.˙1/ D 0; W > 0 in .1; 1/; W.˙1/ D 0 and W 00 .˙1/ > 0:

(4.1)

1

We would like to thank Alberto Farina who, during a summer-school in Cortona (2014), gave a beautiful introduction on phase transitions in the classical case.

© Springer International Publishing Switzerland 2016 C. Bucur, E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana 20, DOI 10.1007/978-3-319-28739-3_4

67

68

4 Nonlocal Phase Transitions

The classical example is W.u/ WD

.u2  1/2 : 4

(4.2)

On the other hand, the functional in E0 produces an ambiguous outcome, since any function u that attains only the values ˙1 is a minimizer for the energy. That is, the energy functional in E0 alone cannot detect any geometric feature of the interface. To avoid this, one is led to consider an additional energy term that penalizes the formation of unnecessary interfaces. The typical energy functional provided by this procedure has the form Z





"2 E .u/ WD W u.x/ dx C 2 ˝

Z

jru.x/j2 dx:

(4.3)

˝

In this way, the potential energy that forces the pure phases is compensated by a small term, that is due to the elastic effect of the reaction of the particles. As a curiosity, we point out that in the classical mechanics framework, the analogue of (4.3) is a Lagrangian action of a particle, with n D 1, x representing a time coordinate and u.x/ the position of the particle at time x. In this framework the term involving the square of the derivative of u has the physical meaning of a kinetic energy. With a slight abuse of notation, we will keep referring to the gradient term in (4.3) as a kinetic energy. Perhaps a more appropriate term would be elastic energy, but in concrete applications also the potential may arise from elastic reactions, therefore the only purpose of these names in our framework is to underline the fact that (4.3) occurs as a superposition of two terms, a potential one, which only depends on u, and one, which will be called kinetic, which only depends on the variation of u (and which, in principle, possesses no real “kinetic” feature). The energy minimizers will be smooth functions, taking values between 1 and 1, forming layers of interfaces of "-width. If we send " ! 0, the transition layer will tend to a minimal surface. To better explain this, consider the energy Z J.u/ D

1 jruj2 C W.u/ dx; 2

(4.4)

whose minimizers solve the Allen-Cahn equation  u C W 0 .u/ D 0:

(4.5)

In particular, for the explicit potential in (4.2), Eq. (4.5) reduces (up to normalizations constants) to  u D u  u3 :

(4.6)

4 Nonlocal Phase Transitions

69

In this setting, the behavior of u in large domains reflects into the behavior of the rescaled function u" .x/ D u x" in B1 . Namely, the minimizers of J in B1=" are the minimizers of J" in B1 , where J" is the rescaled energy functional Z J" .u/ D B1

" 1 jruj2 C W.u/ dx: 2 "

(4.7)

We notice then that J" .u/ 

Z p 2W.u/ jruj dx B1

which, using the Co-area Formula, gives Z J" .u/ 

1

1

p 2W.t/ H n1 .fu D tg/ dt:

The above formula may suggest that the minimizers of J" have the tendency to minimize the .n  1/-dimensional measure of their level sets. It turns out that indeed the level sets of the minimizers of J" converge to a minimal surface as " ! 0: for more details see, for instance, [121] and the references therein. In this setting, a famous De Giorgi conjecture comes into place. In the late 1970s, De Giorgi conjectured that entire, smooth, monotone (in one direction), bounded solutions of (4.6) in the whole of Rn are necessarily one-dimensional, i.e., there exist ! 2 Sn1 and u0 W R ! R such that u.x/ D u0 .!  x/

for any x 2 Rn :

In other words, the conjecture above asks if the level sets of the entire, smooth, monotone (in one direction), bounded solutions are necessarily hyperplanes, at least in dimension n  8. One may wonder why the number eight has a relevance in the problem above. A possible explanation for this is given by the Bernstein Theorem, as we now try to describe. The Bernstein problem asks on whether or not all minimal graphs (i.e. surfaces that locally minimize the perimeter and that are graphs in a given direction) in Rn must be necessarily affine. This is indeed true in dimensions n at most eight. On the other hand, in dimension n  9 there are global minimal graphs that are not hyperplanes (see e.g. [86]). The link between the problem of Bernstein and the conjecture of De Giorgi could be suggested by the fact that minimizers approach minimal surfaces in the limit. In a sense, if one is able to prove that the limit interface is a hyperplane and that this rigidity property gets inherited by the level sets of the minimizers u" (which lie nearby such limit hyperplane), then, by scaling back, one obtains that the level sets of u are also hyperplanes. Of course, this link between the two problems, as

70

4 Nonlocal Phase Transitions

stated here, is only heuristic, and much work is needed to deeply understand the connections between the problem of Bernstein and the conjecture of De Giorgi. We refer to [73] for a more detailed introduction to this topic. We recall that this conjecture by De Giorgi was proved for n  3, see [5, 12, 85]. Also, the case 4  n  8 with the additional assumption that lim u.x0 ; xn / D ˙1;

xn !˙1

for any x0 2 Rn1

(4.8)

was proved in [120]. For n  9 a counterexample can be found in [54]. Notice that, if the above limit is uniform (and the De Giorgi conjecture with this additional assumption is known as the Gibbons conjecture), the result extends to all possible n (see for instance [72, 73] for further details). The goal of the next part of this book is then to discuss an analogue of these questions for the nonlocal case and present related results.

4.1 The Fractional Allen-Cahn Equation The extension of the Allen-Cahn equation in (4.5) from a local to a nonlocal setting has theoretical interest and concrete applications. Indeed, the study of long range interactions naturally leads to the analysis of phase transitions and interfaces of nonlocal type. Given an open domain ˝ Rn and the double well potential W (as in (4.2)), our goal here is to study the fractional Allen-Cahn equation ./s u C W 0 .u/ D 0 in

˝;

for s 2 .0; 1/ (when s D 1, this equation reduces to (4.5)). The solutions are the critical points of the nonlocal energy Z

 1 E .u; ˝/ WD W u.x/ dx C 2 ˝

“ R2n n.˝ C /2

ju.x/  u.y/j2 dx dy; jx  yjnC2s

(4.9)

up to normalization constants that we omitted for simplicity. The reader can compare (4.9) with (4.3). Namely, in (4.9) the kinetic energy is modified, in order to take into account long range interactions. That is, the new kinetic energy still depends on the variation of the phase parameter. But, in this case, far away changes in phase may influence each other (though the influence is weaker and weaker towards infinity). Notice that in the nonlocal framework, we prescribe the function on ˝ C  ˝ C and consider the kinetic energy on the remaining regions (see Fig. 4.1). The prescription of values in ˝ C  ˝ C reflects into the fact that the domain of

4.1 The Fractional Allen-Cahn Equation

71

Fig. 4.1 The kinetic energy

integration of the kinetic integral in (4.9) is R2n n .˝ C /2 . Indeed, this is perfectly compatible with the local case in (4.3), where the domain of integration of the kinetic term was simply ˝. To see this compatibility, one may think that the domain of integration of the kinetic energy is simply the complement of the set in which the values of the functions are prescribed. In the local case of (4.3), the values are prescribed on @˝, or, one may say, in ˝ C : then the domain of integration of the kinetic energy is the complement of ˝ C , which is simply ˝. In analogy with that, in the nonlocal case of (4.9), the values are prescribed on ˝ C  ˝ C D .˝ C /2 , i.e. outside ˝ for both the variables x and y. Then, the kinetic integral is set on the complement of .˝ C /2 , which is indeed R2n n .˝ C /2 . Of course, the potential energy has local features, both in the local and in the nonlocal case, since in our model the nonlocality only occurs in the kinetic interaction, therefore the potential integrals are set over ˝ both in (4.3) and in (4.9). For the sake of shortness, given disjoint sets A, B  Rn we introduce the notation Z Z u.A; B/ WD A

B

ju.x/  u.y/j2 dx dy; jx  yjnC2s

and we write the new kinetic energy in (4.9) as K .u; ˝/ D

1 u.˝; ˝/ C u.˝; ˝ C /: 2

(4.10)

Let us define the energy minimizers and provide a density estimate for the minimizers.

72

4 Nonlocal Phase Transitions

Definition 4.1.1 The function u is a minimizer for the energy E in BR if E .u; BR /  E .v; BR / for any v such that u D v outside BR . The energy of the minimizers satisfy the following uniform bound property on large balls. Theorem 4.1.2 Let u be a minimizer in BRC2 for a large R, say R  1. Then lim

R!C1

1 E .u; BR / D 0: Rn

(4.11)

More precisely,

E .u; BR / 

8 ˆ CRn1 ˆ
Rn1 . These estimates are optimal (we refer to 2 [125] for further details). Proof We introduce at first some auxiliary functions (see Fig. 4.2). Let n o .x/ WD 1 C 2 min .jxj  R  1/C ; 1 ;

n o v.x/ WD min u.x/; .x/ ;

n o d.x/ WD max .R C 1  jxj/; 1 : Then, for jx  yj  d.x/ we have that j .x/ 

.y/j 

2jx  yj : d.x/

(4.12)

Indeed, if jxj  R, then d.x/ D R C 1  jxj and jyj  jx  yj C jxj  d.x/ C jxj  R C 1; thus .x/ D .y/ D 0 and the inequality is trivial. Else, if jxj  R, then d.x/ D 1, and so the inequality is assured by the Lipschitz continuity of (with 2 as the Lipschitz constant). Also, we prove that we have the following estimates for the function d:

 8 1 n1 ˆ CR if s 2 ; 1 ; ˆ Z 2 < 2s 1 n1 (4.13) d.x/ dx  CR log R if s D 2 ; 

ˆ BRC2 ˆ :CRn2s 1 if s 2 0; : 2

4.1 The Fractional Allen-Cahn Equation

73

, v and d

Fig. 4.2 The functions

To prove this, we observe that in the ring BRC2 nBR , we have d.x/ D 1. Therefore, the contribution to the integral in (4.13) that comes from the ring BRC2 n BR is bounded by the measure of the ring, and so it is of order Rn1 , namely Z BRC2 nBR

d.x/2s dx D jBRC2 n BR j  CRn1 ;

(4.14)

for some C > 0. We point out that this order is always negligible with respect to the right hand side of (4.13). Therefore, to complete the proof of (4.13), it only remains to estimate the contribution to the integral coming from BR . For this, we use polar coordinates and perform the change of variables t D =.R C 1/. In this way, we obtain that Z d.x/ BR

2s

Z

n1 d 2s 0 .R C 1  / Z 1 1 RC1 tn1 .1  t/2s dt D C .R C 1/n2s R

dx D C

Z  C .R C 1/

0

1 1 RC1

n2s 0

.1  t/2s dt;

for some C > 0. Now we observe that 8Z 1 ˆ ˆ .1  t/2s dt D C if ˆ ˆ ˆ Z 1 1 0 < 1 ˇ RC1 1 .1  t/2s dt   log.1  t/ˇˇ RC1  log R if ˆ 0 ˆ ˆ ˇ1 0 1 ˆ ˆ : .1t/12s ˇˇ RC1  CR2s1 if 12s 0



s 2 0; 12 ; s D 12 ;

 s 2 12 ; 1 :

74

4 Nonlocal Phase Transitions

The latter two formulas and (4.14) imply (4.13). Now, we define the set A WD fv D

g

and notice that BRC1  A  BRC2 . We prove that for any x 2 A and any y 2 AC n jv.x/  v.y/j  max ju.x/  u.y/j; j .x/ 

o .y/j :

(4.15)

Indeed, for x 2 A and y 2 AC we have that v.x/ D

.x/  u.x/

and v.y/ D u.y/ 

.y/;

therefore v.x/  v.y/  u.x/  u.y/

and v.y/  v.x/ 

.y/  .x/;

which establishes (4.15). This leads to v.A; AC /  u.A; AC / C

.A; AC /:

(4.16)

Notice now that E .u; BRC2/  E .v; BRC2 / since u is a minimizer in BRC2 and v D u outside BRC2 . We have that E .u; BRC2 / D D

1 u.BRC2; BRC2 / C u.BRC2; BC RC2 / C 2

Z W.u/ dx BRC2

1 u.A; A/ C u.A; AC / 2 1 C u.BRC2 n A; BRC2 n A/ C u.BRC2 n A; BC RC2 / 2 Z Z C W.u/ dx C W.u/ dx: A

BRC2 nA

4.1 The Fractional Allen-Cahn Equation

75

Since u and v coincide on AC , by using the inequality (4.16) we obtain that 0  E .v; BRC2 /  E .u; BRC2/

Z

 1 1 W.v/  W.u/ dx v.A; A/  u.A; A/ C v.A; AC /  u.A; AC / C 2 2 A Z

 1 1  v.A; A/  u.A; A/ C .A; AC / C W.v/  W.u/ dx: 2 2 A

D

Moreover, v D 1 u.A; A/ C 2

on A and we have that Z

1 W.u/ dx  2 A

.A; A/ C

Z

C

.A; A / C

W. / dx D E . ; A/; A

and therefore, since BRC1  A  BRC2 , 1 u.BRC1 ; BRC1 / C 2

Z W.u/ dx  E . ; BRC2 /:

(4.17)

BRC1

We estimate now E . ; BRC2 /. For a fixed x 2 BRC2 we observe that Z

j .x/  .y/j2 dy jx  yjnC2s Rn Z Z j .x/  .y/j2 j .x/  .y/j2 D dy C dy nC2s jx  yj jx  yjnC2s jxyjd.x/ jxyjd.x/   Z Z 1 n2sC2 n2s C jx  yj dy C jx  yj dy ; d.x/2 jxyjd.x/ jxyjd.x/

where we have used (4.12) and the boundedness of we have that Z Rn

. Passing to polar coordinates,

  Z 1 Z d.x/ j .x/  .y/j2 1 2sC1 2s1 dy  C  d C  d jx  yjnC2s d.x/2 0 d.x/ D Cd.x/2s :

Recalling that

.x/ D 1 on BRC1 and W.1/ D 0, we obtain that Z

Z

Z j .x/  .y/j2 dy dx C W. / dx jx  yjnC2s BRC2 Rn BRC2 Z Z  d.x/2s dx C W. / dx:

E . ; BRC2 / D

BRC2

BRC2 nBRC1

76

4 Nonlocal Phase Transitions

Therefore, making use of (4.13),

E . ; BRC2 / 

8 ˆ CRn1 ˆ < CRn1 log R ˆ ˆ :CRn2s





1 2; 1 ; D 12 ;  2 0; 12 :

if

s2

if

s

if

s

(4.18)

For what regards the right hand-side of inequality (4.17), we have that 1 u.BRC1; BRC1 / C 2

Z

1 u.BR ; BR / C u.BR ; BRC1 n BR / 2 Z C W.u/ dx:

W.u/ dx  BRC1

(4.19)

BR

We prove now that u.BR ; BC RC1 / 

Z

d.x/2s dx:

(4.20)

BRC2

For this, we observe that if x 2 BR , then d.x/ D R C 1  jxj. So, if x 2 BR and y 2 BC RC1 , then jx  yj  jyj  jxj  R C 1  jxj D d.x/: Therefore, by changing variables z D x  y and then passing to polar coordinates, we have that Z Z /  4 dx jzjn2s dz u.BR ; BC RC1 BC d.x/

BR

Z C

Z

1

dx Z

BR

2s1 d

d.x/

d.x/2s dx:

DC BR

This establishes (4.20). Hence, by (4.13) and (4.20), we have that 8 ˆ CRn1 ˆ < u.BR ; BC d.x/2s dx  CRn1 log R RC1 /  ˆ BRC2 ˆ :CRn2s Z





1 ;1 ; 2 1 D 2 ;  2 0; 12 :

if

s2

if

s

if

s

(4.21)

4.1 The Fractional Allen-Cahn Equation

77

We also observe that, by adding u.BR ; BC RC1 / to inequality (4.19), we obtain that 1 u.BRC1 ; BRC1 / C 2

Z BRC1

W.u/ dx C u.BR ; BC RC1 /

1  u.BR ; BR / C u.BR ; BRC1 n BR / C 2

Z BR

W.u/ dx C u.BR ; BC RC1 /

D E .u; BR /: This and (4.17) give that E .u; BR /  E . ; BRC2 / C u.BR ; BC RC1 /: Combining this with the estimates in (4.18) and (4.21), we obtain the desired result. Another type of estimate can be given in terms of the level sets of the minimizers (see Theorem 1.4 in [125]). Theorem 4.1.3 Let u be a minimizer of E in BR . Then for any 1 ; 2 2 .1; 1/ such that u.0/ > 1 we have that there exist R and C > 0 such that ˇ ˇ ˇ ˇ ˇfu > 2 g \ BR ˇ  CRn if R  R.1 ; 2 /. The constant C > 0 depends only on n, s and W and R.1 ; 2 / is a large constant that depends also on 1 and 2 . The statement of Theorem 4.1.3 says that the level sets of minimizers always occupy a portion of a large ball comparable to the ball itself. In particular, both phases occur in a large ball, and the portion of the ball occupied by each phase is comparable to the one occupied by the other. Of course, the simplest situation in which two phases split a ball in domains with comparable, and in fact equal, size is when all the level sets are hyperplanes. This question is related to a fractional version of a classical conjecture of De Giorgi and to nonlocal minimal surfaces, that we discuss in the following Sect. 4.2 and Chap. 5. Let us try now to give some details on the proof of the Theorem 4.1.3 in the particular case in which s is in the range .0; 1=2/. The more general proof for all s 2 .0; 1/ can be found in [125], where one uses some estimates on the Gagliardo norm. In our particular case we will make use of the Sobolev inequality that we introduced in (2.20). The interested reader can see [122] for a more exhaustive explanation of the upcoming proof.

78

4 Nonlocal Phase Transitions

Proof (Proof of Theorem 4.1.3) Let us consider a smooth function w such that w D 1 on BC R (we will take in sequel w to be a particular barrier for u), and define v.x/ WD minfu.x/; w.x/g:  C n n C Since juj  1, we have that v D u in BC R . Calling D D .R  R / n BR  BR we have from definition (4.10) that K .u  v;BR / C K .v; BR /  K .u; BR / “ 1 j.u  v/.x/  .u  v/.y/j2 C jv.x/  v.y/j2 ju.x/  u.y/j2 D dx dy: 2 D jx  yjnC2s Using the identity ja  bj2 C b2  a2 D 2b.b  a/ with a D u.x/  u.y/ and b D v.x/  v.y/ we get K .u  v; BR / C K .v; BR /  K .u; BR / “ ..u  v/.x/  .u  v/.y// .v.y/  v.x// D dx dy: jx  yjnC2s D n n Since u v D 0 on BC R we can extend the integral to the whole space R R , hence

K .u  v; BR / C K .v; BR /  K .u; BR / “ ..u  v/.x/  .u  v/.y// .v.y/  v.x// dx dy: D jx  yjnC2s n n R R Then by changing variables and using the anti-symmetry of the integrals, we notice that “ ..u  v/.x/  .u  v/.y// .v.y/  v.x// dx dy jx  yjnC2s BR BR “ .u  v/.x/ .v.y/  v.x// dx dy D jx  yjnC2s BR BR “ .u  v/.y/ .v.y/  v.x//  dx dy jx  yjnC2s BR BR “ .u  v/.x/ .v.y/  v.x// D2 dx dy jx  yjnC2s BR BR

4.1 The Fractional Allen-Cahn Equation

79

and “

..u  v/.x/  .u  v/.y// .v.y/  v.x// dx dy jx  yjnC2s BR BC R “ ..u  v/.x/  .u  v/.y// .v.y/  v.x// C dx dy jx  yjnC2s BC R BR “ .u  v/.x/ .v.y/  v.x// D dx dy jx  yjnC2s C BR BR “ .u  v/.y/ .v.y/  v.x//  dx dy jx  yjnC2s BC R BR “ .u  v/.x/ .v.y/  v.x// D2 dx dy: jx  yjnC2s C BR B R

Therefore K .u  v; BR / C K .v; BR /  K .u; BR / “ .u.x/  v.x// .v.y/  v.x// dx dy D2 jx  yjnC2s Rn Rn Z  Z v.y/  v.x/ D2 .u.x/  v.x// dy dx nC2s Rn Rn jx  yj Z  Z v.y/  w.x/ D2 .u.x/  w.x// dy dx nC2s BR \fu>vDwg Rn jx  yj Z  Z w.y/  w.x/ .u  w/.x/ dy dx 2 nC2s BR \fu>vDwg Rn jx  yj Z D2 .u  w/.x/ ../s w/ .x/ dx: BR \fu>wg

Hence K .u  v; BR / K .u; BR /  K .v; BR / C 2

Z BR \fu>wg

.u  w/ ../s w/ dx:

80

4 Nonlocal Phase Transitions

By adding and subtracting the potential energy, we have that K .u  v; BR /

Z

E .u; BR /  E .v; BR / C

W.v/  W.u/ dx BR

Z C2

BR \fu>wg

and since u is minimal in BR , Z K .u  v; BR / 

.u  w/ ../s w/ dx

W.w/  W.u/ dx

BR \fu>wDvg

Z

C2

(4.22) .u  w/ ../ w/ dx: s

BR \fu>wg

We deduce from the properties in (4.1) of the double-well potential W that there exists a small constant c > 0 such that W.t/  W.r/  c.1 C r/.t  r/ C c.t  r/2 W.r/  W.t/ 

1Cr c

when  1  r  t  1 C c when  1  r  t  1:

We fix the arbitrary constants 1 and 2 , take c small as here above. Let then ? WD minf1 ; 2 ; 1 C cg: It follows that Z W.w/  W.u/dx BR \fu>wg

Z

Z W.w/  W.u/dx C

D BR \f? >u>wg

Z

c BR \f? >u>wg

W.w/  W.u/dx BR \fu>maxf? ;wgg

Z

.1  w/.u  w/ dx  c

BR \f? >u>wg

.u  w/2 dx

Z 1 C .1 C w/ dx c BR \fu>maxf? ;wgg Z Z 1 .1  w/.u  w/ dx C .1 C w/ dx: c c BR \fu>maxf? ;wgg BR \f? >u>wg (4.23)

4.1 The Fractional Allen-Cahn Equation

81

Therefore, in (4.22) we obtain that Z K .u  v; BR /   c

BR \f? >u>wg

1 C c C2

.1  w/.u  w/ dx

Z Z

BR \fu>maxf? ;wgg

BR \fu>wg

.1 C w/ dx

(4.24)

.u  w/ ../s w/ dx:

We introduce now a useful barrier in the next Lemma (we just recall here Lemma 3.1 in [125] – there the reader can find how this barrier is build): Lemma 4.1 Given any   0 there exists a constant C > 1 (possibly depending on n; s and ) such that: for any R  C there exists a rotationally symmetric function w 2 C Rn ; Œ1 C CR2s ; 1 with w D 1 in BC R and such that for any x 2 BR one has that 1 .R C 1  jxj/2s  1 C w.x/  C.R C 1  jxj/2s C

and

./s w.x/  .1 C w.x//:

(4.25) (4.26)

Taking w as the barrier introduced in the above Lemma, thanks to (4.24) and to the estimate in (4.26), we have that Z K .u  v; BR /   c .1 C w/.u  w/ dx BR \f? >u>wg

C

1 c

Z

BR \fu>maxf? ;wgg

.1 C w/ dx

Z

C 2

BR \fu>wg

.u  w/.1 C w/ dx:

Let then  D 2c , and we are left with Z K .u  v; BR /  c C  C1

BR \fu>maxf? ;wgg

1 c Z

.u  w/.1 C w/ dx

Z

BR \fu>maxf? ;wgg

BR \fu>maxf? ;wgg

.1 C w/ dx

.1 C w/ dx;

82

4 Nonlocal Phase Transitions

with C1 depending on c (hence on W). Using again Lemma 4.1, in particular the right hand side inequality in (4.25), we have that Z K .u  v; BR /  C1  C

BR \fu>maxf? ;wgg

.R C 1  jxj/2s :

We set V.R/ WD jBR \ fu > ? gj

(4.27)

and the Co-Area formula then gives Z K .u  v; BR /  C2

R 0

.R C 1  t/2s V 0 .t/ dt;

(4.28)

where C2 possibly depends on n; s; W. We use now the Sobolev inequality (2.20) for p D 2, applied to u  v (recalling that the support of u  v is a subset of BR ) to obtain that “ K .u  v; BR / D K .u  v; R / D n

Q  vk Cku

2

2n

L n2s .Rn /

Rn Rn

j.u  v/.x/  .u  v/.y/j2 dx dy jx  yjnC2s

Q  vk2 D Cku

2n

L n2s .BR /

:

From (4.25) one has that w.x/  C.R C 1  jxj/2s  1: We fix K large enough so as to have R  2K and in BRK w.x/  C.1 C K/2s  1  1 C

1 C ? : 2

Therefore in BRK \ fu > ? g we have that ju  vj  u  w  u C 1 

1 C ? 1 C ?  : 2 2

(4.29)

4.1 The Fractional Allen-Cahn Equation

83

Using definition (4.27), this leads to ku  vk2

Z

2n L n2s

.BR /

2n

D

ju  vj n2s dx

 n2s n

BR



1 C ?  2

2n Z  n2s

 n2s n dx BRK \fu>? g

C3 V.R  K/

n2s n

:

In (4.29) we thus have K .u  v; BR /  CQ 3 V.R  K/

n2s n

and from (4.28) it follows that C4 V.R  K/

n2s n

Z

R

 0

.R C 1  t/2s V 0 .t/ dt:

Let R    2K. Integrating the latter integral from  to  n2s C4 V.  K/ n C4 2 Z 

Z

3 2

V.R  K/

D

3 2

Z

D 0

R 0

3 2

0

Z

dR



0

Z

n2s n

3 we have that 2

3 2

.R C 1  t/ Z

V 0 .t/

0

V 0 .t/

3 2

3 2

2s

V .t/ dt

dR !

.R C 1  t/2s dR

C1t

12s



3 C1t 2 noticing that the function V is nondecreasing, Z

2 0

12s

dt:

 1  .2/12s , hence,

V 0 .t/ dt

C5 12s V.2/:

dt

1

1  2s

Since 1  2s > 0, one has for large  that

 n2s V.  K/ n C5 12s 2



0

84

4 Nonlocal Phase Transitions

Therefore 2s V.  K/

n2s n

 2C5 V.2/:

(4.30)

Now we use an inductive argument as in Lemma 3.2 in [125], that we recall here: Lemma 4.2 Let ; 2 .0; 1/;  2 .; 1/ and ; R0 ; C 2 .1; 1/. Let VW .0; 1/ ! .0; 1/be a nondecreasing function. For any r 2 ŒR0 ; 1/, let log V.r/ . Suppose that V.R0 / > and ˛.r/ WD min 1; log r r ˛.r/V.r/

 

 CV. r/;

for any r 2 ŒR0 ; 1/. Then there exist c 2 .0; 1/ and R? 2 ŒR0 ; 1/, possibly depending on ; ; ; R0 ; C such that V.r/ > cr ; for any r 2 ŒR? ; 1/. For R large, one obtains from (4.30) and Lemma 4.2 that V.R/  c0 Rn ; for a suitable c0 2 .0; 1/. Let now  ? WD maxf1 ; 2 ; 1 C cg: We have that jfu > ? g \ BR j C jf? < u <  ? g \ BR j Djfu > ? g \ BR j

(4.31)

DV.R/  c0 R : n

Moreover, from (4.1.2) we have that for some c > 0 E .u; BR /  cRn2s ; therefore Z cR

n2s

E .u; BR /  

inf

t2.? ; ? /

f?  ? g \ BR j  CRn ; with C possibly depending on n; s; W. This concludes the proof of Theorem 4.1.3 in the case s 2 .0; 1=2/.

4.2 A Nonlocal Version of a Conjecture by De Giorgi In this section we consider the fractional counterpart of the conjecture by De Giorgi that was discussed before in the classical case. Namely, we consider the nonlocal Allen-Cahn equation ./s u C W.u/ D 0 in

Rn ;

where W is a double-well potential, and u is smooth, bounded and monotone in one direction, namely juj  1 and @xn u > 0. We wonder if it is also true, at least in low dimension, that u is one-dimensional. In this case, the conjecture was initially proved for n D 2 and s D 12 in [26]. In the case n D 2, for any s 2 .0; 1/, the result is proved using the harmonic extension of the fractional h i Laplacian in [25] and [132]. For n D 3, the proof can be found in [23] for s 2 12 ; 1 . The conjecture is still open i h for n D 3 and s 2 0; 12 and for n  4. Also, the Gibbons conjecture (that is the De Giorgi conjecture with the additional condition that limit in (4.8) is uniform) is also true for any s 2 .0; 1/ and in any dimension n, see [74]. To keep the discussion as simple as possible, we focus here on the case n D 2 and any s 2 .0; 1/, providing an alternative proof that does not make use of the harmonic extension. This part is completely new and not available in the literature. The proof is indeed quite general and it will be further exploited in [42]. We define (as in (4.10)) the total energy of the system to be Z E .u; BR / D KR .u/ C

W.u/dx;

(4.32)

BR

where the kinetic energy is 1 KR .u/ WD 2

“ QR

ju.x/  u.Nx/j2 dx dNx; jx  xN jnC2s

(4.33)

86

4 Nonlocal Phase Transitions

2 n n and QR WD R2n n .BC R / D .BR  BR / [ .BR  .R n BR // [ ..R n BR /  BR /. We recall that the kinetic energy can also be written as

KR .u/ D

1 u.BR ; BR / C u.BR ; BC R /; 2

(4.34)

where for two sets A; B Z Z u.A; B/ D A

B

ju.x/  u.Nx/j2 dx dNx: jx  xN jnC2s

(4.35)

The main result of this section is the following. Theorem 4.2.1 Let u be a minimizer of the energy defined in (4.32) in any ball of R2 . Then u is 1-D, i.e. there exist ! 2 S1 and u0 W R ! R such that u.x/ D u0 .!  x/

for any x 2 R2 :

The proof relies on the following estimate for the kinetic energy, that we prove by employing a domain deformation technique. Lemma 4.3 Let R > 1, ' 2 C01 .B1 /. Also, for any y 2 Rn , let R;C .y/ WD y C '

y R

e1 and R; .y/ WD y  '

y R

e1 :

(4.36)

Then, for large R, the maps R;C and R; are diffeomorphisms on Rn . Furthermore, 1 if we define uR;˙ .x/ WD u.R;˙ .x//, we have that KR .uR;C / C KR .uR; /  2KR .u/ 

C KR .u/; R2

(4.37)

for some C > 0. Proof First of all, we compute the Jacobian of R;˙ . For this, we write R;C;i to denote the ith component of the vector R;C D .R;C;1 ;    ; R;C;n / and we observe that

y 

y @

1 @R;C;i .y/ yi ˙ ' ıi1 D ıij ˙ @j ' ıi1 : D @yj @yj R R R

(4.38)

The latter term is bounded by O.R1 /, and this proves that R;˙ is a diffeomorphism if R is large enough. For further reference, we point out that if JR;˙ is the Jacobian determinant of R;˙ , then the change of variable x WD R;˙ .y/;

xN WD R;˙ .Ny/

(4.39)

4.2 A Nonlocal Version of a Conjecture by De Giorgi

87

gives that dx dNx D JR;˙ .y/ JR;˙ .Ny/ dy dNy 

1 

1 

y

yN 

1 1 1 ˙ @1 ' @1 ' CO 2 C O 2 dydNy D 1˙ R R R R R R  





1 1 1 y yN D 1 ˙ @1 ' ˙ @1 ' C O 2 dy dNy; R R R R R thanks to (4.38). Therefore juR;˙ .x/  uR;˙ .Nx/j2 dx dNx jx  xN jnC2s ˇ ˇ ˇu. 1 .x//  u. 1 .Nx//ˇ2 R;˙ R;˙ D  1 1 jR;˙ .x/  R;˙ .Nx/jnC2s

jx  xN j2 1 1 jR;˙ .x/  R;˙ .Nx/j2

! nC2s 2 dx dNx

0ˇ ˇ2 1 nC2s 2 ˇ ˇ ju.y/  u.Ny/j2 B ˇR;˙ .y/  R;˙ .Ny/ˇ C D @ A jy  yN jnC2s jy  yN j2 !

1

y 1

yN  1 ˙ @1 ' CO 2  1 ˙ @1 ' dy dNy: R R R R R (4.40) Now, for any y, yN 2 Rn we calculate ˇ ˇ2 ˇ ˇ ˇR;˙ .y/  R;˙ .Ny/ˇ   ˇ

yN  ˇ2 y ˇ ˇ ' e1 ˇ D ˇ.y  yN / ˙ ' R R ˇ   

yN ˇˇ2

 ˇ y ˇ ˙ 2 ' y  ' yN D jy  yN j2 C ˇˇ' ' .y1  yN 1 /: R R ˇ R R

(4.41)

Notice also that ˇ 

yN ˇˇ 1 ˇ y ˇ' ˇ  k'kC1 .Rn / jy  yN j;  ' ˇ R R ˇ R hence (4.41) becomes ˇ ˇ2 ˇ ˇ ˇR;˙ .y/  R;˙ .Ny/ˇ jy  yN j2

D 1 C ˙

(4.42)

88

4 Nonlocal Phase Transitions

where

˙ WD

ˇ 

ˇˇ2 ˇ y yN ˇ ˇ'  ' ˇ R R ˇ jy  yN j2

˙2

 

 ' Ry  ' RyN .y1  yN 1 / jy  yN j2

DO

1 : R

(4.43)

As a consequence 0ˇ ˇ2 1 nC2s 2 ˇ ˇ y/ˇ ˇ R;˙ .y/  R;˙ .N nC2s n C 2s B C ˙ C O.2˙ /: D .1 C ˙ / 2 D 1  @ A 2 jy  yN j 2 We plug this information into (4.40) and use (4.43) to obtain juR;˙ .x/  uR;˙ .Nx/j2 dx dNx jx  xN jnC2s 

1  ju.y/  u.Ny/j2 n C 2s ˙ C O 2 D  1 jy  yN jnC2s 2 R 

y 1

yN 

1  1  1 ˙ @1 ' dy dNy ˙ @1 ' CO 2 R R R R R " 

y 1

yN  ju.y/  u.Ny/j2 n C 2s 1 ˙ C ˙ @1 ' ˙ @1 ' D  1 jy  yN jnC2s 2 R R R R #

1 CO 2 dy dNy: R Using this and the fact that

C C  D 2

ˇ 

ˇˇ2 ˇ y yN ˇ ˇ' ˇ R ' R ˇ jy 

yN j2

DO

1 ; R2

thanks to (4.42), we obtain juR;C .x/  uR;C .Nx/j2 juR; .x/  uR; .Nx/j2 C dx dNx jx  xN jnC2s jx  xN jnC2s 

1  ju.y/  u.Ny/j2 D  2 C O dy dNy: jy  yN jnC2s R2

4.2 A Nonlocal Version of a Conjecture by De Giorgi

89

Thus, if we integrate over QR we find that “ KR .uR;C / C KR .uR;C / D 2KR .u/ C

O QR

1  ju.x/  u.Nx/j2 dx dNx: R2 jx  xN jnC2s

This establishes (4.37). Proof (Proof of Theorem 4.2.1) We organize this proof into four steps. Step 1. A geometrical consideration In order to prove that the level sets are flat, it suffices to prove that u is monotone in any direction. Indeed, if u is monotone in any direction, the level set fu D 0g is both convex and concave, thus it is flat. Step 2. Energy estimates Let ' 2 C01 .B1 / such that ' D 1 in B1=2 , and let e D .1; 0/. We define as in Lemma 4.3

y

y e and R; .y/ WD y  ' e; R;C .y/ WD y C ' R R 1 which are diffeomorphisms for large R, and the functions uR;˙ .x/ WD u.R;C .x//. Notice that

uR;C .y/ D u.y/

for y 2 BC R

(4.44)

uR;C .y/ D u.y  e/

for y 2 BR=2 :

(4.45)

By computing the potential energy, it is easy to see that Z

Z

Z

W.uR;C .x// dx C BR

W.uR; .x// dx  2 BR



C R2

W.u.x// dx BR

Z W.u.x// dx: BR

Using this and (4.37), we obtain the following estimate for the total energy E .uR;C ; BR / C E .uR; ; BR /  2E .u; BR / 

C E .u; BR /: R2

(4.46)

Also, since uR;˙ D u in BC R , we have that E .u; BR /  E .uR; ; BR /: This and (4.46) imply that E .uR;C ; BR /  E .u; BR / 

C E .u; BR /: R2

(4.47)

90

4 Nonlocal Phase Transitions

As a consequence of this estimate and (4.11), it follows that lim

R!C1

 E .uR;C ; BR /  E .u; BR / D 0:

(4.48)

Step 3. Monotonicity We claim that u is monotone. Suppose by contradiction that u is not monotone. That is, up to translation and dilation, we suppose that the value of u at the origin stays above the values of e and e, with e WD .1; 0/, i.e. u.0/ > u.e/ and u.0/ > u.e/: Take R to be large enough, say R > 8. Let now ˚  ˚  vR .x/ WD min u.x/; uR;C .x/ and wR .x/ WD max u.x/; uR;C .x/ :

(4.49)

By (4.44) we have that vR D wR D u outside BR . Then, since u is a minimizer in BR and wR D u outside BR , we have that E .wR ; BR /  E .u; BR /:

(4.50)

Moreover, the sum of the energies of the minimum and the maximum is less than or equal to the sum of the original energies: this is obvious in the local case, since equality holds, and in the nonlocal case the proof is based on the inspection of the different integral contributions, see e.g. formula (38) in [114]. So we have that E .vR ; BR / C E .wR ; BR /  E .u; BR / C E .uR;C ; BR / hence, recalling (4.50), E .vR ; BR /  E .uR;C ; BR /:

(4.51)

We claim that vR is not identically neither u, nor uR;C . Indeed, since u.0/ D uR;C .e/ and u.e/ D uR;C .0/ we have that ˚  ˚  vR .0/ D min u.0/; uR;C .0/ D min u.0/; u.e/ D u.e/ D uR;C .0/ < u.0/ and ˚  ˚  vR .e/ D min u.e/; uR;C .e/ D min u.e/; u.0/ D u.e/ < u.0/ D uR;C .e/:

4.2 A Nonlocal Version of a Conjecture by De Giorgi

91

By continuity of u and uR;C , we have that vR D uR;C < u in a neighborhood of 0 and vR D u < uR;C in a neighborhood of e:

(4.52)

We focus our attention on the energy in the smaller ball B2 . We claim that vR is not minimal for E .; B2 /. Indeed, if vR were minimal in B2 , then on B2 both vR and u would satisfy the same equation. However, vR  u in R2 by definition and vR D u in a neighborhood of e by the second statement in (4.52). The Strong Maximum Principle implies that they coincide everywhere, which contradicts the first line in (4.52). Hence vR is not a minimizer in B2 . Let then vR be a minimizer of E .; B2 /, that agrees with vR outside the ball B2 , and we define the positive quantity ıR WD E .vR ; B2 /  E .vR ; B2 /:

(4.53)

We claim that as R goes to infinity, ıR remains bounded away from zero.

(4.54)

To prove this, we assume by contradiction that lim ıR D 0:

(4.55)

R!C1

Consider uQ to be the translation of u, that is uQ .x/ WD u.x  e/. Let also ˚  m.x/ WD min u.x/; uQ .x/ : We notice that in BR=2 we have that uQ .x/ D uR;C .x/. This and (4.49) give that m D vR in BR=2 .

(4.56)

Also, from (4.52) and (4.56), it follows that m cannot be identically neither u nor uQ , and m < u in a neighborhood of 0

and

m D u in a neighborhood of e:

(4.57)

Let z be a competitor for m in the ball B2 , that agrees with m outside B2 . We take a cut-off function 2 C01 .Rn / such that D 1 in BR=4 , D 0 in BC R=2 . Let zR .x/ WD

 .x/z.x/ C 1 

.x/ vR .x/:

92

4 Nonlocal Phase Transitions

Then we have that zR D z on BR=4 and zR D vR on BC R=2 .

(4.58)

In addition, by (4.56), we have that z D m D vR in BR=2 n B2 . So, it follows that zR .x/ D

.x/vR .x/ C .1 

.x//vR .x/ D vR .x/ D z.x/

on BR=2 n B2 :

This and (4.58) imply that zR D vR on BC 2 . We summarize in the next lines these useful identities (see also Fig. 4.3). uR;C D uQ ;

m D vR ;

in BR=2 n B2

uR;C D uQ ;

vR

in BR n BR=2

vR D vR D zR ;

in BC R

uR;C D u D vR D vR D zR ;

in B2

z D zR

D vR D m D z D zR mDz m D z:

We compute now E .m ; B2 /  E .z; B2 / D E .m; B2 /  E .vR ; B2 / C E .vR ; B2 /  E .zR ; B2 / C E .zR ; B2 /  E .z; B2 /: By the definition of ıR in (4.53), we have that E .m ; B2 /  E .z; B2 / D E .m; B2 /  E .vR ; B2 / C ıR C E .vR ; B2 /  E .zR ; B2 / C E .zR ; B2 /  E .z; B2 /: (4.59)

Fig. 4.3 Energy estimates

4.2 A Nonlocal Version of a Conjecture by De Giorgi

93

Using the formula for the kinetic energy given in (4.34) together with (4.35) we have that E .m ; B2 /  E .vR ; B2 / D

1 m.B2 ; B2 / C m.B2 ; BC 2 /C 2

Z

 W m.x/ dx B2

1  vR .B2 ; B2 /  vR .B2 ; BC 2 / 2

Z

 W vR .x/ dx: B2

Since m D vR on BR=2 (recall (4.56)), we obtain E .m ; B2 /  E .vR ; B2 / Z Z jm.x/  m.y/j2  jm.x/  vR .y/j2 D dx dy : jx  yjnC2s B2 BC R=2 Notice now that m and vR are bounded on Rn (since so is u). Also, if x 2 B2 and y 2 BC R=2 we have that jx  yj  jyj  jxj  jyj=2 if R is large. Accordingly, Z

Z

E .m; B2 /  E .vR ; B2 /  C

dx B2

1 C

BR=2

jyjnC2s

dy  CR2s ;

(4.60)

up to renaming constants. Similarly, zR D z on BR=2 and we have the same bound E .zR ; B2 /  E .z; B2 /  CR2s :

(4.61)

Furthermore, since vR is a minimizer for E .; B2 / and vR D zR outside of B2 , we have that E .vR ; B2 /  E .zR ; B2 /  0: Using this, (4.60) and (4.61) in (4.59), it follows that E .m; B2 /  E .z; B2 /  CR2s C ıR : Therefore, by sending R ! C1 and using again (4.55), we obtain that E .m; B2 /  E .z; B2 /:

(4.62)

We recall that z can be any competitor for m, that coincides with m outside of B2 . Hence, formula (4.62) means that m is a minimizer for E .; B2 /. On the other hand, u is a minimizer of the energy in any ball. Then, both u and m satisfy the same equation in B2 . Moreover, they coincide in a neighborhood of e, as stated in the second line

94

4 Nonlocal Phase Transitions

of (4.57). By the Strong Maximum Principle, they have to coincide on B2 , but this contradicts the first statement of (4.57). The proof of (4.54) is thus complete. Now, since vR D vR on BC 2 , from definition (4.53) we have that ıR D E .vR ; BR /  E .vR ; BR /: Also, E .vR ; BR /  E .u; BR /, thanks to the minimizing property of u. Using these pieces of information and inequality (4.51), it follows that ıR  E .uR;C ; BR /  E .u; BR /: Now, by sending R ! C1 and using (4.54), we have that lim E .uR;C ; BR /  E .u; BR / > 0;

R!C1

which contradicts (4.48). This implies that indeed u is monotone, and this concludes the proof of this Step. Step 4. Conclusions In Step 3, we have proved that u is monotone, in any given direction e. Then, Step 1 gives the desired result. This concludes the proof of Theorem 4.2.1. We remark that the exponent two in the energy estimate (4.37) is related to the expansions of order two and not to the dimension of the space. Indeed, the energy estimates hold for any n. However, the two power in the estimate (4.37) allows us to prove the fractional version of De Giorgi conjecture only in dimension two. In other words, the proof of Theorem 4.2.1 is not applicable for n > 2. One can verify this by checking the limit in (4.48)

lim

R!C1

 E .uR;C ; BR /  E .u; BR / D 0;

which was necessary for the Proof of Theorem 4.2.1 in the case n D 2. We know from Theorem 4.1.2 that lim

R!C1

C E .u; BR / D 0: Rn

Confronting this result with inequality (4.47) E .uR;C ; BR /  E .u; BR / 

C E .u; BR /; R2

we see that we need to have n D 2 in order for the the limit in (4.48) to be zero. Of course, the one-dimensional symmetry property in Theorem 4.2.1 is inherited by the spatial homogeneity of the equation, which is translation and rotation invariant. In the case, for instance, in which the potential also depends on the space

4.2 A Nonlocal Version of a Conjecture by De Giorgi

95

variable, the level sets of the (minimal) solutions may curve, in order to adapt themselves to the spatial inhomogeneity. Nevertheless, in the case of periodic dependence, it is possible to construct minimal solutions whose level sets are possibly not planar, but still remain at a bounded distance from any fixed hyperplane. As a typical result in this direction, we recall the following one (we refer to [44] for further details on the argument) (Fig. 4.4): Theorem 4.2.2 Let QC > Q > 0 and Q W Rn ! ŒQ ; QC . Suppose that Q.x C k/ D Q.x/ for any k 2 Zn . Let us consider, in any ball BR , the energy defined by 1 E .u; BR / D KR .u/ C 4

Z

Q.x/ .1  u2 /2 dx; BR

where the kinetic energy KR .u/ is defined as in (4.33). Then, there exists a constant M > 0, such that, given any ! 2 @B1 , there exists a minimal solution u! of ./s u! .x/ D Q.x/ .u! .x/  u3! .x//

for any x 2 Rn

9 for which the level sets fju! j  10 g are contained in the strip fx 2 Rn s.t. j!  xj  Mg. Moreover, if ! is rotationally dependent, i.e. if there exists ko 2 Zn such that !  ko D 0, then u! is periodic with respect to !, i.e.

u! .x/ D u! .y/ for any x, y 2 Rn such that x  y D k and !  k D 0.

Fig. 4.4 Minimal solutions in periodic medium

Chapter 5

Nonlocal Minimal Surfaces

In this chapter, we introduce nonlocal minimal surfaces and focus on two main results, a Bernstein type result in any dimension and the non-existence of nontrivial s-minimal cones in dimension 2. Moreover, some boundary properties will be discussed at the end of this chapter. For a preliminary introduction to some properties of the nonlocal minimal surfaces, see [135]. Let ˝ Rn be an open bounded domain, and E Rn be a measurable set, fixed outside ˝. We will consider for s 2 .0; 1=2/ minimizers of the H s norm jjE jj2H s D

Z

Z Rn

jE .x/  E .y/j2 dx dy jx  yjnC2s

Rn

Z

Z

D2 Rn

Rn

E .x/EC .y/ dx dy: jx  yjnC2s

Notice that only the interactions between E and EC contribute to the norm. In order to define the fractional perimeter of E in ˝, we need to clarify the contribution of ˝ to the H s norm here introduced. Namely, as E is fixed outside ˝, we aim at minimizing the “˝-contribution” to the norm among all measurable sets that “vary” inside ˝. We consider thus interactions between E \ ˝ and EC and between E n ˝ and ˝ n E, neglecting the data that is fixed outside ˝ and that does not contribute to the minimization of the norm (see Fig. 5.1). We define the interaction I.A; B/ of two disjoint subsets of Rn as Z Z I.A; B/ WD Z

A

B

Z

dx dy jx  yjnC2s

D Rn

Rn

A .x/B .x/ dx dy: jx  yjnC2s

© Springer International Publishing Switzerland 2016 C. Bucur, E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana 20, DOI 10.1007/978-3-319-28739-3_5

(5.1)

97

98

5 Nonlocal Minimal Surfaces

Fig. 5.1 Fractional perimeter

Then (see [28]), one defines the nonlocal s-perimeter functional of E in ˝ as Pers .E; ˝/ WD I.E \ ˝; EC / C I.E n ˝; ˝ n E/:

(5.2)

Equivalently, one may write Pers .E; ˝/ D I.E \ ˝; ˝ n E/ C I.E \ ˝; EC n ˝/ C I.E n ˝; ˝ n E/: Definition 5.1 Let ˝ be an open domain of Rn . A measurable set E Rn is sminimal in ˝ if Pers .E; ˝/ is finite and if, for any measurable set F such that E n ˝ D F n ˝, we have that Pers .E; ˝/  Pers .F; ˝/: A measurable set is s-minimal in Rn if it is s-minimal in any ball Br , where r > 0. When s ! 12 , the fractional perimeter Pers approaches the classical perimeter, see [21]. See also [45] for the precise limit in the class of functions with bounded variations, [34, 35] for a geometric approach towards regularity and [6, 117] for an approach based on -convergence. See also [136] for a different proof and Theorem 2.22 in [103] and the references therein for related discussions. A simple, formal statement, up to renormalizing constants, is the following: Theorem 5.2 Let R > 0 and E be a set with finite perimeter in BR . Then lim

s! 12

for almost any r 2 .0; R/.

1 2

  s Pers .E; Br / D Per .E; Br /

5 Nonlocal Minimal Surfaces

99

The behavior of Pers as s ! 0 is slightly more involved. In principle, the limit as s ! 0 of Pers is, at least locally, related to the Lebesgue measure (see e.g. [106]). Nevertheless, the situation is complicated by the terms coming from infinity, which, as s ! 0, become of greater and greater importance. More precisely, it is proved in [58] that, if Perso .E; ˝/ is finite for some so 2 .0; 1=2/, and the limit Z ˇE WD lim 2s s!0

EnB1

dy jyjnC2s

(5.3)

exists, then lim 2s Pers .E; ˝/ D .j@B1 j  ˇE / jE \ ˝j C ˇE j˝ n Ej:

s!0

(5.4)

We remark that, using polar coordinates, Z 0  ˇE  lim 2s s!0

Rn nB1

dy D lim 2s j@B1 j s!0 jyjnC2s

Z

C1 1

12s d D j@B1 j;

therefore ˇE 2 Œ0; j@B1 j plays the role of a convex interpolation parameter in the right hand-side of (5.4) (up to normalization constants). In this sense, formula (5.4) may be interpreted by saying that, as s ! 0, the sperimeter concentrates itself on two terms that are “localized” in the domain ˝, namely jE \ ˝j and j˝ n Ej. Nevertheless, the proportion in which these two terms count is given by a “strongly nonlocal” interpolation parameter, namely the quantity ˇE in (5.3) which “keeps track” of the behavior of E at infinity. As a matter of fact, to see how ˇE is influenced by the behavior of E at infinity, one can compute ˇE for the particular case of a cone. For instance, if ˙  @B1 , j˙j DW b 2 Œ0; 1 , and E is the cone over ˙ (that is E WD ftp; p 2 ˙; t  0g), with j@B 1j we have that Z ˇE D lim 2s j˙j s!0

C1 1

12s d D j˙j D b j@B1 j;

that is ˇE gives in this case exactly the opening of the cone. We also remark that, in general, the limit in (5.3) may not exist, even for smooth sets: indeed, it is possible that the set E “oscillates” wildly at infinity, say from one cone to another one, leading to the non-existence of the limit in (5.3). Moreover, we point out that the existence of the limit in (5.3) is equivalent to the existence of the limit in (5.4), except in the very special case jE \ ˝j D j˝ n Ej, in which the limit in (5.4) always exists. That is, the following alternative holds true: • if jE \ ˝j ¤ j˝ n Ej, then the limit in (5.3) exists if and only if the limit in (5.4) exists,

100

5 Nonlocal Minimal Surfaces

• if jE \ ˝j D j˝ n Ej, then the limit in (5.4) always exists (even when the one in (5.3) does not exist), and lim 2s Pers .E; ˝/ D j@B1 j jE \ ˝j D j@B1 j j˝ n Ej:

s!0

The boundaries of s-minimal sets are referred to as nonlocal minimal surfaces. In [28] it is proved that s-minimizers satisfy a suitable integral equation (see in particular Theorem 5.1 in [28]), that is the Euler-Lagrange equation corresponding to the s-perimeter functional Pers . If E is s-minimal in ˝ and @E is smooth enough, this Euler-Lagrange equation can be written as Z Rn

E .x0 C y/  Rn nE .x0 C y/ dy D 0; jyjnC2s

(5.5)

for any x0 2 ˝ \ @E. Therefore, in analogy with the case of the classical minimal surfaces, which have zero mean curvature, one defines the nonlocal mean curvature of E at x0 2 @E as Z HEs .x0 /

WD Rn

E .y/  EC .y/ dy: jy  x0 jnC2s

(5.6)

In this way, Eq. (5.5) can be written as HEs D 0 along @E. It is also suggestive to think that the function Q E WD E  EC averages out to zero at the points on @E, if @E is smooth enough, since at these points the local contribution of E compensates the one of EC . Using this notation, one may take the liberty of writing HEs .x0 / D D D

1 2 1 2

Z Z

Rn

Q E .x0 C y/ C Q E .x0  y/ dy jyjnC2s

Rn

Q E .x0 C y/ C Q E .x0  y/  2Q E .x0 / dy jyjnC2s

./s Q E .x0 / ; C.n; s/

using the notation of (1.1). Using this suggestive representation, the Euler-Lagrange equation in (5.5) becomes ./s Q E D 0 along @E. We refer to [3] for further details on this argument. It is also worth recalling that the nonlocal perimeter functionals find applications in motions of fronts by nonlocal mean curvature (see e.g. [32, 40, 91]), problems in which aggregating and disaggregating terms compete towards an equilibrium (see

5.1 Graphs and s-Minimal Surfaces

101

e.g. [78] and [56]) and nonlocal free boundary problems (see e.g. [29] and [61]). See also [106] and [139] for results related to this type of problems. In the classical case of the local perimeter functional, it is known that minimal surfaces are smooth in dimension n  7. Moreover, if n  8 minimal surfaces are smooth except on a small singular set of Hausdorff dimension n  8. Furthermore, minimal surfaces that are graphs are called minimal graphs, and they reduce to hyperplanes if n  8 (this is called the Bernstein property, which was also discussed at the beginning of the Chap. 4). If n  9, there exist global minimal graphs that are not affine (see e.g. [86]). Differently from the classical case, the regularity theory for s-minimizers is still quite open. We present here some of the partial results obtained in this direction: Theorem 5.3 In the plane, s-minimal sets are smooth. More precisely: (a) If E is an s-minimal set in ˝ R2 , then @E \ ˝ is a C1 -curve. (b) Let E be s-minimal in ˝ Rn and let ˙E @E \ ˝ denote its singular set. Then H d .˙E / D 0 for any d > n  3. See [124] for the proof of this results (as a matter of fact, in [124] only C1;˛ regularity is proved, but then [9] proved that s-minimal sets with C1;˛ -boundary are automatically C1 ). Further regularity results of the s-minimal surfaces can be 1 found in [35]. There, a regularity theory when s is near is stated, as we see in the 2 following Theorem:

1 1 Theorem 5.4 There exists 0 2 0; such that if s   0 , then 2 2 (a) if n  7, any s-minimal set is of class C1 , (b) if n D 8 any s-minimal surface is of class C1 except, at most, at countably many isolated points, (c) any s-minimal surface is of class C1 outside a closed set ˙ of Hausdorff dimension n  8.

5.1 Graphs and s-Minimal Surfaces We will focus the upcoming material on two interesting results related to graphs: a Bernstein type result, namely the property that an s-minimal graph in RnC1 is flat (if no singular cones exist in dimension n); we will then prove that an s-minimal surface whose prescribed data is a subgraph, is itself a subgraph. The first result is the following theorem: Theorem 5.1.1 Let E D f.x; t/ 2 Rn  R s.t. t < u.x/g be an s-minimal graph, and assume there are no singular cones in dimension n (that is, if K Rn is an s-minimal cone, then K is a half-space). Then u is an affine function (thus E is a half-space).

102

5 Nonlocal Minimal Surfaces

To be able to prove Theorem 5.1.1, we recall some useful auxiliary results. In the following lemma we state a dimensional reduction result (see Theorem 10.1 in [28]). Lemma 5.1 Let E D F  R. Then if E is s-minimal if and only if F is s-minimal. We define then the blow-up and blow-down of the set E are, respectively E0 WD lim Er r!0

and E1 WD lim Er ; r!C1

where Er D

E : r

A first property of the blow-up of E is the following (see Lemma 3.1 in [79]). Lemma 5.2 If E1 is affine, then so is E. We recall also a regularity result for the s-minimal surfaces (see [79] and [9] for details and proof). Lemma 5.3 Let E be s-minimal. Then: (a) If E is Lipschitz, then E is C1;˛ . (b) If E is C1;˛ , then E is C1 . We give here a sketch of the proof of Theorem 5.1.1 (see [79] for all the details). Proof (Sketch of the proof of Theorem 5.1.1) If E RnC1 is an s-minimal graph, then the blow-down E1 is an s-minimal cone (see Theorem 9.2 in [28] for the proof of this statement). By applying the dimensional reduction argument in Lemma 5.1 we obtain an s-minimal cone in dimension n. According to the assumption that no singular s-minimal cones exist in dimension n, it follows that necessarily E1 can be singular only at the origin. We consider a bump function w0 2 C1 .R; Œ0; 1 / such that  w0 .t/ D 0 in

 1; 

w0 .t/ D 1 in

2 3 ; 5 5



1 4



 [

3 ; C1 4



w.t/ D w0 .jtj/: The blow-down of E is ˚  E1 D .x0 ; xnC1 / s.t. xnC1  u1 .x0 / : For a fixed  2 @B1 , let  ˚  Ft WD .x0 ; xnC1 / s.t. xnC1  u1 x0 C tw.x0 /  t

5.1 Graphs and s-Minimal Surfaces

103

be a family of sets, where t 2 .0; 1/ and  > 0. Then for  small, we have that F1 is below E1 .

(5.7)

Indeed, suppose by contradiction that this is not true. Then, there exists k ! 0 such that  u1 x0k C k w.x0k /  1  u1 .x0k /:

(5.8)

But x0k 2 suppw, which is compact, therefore x01 WD lim x0k belongs to the support k!C1

of w, and w.x01 / is defined. Then, by sending k ! C1 in (5.8) we have that u1 .x01 /  1  u1 .x01 /; which is a contradiction. This establishes (5.7). Now consider the smallest t0 2 .0; 1/ for which Ft is below E1 . Since E1 is a graph, then Ft0 touches E1 from below in one point X0 D .x00 ; x0nC1 /, where x00 2 suppw. Since E1 is s-minimal, we have that the nonlocal mean curvature (defined in (5.6)) of the boundary is null. Also, since Ft0 is a C2 diffeomorphism of E1 we have that HFs t .p/ ' t0 ; 0

(5.9)

and there is a region where E1 and Ft0 are well separated by t0 , thus ˇ  ˇ ˇ E1 n Ft \ B3 n B2 ˇ  ct0 ; 0 for some c > 0. Therefore, we see that HFs t .p/ D HFs t .p/  HEs .p/  ct0 : 0

0

This and (5.9) give that t0  ct0 , for some c > 0 (up to renaming it). If  is small enough, this implies that t0 D 0. In particular, we have proved that there exists  > 0 small enough such that, for any t 2 .0; 1/ and any  2 @B1 , we have that  u1 x0 C tw.x0 /  t  u1 .x0 /: This implies that  u1 x0 C tw.x0 /  u1 .x0 / 1  ; t 

104

5 Nonlocal Minimal Surfaces

hence, letting t ! 0, we have that ru1 .x0 /w.x0 / 

1 ; for any x 2 Rn n f0g; and  2 B1 : 

We recall now that w D 1 in B3=5 n B2=5 and  is arbitrary in @B1 . Hence, it follows that jru1 .x/j 

1 ; for any x 2 B3=5 n B2=5 : 

Therefore u1 is globally Lipschitz. By the regularity statement in Lemma 5.3, we have that u1 is C1 . This says that u is smooth also at the origin, hence (being a cone) it follows that E1 is necessarily a half-space. Then by Lemma 5.2, we conclude that E is a half-space as well. We introduce in the following theorem another interesting property related to sminimal surfaces, in the case in which the fixed given data outside a domain is a subgraph. In that case, the s-minimal surface itself is a subgraph. Indeed: Theorem 5.1.2 Let ˝0 be an open and bounded subset of Rn1 with boundary of class C1;1 and let ˝ WD ˝0  R. Let E be an s-minimal set in ˝. Assume that E n ˝ D fxn < u.x0 /; x0 2 Rn1 n ˝0 g

(5.10)

for some continuous function uW Rn1 ! R. Then E \ ˝ D fxn < v.x0 /; x0 2 ˝0 g for some function vW Rn1 ! R. The reader can see [64], where this theorem and the related results are proved; here, we only state the preliminary results needed for our purposes and focus on the proof of Theorem 5.1.2. The proof relies on a sliding method, more precisely, we take a translation of E in the nth direction, and move it until it touches E from above. If the set E \ ˝ is a subgraph, then, up to a set of measure 0, the contact between the translated E and E, will be E itself. However, since we have no information on the regularity of the minimal surface, we need at first to “regularize” the set by introducing the notions of supconvolution and subconvolution. With the aid of a useful result related to the sub/supconvolution of an s-minimal surface, we proceed then with the proof of the Theorem 5.1.2. The supconvolution of a set E  Rn (Fig. 5.2) is given by ]

Eı WD

[ x2E

Bı .x/:

5.1 Graphs and s-Minimal Surfaces

105

Fig. 5.2 The supconvolution of a set

In an equivalent way, the supconvolution can be written as [

]

Eı D

.E C v/:

v2Rn jvjı

Indeed, we consider ı > 0 and an arbitrary x 2 E. Let y 2 Bı .x/ and we define v WD y  x. Then jvj  jy  xj  ı

and y D x C v 2 E C v:

Therefore Bı .x/  E C v for jvj  ı. In order to prove the inclusion in the opposite direction, one notices that taking y 2 E C v with jvj  ı and defining x WD y  v, it follows that jx  yj D jvj  ı: Moreover, x 2 .E C v/  v D E and the inclusion E C v 2 Bı .x/ is proved. On the other hand, the subconvolution is defined as 

] Eı[ WD Rn n .Rn n E/ı : Now, the idea is that the supconvolution of E is a regularized version of E whose nonlocal minimal curvature is smaller than the one of E, i.e.: Z

Rn nE] .y/  E] .y/ ı

Rn

jx 

ı

yjnC2s

Z dy  Rn

Rn nE .y/  E .y/ dy  0; jQx  yjnC2s

(5.11)

106

5 Nonlocal Minimal Surfaces ]

for any x 2 @Eı , where xQ WD x  v 2 @E for some v 2 Rn with jvj D ı. Then, by ] construction, the set E C v lies in Eı , and this implies (5.11). Similarly, one has that the opposite inequality holds for the subconvolution of E, for any x 2 @Eı[ Z

Rn nE[ .y/  E[ .y/ ı

jx  yjnC2s

Rn

ı

dy  0;

(5.12)

By (5.11) and (5.12), we obtain: ]

Proposition 5.1.3 Let E be an s-minimal set in ˝. Let p 2 @Eı and assume that ] Bı .p/  ˝. Assume also that Eı is touched from above by a translation of Eı[ , n namely there exists ! 2 R such that ]

Eı  Eı[ C ! and ]

p 2 .@Eı / \ .@Eı[ C !/: Then ]

Eı D Eı[ C !: Proof (Proof of Theorem 5.1.2) One first remark is that the s-minimal set does not have spikes which go to infinity: more precisely, one shows that ˝0  .1; M/  E \ ˝  ˝0  .1; M/

(5.13)

for some M  0. The proof of (5.13) can be performed by sliding horizontally a large ball, see [64] for details. After proving (5.13), one can deal with the core of the proof of Theorem 5.1.2. The idea is to slide E from above until it touches itself and analyze what happens at the contact points. For simplicity, we will assume here that the function u is uniformly continuous (if u is only continuous, the proof needs to be slightly modified since the subconvolution and supconvolution that we will perform may create new touching points at infinity). At this purpose, we consider Et D E C ten for t  0. Notice that, by (5.13), if t  2M, then E  Et . Let then t be the smallest for which the inclusion E  Et holds. We claim that t D 0. If this happens, one may consider v D inff s.t. .x; / 2 EC g and, up to sets of measure 0, E \ ˝0 is the subgraph of v.

5.1 Graphs and s-Minimal Surfaces

107

Fig. 5.3 Sliding E until it touches itself at an interior point

The proof is by contradiction, so let us assume that t > 0. According to (5.10), the set E n ˝ is a subgraph, hence the contact points between @E and @Et must lie in ˝ 0  R. Namely, only two possibilities may occur: the contact point is interior (it belongs to ˝0  R/, or it is at the boundary (on @˝0  R). So, calling p the contact point, one may have1 that either p 2 ˝0  R p 2 @˝0  R:

or

(5.14) (5.15)

We deal with the first case in (5.14) (an example of this behavior is depicted in ] Fig. 5.3). We consider Eı and Eı[ to be the supconvolution, respectively the subconvolution of E. We then slide the subconvolution until it touches the supconvolution. More precisely, let  > 0 and we take a translation of the subconvolution, Eı[ C en . ] For  large, we have that Eı  Eı[ C en and we consider ı to be the smallest for

1

As a matter of fact, the number of contact points may be higher than one, and even infinitely many contact points may arise. So, to be rigorous, one should distinguish the case in which all the contact points are interior and the case in which at least one contact point lies on the boundary. Moreover, since the surface may have vertical portions along the boundary of the domain, one needs to carefully define the notion of contact points (roughly speaking, one needs to take a definition for which the vertical portions which do not prevent the sliding are not in the contact set). Finally, in case the contact points are all interior, it is also useful to perform the sliding method in a slightly reduced domain, in order to avoid that the supconvolution method produces new contact points at the boundary (which may arise from vertical portions of the surfaces). Since we do not aim to give a complete proof of Theorem 5.1.2 here, but just to give the main ideas and underline the additional difficulty, we refer to [64] for the full details of these arguments.

108

5 Nonlocal Minimal Surfaces

which such inclusion holds. We have (since t is positive by assumption) that ı 

t > 0: 2

]

Moreover, for ı small, the sets @Eı and @.Eı[ C ı en / have a contact point which, according to (5.14), lies in ˝0  R. Let pı be such a point, so we may write ]

pı 2 .@Eı / \ @.Eı[ C ı en /

and

pı 2 ˝0  R:

Then, for ı small (notice that Bı .p/  ˝), Proposition 5.1.3 yields that ]

Eı D Eı[ C ı en : Considering ı arbitrarily small, one obtains that E D E C 0 e n ;

with 0 > 0:

But E is a subgraph outside of ˝, and this provides a contradiction. Hence, the claim that t D 0 is proved. Let us see that we also obtain a contradiction when supposing that t > 0 and that the second case (5.15) holds. Let p D .p0 ; pn /

and

p 2 .@E/ \ .@Et /:

Now, if one takes sequences ak 2 @E and bk 2 @Et , both that tend to p as k goes to infinity, since E n ˝ is a subgraph and t > 0, necessarily ak ; bk belong to ˝. Hence p 2 .@E/ \ ˝ \ .@Et / \ ˝:

(5.16)

Thanks to Definition 2.3 in [28], one obtains that E is a variational subsolution in a neighborhood of p. In other words, if A  E \ ˝ and p 2 A, then 0  Pers .E; ˝/  Pers .E n A; ˝/ D I.A; EC /  I.A; E n A/ (we recall the definition of I in (5.1) and of the fractional perimeter Pers in (5.2)). According to Theorem 5.1 in [28], this implies in a viscosity sense (i.e. if E is touched at p from outside by a ball), that Z Rn

E .y/  Rn nE .y/ dy  0: jp  yjnC2s

(5.17)

5.1 Graphs and s-Minimal Surfaces

109

Fig. 5.4 Sliding E until it touches itself at a boundary point

In order to obtain an estimate on the fractional mean curvature in the strong sense, we consider the translation of the point p as follows: pt D p  ten D .p0 ; pn  t/ D .p0 ; pn;t /: Since t > 0, one may have that either pn ¤ u.p0 /, or pn;t ¤ u.p0 /. These two possibilities can be dealt with in a similar way, so we just continue with the proof in the case pn ¤ u.p0 / (as is also exemplified in Fig. 5.4). Taking r > 0 small, the set Br .p/ n ˝ is contained entirely in E or in its complement. 1 Moreover, one has from [27] that @E \ Br .p/ is a C1; 2 Cs -graph in the direction of 1 the normal to ˝ at p. That is: in Fig. 5.4 the set E is C1; 2 Cs , hence in the vicinity of p D .p0 ; pn /, it appears to be sufficiently smooth. So, let .p/ D . 0 .p/; n .p// be the normal in the interior direction, then up to a rotation and since ˝ is a cylinder (hence n .p/ D 0), we can write .p/ D e1 . 1 Therefore, there exists a function  of class C1; 2 Cs such that p1 D  .p2 ; : : : ; pn / and, in the vicinity of p, we can write @E as the graph G D fx1 D  .x2 ; : : : ; xn /g. Given (5.16), we deduce that there exists a sequence pk 2 G such that pk 2 ˝ and pk ! p as k ! 1. From this it follows that there exists a sequence of points pk ! p such that 1

@E in the vicinity of pk is a graph of class C1; 2 Cs

(5.18)

and Z Rn

E .y/  EC .y/ dy D 0: jpk  yjnC2s

(5.19)

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From (5.18) and (5.19), and using a pointwise version of the Euler-Lagrange equation (see [64] for details), we have that Z Rn

E .y/  EC .y/ dy D 0: jp  yjnC2s

Now, E Et for t strictly positive, hence Z

Et .y/  EtC .y/ Rn

jp  yjnC2s

dy > 0:

(5.20)

Moreover, we have that the set @Et \ B 4r .p/ must remain on one side of the graph G, namely one could have that Et \ B 4r .p/  fx1   .x2 ; : : : ; xn /g or Et \ B 4r .p/  fx1   .x2 ; : : : ; xn /g: Given again (5.16), we deduce that there exists a sequence pQ k 2 @Et \ ˝ such that pQ k ! p as k ! 1 and @Et \ ˝ in the vicinity of pQ k is touched by a surface lying 1 in Et , of class C1; 2 Cs . Then Z

Et .y/  EtC .y/ Rn

jQpk  yjnC2s

dy  0:

Hence, making use of a pointwise version of the Euler-Lagrange equation (see [64] for details), we obtain that Z

Et .y/  EtC .y/ Rn

jp  yjnC2s

dy  0:

But this is a contradiction with (5.20), and this concludes the proof of Theorem 5.1.2. On the one hand, one may think that Theorem 5.1.2 has to be well-expected. On the other hand, it is far from being obvious, not only because the proof is not trivial, but also because the statement itself almost risks to be false, especially at the boundary. Indeed we will see in Theorem 5.3.2 that the graph property is close to fail at the boundary of the domain, where the sminimal surfaces may present vertical tangencies and stickiness phenomena (see Fig. 5.11).

5.2 Non-existence of Singular Cones in Dimension 2

111

5.2 Non-existence of Singular Cones in Dimension 2 We now prove the non-existence of singular s-minimal cones in dimension 2, as stated in the next result (from this, the more general statement in Theorem 5.3 follows after a blow-up procedure): Theorem 5.2.1 If E is an s-minimal cone in R2 , then E is a half-plane. We remark that, as a combination of Theorems 5.1.1 and 5.2.1, we obtain the following result of Bernstein type: Corollary 5.2.2 Let E D f.x; t/ 2 Rn  R s.t. t < u.x/g be an s-minimal graph, and assume that n 2 f1; 2g. Then u is an affine function. Let us first consider a simple example, given by the cone in the plane n o K WD .x; y/ 2 R2 s.t. y2 > x2 ; see Fig. 5.5. Proposition 5.2.3 The cone K depicted in Fig. 5.5 is not s-minimal in R2 . Notice that, by symmetry, one can prove that K satisfies (5.5) (possibly in the viscosity sense). On the other hand, Proposition 5.2.3 gives that K is not s-minimal. This, in particular, provides an example of a set that satisfies the Euler-Lagrange equation in (5.5), but is not s-minimal (i.e., the Euler-Lagrange equation in (5.5) is implied by, but not necessarily equivalent to, the s-minimality property). Proof (Proof of Proposition 5.2.3) The proof of the non-minimality of K is due to an original idea by Luis Caffarelli. Suppose by contradiction that the cone K is minimal in R2 . We add to K a small square adjacent to the origin (see Fig. 5.6), and call K 0 the set obtained. Then K and K 0 have the same s-perimeter. This is due to the interactions considered in Fig. 5.5 The cone K

112

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Fig. 5.6 Interaction of M with A; B; C; D; A0 ; B0 ; C0 ; D0

the s-perimeter functional and the unboundedness of the regions. We remark that in Fig. 5.6 we represent bounded regions, of course, sets A; B; C; D; A0 ; B0 ; C0 and D0 are actually unbounded. Indeed, we notice that in the first image, the white square M interacts with the dark regions A; B; C; D, while in the second the now dark square M interacts with the regions A0 ; B0 ; C0 ; D0 , and all the other interactions are unmodified. Therefore, the difference between the s-perimeter of K and that of K 0 consists only of the interactions I.A; M/CI.B; M/CI.C; M/CI.D; M/I.A0 ; M/I.B0 ; M/I.C0 ; M/ I.D0 ; M/. But A [ B D A0 [ B0 and C [ D D C0 [ D0 (since these sets are all unbounded), therefore the difference is null, and the s-perimeter of K is equal to that of K 0 . Consequently, K 0 is also s-minimal, and therefore it satisfies the EulerLagrange equation in (5.5) at the origin. But this leads to a contradiction, since the dark region now contributes more than the white one, namely Z R2

K 0 .y/  R2 nK 0 .y/ dy > 0: jyj2Cs

Thus K cannot be s-minimal, and this concludes our proof. This geometric argument cannot be extended to a more general case (even, for instance, to a cone in R2 made of many sectors, see Fig. 5.7). As a matter of fact, the proof of Theorem 5.2.1 will be completely different than the one of Proposition 5.2.3 and it will rely on an appropriate domain perturbation argument. The proof of Theorem 5.2.1 that we present here is actually different than the original one in [124]. Indeed, in [124], the result was proved by using the harmonic extension for the fractional Laplacian. Here, the extension will not be used; furthermore, the proof follows the steps of Theorem 4.2.1 and we will recall here just the main ingredients.

5.2 Non-existence of Singular Cones in Dimension 2

113

Fig. 5.7 Cone in R2

Proof (Proof of Theorem 5.2.1) The idea of the proof is the following: if E R2 is an s-minimal cone, then let EQ be a perturbation of the set E which coincides with a translation of E in BR=2 and with E itself outside BR . Then the difference between the energies of EQ and E tends to 0 as R ! C1. This implies that also the energy of E \ EQ is arbitrarily close to the energy of E. On the other hand if E is not a half-plane, the set EQ \ E can be modified locally to decrease its energy by a fixed small amount and we reach a contradiction. The details of the proof go as follows. Let u WD E  R2 nE : From definition (4.35) we have that u.BR ; BR / D 2I.E \ BR ; BR n E/ and C u.BR ; BC R / D I.BR \ E; E n BR / C I.BR n E; E n BR /;

thus Pers .E; BR / D KR .u/;

(5.21)

where KR .u/ is given in (4.33) and Pers .E; BR / is the s-perimeter functional defined in (5.2). Then E is s-minimal if u is a minimizer of the energy KR in any ball BR , with R > 0. Now, we argue by contradiction, and suppose that E is an s-minimal cone different from the half-space. Up to rotations, we may suppose that a sector of E

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5 Nonlocal Minimal Surfaces

has an angle smaller than  and is bisected by e2 . Thus there exists M  1 and p 2 E \ BM on the e2 -axis such that p ˙ e1 2 R2 n E (see Fig. 5.7). We take ' 2 C01 .B1 /, such that '.x/ D 1 in B1=2 . For R large (say R > 8M), we define

y e1 : R;C .y/ WD y C ' R We point out that, for R large, R;C is a diffeomorphism on R2 . 1 Furthermore, we define uC R .x/ WD u.R;C .x//. Then uC R .y/ D u.y  e1 / and uC R .y/ D u.y/

for p 2 B2M for p 2 R2 n BR :

We recall the estimate obtained in (4.37), that, combined with the minimality of u, gives KR .uC R /  KR .u/ 

C KR .u/: R2

But u is a minimizer in any ball, and by the energy estimate in Theorem 4.1.2 we have that 2s KR .uC : R /  KR .u/  CR

This implies that lim KR .uC R /  KR .u/ D 0:

(5.22)

R!C1

Let now vR .x/ WD minfu.x/; uC R .x/g

and

wR .x/ WD maxfu.x/; uC R .x/g:

We claim that vR is not identically u nor uC R . Indeed uC R .p/ D u.p  e1 / D .E  R2 nE /.p  e1 / D 1 u.p/ D .E  R2 nE /.p/ D 1: On the other hand, uC R .p C e1 / D u.p/ D 1 and u.p C e1 / D .E  R2 nE /.p C e1 / D 1:

and

5.2 Non-existence of Singular Cones in Dimension 2

115

By the continuity of u and uC R , we obtain that vR D uC R < u in a neighborhood of p

(5.23)

vR D u < uC R in a neighborhood of p C e1 :

(5.24)

and

Now, by the minimality property of u, KR .u/  KR .vR /: Moreover (see e.g. formula (38) in [114]), KR .vR / C KR .wR /  KR .u/ C KR .uC R /: The latter two formulas give that KR .vR /  KR .uC R /:

(5.25)

vR is not minimal for K2M

(5.26)

We claim that

with respect to compact perturbations in B2M . Indeed, assume by contradiction that vR is minimal, then in B2M both vR and u would satisfy the same equation. Recalling (5.24) and applying the Strong Maximum Principle, it follows that u D vR in B2M , which contradicts (5.23). This establishes (5.26). Now, we consider a minimizer uR of K2M among the competitors that agree with vR outside B2M . Therefore, we can define ıR WD K2M .vR /  K2M .uR /: In light of (5.26), we have that ıR > 0. The reader can now compare Step 3 in the proof of Theorem 4.2.1. There we proved that ıR remains bounded away from zero as R ! C1. Furthermore, since uR and vR agree outside B2M we obtain that KR .uR / C ıR D KR .vR /:

(5.27)

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5 Nonlocal Minimal Surfaces

Using this, (5.25) and the minimality of u, we obtain that ıR D KR .vR /  KR .uR /  KR .uC R /  KR .u/: Now we send R to infinity, recall (5.22) and (5.27), and we reach a contradiction. Thus, E is a half-space, and this concludes the proof of Theorem 5.2.1. As already mentioned, the regularity theory for s-minimal sets is still widely open. Little is known beyond Theorems 5.3 and 5.4, so it would be very interesting to further investigate the regularity of s-minimal surfaces in higher dimension and for small s. It is also interesting to recall that if the s-minimal surface E is a subgraph of some function u W Rn1 ! R (at least in the vicinity of some point x0 D .x00 ; u.x00 // 2 @E) then the Euler-Lagrange (5.5) can be written directly in terms of u. For instance (see formulas (49) and (50) in [9]), under appropriate smoothness assumptions on u, formula (5.5) reduces to Z

Rn nE .x0 C y/  E .x0 C y/ dy jyjnC2s   0 Z .y0 / u.x0 C y0 /  u.x00 / F dy0 C  .x00 /; D 0 0 jy j jy jn1C2s Rn1

0D

Rn

for suitable F and  , and a cut-off function  supported in a neighborhood of x00 . Regarding the regularity problems of the s-minimal surfaces, let us mention the recent papers [47] and [48]. Among other very interesting results, it is proved there that suitable singular cones of symmetric type are unstable up to dimension 6 but become stable in dimension 7 for small s (these cones can be seen as the nonlocal analogue of the Lawson cones in the classical minimal surface theory, and the stability property is in principle weaker than minimality, since it deals with the positivity of the second order derivative of the functional). This phenomenon may suggest the conjecture that the s-minimal surfaces may develop singularities in dimension 7 and higher when s is sufficiently small. In [48], interesting examples of surfaces with vanishing nonlocal mean curvature are provided for s sufficiently close to 1=2. Remarkably, the surfaces in [48] are the nonlocal analogues of the catenoids, but, differently from the classical case (in which catenoids grow logarithmically), they approach a singular cone at infinity, see Fig. 5.8. Also, these nonlocal catenoids are highly unstable from the variational point of view, since they possess infinite Morse index (differently from the standard catenoid, which has Morse index equal to one, i.e. it is, roughly speaking, a minimizer in any functional direction with the exception of one). Moreover, in [48], there are also examples of surfaces with vanishing nonlocal mean curvature that can be seen as the nonlocal analogues of two parallel hyperplanes. Namely, for s sufficiently close to 1=2, there exists a surface of revolution made of two sheets which are the graph of a radial function f D ˙f .r/. When r

5.2 Non-existence of Singular Cones in Dimension 2

117

Fig. 5.8 A nonlocal catenoid

Fig. 5.9 A two-sheet surface with vanishing fractional mean curvature

is small, f is of the order of 1 C . 12  s/r2 , but for large r it becomes of the order q of 12  s  r. That is, the two sheets “repel each other” and produce a linear growth at infinity. When s approaches 1=2 the two sheets are locally closer and closer to two parallel hyperplanes, see Fig. 5.9. The construction above may be extended to build families of surfaces with vanishing nonlocal mean curvature that can be seen as the nonlocal analogue of k parallel hyperplanes, for any k 2 N. These k-sheet surfaces can be seen as the bifurcation, as s is close to 1=2, of the parallel hyperplanes fxn D ai g, for i 2 f1; : : : ; kg, where the parameters ai satisfy the constraints a1 >    > ak ;

k X iD1

ai D 0

(5.28)

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5 Nonlocal Minimal Surfaces

and the balancing relation ai D 2

X .1/iCjC1 : ai  aj 1jn

(5.29)

j¤i

It is actually quite interesting to observe that solutions of (5.29) correspond to (nondegenerate) critical points of the functional E.a1 ; : : : ; ak / WD

k X 1X 2 ai C .1/iCj log jai  aj j 2 iD1 1jn j¤i

among all the k-ples .a1 ; : : : ; ak / that satisfy (5.28). These bifurcation techniques rely on a careful expansion of the fractional perimeter functional with respect to normal perturbations. That is, if E is a (smooth) set with vanishing fractional mean curvature, and h is a smooth and compactly supported perturbation, one can define, for any t 2 R, Eh .t/ WD fx C th.x/.x/; x 2 @Eg; where .x/ is the exterior normal of E at x. Then, the second variation of the perimeter of Eh .t/ at t D 0 is (up to normalization constants) Z Z

@E

h.y/  h.x/ dH n1 .y/ C h.x/ jx  yjnC2s

@E

h.y/  h.x/ dH n1 .y/ C h.x/ jx  yjnC2s

D



Z Z

@E

@E

.x/  .y/  .x/ dH n1 .y/ jx  yjnC2s

1  .x/  .y/ dH n1 .y/: jx  yjnC2s

Notice that the latter integral is non-negative, since .x/  .y/  1. The quantity above, in dependence of the perturbation h, is called, in jargon, “Jacobi operator”. It encodes an important geometric information, and indeed, as s ! 1=2, it approaches the classical operator @E h C jA@E j2 h; where @E is the Laplace-Beltrami operator along the hypersurface @E and jA@E j2 is the sum of the squares of the principal curvatures. Other interesting sets that possess constant nonlocal mean curvature with the structure of onduloids have been recently constructed in [49] and [24]. This type of sets are periodic in a given direction and their construction has perturbative nature (indeed, the sets are close to a slab in the plane). It is interesting to remark that the planar objects constructed in [24] have no counterpart in the local framework, since hypersurfaces of constant classical mean

5.3 Boundary Regularity

119

curvature with an onduloidal structure only exist in Rn with n  3: once again, this is a typical nonlocal effect, in which the nonlocal mean curvature at a point is influenced by the global shape of the set. While unbounded sets with constant nonlocal mean curvature and interesting geometric features have been constructed in [24, 48], the case of smooth and bounded sets is always geometrically trivial. As a matter of fact, it has been recently proved independently in [24] and [43] that bounded sets with smooth boundary and constant mean curvature are necessarily balls (this is the analogue of a celebrated result by Alexandrov for surfaces of constant classical mean curvature).

5.3 Boundary Regularity The boundary regularity of the nonlocal minimal surfaces is also a very interesting, and surprising, topic. Indeed, differently from the classical case, nonlocal minimal surfaces do not always attain boundary data in a continuous way (not even in low dimension). A possible boundary behavior is, on the contrary, a combination of stickiness to the boundary and smooth separation from the adjacent portions. Namely, the nonlocal minimal surfaces may have a portion that sticks at the 1 boundary and that separates from it in a C1; 2 Cs -way. As an example, we can consider, for any ı > 0, the spherical cap  Kı WD B1Cı n B1 \ fxn < 0g; and obtain the following stickiness result: Theorem 5.3.1 There exists ı0 > 0, depending on n and s, such that for any ı 2 .0; ı0 , we have that the s-minimal set in B1 that coincides with Kı outside B1 is Kı itself. That is, the s-minimal set with datum Kı outside B1 is empty inside B1 . The stickiness property of Theorem 5.3.1 is depicted in Fig. 5.10. Fig. 5.10 Stickiness properties of Theorem 5.3.1

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5 Nonlocal Minimal Surfaces

Fig. 5.11 Stickiness properties of Theorem 5.3.2

Other stickiness examples occur at the sides of slabs in the plane. For instance, given M > 1, one can consider the s-minimal set EM in .1; 1/  R with datum C  outside .1; 1/  R given by the “jump” set JM WD JM [ JM , where  WD .1; 1  .1; M/ JM C WD Œ1; C1/  .1; M/: and JM

Then, if M is large enough, the minimal set EM sticks at the boundary of the slab: Theorem 5.3.2 There exist Mo > 0, Co > 0, depending on s, such that if M  Mo then 1C2s

c Œ1; 1/  ŒCo M 2C2s ; M  EM

and .1; 1  ŒM; Co M

1C2s 2C2s

 EM :

(5.30) (5.31)

The situation of Theorem 5.3.2 is described in Fig. 5.11. We mention that the “strange” exponent 1C2s 2C2s in (5.30) and (5.31) is optimal. For the detailed proof of Theorems 5.3.1 and 5.3.2, and other results on the boundary behavior of nonlocal minimal surfaces, see [63]. Here, we limit ourselves to give some heuristic motivation and a sketch of the proofs. As a motivation for the (somehow unexpected) stickiness property at the boundary, one may look at Fig. 5.10 and argue like this. In the classical case, corresponding to s D 1=2, independently on the width ı, the set of minimal perimeter in B1 will always be the half-ball B1 \ fxn < 0g.

5.3 Boundary Regularity

121

Now let us take s < 1=2. Then, the half-ball B1 \ fxn < 0g cannot be an sminimal set, since the nonlocal mean curvature, for instance, at the origin cannot vanish. Indeed, the origin “sees” the complement of the set in a larger proportion than the set itself. More precisely, in B1 (or even in B1Cı ) the proportion of the set is the same as the one of the complement, but outside B1Cı the complement of the set is dominant. Therefore, to “compensate” this lack of balance, the s-minimal set for s < 1=2 has to bend a bit. Likely, the s-minimal set in this case will have the tendency to become slightly convex at the origin, so that, at least nearby, it sees a proportion of the set which is larger than the proportion of the complement (we recall that, in any case, the proportion of the complement will be larger at infinity, so the set needs to compensate at least near the origin). But when ı is very small, it turns out that this compensation is not sufficient to obtain the desired balance between the set and its complement: therefore, the set has to “stick” to the halfsphere, in order to drop its constrain to satisfy a vanishing nonlocal mean curvature equation. Of course some quantitative estimates are needed to make this argument work, so we describe the sketch of the rigorous proof of Theorem 5.3.1 as follows. Proof (Sketch of the proof of Theorem 5.3.1) First of all, one checks that for any fixed  > 0, if ı > 0 is small enough, we have that the interaction between B1 and B1Cı n B1 is smaller than . In particular, by comparing with a competitor that is empty in B1 , by minimality we obtain that Pers .Eı ; B1 /  ;

(5.32)

where we have denoted by Eı the s-minimal set in B1 that coincides with Kı outside B1 . Then, one checks that the boundary of Eı can only lie in a small neighborhood of @B1

(5.33)

if ı is sufficiently small. Indeed, if, by contradiction, there were points of @Eı at distance larger than from @B1 , then one could find two balls of radius comparable to , whose centers lie at distance larger than =2 from @B1 and at mutual distance smaller than , and such that one ball is entirely contained in B1 \ Eı and the other ball is entirely contained in B1 nEı (this is due to a Clean Ball Condition, see Corollary 4.3 in [28]). As a consequence, Pers .Eı ; B1 / is bounded from below by the interaction of these two balls, which is at least of the order of n2s . Then, we obtain a contradiction with (5.32) (by choosing  much smaller than n2s , and taking ı sufficiently small). This proves (5.33). From this, it follows that the whole set Eı must lie in a small neighborhood of @B1 .

(5.34)

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5 Nonlocal Minimal Surfaces

Indeed, if this were not so, by (5.33) the set Eı must contain a ball of radius, say 1=2. Hence, Pers .Eı ; B1 / is bounded from below by the interaction of this ball against fxn > 0g n B1 , which would produce a contribution of order one, which is in contradiction with (5.32). Having proved (5.34), one can use it to complete the proof of Theorem 5.3.1 employing a geometric argument. Namely, one considers the ball B , which is outside Eı for small  > 0, in virtue of (5.34), and then enlarges  until it touches @Eı . If this contact occurs at some point p 2 B1 , then the nonlocal mean curvature of Eı at p must be zero. But this cannot occur (indeed, we know by (5.34) that the contribution of Eı to the nonlocal mean curvature can only come from a small neighborhood of @B1 , and one can check, by estimating integrals, that this is not sufficient to compensate the outer terms in which the complement of Eı is dominant). As a consequence, no touching point between B and @Eı can occur in B1 , which shows that Eı is void inside B1 and completes the proof of Theorem 5.3.1. As for the proof of Theorem 5.3.2, the main arguments are based on sliding a ball of suitably large radius till it touches the set, with careful quantitative estimates. Some of the details are as follows (we refer to [63] for the complete arguments). Proof (Sketch of the proof of Theorem 5.3.2) The first step is to prove a weaker form of stickiness as the one claimed in Theorem 5.3.2. Namely, one shows that c Œ1; 1/  Œco M ; M  EM

and .1; 1  ŒM; co M  EM ;

(5.35) (5.36)

for some co 2 .0; 1/. Of course, the statements in (5.30) and (5.31) are stronger than the ones in (5.35) and (5.36) when M is large, since 1C2s 2C2s < 1, but we will then obtain them later in a second step. To prove (5.35), one takes balls of radius co M and centered at fx2 D tg, for any t 2 Œco M; M . One slides these balls from left to right, till one touches @EM . When M is large enough (and co small enough) this contact point cannot lie in fjx1 j < 1g. This is due to the fact that at least the sliding ball lies outside EM , and the whole fx2 > Mg lies outside EM as well. As a consequence, these contact points see a proportion of EM smaller than the proportion of the complement (it is true that the whole of fx2 < Mg lies inside EM , but this contribution comes from further away than the ones just mentioned, provided that co is small enough). Therefore, contact points cannot satisfy a vanishing mean curvature equation and so they need to lie on the boundary of the domain (of course, careful quantitative estimates are necessary here, see [63], but we hope to have given an intuitive sketch of the computations needed). In this way, one sees that all the portion Œ1; 1/  Œco M ; M is clean from the set EM and so (5.35) is established (and (5.36) can be proved similarly). Once (5.35) and (5.36) are established, one uses them to obtain the strongest form expressed in (5.30) and (5.31). For this, by (5.35) and (5.36), one has only to

5.3 Boundary Regularity

123 1C2s

take care of points in fjx2 j 2 ŒCo M 2C2s ; co M g. For these points, one can use again a sliding method, but, instead of balls, one has to use suitable surfaces obtained by appropriate portions of balls and adapt the calculations in order to evaluate all the contributions arising in this way. The computations are not completely obvious (and once again we refer to [63] for full details), but the idea is, once again, that contact points that are in the set 1C2s fjx2 j 2 ŒCo M 2C2s ; co M g cannot satisfy the balancing relation prescribed by the vanishing nonlocal mean curvature equation. The stickiness property discussed above also has an interesting consequence in terms of the “geometric stability” of the flat s-minimal surfaces. For instance, rather surprisingly, the flat lines in the plane are “geometrically unstable” nonlocal minimal surfaces, in the sense that an arbitrarily small and compactly supported perturbation can produce a stickiness phenomenon at the boundary of the domain. Of course, the smaller the perturbation, the smaller the stickiness phenomenon, but it is quite relevant that such a stickiness property can occur for arbitrarily small (and “nice”) perturbations. This means that s-minimal flat objects, in presence of a perturbation, may not only “bend” in the center of the domain, but rather “jump” at boundary points as well. To state this phenomenon in a mathematical framework, one can consider, for fixed ı > 0 the planar sets H WD R  .1; 0/; F WD .3; 2/  Œ0; ı/ and FC WD .2; 3/  Œ0; ı/: One also fixes a set F which contains H[F [FC and denotes by E be the s-minimal set in .1; 1/  R among all the sets that coincide with F outside .1; 1/  R. Then, this set E sticks at the boundary of the domain, according to the next result: Theorem 5.3.3 Fix 0 > 0 arbitrarily small. Then, there exists ı0 > 0, possibly depending on 0 , such that, for any ı 2 .0; ı0 , 2C 0

E  .1; 1/  .1; ı 12s : The stickiness/instability property in Theorem 5.3.3 is depicted in Fig. 5.12. We remark that Theorem 5.3.3 gives a rather precise quantification of the size of the stickiness in terms of the size of the perturbation: namely the size of the stickiness 0 , in Theorem 5.3.3 is larger than the size of the perturbation to the power ˇ WD 2C 12s for any 0 > 0 arbitrarily small. Notice that ˇ ! C1 as s ! 1=2, consistently with the fact that classical minimal surfaces do not stick at the boundary. The proof of Theorem 5.3.3 is based on the construction of suitable auxiliary barriers (see Fig. 5.13). These barriers are used to detach a portion of the set in a neighborhood of the origin and their construction relies on some compensations

124

5 Nonlocal Minimal Surfaces

Fig. 5.12 The stickiness/instability property in Theorem 5.3.3, with ˇ WD

2C 0 12s

Fig. 5.13 Auxiliary barrier for the proof of Theorem 5.3.3

of nonlocal integral terms. In a sense, the building blocks of these barriers are “self-sustaining solutions” that can be seen as the geometric counterparts of the s-harmonic function xsC discussed in Sect. 2.4. Indeed, roughly speaking, like the function xsC , these barriers “see” a proportion of the set in fx1 < 0g larger than what is produced by their tangent plane, but a proportion smaller than that at infinity, due to their sublinear behavior. Once again, the computations needed to check such a balancing conditions are a bit involved, and we refer to [63] for the complete details.

5.3 Boundary Regularity

125

To conclude this chapter, we make a remark on the connection between solutions of the fractional Allen-Cahn equation and s-minimal surfaces. Namely, a suitably scaled version of the functional in (4.9) -converges to either the classical perimeter or the nonlocal perimeter functional, depending on the fractional parameter s. The -convergence is a type of convergence of functionals that is compatible with the minimization of the energy, and turns out to be very useful when dealing with variational problems indexed by a parameter. This notion was introduced by De Giorgi, see e.g. [50] for details. In the nonlocal case, some care is needed to introduce the “right” scaling of the functional, which comes from the dilation invariance of the space coordinates and possesses a nontrivial energy in the limit. For this, one takes first the rescaled energy functional Z J" .u; ˝/ WD "2s K .u; ˝/ C W.u/ dx; ˝

where K is the kinetic energy defined in (4.10). Then, one considers the functional 8 ˆ "2s J" .u; ˝/ if s 2 .0; 1=2/, ˆ ˆ ˆ ˆ < F" .u; ˝/ WD j" log "j1 J" .u; ˝/ if s D 1=2, ˆ ˆ ˆ ˆ ˆ : if s 2 .1=2; 1/. "1 J" .u; ˝/ The limit functional of F" as " ! 0 depends on s. Namely, when s 2 .0; 1=2/, the limit functional is (up to dimensional constants that we neglect) the fractional perimeter, i.e.  F.u; ˝/ WD

Pers .E; ˝/ if uj˝ D E  EC , for some set E ˝ C1 otherwise.

(5.37)

On the other hand, when s 2 Œ1=2; 1/, the limit functional of F" is (again, up to normalizing constants) the classical perimeter, namely  F.u; ˝/ WD

Per.E; ˝/ if uj˝ D E  EC , for some set E ˝ C1 otherwise,

(5.38)

That is, the following limit statement holds true: Theorem 5.3.4 Let s 2 .0; 1/. Then, F" -converges to F, as defined in either (5.37) or (5.38), depending on whether s 2 .0; 1=2/ or s 2 Œ1=2; 1/. For precise statements and further details, see [123]. Additionally, we remark that the level sets of the minimizers of the functional in (4.9), after a homogeneous scaling in the space variables, converge locally uniformly to minimizers either of the

126

5 Nonlocal Minimal Surfaces

fractional perimeter (if s 2 .0; 1=2/) or of the classical perimeter (if s 2 Œ1=2; 1/): that is, the “functional” convergence stated in Theorem 5.3.4 has also a “geometric” counterpart: for this, see Corollary 1.7 in [125]. One can also interpret Theorem 5.3.4 by saying that a nonlocal phase transition possesses two parameters, " and s. When " ! 0, the limit interface approaches a minimal surface either in the fractional case (when s 2 .0; 1=2/) or in the classical case (when s 2 Œ1=2; 1/). This bifurcation at s D 1=2 somehow states that for lower values of s the nonlocal phase transition possesses a nonlocal interface in the limit, but for larger values of s the limit interface is characterized only by local features (in a sense, when s 2 .0; 1=2/ the “surface tension effect” is nonlocal, but for s 2 Œ1=2; 1/ this effect localizes). It is also interesting to compare Theorems 5.2 and 5.3.4, since the bifurcation at s D 1=2 detected by Theorem 5.3.4 is perfectly compatible with the limit behavior of the fractional perimeter, which reduces to the classical perimeter exactly for this value of s, as stated in Theorem 5.2.

Chapter 6

A Nonlocal Nonlinear Stationary Schrödinger Type Equation

The type of problems introduced in this chapter are connected to solitary solutions of nonlinear dispersive wave equations (such as the Benjamin-Ono equation, the Benjamin-Bona-Mahony equation and the fractional Schrödinger equation). In this chapter, only stationary equations are studied and we redirect the reader to [137, 138] for the study of evolutionary type equations. Let n  2 be the dimension of the reference space, s 2 .0; 1/ be the fractional parameter, and > 0 be a small parameter. We consider the so-called fractional Sobolev exponent 8 < 2n 2s WD n  2s : C1

for n  3; or n D 2 and s 2 .0; 1=2/ for n D 1 and s 2 .0; 1=2

and introduce the following nonlocal nonlinear Schrödinger equation (

2s ./s u C u D up

in ˝ Rn

uD0

in Rn n ˝;

(6.1)

  n C 2s .  1/, namely when p 2 1; in the subcritical case p 2 n  2s This equation arises in the study of the fractional Schrödinger equation when looking for standing waves. Namely, the fractional Schrödinger equation considers solutions  D  .x; t/ W Rn  R ! C of .1; 2s

 i¯@t  D ¯2s ./s C V ;

(6.2)

where s 2 .0; 1/, ¯ is the reduced Planck constant and V D V.x; t; j j/ is a potential. This equation is of interest in quantum mechanics (see e.g. [101] and the appendix in [46] for details and physical motivations). Roughly speaking, the © Springer International Publishing Switzerland 2016 C. Bucur, E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana 20, DOI 10.1007/978-3-319-28739-3_6

127

128

6 A Nonlocal Nonlinear Stationary Schrödinger Type Equation

quantity j .x; t/j2 dx represents the probability density of finding a quantum particle in the space region dx and at time t. The simplest solutions of (6.2) are the ones for which this probability density is independent of time, i.e. j .x; t/j D u.x/ for some u W Rn ! Œ0; C1/. In this way, one can write  as u times a phase that oscillates (very rapidly) in time: that is one may look for solutions of (6.2) of the form  .x; t/ WD u.x/ ei!t=¯ ; for some frequency ! 2 R. Choosing V D V.j j/ D j jp1 D up1 , a substitution into (6.2) gives that

 ¯2s ./s u C !u  up ei!t=¯ D ¯2s ./s   i¯@t  C V D 0; which is (6.1) (with the normalization convention ! WD 1 and WD ¯). The goal of this chapter is to construct solutions of problem (6.1) that concentrate at interior points of the domain ˝ for sufficiently small values of . We perform a blow-up of the domain, defined as 1 ˝ WD ˝ D



 x ;x 2 ˝ :

We can also rescale the solution of (6.1) on ˝ , u .x/ D u. x/: The problem (6.1) for u then reads (

./s u C u D up

in ˝

uD0

in Rn n ˝ :

(6.3)

When ! 0, the domain ˝ invades the whole of the space. Therefore, it is also natural to consider (as a first approximation) the equation on the entire space ./s u C u D up in Rn :

(6.4)

The first result that we need is that there exists an entire positive radial least energy solution w 2 H s .Rn / of (6.4), called the ground state solution. Here follow some relevant results on this. The interested reader can find their proofs in [83]. 1. The ground state solution w 2 H s .Rn / is unique up to translations. 2. The ground state solution w 2 H s .Rn / is nondegenerate, i.e., the derivatives Di w are solutions to the linearized equation ./s Z C Z D pZ p1 :

(6.5)

6 A Nonlocal Nonlinear Stationary Schrödinger Type Equation

129

3. The ground state solution w 2 H s .Rn / decays polynomially at infinity, namely there exist two constants ˛; ˇ > 0 such that ˛jxj.nC2s/  u.x/  ˇjxj.nC2s/ : Unlike the fractional case, we remark that for the (classical) Laplacian, at infinity the ground state solution decays exponentially fast. We also refer to [82] for the one-dimensional case. The main theorem of this chapter establishes the existence of a solution that concentrates at interior points of the domain for sufficiently small values of . This concentration phenomena is written in terms of the ground state solution w. Namely, the first approximation for the solution is exactly the ground state w, scaled and concentrated at an appropriate point  of the domain. More precisely, we have: Theorem 6.1 If is sufficiently small, there exist a point  2 ˝ and a solution U of the problem (6.1) such that ˇ ˇˇ

ˇ ˇU .x/  w x   ˇ  C nC2s ; ˇ ˇ and dist.; @˝/  ı > 0. Here, C and ı are constants independent of or ˝, and the function w is the ground state solution of problem (6.4). The concentration point  in Theorem 6.1 is influenced by the global geometry of the domain. On the one hand, when s D 1, the point  is the one that maximizes the distance from the boundary. On the other hand, when s 2 .0; 1/, such simple characterization of  does not hold anymore: in this case,  turns out to be asymptotically the maximum of a (complicated, but rather explicit) nonlocal functional: see [46] for more details. We state here the basic idea of the proof of Theorem 6.1 (we refer again to [46] for more details). Proof (Sketch of the proof of Theorem 6.1) In this proof, we make use of the Lyapunov-Schmidt procedure. Namely, rather than looking for the solution in an infinite-dimensional functional space, one decomposes the problem into two orthogonal subproblems. One of these problem is still infinite-dimensional, but it has the advantage to bifurcate from a known object (in this case, a translation of the ground state). Solving this auxiliary subproblem does not provide a true solution of the original problem, since a leftover in the orthogonal direction may remain. To kill this remainder term, one solves a second subproblem, which turns out to be finite-dimensional (in our case, this subproblem is set in Rn , which corresponds to the action of the translations on the ground state). A structural advantage of the problem considered lies in its variational structure. Indeed, Eq. (6.3) is the Euler-Lagrange equation of the energy functional I .u/ D

1 2

Z ˝

 ./s u.x/ C u.x/ u.x/ dx 

1 pC1

Z ˝

upC1 .x/ dx

(6.6)

130

6 A Nonlocal Nonlinear Stationary Schrödinger Type Equation

for any u 2 H0s .˝ / WD fu 2 H s .Rn / s.t. u D 0 a.e. in Rn n ˝ g. Therefore, the problem reduces to finding critical points of I . To this goal, we consider the ground state solution w and for any  2 Rn we let w WD w.x  /. For a given  2 ˝ a first approximation uN  for the solution of problem (6.3) can be taken as the solution of the linear problem (

p

./s u C u D w

in ˝ ;

u D 0

in Rn n ˝ :

(6.7)

The actual solution will be obtained as a small perturbation of uN  for a suitable point  D . /. We define the operator L WD ./s C I, where I is the identity and we notice that L has a unique fundamental solution that solves L D ı0

in Rn :

The Green function G of the operator L in ˝ satisfies (

L G .x; y/ D ıy .x/

if x 2 ˝ ;

G .x; y/ D 0

if x 2 Rn n ˝ :

(6.8)

It is convenient to introduce the regular part of G , which is often called the Robin function. This function is defined by H .x; y/ WD .x  y/  G .x; y/

(6.9)

and it satisfies, for a fixed y 2 Rn , (

L H .x; y/ D 0

if x 2 ˝ ;

H .x; y/ D .x  y/

if x 2 Rn n ˝ :

Then Z u .x/ D

˝

u .y/ı0 .x  y/ dy;

and by (6.8) Z u .x/ D

˝

u .y/L G .x; y/ dy:

(6.10)

6 A Nonlocal Nonlinear Stationary Schrödinger Type Equation

131

The operator L is self-adjoint and thanks to the above identity and to Eq. (6.7) it follows that Z u .x/ D L u .y/G .x; y/ dy Z

˝

p

D ˝

w .y/G .x; y/ dy:

So, we use (6.9) and we obtain that Z Z p u .x/ D w .y/ .x  y/ dy  ˝

p

˝

w .y/H .x; y/ dy:

Now we notice that, since w is solution of (6.4) and is the fundamental solution of L , we have that Z Z p w .y/ .x  y/ dy D L w .y/ .x  y/ dy Rn

Z

Rn

D Rn

w .y/L .x  y/ dy

D w .x/: Therefore we have obtained that Z Z p u .x/ D w .x/  w .y/ .x  y/ dy  Rn n˝

p

˝

w .y/H .x; y/ dy:

(6.11)

Now we can insert (6.11) into the energy functional (6.6) and expand the errors in ı with ı fixed and small, the energy of u is a powers of . For dist.; @˝ /  perturbation of the energy of the ground state w and one finds (see Theorem 4.1 in [46]) that 1 I .u / D I.w/ C H ./ C O. nC4s /; 2

(6.12)

where Z H ./ WD

Z p

˝

˝

p

H .x; y/w .x/w .y/ dx dy

and I is the energy computed on the whole space Rn , namely I.u/ D

1 2

Z Rn

 ./s u.x/ C u.x/ u.x/ dx 

1 pC1

Z Rn

upC1 .x/ dx:

(6.13)

132

6 A Nonlocal Nonlinear Stationary Schrödinger Type Equation

In particular, I .u / agrees with a constant (the term I.w/), plus a functional over a finite-dimensional space (the term H ./, which only depends on  2 Rn ), plus a small error. We remark that the solution u of Eq. (6.7) which can be obtained from (6.11) does not provide a solution for the original problem (6.3) (indeed, it only solves (6.7)): for this, we look for solutions u of (6.3) as perturbations of u , in the form u WD u C :

(6.14)

are considered among those vanishing outside ˝ and @w orthogonal to the space Z D Span.Z1 ; : : : ; Zn /, where Zi D are solutions @xi of the linearized Eq. (6.5). This procedure somehow “removes the degeneracy”, namely we look for the corrector in a set where the linearized operator is invertible. This makes it possible, fixed any  2 Rn , to find D  such that the function u , as defined in (6.14) solves the equation

The perturbation functions

p

./s u C u D u C

n X

ci Zi in ˝ :

(6.15)

iD1

That is, u is solution of the original Eq. (6.3), up to an error that lies in the tangent space of the translations (this error is exactly the price that we pay in order to solve the corrector equation for on the orthogonal of the kernel, where the operator is nondegenerate). As a matter of fact (see Theorem 7.6 in [46] for details) one can see that the corrector D  is of order nC2s . Therefore, one can compute I .u / D I .u C  / as a higher order perturbation of I .u /. From (6.12), one obtains that 1 I .u / D I.w/ C H ./ C O. nC4s /; 2

(6.16)

see Theorem 7.17 in [46] for details. Since this energy expansion now depends only on  2 Rn , it is convenient to define the operator J W ˝ ! R as J ./ WD I .u /: This functional is often called the reduced energy functional. From (6.16), we conclude that 1 J ./ D I.w/ C H ./ C O. nC4s /: 2

(6.17)

The reduced energy J plays an important role in this framework since critical points of J correspond to true solutions of the original Eq. (6.3). More precisely (see

6 A Nonlocal Nonlinear Stationary Schrödinger Type Equation

133

Lemma 7.16 in [46]) one has that ci D 0 for all i D 1; : : : ; n in (6.15) if and only if @J ./ D 0: @

(6.18)

In other words, when approaches 0, to find concentration points, it is enough to find critical points of J, which is a finite-dimensional problem. Also, critical points for J come from critical points of H , up to higher orders, thanks to (6.17). The issue is thus to prove that H does possess critical points and that these critical points survive after the small error of size nC4s : in fact, we show that H possesses a minimum, which is stable for perturbations. For this, one needs a bound for the Robin function H from above and below. To this goal, one builds a barrier function ˇ defined for  2 ˝ and x 2 Rn as Z ˇ .x/ WD

Rn n˝

.z  / .x  z/ dz:

Using this function in combination with suitable maximum principles, one obtains the existence of a constant c 2 .0; 1/ such that cH .x; /  ˇ .x/  c1 H .x; /; for any x 2 Rn and any  2 ˝ with dist.; @˝ / > 1, see Lemma 2.1 in [46]. From this it follows that H ./ ' d.nC4s/ ;

(6.19)

for all points  2 ˝ such that d 2 Œ5; ı= . So, one considers the domain ˝ ;ı of the points of ˝ that lie at distance more than ı= from the boundary of ˝ . By (6.19), we have that H ./ '

nC4s for any  2 @˝ ;ı : ı nC4s

(6.20)

Also, up to a translation, we may suppose that 0 2 ˝. Thus, 0 2 ˝ and its distance from @˝ is of order 1= (independently of ı). In particular, if ı is small enough, we have that 0 lies in the interior of ˝ ;ı , and (6.19) gives that H .0/ ' nC4s : By comparing this with (6.20), we see that H has an interior minimum in ˝ ;ı . The value attained at this minimum is of order nC4s , and the values attained at the boundary of ˝ ;ı are of order ı n4s nC4s , which is much larger than nC4s , if ı is

134

6 A Nonlocal Nonlinear Stationary Schrödinger Type Equation

Fig. 6.1 Geometric interpretation

small enough. This says that the interior minimum of H in ˝ ;ı is nondegenerate and it survives to any perturbation of order nC4s , if ı is small enough. This and (6.17) imply that J has also an interior minimum at some point  in ˝ ;ı . By construction, this point  satisfies (6.18), and so this completes the proof of Theorem 6.1. The variational argument in the proof above (see in particular (6.18)) has a classical and neat geometric interpretation. Namely, the “unperturbed” functional (i.e. the one with D 0) has a very degenerate geometry, since it has a whole manifold of minimizers with the same energy: this manifold corresponds to the translation of the ground state w, namely it is of the form M0 WD fw ;  2 Rn g and, therefore, it can be identified with Rn . For topological arguments, this degenerate picture may constitute a serious obstacle to the existence of critical points for the “perturbed” functional (i.e. the one with ¤ 0). As an obvious example, the reader may think of the function of two variables f W R2 ! R given by f .x; y/ WD x2 C y. When D 0, this function attains its minimum along the manifold fx D 0g, but all the critical points on this manifold are “destroyed” by the perturbation when ¤ 0 (indeed rf .x; y/ D .2x; / never vanishes). In the situation described in the proof of Theorem 6.1, this pathology does not occur, thanks to the nondegeneracy provided in [83]. Indeed, by the nondegeneracy of the unperturbed critical manifold, when ¤ 0 one can construct a manifold, diffeomorphic to the original one (in our case of the form M WD fu C ./;  2 Rn g), that enjoys the special feature of “almost annihilating” the gradient of the functional, up to vectors parallel to the original manifold M0 (this is the meaning of formula (6.15)) (see also Fig. 6.1).

6 A Nonlocal Nonlinear Stationary Schrödinger Type Equation

135

Then, if one finds a minimum of the functional constrained to M , the theory of Lagrange multipliers (at least in finite dimension) would suggest that the gradient is normal to M . That is, the gradient of the functional is, simultaneously, parallel to M0 and orthogonal to M . But since M is diffeomorphically close to M0 , the only vector with this property is the null vector, hence this argument provides the desired critical point. We also recall that the fractional Schrödinger equation is related to a nonlocal canonical quantization, which in turn produces a nonlocal Uncertainty Principle. In the classical setting, one considers the momentum/position operators, which are defined in Rn , by Pk WD i¯@k and Qk WD xk

(6.21)

for k 2 f1; : : : ; ng. Then, the Uncertainty Principle states that the operators P D .P1 ; : : : ; Pn / and Q D .Q1 ; : : : ; Qn / do not commute (which makes it practically impossible to measure simultaneously both momentum and position). Indeed, in this case a simple computation shows that ŒQ; P WD

n X ŒQk ; Pk D i¯n:

(6.22)

kD1

The nonlocal analogue of this quantization may be formulated by introducing a nonlocal momentum, i.e. by replacing the operators in (6.21) by Psk WD i¯s @k ./

s1 2

and Qk WD xk :

(6.23)

In this case, using that the Fourier transform of the product is the convolution of the Fourier transforms, one has that  .Oxk  g/./ D F xk F 1 g.x/ ./ Z Z dx dy e2ix.y/ xk g.y/ D Rn

1 D 2i D D

i 2 i 2

Rn

Z

Z

dx Z

Rn

Z

Rn

dx Z

Rn

Rn

Rn

dy @yk e2ix.y/ g.y/

dy e2ix.y/ @k g.y/

dx e2ix F 1 .@k g/.x/

 i F F 1 .@k g/ ./ 2 i D @k g./; 2

D

(6.24)

136

6 A Nonlocal Nonlinear Stationary Schrödinger Type Equation

for any test function g. In addition, F .Psk f / D .2/s ¯s k jjs1 fO : Therefore, given any test function , using this with f WD also (6.24) with g WD F .Psk / and g WD O , we obtain that

and f WD xk , and

 F Qk Psk .x/  Psk Qk .x/  D F xk Psk .x/  Psk .xk .x// D xO k  F .Psk .x//  F .Psk .xk .x/// D D D D D

i @k F .Psk /./  .2/s ¯s k jjs1 F .xk .x//./ 2  .2/s1 i¯s @k k jjs1 O ./  .2/s ¯s k jjs1 xO k  O ./  .2/s1 i¯s @k k jjs1 O ./  .2/s1 i¯s k jjs1 @k O ./  .2/s1 i¯s @k k jjs1 O ./  .2/s1 i¯s jjs1 C .s  1/k2 jjs3 O ./:

Consequently, by summing up,  F ŒQ; Ps / D .2/s1 i¯s jjs1 .n C s  1/ O ./: So, by taking the anti-transform, ŒQ; Ps

 Di¯s .n C s  1/ F 1 .2jj/s1 O Di¯s .n C s  1/ ./

s1 2

:

(6.25)

Notice that, as s ! 1, this formula reduces to the the classical Heisenberg Uncertainty Principle in (6.22).

6.1 From the Nonlocal Uncertainty Principle to a Fractional Weighted Inequality Now we point out a simple consequence of the Uncertainty Principle in formula (6.25), which can be seen as a fractional Sobolev inequality in weighted spaces. The result (which boils down to known formulas as s ! 1) is the following:

6.1 From the Nonlocal Uncertainty Principle to a Fractional Weighted Inequality

137

Proposition 6.1 For any u 2 S .Rn /, we have that 2  s1    ./ 4 u  2

L .Rn /



    2 s1      jxju  2 n  r./ 2 u  2 n : L .R / L .R / nCs1

Proof The proof is a general argument in operator theory. Indeed, suppose that there are two operators S and A, acting on a space with a scalar Hermitian product. Assume that S is self-adjoint and A is anti-self-adjoint, i.e. hu; Sui D hSu; ui and hu; Aui D hAu; ui; for any u in the space. Then, for any  2 R,

 k.A C S/uk2 D kAuk2 C 2 kSuk2 C  hAu; Sui C hSu; Aui D kAuk2 C 2 kSuk2 C h.SA  AS/u; ui: Now we apply this identity in the space C01 .Rn / L2 .Rn /, taking S WD Qk D xk s1 and A WD iPsk D ¯s @k ./ 2 (recall (6.23) and notice that iPsk is anti-self-adjoint, thanks to the integration by parts formula). In this way, and using (6.25), we obtain that 0

n X

k.iPsk C Qk /uk2L2 .Rn /

kD1

D

n h X

kPsk uk2L2 .Rn / C 2 kQk uk2L2 .Rn / C ihŒQk ; Psk u; uiL2 .Rn /

kD1

2  s1   D ¯2s  r./ 2 u  2

L .Rn /

 2   C 2  jxju  2

L .Rn /

s1

Ci2  .n C s  1/ ¯s h./ 2 u; uiL2 .Rn / 2   2 s1     D ¯2s  r./ 2 u  2 n C 2  jxju  2 L .R /

 2 s1    .n C s  1/ ¯s  ./ 4 u  2

L .Rn /

L .Rn /

:

Now, if u 6 0, we can optimize this identity by choosing 2  s1   .n C s  1/ ¯s  ./ 4 u  2 n L .R /  WD  2   2  jxju  2 n L .R /

i

138

6 A Nonlocal Nonlinear Stationary Schrödinger Type Equation

and we obtain that  2 s1   0  ¯2s  r./ 2 u  2 L

4  s1   .n C s  1/2 ¯2s  ./ 4 u  2 n L .R /  ;  2 .Rn /   4  jxju  2 n L .R /

which gives the desired result.

Appendix A

Alternative Proofs of Some Results

We present in this Appendix alternative proofs of some results already introduced in this book. In the first section, we give two different proofs that the function xsC is s-harmonic on the positive half-line RC . We then alternatively compute some constants related to the fractional Laplacian.

A.1 Another Proof of Theorem 2.4.1 Here we present a different proof of Theorem 2.4.1, based on the Fourier transforms of homogeneous distributions. This proof is the outcome of a pleasant discussion with Alexander Nazarov. Proof (Alternative proof of Theorem 2.4.1) We are going to use the Fourier transform of jxjq in the sense of distribution, with q 2 C n Z. Namely (see e.g. Lemma 2.23 on page 38 of [96]) F .jxjq / D Cq jj1q ;

(A.1)

with Cq WD 2.2/q1 .1 C q/ sin

q : 2

(A.2)

We remark that the original value of the constant Cq in [96] is here multiplied by a .2/q1 term, in order to be consistent with the Fourier normalization that we have introduced. We observe that the function R 3 x 7! jxjq is locally integrable only when q > 1, so it naturally induces a distribution only in this range of the parameter q (and, similarly, the function R 3  7! jj1q is locally integrable only when q < 0): therefore, to make sense of the formulas above in a wider range of parameters q it is necessary to use analytic continuation and a special procedure that © Springer International Publishing Switzerland 2016 C. Bucur, E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana 20, DOI 10.1007/978-3-319-28739-3

139

140

Appendix A

is called regularization: see e.g. page 36 in [96] (as a matter of fact, we will do a procedure of this type explicitly in a particular case in (A.11)). Since R 3 x 7! jxjq is even, we can write (A.1) also as F 1 .jjq / D Cq jxj1q :

(A.3)

We observe that, by elementary trigonometry, sin

.s  1/ .s  2/ s .s C 1/ D  sin and sin D sin : 2 2 2 2

Moreover, .2 C s/ D .1 C s/ .1 C s/ and .s/ D .s  1/ .s  1/: Hence, by (A.2), 1s 1s .s C 1/ .s  2/  CsC1 Cs2 D  4.2/2s1 .2 C s/ .s  1/ sin sin 1Cs 1Cs 2 2 D 4 .1 C s/ .s/ sin

.s  1/ s sin D Cs Cs1 : 2 2

(A.4)

Moreover, jxjs C

1 @x jxjsC1 D 2xsC : sC1

So, taking the Fourier transform and using (A.1) with q WD s and q WD s C 1, we obtain that 2F .xsC / D F .jxjs / C

 1 F @x jxjsC1 sC1

2i F .jxjsC1 / sC1 2i D Cs jj1s C CsC1 jj2s : sC1 D F .jxjs / C

As a consequence, 2jj2s F .xsC / D Cs jj1Cs C

2i CsC1 jj2Cs : sC1

Appendix A

141

Hence, recalling (6.24),

 2CsC1 i 1 F ./  F 1 .jj2Cs / 2F 1 jj2s F .xsC / D Cs F 1 .jj1Cs / C sC1 2CsC1 i i  @x F 1 .jj2Cs / D Cs F 1 .jj1Cs /  sC1 2 CsC1 @x F 1 .jj2Cs /: D Cs F 1 .jj1Cs / C sC1 Accordingly, exploiting (A.3) with q WD 1 C s and q WD 2 C s,

 CsC1 Cs2 @x jxj1s 2F 1 jj2s F .xsC / D Cs Cs1 jxjs C sC1 1s  CsC1 Cs2 x jxj1s : D Cs Cs1 jxjs C 1Cs So, recalling (A.4),



  2F 1 jj2s F .xsC / D Cs Cs1 jxjs  x jxj1s : This and (2.7) give that

 ./s .xsC / D CQ s jxjs  x jxj1s ; for some CQ s . In particular, the quantity above vanishes when x > 0, thus providing an alternative proof of Theorem 2.4.1. Yet another proof of Theorem 2.4.1 can be obtained in a rather short, but technically quite advanced way, using the Paley-Wiener theory in the distributional setting. The sketch of this proof goes as follows: Proof (Alternative proof of Theorem 2.4.1) The function h WD xsC is homogeneous of degree s. Therefore its (distributional) Fourier transform F h is homogeneous of degree 1  s (see Lemma 2.21 in [96]). Moreover, h is supported in fx  0g, therefore F h can be continued to an analytic function in C WD fz 2 C s.t. =z < 0g (see Theorem 2 in [129]). Therefore, g./ WD jjs F h./ is homogeneous of degree 1 and is the trace of a function that is analytic in C . In particular, for any y < 0, we have that g.iy/ D

c g.i/ D ; y iy

142

Appendix A

where c WD ig.i/. That is, g.z/ coincides with whole of C , by analytic continuation. That is, in the sense of distributions, g./ D

c z

on a half-line, and then in the

c ;   i0

for any  2 R. Now we recall the Sokhotski Formula (see e.g. (3.10) in [16]), according to which 1 1 D iı C P.V. ;  ˙ i0  where ı is the Dirac’s Delta and the identity holds in the sense of distributions. By considering this equation with the two sign choices and then summing up, we obtain that 1 1  D 2iı:  C i0   i0 As a consequence g./ D

c c D C 2icı:   i0  C i0

Therefore jj2s F h./ D jjs g./ D

c jjs C 2icjjs ı:  C i0

Of course, as a distribution, jjs ı D 0, since the evaluation of jjs at  D 0 vanishes, therefore we can write `./ WD jj2s F h./ D

c jjs :  C i0

Accordingly, ` is homogeneous of degree 1 C s and it is the trace of a function analytic in CC WD fz 2 C s.t. =z > 0g. Consequently, F 1 ` is homogeneous of degree s (see again Lemma 2.21 in [96]), and it is supported in fx  0g (see again Theorem 2 in [129]). Since ./s xsC coincides (up to multiplicative constants) with F 1 `, we have just s s shown that ./s xsC D co xs  , for some co 2 R, and so in particular ./ xC vanishes in fx > 0g.

Appendix A

143

A.2 Another Proof of Lemma 2.3 For completeness, we provide here an alternative proof of Lemma 2.3 that does not use the theory of the fractional Laplacian. Proof (Alternative proof of Lemma 2.3) We first recall some basic properties of the modified Bessel functions (see e.g. [4]). First of all (see formula 9.1.10 on page 360 of [4]), we have that Js .z/ WD

C1 zs .1/k z2k zs X D C O.jzj2Cs / 2s kD0 22k kŠ .s C k C 1/ 2s .1 C s/

as jzj ! 0. Therefore (see formula 9.6.3 on page 375 of [4]), Is .z/ WD e D e D

is 2

is 2

i

Js .e 2 z/



 e 2 zs 2Cs C O.jzj / 2s .1 C s/ is

zs C O.jzj2Cs /; 2s .1 C s/

as jzj ! 0. Using this and formula 9.6.2 on page 375 of [4],

  Is .z/  Is .z/ 2 sin.s/   zs zs  2s  C O.jzj / : D 2 sin.s/ 2s .1  s/ 2s .1 C s/

Ks .z/ WD

Thus, recalling Euler’s reflection formula .1  s/ .s/ D

 ; sin.s/

and the relation .1 C s/ D s .s/, we obtain .1  s/ .s/ Ks .z/ D 2 D



zs zs  C O.jzj2s / 2s .1  s/ 2s .1 C s/

.s/ zs .1  s/ zs C O.jzj2s /;  21s 21Cs s



144

Appendix A

as jzj ! 0. We use this and formula (3.100) in [105] (or page 6 in [112]) and get that, for any small a > 0, Z

C1

1

cos.2t/ 1

.t2 C a2 /sC 2

0

dt D

 sC 2  Ks .2a/ as s C 12

  1  sC 2 .1  s/  s as .s/ 2s C O.a / D s   2s a s C 12 2 s as 1

(A.5)

1

 2 .s/  2sC 2 .1  s/ C O.a22s /:  D 2s   2a s C 12 2s s C 12 Now we recall the generalized hypergeometric functions m Fn (see e.g. page 211 in [112]): as a matter of fact, we just need that for any b, c, d > 0, 1 F2 .bI c; dI 0/

D

.b/ .c/ .d/  D 1: .b/ .c/ .d/

We also recall the Beta function relation B.˛; ˇ/ D

.˛/ .ˇ/ ; .˛ C ˇ/

(A.6)

see e.g. formula 6.2.2 in [4]. Therefore using formula (3.101) in [105] (here with y WD 0,  WD 0 and WD s  12 , or see page 10 in [112]), Z

C1

dt 1

.a2 C t2 /sC 2

0

Then, we recall that Z

C1 0

1 2

    1 1 1 a2s B ; s 1 F2 I 1  s; I 0 D 2 2 2 2 1 .s/ : D 2s 2  1 2a 2 C s 1

D  2 , so, making use of (A.5), for any small a > 0,

1  cos.2t/ 1

.t2 C a2 /sC 2

1

 2sC 2 .1  s/ C O.a22s /:  dt D 2s s C 12

Therefore, sending a ! 0 by the Dominated Convergence Theorem we obtain Z 0

C1

1  cos.2t/ dt D lim a!0 t1C2s

Z

C1 0

1  cos.2t/ 1

.t2 C a2 /sC 2

1

 2sC 2 .1  s/ :  dt D 2s s C 12 (A.7)

Appendix A

145

Now we recall the integral representation of the Beta function (see e.g. formulas 6.2.1 and 6.2.2 in [4]), namely

 n1

1

 Z C1  n3 Cs n1 1  2 n d: ; Cs D DB n 2 2 2 Cs .1 C / 2 Cs 0

2

2

(A.8)

We also observe that in any dimension N the .N  1/-dimensional measure of the unit sphere is

N 2 N 2 C1

N

.

Z

/

, (see e.g. [89]). Therefore N

dY RN

.1 C jYj2 /

NC1C2s 2

D

N 2 N 2 C1

Z 0

C1

N1 .1 C 2 /

NC1C2s 2

d:

In particular, taking N WD n  1 and using the change of variable 2 DW , Z

n1 Z C1 n2 .n  1/  2  D d nC2s nC2s n1 2 C1 0 Rn1 .1 C jj2 / 2 .1 C 2 / 2 n1 Z C1 n3 .n  1/  2  2  D d: nC2s 2 n1 0 .1 C / 2 2 C1

d

By comparing this with (A.8), we conclude that Z

n1

d

Rn1

.1 C jj2 /

nC2s 2

.n  1/  2  n1  D 2 2 C1  n1  2 12 C s  : D n2 C s

 n1

 1 Cs  n 2 2 Cs 2

Accordingly, with the change of variable  WD j!1 j1 .!2 ; : : : ; !n /, Z Rn

1  cos.2!1 / d! j!jnC2s ! Z Z 1  cos.2!1 / D d!2 : : : d!n d!1 nC2s R Rn1 .! 2 C ! 2 C    C ! 2 / 2 n 1 2 ! Z Z 1  cos.2!1 / D d d!1 nC2s R Rn1 j!1 j1C2s .1 C jj2 / 2

146

Appendix A

Z Cs 1  cos.2!1 / D d!1 j!1 j1C2s 2 Cs R  Z C1 n1 2 2 12 C s 1  cos.2t/ n D dt: t1C2s 2 Cs 0 

n1 2

n

1 2

Hence, recalling (A.7), Z Rn

 n1 1 2 2 12 C s  2sC 2 .1  s/ 1  cos.2!1 /   d! D  j!jnC2s n2 C s 2s s C 12 n

D

 2 C2s .1  s/  ; s n2 C s

as desired. To complete the picture, we also give a different proof of (A.7) which does not use the theory of special functions, but the Fourier transform in the sense of distributions and the unique analytic continuation: Proof (Alternative proof of (A.7)) We recall (see1 e.g. pages 156–157 in [141]) that for any  2 C with 0, we point out that Z R

jyj1 .R;R/ .y/ dy D 2

Z

R

y1 dy D

0

2R ; 

and we obtain that  Z Z  jyj1 .x/ cos.2xy/  .R;R/ .y/ dx dy R

R

 Z 1 Z  2  2 2R  D jxj .x/ dx  .x/ dx:   R 1 R 2

(A.11)

This formula holds true, in principle, for  2 C, with