Nonlocal Perimeter, Curvature and Minimal Surfaces for Measurable Sets [1st ed.] 978-3-030-06242-2;978-3-030-06243-9

Table of contents :
Front Matter ....Pages i-xviii
Nonlocal Perimeter (José M. Mazón, Julio Daniel Rossi, J. Julián Toledo)....Pages 1-17
Nonlocal Isoperimetric Inequality (José M. Mazón, Julio Daniel Rossi, J. Julián Toledo)....Pages 19-28
Nonlocal Minimal Surfaces and Nonlocal Curvature (José M. Mazón, Julio Daniel Rossi, J. Julián Toledo)....Pages 29-44
Nonlocal Operators (José M. Mazón, Julio Daniel Rossi, J. Julián Toledo)....Pages 45-52
Nonlocal Cheeger and Calibrable Sets (José M. Mazón, Julio Daniel Rossi, J. Julián Toledo)....Pages 53-80
Nonlocal Heat Content (José M. Mazón, Julio Daniel Rossi, J. Julián Toledo)....Pages 81-106
A Nonlocal Mean Curvature Flow (José M. Mazón, Julio Daniel Rossi, J. Julián Toledo)....Pages 107-118
Back Matter ....Pages 119-123

Citation preview

Frontiers in Mathematics

José M. Mazón Julio Daniel Rossi J. Julián Toledo

Nonlocal Perimeter,

Curvature and Minimal Surfaces

for Measurable Sets

Frontiers in Mathematics

Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta) William Y. C. Chen (Nankai University, Tianjin) Benoît Perthame (Sorbonne Université, Paris) Laurent Saloff-Coste (Cornell University, Ithaca) Igor Shparlinski (The University of New South Wales, Sydney) Wolfgang Sprößig (TU Bergakademie Freiberg) Cédric Villani (Institut Henri Poincaré, Paris)

More information about this series at http://www.springer.com/series/5388

José M. Mazón • Julio Daniel Rossi • J. Julián Toledo

Nonlocal Perimeter, Curvature and Minimal Surfaces for Measurable Sets

José M. Mazón Departamento de Análisis Matemático Universitat de València Valencia, Spain

Julio Daniel Rossi Departamento de Matemáticas Universidad de Buenos Aires Buenos Aires, Argentina

J. Julián Toledo Departamento de Análisis Matemático Universitat de València Val`encia, Spain

ISSN 1660-8046 ISSN 1660-8054 (electronic) Frontiers in Mathematics ISBN 978-3-030-06242-2 ISBN 978-3-030-06243-9 (eBook) https://doi.org/10.1007/978-3-030-06243-9 Library of Congress Control Number: 2018966514 Mathematics Subject Classification (2010): 45C99, 28A75, 49Q05, 45M05, 35K08 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

J. M. Mazón dedicates this book to Claudia J. D. Rossi dedicates this book to Cecilia J. Toledo dedicates this book to his parents, Julián and Feliciana

Preface

The goal in this monograph is to present recent results concerning nonlocal perimeter, curvature, and minimal surface that can be used for measurable sets. The concept of perimeter and curvature of planar forms goes back (at least) to the ancient Greece where mathematicians developed various geometric tools and ideas to try to give sense to different measures (area, length, etc.). In this context, the perimeter of a planar set was defined as the length of the curve that surrounds the set. Nowadays, there are modern concepts of perimeter and curvature that extend this intuitive idea and make it applicable to a wider class of sets. This modern point of view starts with Caccioppoli’s seminal works and continued with the ideas developed by De Giorgi in the 1950s. The usual concepts of perimeter, curvature, and minimal surfaces require some regularity of the involved sets. For instance, to define the perimeter in geometric measure theory one asks for its characteristic function to be in the space BV (bounded variation functions); see [6, 57]. A fundamental result by De Giorgi and Federer shows that if we use this definition the perimeter coincides with the (N − 1)-dimensional Hausdorff measure of a certain subset of the topological boundary, so that consistency with the naïve idea of being the length of the boundary is guaranteed. With this notion, not every bounded measurable set has finite perimeter. In the same spirit, when one wants to compute the curvature of the boundary of a set using classical tools one asks for this boundary to be twice differentiable. Here we focus on a new version of perimeter and curvature that can be applied (and gives a finite number) to any measurable set with finite measure. We follow our recent results in [58, 59]. The main idea to obtain a notion of perimeter that can be used to a large class of sets is to realize that the usual notions of perimeter and curvature involve the computation of local quantities. Here we work with a nonlocal analogous to these quantities. In this direction, recently, as analogous objects to the standard heat equation, nonlocal diffusion problems were proposed, like  ut (x, t) = J ∗ u(x, t) − u(x, t) =

RN

J (x − y)(u(y, t) − u(x, t)) dy,

with a convolution kernel J that is assumed to be continuous, radially symmetric and non increasing, nonnegative, compactly supported verifying J = 1 (some of these conditions vii

viii

Preface

can be relaxed according to the result one wants to prove). This nonlocal equation shares many properties with its local counterpart, the heat equation; for example, bounded stationary solutions are constants, there is a maximum principle, and there is infinite speed of propagation (compactly supported nonnegative and nontrivial initial conditions give rise to solutions that are strictly positive everywhere). In addition, the decay rate as t goes to infinity for both problems is the same. However, there is a major difference: the heat equation has an instantaneous smoothing effect; solutions become C ∞ for every positive time even if the initial condition is not differentiable; on the other hand, solutions to the nonlocal evolution equation are as smooth as the initial data are (there is no smoothing effect in this case). This fact can be explained looking at the fundamental solution, that is smooth (a Gaussian) for the heat equation, but is a smooth function plus a delta for the nonlocal problem; see the recent book [12]. This lack of regularizing effect is closely related to the fact that the right-hand side of the equation, J ∗ u − u, makes sense for nonsmooth functions u. Concerning the perimeter of a set, as we have mentioned, a usual definition in geometric measure theory is to compute the total variation of the characteristic function of the set,  P er(E) =

RN

|DχE |.

Looking at this approach it seems reasonable to take the limit as p  1 in the p-Laplacian type energy of a function u to obtain 

 lim

p1 RN

|Du|p =

RN

|Du|.

In the nonlocal setting, there is an analogous to the p-energy given by 1 2p



 RN

RN

J (x − y)|u(y) − u(x)|p dy dx.

If we take the limit as p  1 in this expression, we formally arrive to 1 2



 RN

RN

J (x − y)|u(y) − u(x)| dy dx,

that is a nonlocal quantity analogous to the total variation. Now, if we evaluate this expression in the characteristic function of a set, u = χE , we get 1 2

 



 RN

RN

J (x − y)|χ E (y) − χ E (x)| dx dy = E

Rn \E

J (x − y) dy dx.

Preface

ix

Let E ⊂ RN be a measurable set; the nonlocal J-perimeter of E is defined by 

  PJ (E) :=

RN \E

E

J (x − y) dy

dx.

Note that if |E| < +∞, we have   PJ (E) = |E| −

J (x − y) dy dx. E

E

To simplify the exposition, we are confining ourselves to study subsets of the Euclidean space RN ; however, a large part of the results presented here can be extended to metric spaces with a measure (in this more general case, we have to consider a kernel J (x, y)). This definition of perimeter is nonlocal in the sense that it is determined by the behavior of E in a neighborhood of the boundary ∂E. The quantity PJ (E) measures the interaction between points of E and E c via the interaction density J (x − y). For J compactly supported in a ball Br (0), the interaction is possible when the points x ∈ E and y ∈ E c := RN \ E are both close to the boundary ∂E:  PJ (E) =

 {x∈E : d(x,E c ) 0 for every small δ. So, by analogy with the classical case, we consider the left-hand side as a nonlocal mean curvature, denoted by  J (x) H∂E

:=

RN

J (x − y) (χ E c (y) − χ E (y)) dy.

Preface

xi

J (x) makes sense for every x ∈ RN and not only for points x on Note that this curvature H∂E the boundary of the set E. For this nonlocal curvature, we can also show an approximation result to recover the usual notion of mean curvature. In fact, when E ⊂ RN is a smooth set with C 2 boundary, for every x ∈ ∂E, we have

lim ↓0

CJ J H (x) = (N − 1)H∂E (x),  ∂E

where H∂E (x) is the (local) mean curvature of ∂E at x. Once we have a notion of curvature, one can study a mean curvature flow, that is, the evolution of a set when we prescribe that the normal velocity of a point on the boundary is proportional to the curvature of the set at that point. To study such a flow we need to introduce a normal vector defined for nonsmooth sets. After this is done we are able to show that, as happens for the local mean curvature flow, a ball evolves through this flow as a collection of balls that shrink to a point in finite time. The study of nonlocal perimeters is motivated by both theory and applications as we will briefly describe below. Note that in applications, it is also natural to consider interactions that are not homogeneous or rotationally invariant (but here we restrict ourselves to this class of kernels to simplify the exposition). The first application that we present is related to image processing and bitmaps; see [33]. Let us consider the framework of BMP-type images with square pixels of (small) size δ > 0 (suppose that 1/δ ∈ N for simplicity). As an example, let us consider a picture of a square of side of length one, with sides at 45 √ degrees with respect to the √ √ orientation of√the pixels. Take the square with vertices at (1/ 2, 0), (0, 1/ 2), (−1/ 2, 0), (0, −1/ 2) and pixels of the form (kδ, (k + 1)δ) × (j δ, (j + 1)δ). Let us look at the “computational image” composed by the pixels (small squares whose sides are of length δ) that intersects the square and let us compare it with the original square. Note that the “discretization” of the image is composed by a large number of pixels. In this configuration, the classical perimeter functional provides a rather inaccurate tool to analyze the perimeter of the original square, no matter how small the pixels are, i.e., no matter how good is the image resolution. Indeed, the perimeter of the ideal square is 4, while the perimeter √ of the picture displayed by the monitor (the one composed by the pixels) is always 4 2 (independently √ of the size of δ). Hence, the classical perimeter is always producing an error by a factor 2, even in cases of extremely high resolution. Instead, the nonlocal perimeter studied here or other nonlocal perimeters would provide a much better approximation of the classical perimeter of the ideal square in the case of high image resolution. Indeed, note that both the nonlocal perimeter and the nonlocal mean curvature at a point x are continuous as functions of E in the following sense: if En → E in the sense that |En E| → 0 then PJ (En ) → PJ (E)

and

J J H∂E (x) → H∂E (x). n

xii

Preface

Hence, as the resolution of the image gets better (δ becomes very small) the measure of the difference of the sets (the real set vs the union of the pixels) goes to zero (the symmetric difference has measure of order 4δ) and then the nonlocal perimeter of the computational image gets very close to the nonlocal perimeter of the original square. Therefore, in this case, the nonlocal perimeter provides more precise information than the classical one. For applications of nonlocal operator to image processing, see for instance [13, 14, 41, 50, 53]. Another motivation for the study of nonlocal minimal surfaces comes from models describing phase-transitions problems with long-range interactions; see [25–28, 67]. In these models, the minimal surface appears in a limit in which the zone that connects the two stable states concentrates along a surface. Let us now summarize the contents of this book. After a brief section in which we fix the notation that will be used along the whole book, the content of the different chapters can be described as follows: • In Chap. 1, we introduce the nonlocal perimeter, compare it with its local counterpart, and show that the local perimeter can be recovered from the nonlocal one by rescaling the kernel. • In Chap. 2, we analyze some properties of the nonlocal perimeter and prove a nonlocal isoperimetric inequality and a nonlocal co-area formula. • In Chap. 3, we introduce nonlocal minimal surfaces and the nonlocal curvature. Here we also include the proof that the local curvature can be obtained as a limit of the nonlocal curvature when the kernel is rescaled. • Chapter 4, is devoted to develop a nonlocal functional setting that involves the nonlocal analogous to bounded variation functions and its relation with the nonlocal perimeter. • In Chap. 5, we deal with nonlocal Cheeger and calibrable sets. • In Chap. 6, we describe the nonlocal heat content of a set and study its behavior for small t. In this description, the nonlocal perimeter plays a crucial role. • Finally, in Chap. 7 we introduce a nonlocal normal vector and, associated with this normal vector, we state a nonlocal mean curvature flow problem. The bibliography of this monograph does not escape the usual rule of being incomplete. In general, we have listed those papers which are closer to the topics discussed here. But, even for those papers, the list is far from being exhaustive and we apologize for any possible omission. Valencia, Spain Buenos Aires, Argentina Valéncia, Spain October 2018

José M. Mazón Julio Daniel Rossi J. Julián Toledo

Acknowledgment

The authors have been partially supported by the Spanish MEC and FEDER, project MTM2015-70227-P.

xiii

Notations

The ambient space will be the Euclidean space RN . Throughout the book we will use the following notations: Br (x) denotes the ball centered at x ∈ RN of radius r > 0. Br denotes the ball centered at 0 ∈ RN of radius r > 0. Br (x) and Br denote the closure of such balls. The measure of B1 is denoted by ωN : ωN = |B1 |. The Hausdorff (N − 1)-dimensional measure of a set A ⊂ RN is denoted by HN−1 (A). The boundary of ∂B1 is denoted by S N−1 , for which  HN−1 S N−1 = NωN . We will write (r, 0) to denote the point (r, 0, 0, . . . , 0) ∈ RN , r ∈ R.

xv

Contents

1

Nonlocal Perimeter .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.1 The Classical Concept of Perimeter .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.2 Nonlocal Perimeter for Non-singular Kernels . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.3 Rescaling the Nonlocal Perimeter .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

1 1 6 13

2

Nonlocal Isoperimetric Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.1 Nonlocal Isoperimetric Inequality .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.2 Nonlocal Coarea Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

19 19 27

3

Nonlocal Minimal Surfaces and Nonlocal Curvature .. . . . . . . . . . .. . . . . . . . . . . . . . . 3.1 Nonlocal Minimal Surfaces and Nonlocal Curvature . . . . . . . . .. . . . . . . . . . . . . . . 3.2 Rescaling the Nonlocal Curvature.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 3.3 Isoperimetric Inequalities and Curvature . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

29 29 33 38

4

Nonlocal Operators .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 4.1 A Characterization of the Nonlocal Perimeter .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 4.2 Nonlocal 1-Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

45 45 47

5

Nonlocal Cheeger and Calibrable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.1 Nonlocal Cheeger Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.2 Nonlocal Calibrable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.3 Rescaling . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

53 53 60 78

6

Nonlocal Heat Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.1 The Classical Heat Content .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.2 The Heat Content for Nonlocal Diffusion with Nonsingular Kernels . . . . . . . 6.2.1 Definition and a Characterization . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.2.2 The Asymptotic Expansion of the J -heat Content . . . .. . . . . . . . . . . . . . . 6.2.3 Nonlocal Curvature in the Expansion .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.2.4 A Probabilistic Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.2.5 The J -heat Loss of D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.2.6 The J -heat Content and the Nonlocal Isoperimetric Inequality . . . . .

81 81 83 83 86 93 94 97 97

xvii

xviii

Contents

6.3 Convergence to the Heat Content when Rescaling the Kernel .. . . . . . . . . . . . . . 98 6.4 The Spectral Heat Content .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 101 6.5 Lack of Regularity for the Nonlocal Heat Diffusion . . . . . . . . . .. . . . . . . . . . . . . . . 104 7

A Nonlocal Mean Curvature Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 107 7.1 Nonlocal Normal Vector .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 108 7.2 Nonlocal Mean Curvature Flow .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 111

Bibliography . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 119 Index . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 123

1

Nonlocal Perimeter

1.1

The Classical Concept of Perimeter

The word perimeter comes from the Greek peri (around) and meter (measure). A perimeter is usually used with two senses: it is the boundary that surrounds an N-dimensional set, and it is the measure of such boundary. We will see in these two first sections that these two concepts must be well precise. Nevertheless, for a set E ⊂ RN with smooth C 2 boundary its perimeter is clearly defined as Per(E) = HN−1 (∂E), being HN−1 the Hausdorff (N − 1)-dimensional measure. The study of sets of finite perimeter for general sets without smooth boundary goes back to Renato Caccioppoli, who in 1927 defined the measure of the boundary of an open set of the plane using the total variation in the sense of Tonelli. More precisely he says that E is a set of finite perimeter if there exist polyhedral sets En such that En → E locally in measure and sup Area(∂En ) < +∞. n∈N

In 1951 Caccioppoli, by using the generalization of functions of bounded variation to the case of several variables given by Lamberto Cesari in 1936, suggested to study the

© Springer Nature Switzerland AG 2019 J. M. Mazón et al., Nonlocal Perimeter, Curvature and Minimal Surfaces for Measurable Sets, Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-06243-9_1

1

2

1 Nonlocal Perimeter

geometrical properties of the Lebesgue measurable sets E with  E

divφ(x)dx ≤ Kφ∞ ,

∀ φ ∈ C01 (RN , RN ),

(1.1)

whenever a positive real number K exists. Caccioppoli was aware that (1.1) is equivalent to the existence of a vector finite Radon measure μE with 

 divφ(x)dx = E

φ · dμE ,

∀ φ ∈ C01 (RN , RN ),

∂E

(1.2)

and that the total variation |μE | of μE satisfies |μE |(RN ) ≤ K. In 1952, at the Salzburg Congress of the Austrian Mathematical Society, Ennio De Giorgi presented his first results on this subject developing the ideas of Caccioppoli. De Giorgi original definition of perimeter of a measurable subset E ⊂ RN was based on the heat semigroup in RN , because of its regularizing effect, and can be described as follows: Per(E) := lim ∇x T (t)χ E L1 (RN ) , t →0

where χ E denotes the characteristic function of E, and (T (t))t ≥0 is the heat semigroup, that is, if pN : RN × Rn × R → R is the Gauss–Weierstrass kernel, defined by pN (x, y, t) =

1 (4πt)

N 2

e−

|x−y|2 4t

,

then  T (t)u(x) =

RN

pN (x, y, t)u(y)dy,

for every u ∈ L1 (RN ); and he proved that, whenever μE is the Radon vector measure satisfying (1.2), Per(E) = |μE |(RN ). We refer also to Chap. 6 for another look of this interplay with the heat evolution problem in the nonlocal perimeter framework.

1.1 The Classical Concept of Perimeter

3

He also announced the validity of the isoperimetrical inequality for Lebesgue measurable sets, that is, the inequality    N

N−1 min |E|, |RN \ E| ≤ Per(E)

(1.3)

for any measurable set E and any N ≥ 2, and conjectured that

 inf Per(E) : |E| = 1 = Per(B) for B the ball of measure one. The proofs of all these results appear in [38] and [39]. After the death of Cacciopoli, De Giorgi started to call the sets of finite perimeter as Caccioppoli sets. In 1960 Federer and Fleming [43] introduced a new approach to the theory by mean of the normal and integral currents. In his famous book on Geometric Measure Theory [42], Federer showed that Caccioppoli sets are normal currents of dimension N in Ndimensional Euclidean spaces. Now, after the paper of Miranda [60], the easy and more traditional way to study the theory of sets of finite perimeter is the approach that uses functions of bounded variation in the framework of the theory of distributions. Two excellent monographs about this approach are [6, 51]. Let us overview this approach. Let  be an open bounded subset of RN . A function u ∈ L1 () whose partial derivatives in the sense of distributions are measures with finite total variation in  is called a function of bounded variation. The class of such functions will be denoted by BV () (see [6, 51]): BV () = {u ∈ L1 () : |Du|() < ∞}, where |Du| is the total variation of the distributional gradient of u, which turns out to be  |Du|() = sup

u div φ dx : φ ∈ 

C0∞ (, RN ), |φ(x)|

≤ 1 for x ∈  ,

 and will be also denoted by  |Du|. For u ∈ BV (), the gradient Du is a Radon measure that decomposes into its absolutely continuous and singular parts Du = D a u + D s u. Then D a u = ∇u LN , where ∇u is the Radon–Nikodym derivative of the measure Du with respect to the Lebesgue measure LN . A measurable set E ⊂ RN is said to be of finite perimeter in  if the characteristic function χ E of E belongs to BV (), in this case the perimeter of E is defined as Per(E, ) = |D χ E |().

4

1 Nonlocal Perimeter

A measurable set E ⊂ RN is said to be of finite perimeter if χ E ∈ BV (RN ), and we denote Per(E) = Per(E, RN ). Then, we have that  Per(E) = sup

RN

χ E (x)divφ(x)dx : φ ∈ Cc∞ (RN , RN ) φ∞ ≤ 1 .

(1.4)

If u ∈ BV (RN ), the energy functional associated with the total variation is  T V(u) =

RN

|Du|,

(1.5)

that is,  T V(u) = sup

RN

u(x)divφ(x)dx : φ ∈ Cc∞ (RN , RN ) φ∞ ≤ 1 .

Then, for a measurable set E ⊂ RN having finite perimeter,  Per(E) =

RN

|D χ E |.

(1.6)

If E ⊂ RN is a bounded open set with boundary ∂E of class C 2 , then E is a set of finite perimeter and Per(E) = HN−1 (∂E).

(1.7)

In fact, if φ ∈ Cc∞ (RN , RN ) with φ∞ ≤ 1, applying the Gauss–Green formula, we have 

 divφ(x)dx = E

φ · νdHN−1 ≤ HN−1 (∂E), ∂E

where ν is the unit outward normal ν to ∂E. Then, by (1.4), we have Per(E) ≤ HN−1 (∂E). Let us see the another inequality. Since ∂E is of class C 2 , there exists an open set U containing ∂E, such that d(x) := dist(x, E) is of class C 1 on U \ ∂E and ∇d(x) =

x − ξ(x) , d(x)

1.1 The Classical Concept of Perimeter

5

being ξ(x) the unique point of ∂E such that |x − ξ(x)| = d(x). Hence, the unit outward normal ν to ∂E has an extension νˆ ∈ Cc1 (RN ) such that |ˆν | ≤ 1. Therefore, if we take φ = ηνˆ , with η ∈ Cc∞ (RN ), we get 

 divφ(x)dx = E

ηdHN−1 . ∂E

Therefore  Per(E) ≥ sup

η dH

: η∈

N−1

∂E

Cc∞ (RN ),

|η| ≤ 1 = HN−1 (∂E),

and we finish the proof of (1.7). In the case that the set E has no smooth boundary, the topological boundary ∂E of a set of finite perimeter is not a good candidate to measure the perimeter because its Hausdorff measure exceeds, in general, such value. The correct boundary in this context is the reduced boundary introduced by De Giorgi. The reduced boundary of a set of finite perimeter E ⊂ RN , denoted by F E, is defined as the collection of points x at which the limit D χ E (Bρ (x)) ρ↓0 |D χ E |(Bρ (x))

νE (x) := lim exists and has length equal to one, i.e.,

|νE (x)| = 1. The function νE : F E → S N−1 is called the generalized inner normal to E. De Giorgi proved the following generalization of the Gauss–Green formula (see [6, Theorem 3.36] for a proof): 

 divφ(x)dx = − E



νE · φ d|D χ E | ∀φ ∈ Cc1 (, RN ),

for any set E of finite perimeter in , where D χ E = νE |D χ E | is the polar decomposition of D χ E . And he also proved that Per(E) = HN−1 (F E), which is a generalization of the classical formula (1.7) (see [6, Theorem 3.59] for a proof). This result provides geometric intuition for the notion of reduced boundary and confirms that the above definition of sets of finite perimeter is the more natural.

6

1 Nonlocal Perimeter

In 1960 Fleming and Rishel [46] proved the following coarea formula (see [6, Theorem 3.40] for a proof). Let  ⊂ RN be an open set. If u ∈ BV (), then the set {u > t} has finite perimeter in  for L1 -a.e. t ∈ R. Moreover,  |Du|(B) =

+∞

|D χ {u>t } |(B)dt

−∞

(1.8)

and  Du(B) =

+∞ −∞

D χ {u>t } (B)dt,

for any Borel set B ⊂ . As mentioned in the introduction, the concept of nonlocal perimeter was introduced in 1 , 0 < s < 1, the nonlocal perimeter reappeared [17, 36], and for kernels of the form |x|N+s

in [29]. The s-perimeter of E ⊂ RN is defined formally as   Pers (E) =

RN \E

E

1 dxdy, |x − y|N+s

and, if Pers (E) < ∞, it can be rewritten as 1 Pers (E) = 2

1.2



 RN

RN

|χ E (y) − χ E (x)| dxdy. |x − y|N+s

Nonlocal Perimeter for Non-singular Kernels

Let J : RN → [0, +∞] be a radially symmetric non-singular kernel, that is, a measurable, nonnegative, and radially symmetric function verifying  RN

J (z)dz = 1.

This conditions on J will be considered all around the book, but we will impose other conditions as continuity, radially non-increasing, etc., that will be given when necessary. Associated with this kernel J , a nonlocal perimeter is defined for general measurable sets. Let E ⊂ RN be a measurable set, the nonlocal J-perimeter of E is defined by   PJ (E) = E

RN \E

 J (x − y)dy dx.

(1.9)

1.2 Nonlocal Perimeter for Non-singular Kernels

7

Implicitly when the integral is not finite, we are setting PJ (E) = +∞. We have that PJ (E) = PJ (RN \ E). It is easy to see that PJ (E) =

1 2



 RN

RN

J (x − y)|χ E (y) − χ E (x)|dxdy,

(1.10)

and that, if |E| < +∞,   PJ (E) = |E| −

J (x − y)dydx. E

(1.11)

E

Observe that (1.11) is not true in general for the local case where bounded sets can have infinite perimeter. This definition of perimeter is nonlocal in the sense that it is determined by the behavior of E in a neighborhood of the boundary ∂E. The quantity PJ (E) measures the interaction between points of E and RN \ E via the interaction density J (x − y). For J compactly supported in the ball Br , the interaction is possible when the points x ∈ E and y ∈ RN \ E are both close to the boundary ∂E:  PJ (E) =

 {x∈E : d(x,RN \E) 0 consider the rescaled kernel Jr (x) =

1 x  . J rN r

Observe that Jr has the same total mass as J , and verifies  PJr (E) = r N PJ

 1 E , r

(1.22)

for all set E of Lebesgue finite measure. Observe also that MJr = rMJ . Then, under condition (1.19), (1.21) implies that PJr (E) ≤

MJ MJr Per (E) = r Per (E) , 2 2

(1.23)

for E with Lebesgue finite measure. Observe that, from (1.23), if r converges to 0, then PJr (E) also does it, nevertheless we will see in Theorem 1.11 that, for bounded sets of finite perimeter, 1 PJr (E) = c Per(E), r→0 r lim

for an adequate constant c > 0.

1.3

Rescaling the Nonlocal Perimeter

Assume that J satisfies (1.19). Consider the rescaled kernel introduced in Remark 1.5, J (x) =

1 x  , J N 

 > 0.

Let also consider 2

CJ =  RN

J (z)|zN |dz

Observe that CJ =

1 CJ . 

.

14

1 Nonlocal Perimeter

In [12] it is proved that the solutions u of equations of the form  ut (x, t) = CJ

J (x − y)

u(y, t) − u(x, t) dy |u(y, t) − u(x, t)|

converge (up to a subsequence) to the solution of ut = 1 u,   Du . for different boundary conditions, being 1 u = div |Du| Our aim in this section is to study which is the behavior of the nonlocal J -perimeter under rescaling. Example 1.6 Take J = 12 χ [−1,1] . Then, a simple calculation gives PJ ([a, b]) =

  + 2  1 1 − 1 − (b − a) , 2

(1.24)

that is,

PJ ([a, b]) =

⎧ 1 ⎪ ⎪ ⎪ ⎨ 2

if b − a > 1,

⎪   ⎪ ⎪ ⎩ (b − a) 1 − 1 (b − a) if b − a ≤ 1. 2

Let us compute lim→0 CJ PJ (E). Observe that for this particular kernel, we have CJ = 4. On account of (1.22) and (1.24), for  small, we have 4 PJ ([a, b]) = 4PJ  



a b ,  

 2 = Per([a, b]).

Example 1.7 Let us now consider J (x) = w(x)χ BR (x), with w(r) = CR r α . Then 

 RN



J (x)dx = CR

x dx = CR BR



R

= CR NωN 0

R

α

r α+N−1 dr =



 α

r dσ dr 0

∂Br

N|BR |R α CR NωN N+α R . = CR N +α N +α

1.3 Rescaling the Nonlocal Perimeter

Then, if α > −N and CR = R = 1, and α =

− 12 ,

15



N+α N|BR |R α , we have RN

J (x)dx = 1. Consider the case N = 1,

which corresponds to 1 J (x) = √ χ [−1,1] (x). 4 |x|

Then, a simple calculation gives

PJ ([a, b]) =

⎧ 3 2 ⎪ 2 ⎪ ⎪ ⎨ b − a − 3 (b − a) if b − a < 1, ⎪ ⎪ ⎪ ⎩1 3

(1.25)

if b − a ≥ 1.

Now CJ = 6 and, on account of (1.22) and (1.25), for  small, we have 6 CJ PJ ([a, b]) = PJ ([a, b]) = 6PJ  



a b ,  

 = 2 = Per([a, b]).

Therefore, in these two examples we have obtained that lim CJ PJ (E) = Per(E). ↓0

The following results by J. Dávila and A. Ponce will be especially important in order to get the rescaling results. See also [17, 20, 62, 72]. Theorem 1.8 (Dávila [36]) Let B ⊂ RN be open, bounded with a Lipschitz boundary, and let 0 ≤ ρ be the radial functions satisfying 

 RN

ρ (x)dx = 1,

lim

→0 |x|>δ

ρ (x)dx = 0

∀ δ > 0.

(1.26)

Then   lim

→0 B

B

|u(x) − u(y)| ρ (x − y)dxdy = K1,N |x − y|

 |Du|, B

where K1,N

1 = NωN





S N−1

N

, 2 |e · σ |dσ = √ π  N+1 2

|e| = 1.

(1.27)

16

1 Nonlocal Perimeter

Theorem 1.9 (Ponce [63]) Let N ≥ 2. Let B ⊂ RN be open, bounded with a Lipschitz boundary, and let 0 ≤ ρ be the radial functions satisfying (1.26). Let n ↓ 0 as n → +∞. If {un }n ⊂ L1 () is a bounded sequence such that   B

B

|un (x) − un (y)| ρn (x − y)dxdy ≤ K1,N M, |x − y|

where K1,N is given in (1.27) and M is a constant, then {un }n is relatively compact in L1 (B). Moreover, if unj → u in L1 (B), then u ∈ BV (B) and  |Du| ≤ M. B

The previous result holds in dimension N ≥ 2; a counterexample for dimension N = 1 appears in [17]. Remark 1.10 For J satisfying (1.19), MJ 1 CJ = . 2 K1,N

(1.28)

Indeed, denoting J˜(r) = J (x) if |x| = r,  K1,N

MJ CJ = K1,N  2

|w|J (w)dw

RN RN

|wN |J (w)dw

 = K1,N 

∞

∞

0

0

r J˜(r)dσ dr ∂Br

J˜(r)|σ · eN |dσ dr

∂Br

 =

 S N−1

|eN · σ |dσ NωN

NωN

·

∞

J˜(r)r N

∞

0

r N J˜(r)dr

S N−1

0

|eN · σ |dσ dr

= 1.

Theorem 1.11 Assume J satisfies condition (1.19). If u ∈ BV (RN ) has compact support, then  lim CJ T V J (u) = |Du|. ↓0

RN

1.3 Rescaling the Nonlocal Perimeter

17

In particular, if E ⊂ RN is a bounded set of finite perimeter, then CJ N  PJ lim CJ PJ (E) = lim ↓0 ↓0 



E 

 = Per(E).

Proof Since u has compact support, for a large ball B containing supp(u) we can rewrite T V J (u) =

1 2

  J (x − y)|u(y) − u(x)|dydx, B

B

and 

 RN

|Du| =

|Du|. B

Now, CJ T V J (u) =

1 K1,N

  B

B

|u(x) − u(y)| ρ (x − y)dxdy, |x − y|

being ρ (z) =

1 |z| CJ K1,N J (z). 2 

Then, by (1.28),  RN

ρ (z)dz =

1 CJ K1,N 2

 

RN

=

1 CJ K1,N 2

=

1 CJ K1,N MJ 2

RN

|z| 1  z  dz J  N  |z|J (z)dz

= 1, and, for δ > 0,  lim

→0 |z|>δ

ρ (z)dz =

1 CJ K1,N lim →0 2

 |z|> δ

since J satisfies (1.19). Therefore, we can apply Theorem 1.8 to get the result.

|z|J (z)dz = 0,

 

2

Nonlocal Isoperimetric Inequality

2.1

Nonlocal Isoperimetric Inequality

For the nonlocal perimeter, there is also an isoperimetric inequality, and here the main hypothesis used on J is that it is radially nonincreasing. Its proof uses the symmetric decreasing rearrangement, which replaces a given nonnegative function f by a radial function f ∗ . Let us recall briefly the definition and some basic properties of this rearrangement. Let E be a measurable set of finite measure. Its symmetric rearrangement E ∗ is given by the open centred ball whose measure agrees with |E|, that is, E ∗ = Br if r is such that |Br | = ωN r N = |E|. Now for a nonnegative and measurable function f that vanishes at infinity, in the sense that all its positive level sets have finite measure, we define the symmetric decreasing rearrangement f ∗ by symmetrizing its level sets: ∗



∞

f (x) = 0

χ {f (x)>t }∗ dt.

Note that for a radially nonincreasing function, it holds that f ∗ = f and that the previous definitions are consistent in the sense that χ A∗ = (χ A )∗ . We refer to [54, 56] for details.

© Springer Nature Switzerland AG 2019 J. M. Mazón et al., Nonlocal Perimeter, Curvature and Minimal Surfaces for Measurable Sets, Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-06243-9_2

19

20

2 Nonlocal Isoperimetric Inequality

For the rearrangement of functions interacting with a convolution, we have the Riesz rearrangement inequality [56, Theorem 3.7], namely,   f (x)(g ∗ h)(x) dx ≤ f ∗ (x)(g ∗ ∗ h∗ )(x) dx. (2.1) RN

RN

Theorem 2.1 (Isoperimetric Inequality) Assume J is radially nonincreasing. For every measurable set E with finite measure, it holds that PJ (E) ≥ PJ (Br )

(2.2)

where Br is a ball such that |Br | = |E|. Proof Using (2.1), one has   PJ (E) =

RN \E

E

 J (x − y)dy dx

 



= |E| −

J (x − y)dy dx 

= |E| −  ≥ |E| −

E

E

χ E (x)(J ∗ χ E )(x) dx

RN

RN

(χ E )∗ (x)(J ∗ ∗ (χ E )∗ )(x) dx

 = |Br | −

RN

χ Br (x)(J ∗ χ Br )(x) dx

= P J (Br ), and we conclude (2.2) from the fact that |E| = |Br |.

 

We are going to get now the nonlocal version of the relative isoperimetric inequality (1.3) obtained by De Giorgi. We will use the following Poincaré-type inequality given in [11, Proposition 4.1]. Proposition 2.2 Assume that J is radially nonincreasing. Given q ≥ 1 and  a bounded domain in RN , the quantity:

βq−1 (J, , q) =

inf

u∈Lq (),

 u=0

1 2

   

J (x − y)|u(y) − u(x)|q dy dx  |u(x)|q dx 

2.1 Nonlocal Isoperimetric Inequality

21

is strictly positive. Consequently,    q     1 1   βq−1 (J, , q) u − u ≤ J (x − y)|u(y) − u(x)|q dy dx, ||   2   

(2.3)

for every u ∈ Lq (). Proof It is enough to prove that there exists a constant c such that  

1/q

uq ≤ c

J (x − y)|u(y) − u(x)| dydx q

 

    +  u ,

(2.4)



for every u ∈ Lq (). Since J is radially nonincreasing, there exists r > 0 such that J (z) ≥ α > 0 in Br \ {0}. m Since  ⊂ ∪x∈ Br/2 (x), there exists {xi }m i=1 ⊂  such that  ⊂ ∪i=1 Br/2 (xi ). Let 0 < δ < r/2 such that Bδ (xi ) ⊂  for all i = 1, . . . , m. Then, for any xˆi ∈ Bδ (xi ), i = 1, . . . , m,

=

m 

(Br (xˆi ) ∩ ).

(2.5)

i=1

Let us argue by contradiction. Suppose that (2.4) is false. Then, there exists un ∈ Lq (), with un Lq () = 1, and satisfying  

1/q J (x − y)|un (y) − un (x)|q dydx

1≥n  

    +  un 

∀n ∈ N.



Consequently,   J (x − y)|un (y) − un (x)|q dy dx = 0

lim n

(2.6)

 

and  un = 0.

lim n



(2.7)

22

2 Nonlocal Isoperimetric Inequality

Let Fn (x, y) = J (x − y)1/q |un (y) − un (x)| and  fn (x) =

J (x − y)|un (y) − un (x)|q dy. 

From (2.7), it follows that fn → 0

in L1 ().

Passing to a subsequence if necessary, we can assume that fn (x) → 0

∀x ∈  \ D1 ,

D1 null.

(2.8)

On the other hand, by (2.6), we also have that Fn → 0 en Lq ( × ). So, we can suppose, up to a subsequence, Fn (x, y) → 0

∀(x, y) ∈  ×  \ C,

C null.

(2.9)

for all x ∈  \ D2 , the section Cx of C is null.

(2.10)

Let D2 ⊂  be a null set satisfying that,

Let xˆ1 ∈ Bδ (x1 ) \ (D1 ∪ D2 ), then there exists a subsequence, denoted equal, such that un (xˆ1 ) → λ1 ∈ [−∞, +∞]. Consider now xˆ2 ∈ Bδ (x2 ) \ (D1 ∪ D2 ), then up to a subsequence, we can assume un (xˆ2 ) → λ2 ∈ [−∞, +∞]. So, successively, for xˆm ∈ Bδ (xm ) \ (D1 ∪ D2 ), there exists a subsequence, again denoted equal, such that un (xˆm ) → λm ∈ [−∞, +∞].

2.1 Nonlocal Isoperimetric Inequality

23

By (2.9) and (2.10), un (y) → λi

∀y ∈ (Br (xˆi ) ∩ ) \ Cxˆi .

Now, by (2.5),  = (Br (xˆ1 ) ∩ ) ∪ (∪m i=2 (Br (xˆ i ) ∩ )). Hence, since  is a bounded domain, there exists i2 ∈ {2, .., m} such that (Br (xˆ1 ) ∩ ) ∩ (Br (xˆi2 ) ∩ ) = ∅. Therefore, λ1 = λi2 . Let us call i1 = 1. Again, since  = (Br (xˆi1 ) ∩ ) ∪ (Br (xˆi1 ) ∩ ) ∪ (∪i∈{1,...,m}\{i1 ,i2 } (Br (xˆi ) ∩ )), and there exists i3 ∈ {1, . . . , m} \ {i1 , i2 } such that (Br (xˆi1 ) ∩ ) ∪ (Br (xˆi1 ) ∩ ) ∩ (Br (xˆi3 ) ∩ ) = ∅. Consequently, λi1 = λi2 = λi3 . Using the same argument, we get λ1 = λ2 = . . . = λm = λ. If |λ| = +∞, we have shown that |un (y)|q → +∞

for almost every y ∈ ,

which contradicts un Lq () = 1 for all n ∈ N. Hence, λ is finite. On the other hand, by (2.8), fn (xˆi ) → 0, i = 1, . . . , m. Hence, Fn (xˆ1 , .) → 0 in Lq (). Since un (xˆ1 ) → λ, from the above we conclude that un → λ

in Lq (Br (xˆi ) ∩ ).

24

2 Nonlocal Isoperimetric Inequality

Using again a compactness argument, we get un → λ in Lq (). By (2.7), λ = 0, so un → 0 in Lq (), which contradicts un Lq () = 1.

 

If we define 1 (u) = ||

 u(x)dx, 

the Poincaré inequality (2.3), for the particular case q = 1 and  = Bρ (x0 ), can be written as:  |u − (u) | ≤ T V J (u, Bρ (x0 )), β0 (J, Bρ (x0 )) (2.11) Bρ (x0 )

for all u ∈ L1 (Bρ (x0 )), where β0 (J, Bρ (x0 )) =

T V J (u, Bρ (x0 ))  Bρ (x0 ) u=0 |u(z)| dz

inf

u∈L1 (Bρ (x0 )),

Bρ (x0 )

We have that β0 (J, Bρ (x0 )) = β0 (J, Bρ ) does not depend on x0 , and it is easy to see that β0 (J, Bρ ) = β0 (J 1 , B1 ). ρ

(2.12)

Following the proof of [6, Theorem 3.46], we give the following relative isoperimetric inequality. Theorem 2.3 (Relative Isoperimetric Inequality) Assume that J is radially nonincreasing and satisfies condition (1.19). Let N ≥ 2 be an integer. For any set E of finite J -perimeter in RN , either E or RN \ E has finite Lebesgue measure and PJ (BρE ) ≤ PJ (E)

 for |BρE | = min |E|, |RN \ E| .

2.1 Nonlocal Isoperimetric Inequality

25

Proof Applying (2.11) to u = χ E , and having in mind (2.12), we get 2(χ E )Bρ (x)(1 − (χ E )Bρ (x) ) ≤

1 T V J (χ E , Bρ (x)). ωN ρ N β0 (J1/ρ , B1 )

Since (χ E )Bρ (x) ∈ [0, 1] and min{t, 1 − t} ≤ 2t (1 − t) for any t ∈ [0, 1], we obtain   min (χ E )Bρ (x), 1 − (χ E )Bρ (x) ≤

ωN

ρN β

1 T V J (χ E , Bρ (x)), 0 (J1/ρ , B1 )

(2.13)

1 PJ (E). 0 (J1/ρ , B1 )

(2.14)

which implies   min (χ E )Bρ (x) , 1 − (χ E )Bρ (x) ≤

ωN

ρN β

Let us see now that {ρ N β0 (J1/ρ , B1 ) : ρ > 0} is not bounded.

(2.15)

In fact, if there exists M > 0 such that ρ N β0 (J1/ρ , B1 ) ≤ M for all ρ > 0, we have inf 

 B1

u=0,

B1

1 |u|=1 2





J 1 (x − y) B1

B1

ρ

|u(y) − u(x)| dydx ≤ M(1/ρ)N−1 . 1/ρ

Then, we can find uρ such that 

 uρ = 0, B1

|uρ | = 1,

(2.16)

B1

and 1 2



 J 1 (x − y)

B1

B1

ρ

|uρ (y) − uρ (x)| dydx ≤ 2M(1/ρ)N−1 . 1/ρ

From (2.17), by Theorem 1.9, we get ρn → +∞ such that uρn → u

in L1 (B1 ),

u ∈ BV (B1 ),  |Du| = 0. B1

(2.17)

26

2 Nonlocal Isoperimetric Inequality

On the other hand, by (2.16), we have 

 u = 0 and

|u| = 1.

B1

B1

and we arrive to a contradiction. Therefore, (2.15) holds. Then, there exists ρ0 > 0, depending on PJ (E), such that 1 1 PJ (E) < . ωN ρ0 N β0 (J1/ρ0 , B1 ) 2 Hence by (2.14), (χ E )Bρ0 (x) ∈ (0, 1/2) ∪ (1/2, 1). By a continuity argument, either (χ E )Bρ0 (x) ∈ (0, 1/2) for any x ∈ RN or (χ E )Bρ0 (x) ∈ (1/2, 1) for any x ∈ RN . If the first possibility is true, by (2.13), we obtain |E ∩ Bρ0 (x)| ωN ρ0N

= (χ E )Bρ0 (x) ≤

1 ωN ρ0N β0 (J1/ρ0 , B1 )

T V J (χ E , Bρ0 (x)).

Hence, |E ∩ Bρ0 (x)| ≤

T V J (χ E , Bρ0 (x)) . β0 (J1/ρ0 , B1 )

(2.18)

For i = 1, 2, . . . , k, let Fi be a numerable family of disjoint balls of radius ρ0 , such that the union of the members of this family covers LN -almost all of RN . Then, by (2.18), we have |E| ≤

k  

|E ∩ B|

i=1 B∈Fi

≤

≤

1

k  

β0 (J1/ρ0 , B1 )

i=1 B∈Fi

1

k 

β0 (J1/ρ0 , B1 )

i=1

T V J (E, B)

T V J (E, RN ),

2.2 Nonlocal Coarea Formula

27

and so |E| ≤

k β0 (J1/ρ0 , B1 )

PJ (E).

If (χ E )Bρ0 (x) ∈ (1/2, 1) for any x ∈ RN , a symmetric argument shows that |RN \ E| can be estimated as above. This shows that either E or RN \ E has finite Lebesgue measure. Finally, the second part of the result is consequence of the Isoperimetric Inequality given in Theorem 2.1.   In Sect. 3.3, we will give an isoperimetric inequality involving the nonlocal curvature.

2.2

Nonlocal Coarea Formula

Similarly to the coarea formula (1.8) in the local case, we have the following nonlocal coarea formula. Theorem 2.4 (Coarea Formula) For any u ∈ L1 (RN ), let Et (u) = {x ∈ RN : u(x) > t}. Then,  T V J (u) =

+∞ −∞

PJ (Et (u)) dt.

Proof Since  u(x) = 0

∞

 χ Et (u) (x) dt −

0 −∞

(1 − χ Et (u) (x)) dt,

we have  u(y) − u(x) =

+∞

−∞

χ Et (u) (y) − χ Et (u) (x) dt.

Moreover, since u(y) ≥ u(x) implies χ Et (u) (y) ≥ χ Et (u) (x), we obtain that  |u(y) − u(x)| =

+∞

−∞

|χ Et (u) (y) − χ Et (u) (x)| dt.

(2.19)

28

2 Nonlocal Isoperimetric Inequality

Therefore, by Tonelli–Hobson’s Theorem, we get T V J (u) =

1 2

=

1 2 



 RN

RN

J (x − y)|u(y) − u(x)|dxdy 



 RN

RN

J (x − y)

+∞ −∞

 |χ Et (u) (y) − χ Et (u) (x)|dt dxdy

  +∞  1  χ χ = J (x − y)| Et (u) (y) − Et (u) (x)|dxdy dt 2 RN RN −∞  +∞ = PJ (Et (u))dt. −∞

 

3

Nonlocal Minimal Surfaces and Nonlocal Curvature

Recall that if a set E has minimal local perimeter in a bounded set , then it has zero mean curvature at each point of ∂E ∩  (see [51]), and the equation that says that the curvature is equal to zero is the Euler–Lagrange equation associated to the minimization of the perimeter of a set. In order to give a nonlocal version of this result we will also introduce the nonlocal concept of mean curvature.

3.1

Nonlocal Minimal Surfaces and Nonlocal Curvature

Let  be an open bounded subset of RN . We say that a measurable set E ⊂ RN is Jminimal in  if PJ (E, ) ≤ PJ (F, ) for any measurable set F such that F \  = E \ . Proposition 3.1 Let  be an open bounded subset of RN . Then, E ⊂ RN is J -minimal in  if and only if PJ (E) ≤ PJ (F ) for any measurable set F such that F \  = E \ . Proof If E \  = F \ , then RN \ ( ∪ E) = RN \ ( ∪ F ). Therefore, since PJ (A, ) = LJ (A, RN \ A) − LJ (A \ , RN \ (A ∪ )) we get the result.

© Springer Nature Switzerland AG 2019 J. M. Mazón et al., Nonlocal Perimeter, Curvature and Minimal Surfaces for Measurable Sets, Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-06243-9_3

 

29

30

3 Nonlocal Minimal Surfaces and Nonlocal Curvature

We now introduce the concept on nonlocal curvature. Definition 3.2 Let E ⊂ RN be measurable, with N ≥ 2. For a point x ∈ RN we define the J -mean curvature of ∂E at x as  J J (x − y)(χ E (y) − χ RN \E (y))dy. H∂E (x) = − RN

Some observations are in order. First of all, observe that J J H∂E (x) = −H∂(R N \E) (x),

and that J −1 ≤ H∂E (x) ≤ 1. J Note also that H∂E (x) makes perfect sense for every x ∈ RN , not necessary for points in ∂E. This fact will be used later in the book. J (x) Like the usual mean curvature, if ∂E is a smooth boundary, for x ∈ ∂E, H∂E measures in some average sense the deviation of ∂E from its tangent hyperplane at x. Let us point out finally that with our signs choice, the curvature of a ball is positive as it is commonly in geometry texts. Some authors define the curvature with the reverse sign, see, for example, [1, 29].

Definition 3.3 We say that ∂E is a J-minimal surface in a bounded open set  if the set ∂E ∩  satisfies the nonlocal minimal surface equation J (x) = 0 H∂E

for all x ∈ ∂E ∩  such that |E ∩ Bδ (x)| > 0 and |(RN \ E) ∩ Bδ (x)| > 0 for every small δ > 0. As examples of nonlocal minimal surfaces we have the following. By symmetry, any hyperplane {x ∈ RN : xN > 0} is a J -minimal surface in RN . Also, the classical cone in the plane, {(x, y) ∈ R2 : xy > 0}, is a J -minimal surface in R2 .

3.1 Nonlocal Minimal Surfaces and Nonlocal Curvature

31

Theorem 3.4 Let  be an open bounded subset of RN . If E ⊂ RN is a J -minimal set in , then ∂E is a J -minimal surface in . Proof Let E ⊂ RN be a J -minimal set in . Given x0 ∈ ∂E ∩ , there is δ > 0 such that Bδ (x0 ) ⊂ . Now consider Aδ = Bδ (x0 ) ∩ (RN \ E) and Fδ = Aδ ∪ E. Since, Fδ \  = E \ , we have PJ (E) ≤ PJ (Fδ ), that is, 

  E

RN \E



J (x − y)dydx ≤ Fδ

RN \Fδ

J (x − y)dydx.

Now, since Aδ ∩ E = ∅,  

 RN \E

E





J (x − y)dydx = Fδ

RN \E







J (x − y)dydx −

RN \E

Aδ

J (x − y)dydx.

Therefore 

 Aδ

RN \E

J (x − y)dydx ≥ Fδ



RN \E

Fδ



J (x − y)dy −

RN \Fδ

 J (x − y)dy dx



= 





Aδ

=

 

J (x − y)dydx + Aδ

Aδ

J (x − y)dy dx

J (x − y)dydx. E

Aδ

Hence 







J (x − y)dydx ≤ Aδ

Aδ

Aδ

 J (x − y)dy dx.

 RN \E

J (x − y)dy − E

32

3 Nonlocal Minimal Surfaces and Nonlocal Curvature

Now, since x0 ∈ ∂E, we have |Aδ | > 0. Then, dividing by |Aδ |, using that  x →



 RN \E

J (x − y)dy −

J (x − y)dy E

is continuous, and letting δ → 0, we conclude that 



 RN \E

J (x0 − y)dy −

J (x0 − y)dy ≥ 0. E

With a similar procedure, but taking now A˜ δ = Bδ (x0 ) ∩ E and F˜δ = E \ A˜ δ , we arrive to 

 A˜ δ





RN \E

J (x − y)dy −

J (x − y)dy dx ≤ − E





A˜ δ

A˜ δ

J (x − y)dydx.

Then, dividing by |A˜ δ | > 0 and letting δ → 0, we get 



 RN \E

J (x0 − y)dy −

J (x0 − y)dy ≤ 0. E

Therefore, as we wanted to show,  J H∂E (x0 ) = −

RN

J (x0 − y)(χ E (y) − χ RN \E (y))dy = 0.  

Remark 3.5 Let us point out that a similar result to Theorem 3.4 has been obtained in 1 , where the nonlocal curvature at x ∈ ∂E, for E with [1, 29] for the singular kernel |x|N+s smooth boundary, is defined (formally) as  s (x) H∂E

=−

RN

χ E (y) − χ RN \E (y) |x − y|N+s

dy.

Moreover in this case the following rescaling formula holds for smooth sets: s lim (1 − s)H∂E (x) = (N − 1)ωN−1 H∂E (x),

s→1

where H∂E (x) is the (local) mean curvature of ∂E at x, see [1].

3.2 Rescaling the Nonlocal Curvature

3.2

33

Rescaling the Nonlocal Curvature

In this section we will give a convergence result of the nonlocal J -curvature under a rescale of the kernel. First of all we present two examples for illustration. Example 3.6 1. If J (x) =

1 χ |B1 | B1 ,

then for any x ∈ Br (x0 ), we have 

J H∂B (x) = − r (x0 )

=−

1 |B1 |

RN



B1 (x)

J (x − y)(χ Br (x0 ) (y) − χ RN \Br (x0 ) (y))dy

(χ Br (x0 ) (y) − χ RN \Br (x0 ) (y))dy

=−

 1  |B1 (x) ∩ Br (x0 )| − |B1 (x) ∩ (RN \ Br (x0 ))| |B1 |

=−

1 (2|B1 (x) ∩ Br (x0 )| − |B1 (x)|) |B1 |

=−

2|B1 (x) ∩ Br (x0 )| + 1. |B1 |

And for the rescaled kernel, J H∂B (x) = − r (x0 )

1 N



 RN

=−

J

x−y 

 (χ Br (x0 ) (y) − χ RN \Br (x0 ) (y))dy

2|B (x) ∩ Br (x0 )| + 1. |B |

Now, for N = 2, a simple calculus gives, for x ∈ ∂Br (x0 ) and  small,  |B (x) ∩ Br (x0 )| =  2

 − 2r



  2    π + 1− + arcsin − 2r 2r 2

⎡ 2r 2 −  2 π 1− +r 2 ⎣ − 2 2r 2



2r 2 −  2 2r 2

2



⎤  2r 2 −  2 ⎦ . − arcsin 2r 2 

34

3 Nonlocal Minimal Surfaces and Nonlocal Curvature

Hence      2 2  = 1− + arcsin πr 2r π 2r ⎡ ⎤  2  2 2  2r 2 ⎣ π 2r 2 −  2 2r −  2 2r −  2 ⎦ − 2 − 1− − arcsin . π 2 2r 2 2r 2 2r 2

J H∂B (x) r (x0 )

Now, 2

CJ =  R2

J (z)|z2|dz

=

1 |B1 |

2



|z2 |dz

=

3π . 2

B1

Therefore,      2 3 CJ J 3 H∂Br (x0 ) (x) = 1− + arcsin  2r 2r  2r ⎡ ⎤  2  2 2  2r −  2 2r −  2 2r 2 −  2 3r 2 π ⎦. 1− − arcsin − 3 ⎣ − 2  2r 2 2r 2 2r 2 Then, since   3 3 arcsin = →0  2r 2r lim

and ⎡ 2r 2 −  2 3r 2 ⎣ π lim 3 − →0  2 2r 2

 1−

2r 2 −  2 2r 2

2

⎤  2r 2 −  2 ⎦ 2 − arcsin = , 2r 2 r 

we get lim

→0

3 2 1 CJ J 3 H + − = = H∂Br (x0 ) (x) =  ∂Br (x0 ) 2r 2r r r

2. Consider now E as the square E = {(x, y) ∈ R2 : (x, y)∞ ≤ 1} and the same kernel as above. Then, for 0 <  < 1 we have  3π 3  CJ J H∂E (1, 1) = − 3 |E ∩ B (1, 1)| − |(R2 \ E ∩ B (1, 1)| = .  2 4

3.2 Rescaling the Nonlocal Curvature

35

Therefore, CJ J H∂E (1, 1) = +∞. →0  lim

Theorem 3.7 Let N ≥ 2. Assume that J has compact support Br , is continuous almost everywhere, and bounded. Let E ⊂ RN be a smooth set such that ∂E is of class C 2 . Then, for every x ∈ ∂E, we have J (x) = (N − 1)H∂E (x), lim CJ H∂E

(3.1)

↓0

where H∂E (x) is the (local) mean curvature of ∂E at x. Proof It is well-known that curvature may be easily computed in normal coordinates. Namely, suppose ∂E is described as a graph in normal coordinates, meaning that, in an open ball Br0 , ∂E coincides with the graph of a C 2 function ϕ : Br0 ∩ RN−1 → R with ϕ(0) = 0 and ∇ϕ(0) = 0 such that E ∩ Br0 = {(y1, . . . , yN ) : yN < ϕ(y1 , . . . , yN−1 )}. Since D 2 ϕ(0) is a real symmetric matrix, it will admit N − 1 real eigenvalues λ1 , . . . , λN−1 . Minus the arithmetic mean of the eigenvalues is called the mean curvature and we denote it by H∂E (0), namely, H∂E (0) = −

λ1 + · · · + λN−1 . N −1

We will assume that the coordinate axes e1 , . . . , eN−1 are the eigenvectors associated with the eigenvalues of D 2 ϕ(0), λ1 , . . . , λN−1 . We can also assume that r0 < r. Then, for  small enough, J (0) = −CJ CJ H∂E

CJ = − N  =−

CJ N

 RN

J (y)(χ E (y) − χ RN \E (y))dy



J Br

y  

(χ E (y) − χ RN \E (y))dy

 {yN ϕ(y1 ,...,yN−1 )}∩Br

J

dy    y dy . 

36

3 Nonlocal Minimal Surfaces and Nonlocal Curvature

Hence, changing variables as z = y , we get J CJ H∂E (0) = −CJ

 {zN < 1 ϕ(z1 ,...,zN−1 )}∩Br

J (z)dz



−

{zN > 1 ϕ(z1 ,...,zN−1 )}∩Br

 J (z)dz .

And, since J is radially symmetric,  J (0) CJ H∂E

= −CJ

 RN−1 ∩Br

1  ϕ(z1 ,...,zN−1 )

− 1 ϕ(z1 ,...,zN−1 )

J (z)dz.

Now, by Taylor’s expansion, we have ϕ(z1 , . . . , zN−1 ) =

1 2 D ϕ(0)(z1, . . . , zN−1 ) + O( 3 ) 2

N−1 1 = λi  2 zi2 + O( 3 ). 2 i=1

Therefore, J lim CJ H∂E (0) = − lim CJ

→0

→0



 RN−1 ∩Br

1  ϕ(z1 ,...,zN−1 )

− 1 ϕ(z1 ,...,zN−1 )

J (z)dz.

  1 #N−1 2  2 1  2 i=1 λi zi +O( )  #  J (z)dzN dz1 dz2 . . . dzN−1 . 2 2 RN−1 ∩Br  −  12 N−1 i=1 λi zi +O( )

 = − lim CJ →0

(3.2)

Then, since J is continuous almost everywhere and it is radially symmetric, we have zn → J (z1 , z2 , . . . , zN−1 , zN )

is continuous at 0

for almost every (z1 , z2 , . . . , zN−1 ) in RN−1 . Therefore 1 lim →0 



# 2 2  12 N−1 i=1 λi zi +O( )  #  J (z)dzN 2 2 −  12 N−1 i=1 λi zi +O( )

= J (z1 , z2 , . . . zN−1 , 0)

(3.3)

3.2 Rescaling the Nonlocal Curvature

37

for almost every (z1 , z2 , . . . , zN−1 ) in RN−1 . Now since J is bounded in Br , we can pass to the limit in (3.2) to get J lim CJ H∂E (0) →0

N−1 

= −CJ

 λi

i=1

RN−1

J (z1 , . . . , zN−1 , 0)zi2 dz1 . . . dzN−1 ,

that is, on account of the symmetry of J , 

J lim CJ H∂E (0) = −CJ

→0

RN−1

2 J (z1 , . . . , zN−1 , 0)zN−1 dz1 . . . dzN−1

N−1 

λi .

(3.4)

i=1

Now, if we make the change of variables zi = zi , i = 1, . . . , N − 2, zN−1 = r cos θ , and zN = r sin θ , we have, writing zˆ = (z1 , z2 , . . . , zN−2 ), 

 RN

J (z)|zN |dz = =4



+∞  2π

RN−2

0





RN−2

 =2

RN

J (ˆz, r cos θ, r sin θ )r 2 | sin θ |dθ drd zˆ

0 +∞

J (ˆz, r, 0)r 2 drd zˆ

0

J (ˆz, r, 0)r 2 d zˆ dr



=2

RN−1

2 J (z1 , . . . , zN−1 , 0)zN−1 dz1 . . . dzN−1 .

Hence  CJ

RN−1

2 J (z1 , . . . , zN−1 , 0)zN−1 dz1 . . . dzN−1 = 1.

Therefore, from (3.4), we get J lim CJ H∂E (0) = (N − 1)H∂E (0).

→0

  Remark 3.8 Observe, for example, that if J is radially non-increasing, then it is continuous almost everywhere. Let us explain why (3.3) is true. Take x as a point where J is continuous. Since J is radially symmetric, we have that J is also continuous at (x, ˆ 0) ∈ RN N−1 with xˆ ∈ R such that |(x, ˆ 0)| = |x|. Now for ˆ 0)} A = {xˆ ∈ RN−1 : J is continuous at (x, we have that HN−1 (RN−1 \ A) = 0 since, again, J is radially symmetric.

38

3 Nonlocal Minimal Surfaces and Nonlocal Curvature

In the next example we will see that the assumption “∂E is of class C 2 ” in the above result is necessary. Example 3.9 For N = 2. Assume that J (x) = defined by

ϕ(x) =

⎧ ⎨ x2 ⎩

1 χ |B1 | B1 .

Let ϕ : R → R be the function

if x ≥ 0,

−x 2 if x < 0,

and consider the set E = {(x, y) ∈ R2 : y < ϕ(x)}. J Then by symmetry we have H∂E (0, 0) = 0, and consequently J lim CJ H∂E (0, 0) = 0. ↓0

Now, H∂E (0, 0) = 2 (here the curvature is understood as one over the largest radius of a tangent ball). Therefore, (3.1) is not true in this case.

3.3

Isoperimetric Inequalities and Curvature

Assume in this section that J is radially non-increasing. Lemma 3.10 The following relation holds true: d J PJ (Br ) = H∂B (y)Per(Br ), r dr

(3.5)

with y ∈ ∂Br . Proof It is enough to calculate the derivative from the right. So, for h > 0, PJ (Br+h ) − PJ (Br ) = |Br+h | − |Br |   −

 Br+h ×Br+h

J (x − y)dydx −

Br ×Br

 J (x − y)dydx .

3.3 Isoperimetric Inequalities and Curvature

39

Hence, PJ (Br+h ) − PJ (Br ) = |Br+h | − |Br |   −

 



Br+h \Br × Br+h \Br

J (x − y)dydx − 2





Br+h \Br ×Br

 J (x − y)dydx .

Now, we have lim

h→0+

|Br+h | − |Br | = Per(Br ), h

and lim

h→0+

1 h

 



Br+h \Br × Br+h \Br

J (x − y)dydx = 0.

Moreover 1 lim h→0+ h

 



Br+h \Br ×Br

J (x − y)dydx = Per(Br )

J (y) ˜ 1 − H∂B r

2

,

with y˜ ∈ ∂Br . In fact, making a spherical change of coordinates y = g(ρ, σ ) and having in mind the radial symmetry of the curvature, we have  1 lim J (x − y)dydx  h→0+ h Br+h \Br ×Br  r+h   1 = lim ρ N−1 J (x − g(ρ, σ ))dρdσ dx h→0+ h Br S N−1 r   = r N−1 J (x − g(r, σ ))dσ dx Br

= r N−1

S N−1



S N−1

 =r

N−1

=r

N−1

J (x − g(r, σ ))dxdσ Br J 1 − H∂B (g(r, σ )) r

2

S N−1



J 1 − H∂B (y) ˜ r

2

S N−1

= Per(Br ) for y˜ ∈ ∂Br .



J (y) 1 − H∂B ˜ r

2

,

dσ

dσ

40

3 Nonlocal Minimal Surfaces and Nonlocal Curvature

Therefore, we get lim

h→0+

PJ (Br+h ) − PJ (Br ) J = Per(Br )H∂B (y), r h

with y ∈ ∂Br .

 

Corollary 3.11  PJ (Br ) =

J H∂B (x)dx. x

Br

(3.6)

Proof By (3.5), we have d J PJ (Br ) = H∂B (r, 0)Per(Br ). r dr

(3.7)

Integrating in (3.7) we get 

r

PJ (Br ) = NωN 0

 = Br

J ρ N−1 H∂B (ρ, 0)dρ ρ

J H∂B (x)dx. x

  Let E be a finite Lebesgue set. In the local case, the isoperimetric inequality says that Per(BρE ) ≤ Per(E)

for |BρE | = |E|,

(3.8)

with equality holding if and only if E is a ball. Since  ρE =

|E| ωN

1

N

and 1/N

Per(BρE ) = NωN |E|

N−1 N

,

the above inequality can be written as 1/N

NωN |E|

N−1 N

≤ Per(E).

(3.9)

3.3 Isoperimetric Inequalities and Curvature

41

An isoperimetric type inequality for s-perimeters also holds (see, for example, [47]), and it states that Pers (BρE ) ≤ Pers (E)

for |BρE | = |E|,

(3.10)

with equality holding if and only if E is a ball. Since Pers (Br ) = r N−s Pers (B1 ),

(3.11)

the inequality (3.10) can be written as 

|E| Pers (B1 ) ωN

 N−s N

≤ Pers (E).

(3.12)

In our case, we can also obtain a similar inequality to (3.9) and (3.12). In fact, by (3.6)  PJ (Br ) = Br

J H∂B (x)dx. x

Therefore, for |BρE | = |E|,  PJ (BρE ) = NωN

(|E|/ωN )1/N 0

J r N−1 H∂B (r, 0)dr, r

and we can write the nonlocal isoperimetric inequality (2.2), assuming J is radially nonincreasing, as ψJ,N (|E|) ≤ PJ (E),

(3.13)

being ψJ,N the strictly increasing function  ψJ,N (s) = NωN

(s/ωN )1/N 0

J r N−1 H∂B (r, 0)dr. r

(3.14)

But we can rewrite the above three isoperimetric inequalities with another common formulation. Assume J is radially non-increasing. First observe that, thanks to (3.6), we can write (2.2) as 

1 |BρE |



 BρE

J H∂B (x)dx |E| ≤ PJ (E). x

(3.15)

42

3 Nonlocal Minimal Surfaces and Nonlocal Curvature

And this expression is the nonlocal version of the classical isoperimetric inequality (3.9), since it can be written as    1 N −1 dx |E| ≤ Per(E). (3.16) |BρE | BρE x Observe that, if we consider the rescaled kernel J , we have 

1 |BρE |



 BρE

J H∂B (x)dx x

|E| ≤ PJ (E).

(3.17)

Now, assuming the assumptions of Theorems 1.11 and 3.7, we have lim CJ PJ (E) = Per(E) ↓0

and J lim CJ H∂B (x) = (N − 1)H∂Bx (x) = x ↓0

N −1 . x

Therefore, taking limits in (3.17) as  → 0 and applying the dominate convergence theorem, we obtain the isoperimetric inequality (3.16). For the fractional kernel it also holds that d s Pers (Br ) = H∂B (r, 0)Per(Br ). r dr

(3.18)

One can find a proof in [32]. Nevertheless, having in mind (3.11), a simple calculation gives d N −s 1 Pers (Br ) = Pers (B1 )Per(Br ), dr NωN r s and, since s H∂B (r, 0) = r

1 s H (1, 0), r s ∂B1

one gets easily that d N − s Pers (B1 ) s Pers (Br ) = H∂Br (r, 0)Per(Br ). s dr NωN H∂B (1, 0) 1

3.3 Isoperimetric Inequalities and Curvature

43

And the following relation holds true for the fractional perimeter and fractional curvature of the unit ball: N − s Pers (B1 ) = 1, s (1, 0) NωN H∂B 1 or Pers (B1 ) N = H s (1, 0). |B1 | N − s ∂B1 s (1, 0) and Per (B ) are dimensional constants depending on N and s. The curvatures H∂B s 1 1 In [47] (see also [45]), one can find different descriptions of Pers (B1 ). Integrating in (3.18),



r

Pers (Br ) = NωN 0

 = Br

s τ N−1 H∂B (τ, 0)dτ τ

s H∂B (x)dx. x

Hence, the fractional isoperimetric inequality (3.10) can also be written as 

1 |BρE |



 BρE

s H∂B (x)dx x

|E| ≤ Pers (E).

(3.19)

Therefore, all the three isoperimetric inequalities have a common formulation: (3.15), (3.16), and (3.19). Observe that an explicit power of |E| appears in (3.12), like in (3.9). Nevertheless, an explicit calculus of  BρE

J H∂B (x)dx x

(3.20)

as a function of ρE is not clear even for the uniform J on a ball; observe that, for the J -curvature, J1

J H∂B (r, 0) = H∂Br 1 (1, 0), r

and hence (3.20) cannot be written in a so clean way.

44

3 Nonlocal Minimal Surfaces and Nonlocal Curvature

Remark 3.12 In the local case it is well-known that in (3.8) we have equality if and only if E is a ball. The same happens for the fractional isoperimetric inequality (3.12). In the case of radially non-increasing J having compact support BR , as a consequence of [24, Theorem 1], the equality in (3.13) holds if and only if D is a ball of radius r, when r > R2 . In Example 5.8 we show that the J -isoperimetric inequality is an equality for sets that are not balls.

4

Nonlocal Operators

4.1

A Characterization of the Nonlocal Perimeter

Following Gilboa–Osher [50] (see also [14]), we introduce the following nonlocal operators. For a function u : RN → R, we define its nonlocal gradient as the function ∇J u : RN × RN → R defined by: (∇J u)(x, y) = J (x − y)(u(y) − u(x)),

x, y ∈ RN .

And for a function z : RN × RN → R, its nonlocal divergence divJ z : RN → R is defined as:  1 (z(x, y) − z(y, x))J (x − y)dy. (divJ z)(x) = 2 RN Remark 4.1 (An Interpretation of divJ z) Suppose that RN represents a continuous network, with a source uniformly distributed in  ⊂ RN and being RN \  a sink. If the transportation activity is described by z, in such a way that at each point x ∈ , z(x, y)J (x − y) is the incoming quantity of flow from y ∈ RN and z(y, x)J (x − y) is the outcoming flow at y ∈ RN , then  2(divJ z)(x) =

RN

(z(x, y) − z(y, x))J (x − y)dy,

represents the total flow at x.

© Springer Nature Switzerland AG 2019 J. M. Mazón et al., Nonlocal Perimeter, Curvature and Minimal Surfaces for Measurable Sets, Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-06243-9_4

45

46

4 Nonlocal Operators

For p ≥ 1, we define the space:

 p XJ (RN ) = z ∈ L∞ (RN × RN ) : divJ z ∈ Lp (RN ) . 

Observe that XJ∞ (RN ) = L∞ (RN ×RN ). For u ∈ BVJ (RN )∩Lp (RN ) and z ∈ XJ (RN ), 1 ≤ p ≤ ∞, we have the following Green formula:  RN

u(x)(divJ z)(x)dx = −

1 2

p

 RN ×RN

(∇J u)(x, y)z(x, y)dxdy.

(4.1)

In the next result, we characterize T V J and the nonlocal perimeter using the nonlocal divergence operator. Let us denote by sign0 (r) the usual sign function and by sign(r) the multivalued sign function for which sign(0) = [−1, 1]. 

Proposition 4.2 Let 1 ≤ p ≤ ∞. For u ∈ BVJ (RN ) ∩ Lp (RN ), we have  T V J (u) = sup

RN

p u(x)(divJ z)(x)dx : z ∈ XJ (RN ), z∞ ≤ 1 .

(4.2)

If u ∈ BVJ (RN ), then

 sup

RN

u(x)(divJ z)(x)dx : z ∈

p XJ (RN ),

z∞ ≤ 1 = +∞.

In particular, for any measurable set E ⊂ RN , we have  PJ (E) = sup E

(divJ z)(x)dx : z ∈ XJ1 (RN ), z∞ ≤ 1 .



p

Proof Let u ∈ L1 (RN ) ∩ Lp (RN ). Given z ∈ XJ (RN ) with z∞ ≤ 1, applying Green formula (4.1), we have  RN

u(x)(divJ z)(x)dx =

1 2

1 ≤ 2

 

RN ×RN

RN ×RN

(∇J u)(x, y)z(x, y)dxdy |u(y) − u(x)|J (x − y)dxdy

= T V J (u). Therefore,  sup

RN

p u(x)(divJ z)(x)dx : z ∈ XJ (RN ), z∞ ≤ 1 ≤ T V J (u).

4.2 Nonlocal 1-Laplacian

47

On the other hand, if we define zn (x, y) = sign0 (u(y) − u(x))χ Bn (0,0)(x, y), p

then zn ∈ XJ (RN ) with zn ∞ ≤ 1 and 1 T V J (u) = 2

 RN ×RN

1 n→∞ 2

|u(y) − u(x)|J (x − y)dxdy



|u(y) − u(x)|J (x − y)dxdy

= lim

1 n→∞ 2  = lim = lim



Bn (0,0)

RN ×RN

n→∞ RN



≤ sup

RN

(∇J u)(x, y)zn (x, y)dxdy

u(x)(divJ zn )(x)dx

p u(x)(divJ z)(x)dx : z ∈ XJ (RN ), z∞ ≤ 1 .  

Corollary 4.3 For u ∈ L1 (RN ),  T V J (u) = sup

∞

RN

u(x)(divJ z)(x)dx : z ∈ L (R × R ), z∞ ≤ 1 . N

N

In particular, for any measurable set E ⊂ RN with finite measure,  PJ (E) = sup E

4.2

(divJ z)(x)dx : z ∈ L∞ (RN × RN ), z∞ ≤ 1 .

(4.3)

Nonlocal 1-Laplacian

Nonlocal evolution equations of the form:  ut (x, t) =

RN

J (x − y)|u(y, t) − u(x, t)|p−2 (u(y, t) − u(x, t))dy,

p≥1

and variations of it, have been recently widely used to model diffusion processes (see [12] and the references therein).

48

4 Nonlocal Operators

For p = 1, using the nonlocal calculus from Sect. 4.1, we have formally     1 ∇J u ∇J u ∇J u divJ (x) = (x, y) − (y, x) dy J (x − y) |∇J u| 2 RN |∇J u| |∇J u|    1 (u(y) − u(x))J (x − y) − (u(x) − u(y))J (x − y) dy = J (x − y) 2 RN J (x − y)|u(y) − u(x)|  u(y) − u(x) = dy, J (x − y) N |u(y) − u(x)| R 

that we have called nonlocal 1-Laplacian operator in [10] and [11]:  J1 u(x) =

RN

J (x − y)

u(y) − u(x) dy |u(y) − u(x)|

for u ∈ L1 (RN ), x ∈ RN .

Also, if p > 1, we have  divJ 1/p |∇J 1/p u|p−2 ∇J 1/p u (x) =

 RN

J (x − y)|u(y) − u(x)|p−2 (u(y) − u(x))dy,

that we have called nonlocal p-Laplacian operator. In [10] and [11], we have studied these nonlocal operators with different boundary conditions, and from these works we take the following definition. Definition 4.4 Given v ∈ L1 (RN ), we say that u ∈ L1 (RN ) is a solution of −J1 u  v

in RN

if there exists g ∈ L∞ (RN × RN ) with g∞ ≤ 1 verifying g(x, y) = −g(y, x) for (x, y) a.e in RN × RN , J (x − y)g(x, y) ∈ J (x − y)sign(u(y) − u(x)) a.e (x, y) ∈ RN × RN , and  −

RN

J (x − y)g(x, y) dy = v(x)

a.e x ∈ RN .

We point out that, in general, the operator J1 is multivalued (see Remark 5.11). In order to study the Cauchy problem associated with the nonlocal 1-Laplacian, we will see that we can consider it as the gradient flow in L2 (RN ) of the functional T V J . For that,

4.2 Nonlocal 1-Laplacian

49

we consider now the functional: F J : L2 (RN ) →] − ∞, +∞] defined by:

F J (u) =

⎧ 2 N N ⎪ ⎨ T V J (u) if u ∈ L (R ) ∩ BVJ (R ), ⎪ ⎩

+∞

if u ∈ L2 (RN ) \ BVJ (RN ),

which is convex and lower semicontinuous. Following the method used in [9] to get the characterization of the subdifferential of the total variation, we get the following characterization of the subdifferential of the functional F J . $ : L2 (RN ) → [0, ∞] as: Given a functional  : L2 (RN ) → [0, ∞], we define 

$(v) = sup 

with the convention that

0 0

=

0 ∞

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

v(x)w(x)dx

RN

: w ∈ L2 (RN )

(w)

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

$2 ≤  $1 . = 0. Obviously, if 1 ≤ 2 , then 

Theorem 4.5 Let u ∈ L1 (RN ) ∩ L2 (RN ) and v ∈ L2 (RN ). The following assertions are equivalent: (i) v ∈ ∂F J (u); (ii) There exists z ∈ XJ2 (RN ), z∞ ≤ 1, such that v = −divJ z and  RN

u(x)v(x)dx = F J (u);

(iii) There exists z ∈ XJ2 (RN ), z∞ ≤ 1, such that (4.4) holds and 1 F (u) = 2



J

RN ×RN

∇J u(x, y)z(x, y)dxdy;

(4.4)

50

4 Nonlocal Operators

(iv) −J1 u  v in RN ; (v) There exists g ∈ L∞ (RN × RN ) antisymmetric with g∞ ≤ 1 such that  −

RN

J (x − y)g(x, y) dy = v(x)

a.e x ∈ RN ,

(4.5)

and  −

 RN

RN

J (x − y)g(x, y)dy u(x)dx = F J (u).

Proof Since F J is convex, lower semicontinuous and positive homogeneous of degree 1, by Andreu et al. [9, Theorem 1.8], we have  (J (v) ≤ 1, ∂F J (u) = v ∈ L2 (RN ) : F

RN

u(x)v(x)dx = F J (u) .

(4.6)

We define, for v ∈ L2 (RN ),

 (v) = inf z∞ : z ∈ XJ2 (RN ), v = −divJ z .

(4.7)

Observe that  is convex, lower semicontinuous and positive homogeneous of degree 1. Moreover, it is easy to see that, if (v) < ∞, the infimum in (4.7) is attained, i.e. there exists some z ∈ XJ2 (RN ), v = −divJ z and (v) = z∞ . Let us see that (J . =F (J (v) ≤ (v). Thus, we may assume that (v) < ∞. Let If (v) = ∞, then we have F ∞ N N z ∈ L (R × R ) such that v = −divJ z. Then, for w ∈ L2 (RN ), we have  RN

w(x)v(x)dx =

1 2

 RN ×RN

(∇J w)(x, y)z(x, y)dxdy ≤ z∞ F J (w).

(J (v) ≤ z∞ . Now, taking infimum in z, we get Taking supremum in w, we obtain that F ( J F (v) ≤ (v). To prove the opposite inequality, let us denote D = {divJ z : z ∈ XJ2 (RN )}.

4.2 Nonlocal 1-Laplacian

51

Then, by (4.2), we have, for v ∈ L2 (RN ),

$ (v) = sup 

≥ sup

= sup

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎨

w(x)v(x)dx (w)

RN

⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

RN

w(x)v(x)dx (w)

RN

: w ∈ L2 (RN )

: w∈D

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

divJ z(x)v(x)dx : z ∈ XJ2 (RN )

z∞

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

= F J (v). $ Observe that the last term is equal to +∞ if v ∈ L2 (RN ) \ BVJ (RN ). Thus, F J ≤ , $ ( J $ ≤ F . Therefore, which implies by Andreu et al. [9, Proposition 1.6], that  =  (J , and consequently from (4.6), we get  =F  ∂F J (u) = v ∈ L2 (RN ) : (v) ≤ 1,

RN

u(x)v(x)dx = F1J (u)

= v ∈ L2 (RN ) : ∃z ∈ XJ2 (RN ), v = −divJ z, z∞ ≤ 1,



u(x)v(x)dx = F (u) , J

RN

from where it follows the equivalence between (i) and (ii). To get the equivalence between (ii) and (iii), we only need to apply the Green formula (4.1). By the antisymmetry of g, it is easy to see that (iv) and (v) are equivalent. On the other hand, to see that (iii) implies (v), it is enough to take g(x, y) = 12 (z(x, y) − z(y, x)). Finally, to see that (v) implies (ii), it is enough to take z(x, y) = g(x, y) (observe that,   from (4.5), −divJ (g) = v, so g ∈ XJ2 (RN )). Remark 4.6 Observe that if v ∈ ∂F J (u), then any function z satisfying the characterization of Theorem 4.5 also satisfies J (x − y)z(x, y) ∈ J (x − y)sign((u(y) − u(x)) a.e. in RN × RN . But, moreover, we can choose one being antisymmetric.

52

4 Nonlocal Operators

By Theorem 4.5 and following [12, Theorem 7.5], it is easy to prove the following result. Lemma 4.7 ∂F J is an m-completely accretive operator in L2 (RN ). As consequence of Theorem 4.5 and Lemma 4.7, we can give the following existence and uniqueness result for the Cauchy problem: ⎧ ⎨ ut − J1 u  0 ⎩

in (0, T ) × RN

u(0, x) = u0 (x) x ∈ RN .

(4.8)

Theorem 4.8 For every u0 ∈ L2 (RN ), there exists a unique solution of the Cauchy problem (4.8) in (0, T ) for any T > 0, in the following sense: the solution u ∈ W 1,1 (0, T ; L2 (RN )), u(0, ·) = u0 , and for almost all t ∈ (0, T ) ut (·, t) − J1 u(t)  0. Moreover, we have the following contraction principle in any Lq (RN )-space with 1 ≤ q ≤ ∞: u(t) − v(t)q ≤ u0 − v0 q

∀0 < t < T,

for any pair of solutions, u, v, of problem (4.8) with initial data u0 , v0 , respectively. Proof By the theory of maximal monotone operators (see [19]), and having in mind the characterization of the subdifferential of F J , for every u0 ∈ L2 (), there exists a unique strong solution of the abstract Cauchy problem: ⎧ ⎨ u (t) + ∂F J (u(t)  0, ⎩

t ∈ (0, T ),

u(0) = u0 ,

that is exactly the concept of solution given. The contraction principle is consequence of being the operator completely accretive (see [16]).  

5

Nonlocal Cheeger and Calibrable Sets

5.1

Nonlocal Cheeger Sets

Given a non-null, measurable and bounded set  ⊂ RN , we define its J-Cheeger constant by:

PJ (E) : E ⊂ , E measurable with |E| > 0 . = inf |E|

hJ1 ()

As a consequence of (1.14), we have that

PJ (E) |E|

(5.1)

≤ 1. Hence,

hJ1 () ≤ 1. A measurable set E ⊂  achieving the infimum in (5.1) is said to be a J-Cheeger set of . To get a lower bound for hJ1 (), we recall the following Poincaré-type inequality given in [12, Proposition 6.25]. Proposition 5.1 Suppose J is continuous. Given  a bounded domain in RN , p ≥ 1 and ψ ∈ Lp (J \ ), there exists λ(J, , p) > 0 such that 

  |u(x)| dx ≤

λ(J, , p) 

 J (x − y)|uψ (y) − u(x)| dy dx +

p

p

 J

J \

|ψ(y)|p dy

for all u ∈ Lp ().

© Springer Nature Switzerland AG 2019 J. M. Mazón et al., Nonlocal Perimeter, Curvature and Minimal Surfaces for Measurable Sets, Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-06243-9_5

53

54

5 Nonlocal Cheeger and Calibrable Sets

Proof First, let us assume that there exist r, α > 0 such that J (x) ≥ α in Br . Let O0 = {x ∈ J \  : d(x, ) ≤ r/2}, O1 = {x ∈  : d(x, O0 ) ≤ r/2},

 j −1 Oj = x ∈  \ ∪k=1 Ok : d(x, Oj −1 ) ≤ r/2 ,

j = 2, 3, . . .

Observe that we can cover  by a finite number of non-null sets {Oj }ljr=1 . Now, 

  J (x − y)|uψ (y) − u(x)|p dy dx ≥



Oj

 J

Oj−1

J (x − y)|uψ (y) − u(x)|p dy dx,

j = 1, . . . , lr , and  Oj

 Oj−1

J (x − y)|uψ (y) − u(x)|p dy dx ≥

  1 J (x − y)|u(x)|p dy dx 2p Oj Oj−1   J (x − y)|uψ (y)|p dy dx − 

1 = p 2 Oj

Oj

 Oj−1

Oj−1

J (x − y) dy

|u(x)|p dx



 −



Oj−1

 Oj

J (x − y) dx

|uψ (y)|p dy

  1 ≥ p min J (x − y)dy |u(x)|p dx 2 x∈Oj Oj−1 Oj  − |uψ (y)|p dy, Oj−1

since

 RN

J (x)dx = 1. Hence,

   J



 J (x − y)|uψ (y) − u(x)|p dy dx ≥ αj

Oj

|u(x)|p dx −

Oj−1

|uψ (y)|p dy,

5.1 Nonlocal Cheeger Sets

55

where  1 αj = p min J (x − y)dy > 0. 2 x∈Oj Oj−1 Therefore, since uψ (y) = ψ(y) if y ∈ O0 , uψ (y) = u(y) if y ∈ Oj , j = 1, . . . , lr , j Oj ∩ Oi = ∅, for all i = j and | \ ∪jr=1 Oj | = 0, it is easy to see that there exists ˆ λˆ = λ(J, , p) > 0 such that 

|u|p ≤ λˆ 

 

J (x − y)|uψ (y) − u(x)|p dy dx + λˆ

 J

 O0

|ψ|p .

The proof is finished by taking λ(J, , p) = λˆ −1 . In the general case, we have that there exist a ≥ 0 and r, α > 0 such that J (x) ≥ α in the annulus A(0, a, r).

(5.2)

In this case, we proceed as before with the same choice of the sets Oj for j ≥ 0 and

 j −1 O−j = x ∈ J \ ( ∪ ∪k=0 O−k ) : d(x, O−j +1 ) ≤ r/2 ,

j = 1, 2, 3, . . .

Observe that for each Oj , j ≥ 1, there exists Oj e with j e < j and such that |(x + A(0, a, r)) ∩ Oj e | > 0 ∀x ∈ Oj .

(5.3)

With this choice of Oj and taking into account (5.2) and (5.3), as before, we obtain  

 J (x − y)|uψ (y) − u(x)|p dy dx ≥

 J

Oj



 ≥ αj



|u(x)| dx −

Oj e

p

Oj

Oj e

J (x − y)|uψ (y) − u(x)|p dy dx

|uψ (y)|p dy,

j = 1, . . . , lr , where αj = And, we conclude as before.

 1 min J (x − y)dy > 0. 2p x∈Oj Oj e  

56

5 Nonlocal Cheeger and Calibrable Sets

Proposition 5.2 Suppose J is continuous. Then, hJ1 () ≥

λ(J, , p) , 2

where λ(J, , p) is given in Proposition 5.1. Proof From Proposition 5.1, for p = 1 and ψ = 0, we get that λ(J, , 1)|E| ≤ 2PJ (E)

for all measurable E ⊂ ,

and consequently, hJ1 () ≥

λ(J, , 1) . 2  

It is well known (see [49]) that the classical Cheeger constant: h1 () = inf

Per(E) : E ⊂ , |E| > 0 , |E|

for  a bounded smooth domain, is an optimal Poincaré constant, namely, it coincides with the first eigenvalue of the 1-Laplacian:

h1 () = inf

 ⎧ ⎪ ⎪ ⎨ |Du| + ⎪ ⎪ ⎩



|u|dH ∂

uL1 ()

N−1

⎫ ⎪ ⎪ ⎬

: u ∈ BV (), u = 0 . ⎪ ⎪ ⎭

It is also characterized (see [52]) as:

 h1 () = sup h ∈ R : ∃V ∈ L∞ (, RN ), V ∞ ≤ 1, divV ≥ h , which is usually referred to as a continuous version of the Min Cut Max Flow Theorem. In the next result, we obtain nonlocal versions of these two characterizations of the J -Cheeger constant given in (5.1).

5.1 Nonlocal Cheeger Sets

57

Theorem 5.3 Let  be a non-null, measurable and bounded set of RN , then

 hJ1 () = inf T V J (u) : u ∈ BVJ (RN ), u = 0 in RN \ , u1 = 1

T V J (u) N N : u ∈ BVJ (R ), u = 0 in R \ , u = 0 , = inf u1

(5.4)

and

 hJ1 () = sup h ∈ R+ : ∃z ∈ XJ∞ (RN ), z∞ ≤ 1, divJ z ≥ h in  = sup = sup





1 z∞

: divJ z = χ 

1 z∞

 : divJ z = 1 in  .

(5.5)

Proof Given a measurable subset E ⊂  with |E| > 0, we have T V J (χ E ) PJ (E) . = χ E 1 |E| Therefore, inf

T V J (u) : u ∈ BVJ (RN ), u = 0 in RN \ , u = 0 ≤ hJ1 (). u1

On the other hand, by the coarea formula (2.19) and Cavalieri’s formula, given u ∈ BVJ (RN ), with u = 0 in RN \  and u = 0, we have  T V J (u) =

+∞ −∞

 PJ (Et (u)) dt ≥ hJ1 ()

+∞

−∞

|Et (u)| dt = hJ1 ()u1

and taking infimum we get inf

T V J (u) : u ∈ BVJ (RN ), u = 0 in RN \ , u = 0 ≥ hJ1 (). u1

Therefore, (5.4) holds true. Let   A = h ∈ R+ : ∃z ∈ XJ∞ (RN ), z∞ ≤ 1, divJ z ≥ h in 

58

5 Nonlocal Cheeger and Calibrable Sets

and α = sup A (observe that 0 ≤ α ≤ 1). Given h ∈ A and E ⊂  with |E| > 0, applying (4.3), we get 



h|E| =

h dx ≤

divJ z(x)dx ≤ PJ (E).

E

E

Hence, h≤

PJ (E) , |E|

and, taking supremum in h and infimum in E, we obtain that α ≤ hJ1 (). By (5.4), we have

u1 N N = sup : u ∈ BVJ (R ), u = 0 in R \ , u = 0 T V J (u) hJ1 ()

u1 : u ∈ BVJ (RN ), u ≥ 0, u = 0 in RN \ , u = 0 = sup T V J (u)

 N N u(x)dx : T V J (u) ≤ 1, u ∈ BVJ (R ), u = 0 in R \  = sup 1



 = sup u, χ   − (L(u)) : u ∈ L1 (RN ) , being L : L1 (RN ) → L1 (RN × RN ) the linear map: L(u)(x, y) =

1 (u(x) − u(y))J (x − y), 2

and  : L1 (RN × RN ) → [0, +∞] the convex function: (w) =

⎧ ⎨0 ⎩

if wL1 (RN ×RN ) ≤ 1,

+∞ otherwise.

By the Frenchel–Rockafeller duality Theorem [21, Th. 1.12] and having in mind [40, Prop. 5], we have

   sup u, χ   − (L(u)) : u ∈ L1 (RN ) = inf ∗ (z) : L∗ (z) = χ  .

5.1 Nonlocal Cheeger Sets

59

Now, ∗ (z) = sup

 RN ×RN

z(x, y)w(x, y)dxdy − (w) : w ∈ L1 (RN × RN ) = z∞ .

On the other hand, L∗ (z), u = z, L(u) = 1 = 2

 RN ×RN

1 z(x, y) (u(x) − u(y))J (x − y)dxdy 2

 RN ×RN

(z(x, y) − z(y, x))J (x − y)u(x)dxdy = divJ z, u,

that is, L∗ (z) = divJ z. Consequently, 1 hJ1 ()

= inf {z∞ : divJ z = χ  } ,

from where it follows that

1 1 J χ h1 () = sup : divJ z =  ≤ sup : divJ z = 1 in  ≤ α, z∞ z∞  

and we finish the proof of (5.5).

Remark 5.4 It is well known that every bounded domain  ⊂ RN with Lipschitz boundary contains a classical Cheeger set E, that is, a set E ⊂  such that h1 () =

Per(E) . |E|

Furthermore, in [4] it is proved that there is a unique Cheeger set inside any nontrivial convex body in RN , being this Cheeger set convex. On the other hand, in [18] it is proved that for any s ∈ (0, 1), every open and bounded set  ⊂ RN admits and s-Cheeger set, that is, a set E ⊂  such that

Pers (F ) Pers (E) = inf : F ⊂ , |F | > 0 . |E| |F | We will show in Example 5.18 that there are convex sets without a J -Cheeger set. This is due to the lack of compactness when one considers nonsingular kernels.

60

5.2

5 Nonlocal Cheeger and Calibrable Sets

Nonlocal Calibrable Sets

We say that  is J-calibrable if it is a J -Cheeger set of itself, that is, if  is a non-null measurable bounded set and hJ1 () =

PJ () . ||

As a consequence of the Isoperimetric Inequality given in Theorem 2.1, we show that any ball is J -calibrable when J is radially nonincreasing. Proposition 5.5 Assume J is radially nonincreasing. The following properties hold true: 1. For 0 < r < R, PJ (Br (x0 )) PJ (BR (x0 )) > . |Br (x0 )| |BR (x0 )|

(5.6)

2. Any ball BR (x0 ) ⊂ RN is J -calibrable. Proof We can take x0 = 0 as the centre of the ball. First, we prove that a ball BR is J -calibrable if and only if the function (r) =

PJ (Br ) |Br |

verifies that (r) ≥ (R),

∀r ∈ (0, R).

(5.7)

Obviously, the condition is necessary. On the other hand, given E ⊂ BR , by the isoperimetric inequality we have that PJ (Br ) PJ (E) ≥ = (r) |E| |Br | where Br is a ball such that |Br | = |E|. Hence, since we are assuming that the function (r) verifies (5.7), we have PJ (BR ) PJ (E) ≥ (r) ≥ (R) = , |E| |BR | and consequently BR is J -calibrable.

5.2 Nonlocal Calibrable Sets

61

By the above characterization, we need to show that (5.7) holds. Let us prove that in fact the inequality is strict, that is, that (5.6) holds true. From (1.11),   PJ (Br ) = |Br | − J (x − y)dydx. Br

Br

Hence, (5.6) is true if and only if for every 0 < r < R,

F (r) < F (R) where





1 F (r) = |Br |

 J (x − y)dy dx.

Br

Br

Take 0 < r < R. Then, changing variables z = F (r) = =

1 |Br | 1 |BR |

1 < |BR |

Br



Br



J 

BR

we have

 J (x − y)dy dx





R r x,



r R

Br

  z − y dy dz 

J (z − y)dy dz BR

BR

= F (R), since, for any z ∈ BR , Br ( Rr z) ⊂⊂ BR (z) (note that 0 ∈ ∂Br ( Rr z)), which implies    r z − y dy = J J (y)dy R Br Br ( Rr z)  J (y)dy < 

BR (z)

=

J (z − y)dy. BR

  Remark 5.6 We have that  is indeed continuously differentiable in ]0, +∞[. By Lemma 3.10,  (r) = with y ∈ ∂Br arbitrary.

J (y)Per(B )|B | − Per(B )P (B ) H∂B r r r J r r

|Br |2

,

62

5 Nonlocal Cheeger and Calibrable Sets

Therefore, by the characterization given in the proof of Proposition 5.5 and having in J (x) is attained on any point of the boundary of Br , we get mind that clearly ess supH∂B r x∈Br

J Br is J -calibrable ⇐⇒ ess supH∂B (x) ≤ r x∈Br

PJ (Br ) . |Br |

The next result shows that any measurable and non-null set inside a ball of radius J -calibrable when J = |B11 | χ B1 . Proposition 5.7 Let J =

1 χ |B1 | B1 .

1 2

is

If  ⊂ B 1 with || > 0, then  is J -calibrable. 2

Proof Let E ⊂  non-null. For x ∈ E, E ⊂ B1 (x). Then,  

  1 χ B (x − y)dydx PJ (E) = J (x − y) dy dx = |B1 | E RN \E 1 E R\E   1 χ B (x)(y)dydx = |B1 | E RN \E 1  1 = |B1 (x) \ E|dx |B1 | E  1 (|B1 (x)| − |E ∩ B1 (x)|)dx = |B1 | E   |E| , = |E| 1 − |B1 | and   PJ (E) |E| = 1− , |E| |B1 | that is decreasing with |E|. Hence, the Cheeger constant of  is given by:   PJ () || = , hJ1 () = 1 − |B1 | || as we wanted to show.

 

5.2 Nonlocal Calibrable Sets

Example 5.8 If J =

63

1 χ |B1 | B1 ,

by Example 3.6, we have

 ψJ,N (s) = N

(s/ωN )1/N

   r N−1 ωN − 2 B1 (r, 0) ∩ Br  dr,

0

where ψJ,N is defined in (3.14). In the case s < ωN /2N , we have B1 (r, 0) ∩ Br = Br , and therefore  ψJ,N (s) = NωN

(s/ωN )1/N

r

N−1



1 − 2r

N



0

  1 dr = s 1 − s . ωN

Consequently, if |E| ≤ |B 1 |, by (3.13), the J -isoperimetric inequality stays as 2

  1 |E| ≤ PJ (E). |E| 1 − ωN Now by Proposition 5.7, if E ⊂ B 1 , then E is J -calibrable and 2

  1 PJ (E) = |E| 1 − |E| . ωN Therefore, the J -isoperimetric inequality is an equality for sets that are not balls. In the local case, a set  ⊂ RN is called calibrable if

Per() Per(E) = inf : E ⊂ , E with finite perimeter, |E| > 0 . || |E| Remark 5.9 For the local usual perimeter, when  is the union of two disjoint intervals in R,  = (a, b) ∪ (c, d), then the set is calibrable if and only if the two intervals have the same length (otherwise the Cheeger set inside  is the bigger interval). For the nonlocal perimeter with J = 12 χ [−1,1] , we have the following facts. Assume c − b ≥ 1. () (E) with the quotient of PJ|E| for E ⊂ (a, b) ∪ (c, d). We We want to compare PJ|| decompose E as E = E1 ∪ E2 with E1 = E ∩ (a, b) and E2 = E ∩ (c, d). For the case in which the two intervals that compose  have the same length, we have PJ (E) PJ () ≤ || |E| iff PJ (E1 ) + PJ (E2 ) PJ ((a, b)) ≤ , b−a |E1 | + |E2 |

64

5 Nonlocal Cheeger and Calibrable Sets

which is true since, if |Ei | = 0 then, by the Isoperimetric Inequality, PJ (Ei ) PJ ((a, b)) ≤ . b−a |Ei | On the other hand, if b − a > d − c, then PJ ((a, b)) PJ () < |b − a| || and therefore we have that  is not J -calibrable. In [5], it is proved the following characterization of convex calibrable set. Theorem 5.10 ( [5]) Given a bounded convex set  ⊂ RN of class C 1,1 , the following facts are equivalent: (a)  is calibrable.   () χ Du (b) χ  satisfies −1 χ  = Per , being  u = div  1 || |Du| . When  is convex, these statements are also equivalent to: (c) (N − 1)ess supH∂ (x) ≤ x∈∂

Per() . ||

We are going to study the validity of a similar result to the above theorem for the nonlocal case. In the following remark, we will introduce the main idea that is behind the proof for the nonlocal case. Remark 5.11 Let  ⊂ RN be a Borel set and assume there exist a constant λ > 0 and a function τ with τ (x) = 1 in  such that −λτ ∈ J1 χ 

in RN .

Then, there exists g ∈ L∞ (RN × RN ), g(x, y) = −g(y, x) for almost all (x, y) ∈ RN × RN , g∞ ≤ 1, satisfying  RN

J (x − y)g(x, y) dy = −λτ (x)

a.e x ∈ RN .

with J (x − y)g(x, y) ∈ J (x − y)sign(χ  (y) − χ  (x)) a.e. (x, y) ∈ RN × RN .

5.2 Nonlocal Calibrable Sets

65

Then,  λ|| =

RN

λτ (x)χ  (x)dx



=−





RN

RN

J (x − y)g(x, y) dy χ  (x)dx

= T V J (χ  ) = PJ (), and consequently, λ=

PJ () . ||

On the other hand, we observe again that the operator J1 is multivalued. Let us take, for example, J = 12 χ [−1,1] . We have that −f ∈ J1 χ ]−1,1[ ⇐⇒ ∃g antisymmetric, g∞ ≤ 1 satisfying  R

J (x − y)g(x, y) dy = −f (x) a.e x ∈ R.

and J (x − y)g(x, y) ∈ J (x − y)sign(χ [−1,1[ (y) − χ ]−1,1[ (x)) a.e. (x, y) ∈ R × R. Then, by taking for instance g(x, y) = sign0 (χ ]−1,1[ (y) − χ ]−1,1[ (x)), we have, if x ∈ [−1, 1], f (x) =

1 |x|, 2

66

5 Nonlocal Cheeger and Calibrable Sets

and, if x ∈ [−1, 1], ⎧ 1 ⎪ − (x + 2) if − 2 ≤ x ≤ −1, ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎨ f (x) = − 1 (2 − x) if 1 ≤ x ≤ 2, ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 if x ≤ −2 or x ≥ 2. But, by taking ⎧ 1 if y ∈ [−1, 1], x ∈ [−1, 1], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1 if x ∈ [−1, 1], y ∈ [−1, 1], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ in {0 < y < x < 1} ∪ {−1 < x < y < 0}, ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨ 1 g(x, y) = − in {0 < x < y < 1} ∪ {−1 < y < x < 0}, ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ in {1 < y < x} ∪ {y < x < −1}, ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ − in ∪ {1 < x < y} ∪ {x < y < −1}, ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 otherwise, we get a different but interesting representation for J1 χ ]−1,1[ . We get that −

1 PJ (] − 1, 1[) τ = τ ∈ J1 χ ]−1,1[ |] − 1, 1[| 4

with τ (x) =

⎧ ⎨1 ⎩

in RN

if x ∈ [−1, 1],

−(|x| − 2)− , otherwise,

Note that this function τ verifies that τ = 1 in ] − 1, 1[, and this gives, as we will see in the next theorem, that ] − 1, 1[ is J -calibrable. Of course, in this simple case, this was obtained previously by more elementary methods.

5.2 Nonlocal Calibrable Sets

67

The next result is the nonlocal version of the fact that (a) is equivalent to (b) in Theorem 5.10. Theorem 5.12 Let  ⊂ RN be a non-null measurable bounded set.  (i) Assume that  J (x − y)dy ≥ α > 0 for all x ∈ . If  is J -calibrable, then there exists a function τ equal to 1 in  such that −

PJ () τ ∈ J1 χ  ||

in RN .

(5.8)

(ii) If there exists a function τ equal to 1 in  and satisfying (5.8), then  is J -calibrable. Proof We first prove (ii): By hypothesis, there exists g(x, y) = −g(y, x) for almost all (x, y) ∈ RN × RN , g∞ ≤ 1, satisfying  PJ () τ (x) a.e x ∈ RN . J (x − y)g(x, y) dy = − || RN with J (x − y)g(x, y) ∈ J (x − y)sign(χ  (y) − χ  (x)) a.e. (x, y) ∈ RN × RN , and τ is a function such that τ =1

in .

Then, if F is a bounded measurable set, F ⊂ , we have  PJ () PJ () |F | = τ (x)χ F (x)dx || || RN   =− J (x − y)g(x, y)χ F (x) dydx =

1 2



RN



RN

RN

RN

J (x − y)g(x, y)(χ F (y) − χ F (x)) dydx

≤ PJ (F ), Therefore, hJ1 () = and consequently  is J -calibrable.

PJ () , ||

68

5 Nonlocal Cheeger and Calibrable Sets

Let us now prove (i): Let g˜ ∈ L∞ (RN × RN ) be defined as:

g˜ (x, y) =

⎧ ⎪ ⎪0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −1

if x ∈ RN \ , y ∈ RN \ , if x ∈ , y ∈ RN \ ,

⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ g(x) ˆ

if x ∈ RN \ , y ∈ , if x, y ∈ ,

being gˆ a function to be determined. We define τ (x) = − P

1



J ()

RN

||

J (x − y)˜g(x, y)dy,

x ∈ RN .

For x ∈ , we have τ (x) =

|| PJ ()

 RN \

J (x − y)dy − g(x) ˆ

|| PJ ()

 J (x − y)dy. 

Then, taking  () + − PJ||  g(x) ˆ =

RN \

J (x − y)dy ,

J (x − y)dy

x ∈ ,



we have that τ (x) = 1 for all x ∈ . Moreover, for x ∈ RN \ , τ (x) = −

|| PJ ()

 J (x − y)dy ≤ 0. 

We claim now that PJ () τ ∈ ∂F J (0). || Take w ∈ L2 (RN ) with T V J (w) < +∞. Since 

∞

w(x) = 0

 χ Et (w) (x)dt −

0 −∞

(1 − χ Et (w) )(x)dt,

(5.9)

5.2 Nonlocal Calibrable Sets

69

we have 

PJ () PJ () τ (x)w(x)dx = || ||

RN

=

PJ () ||



 RN

τ (x)

−∞

+∞ 



−∞

∞

RN

 χ Et (w) (x)dt dx

τ (x)χ Et (w) (x)dxdt.

Now, using that  is J -calibrable we have that PJ () ||

+∞ 



−∞

RN

τ (x)χ Et (w) (x)dxdt

   PJ () +∞ PJ () +∞ = |Et (w) ∩ |dt + τ (x)dxdt || −∞ || −∞ Et (w)\  +∞   PJ () +∞ ≤ PJ (Et (w) ∩ )dt + τ (x)dxdt. || −∞ Et (w)\ −∞ By Proposition 1.2 and the coarea formula given in Theorem 2.4, we get 

+∞ −∞

PJ (Et (w) ∩ )dt + 

=

+∞ −∞

 −

 =

PJ (Et (w) ∩ )dt +

+∞

+∞

−∞ +∞

−∞



+∞ 

τ (x)dxdt

−∞



−∞

 +

PJ () ||

Et (w)\

+∞ −∞

PJ (Et (w) \ )dt 

2LJ (Et (w) \ , Et (w) ∩ )dt − 2LJ (Et (w) \ , Et (w) ∩ )dt +

+∞ −∞

PJ (Et (w) \ )dt

PJ () ||



+∞  −∞

τ (x)dxdt Et (w)\

PJ (Et (w))dt + I = T V J (w) + I,

with  I =−

+∞

−∞

 PJ (Et (w) \ )dt +

PJ () + ||



+∞ −∞

2LJ (Et (w) \ , Et (w) ∩ )dt

+∞ 

−∞

τ (x)dxdt. Et (w)\

70

5 Nonlocal Cheeger and Calibrable Sets

Hence, if we prove that I ≤ 0, we get  RN

PJ () τ (x)w(x)dx ≤ T V J (w). ||

(5.10)

Now, since PJ (Et (w) \ ) = LJ (Et (w) \ , RN \ (Et (w) \ )) .

= LJ (Et (w) \ , (Et (w) ∩ ) ∪ (RN \ Et (w))), we have  I =−

+∞

−∞

 PJ (Et (w) \ )dt +

+∞ −∞

2LJ (Et (w) \ , Et (w) ∩ )dt

  PJ () +∞ + τ (x)dxdt || −∞ Et (w)\  +∞  N =− LJ (Et (w) \ , R \ Et (w))dt + −∞

+∞ −∞

LJ (Et (w) \ , Et (w) ∩ )dt

  PJ () +∞ τ (x)dxdt || −∞ Et (w)\   +∞   = −J (x − y)dy +

−∞

RN \Et (w)

Et (w)\

  PJ () τ (x) dx dt + J (x − y)dy + || Et (w)∩    (−˜g(x, y) − 1)J (x − y)dy dx dt 

+∞ 

 =

−∞

 +  −

Et (w)\

+∞ 

RN \Et (w)



−∞

Et (w)\

−∞

Et (w)\

+∞ 

  (−˜g(x, y) + 1)J (x − y)dy dx dt

Et (w)∩

  J (x − y)˜g(x, y)dy dx dt.

 Et (w)\

Now, the first integral is negative since g˜ (x, y) ≥ −1 for x ∈ RN \ , the second is zero since g˜ (x, y) = 1 for x ∈ RN \  and y ∈  and the last integral is zero since g˜ = 0 in (RN \ ) × (RN \ ). Therefore, I ≤ 0, and consequently (5.10) holds.

5.2 Nonlocal Calibrable Sets

71

Now, (5.10) implies that (5.9) is true. Then, by Theorem 4.5, we have −

PJ () τ ∈ J1 (0). ||

Thus, there exists g ∈ L∞ (RN × RN ), g(x, y) = −g(y, x) for almost all (x, y) ∈ RN × RN , g∞ ≤ 1, satisfying  RN

J (x − y)g(x, y) dy = −

PJ () τ (x) a.e x ∈ RN . ||

and J (x − y)g(x, y) ∈ J (x − y)sign(0) a.e. (x, y) ∈ RN × RN . Now, multiplying by χ  and integrating by parts we get  PJ () τ (x)χ  (x)dx || RN   J (x − y)g(x, y)χ  (x)dxdy =−

PJ () =

=

1 2



RN



RN

RN

RN

J (x − y)g(x, y)(χ  (y) − χ  (x))dxdy

≤ PJ (), from where it follows that J (x − y)g(x, y) ∈ J (x − y)sign(χ  (y) − χ  (x)) a.e. (x, y) ∈ RN × RN , and consequently, −

PJ () τ ∈ J1 χ  ||

in RN .  

Remark 5.13 It is not clear when  J -calibrable implies that −

PJ () χ  ∈ J1 χ  ||

in RN .

72

5 Nonlocal Cheeger and Calibrable Sets

We now give a result that says that J -calibrability is related to the nonlocal curvature J , which is the nonlocal version of one implication in the equivalence between function H∂E (a) and (c) in Theorem 5.10. Theorem 5.14 Let  ⊂ RN be a bounded set and assume x ∈ . Then, J (x) ≤  is J -calibrable ⇒ ess supH∂ x∈



 J (x

− y)dy > 0 for all

PJ () . ||

Proof By Theorem 5.12, there exists g ∈ L∞ (RN × RN ) antisymmetric with g∞ ≤ 1 and a function τ , equal to 1 in , such that  −

RN

J (x − y)g(x, y) dy =

PJ () τ (x) ||

a.e x ∈ RN ,

(5.11)

and J (x − y)g(x, y) ∈ J (x − y)sign(χ  (y) − χ  (x)) a.e. in RN × RN .

(5.12)

Then, by (5.11), (5.12) and since τ = 1 in , we have  J H∂ (x) =

RN

J (x − y)(χ RN \ (y) − χ  (y))dy

 ≤−

RN

J (x − y)g(x, y) dy ≤

PJ () ||

for a.e. x ∈ .  

In the next example we show that the reverse of Theorem 5.14 is not true in general. Remark 5.15 Observe that J ess supH∂ (x) ≤ x∈

PJ () ||

5.2 Nonlocal Calibrable Sets

73

if and only if 1 ||

 

 J (x − y)dydx ≤ 2 ess inf x∈

 

J (x − y)dy.

(5.13)



Example 5.16 In general, J ess supH∂ (x) ≤ x∈

PJ () does not imply  is J -calibrable. ||

In fact, to provide an example where this happens, take N = 3 and the uniform J = 1 χ N B1 . Let 0 <  < 1, and x0 ∈ R such that x0  > 4. Consider  = B1 ∪ B1+ (x0 ). |B1 |

For shortness, we write B2, = B1+ (x0 ). Then, by Proposition 1.2 and (5.6), we have that PJ ( ) PJ (B1 ) + PJ (B2, ) PJ (B2, ) = > . | | |B1 | + |B2, | |B2, |

Therefore,  is not J -calibrable. Now, let us see that for  small PJ ( ) . | |

J ess supH∂ (x) ≤  x∈

It is easy to see that J J ess supH∂ (x) = ess supH∂B (x).  1 x∈

x∈B1

Let us see that there exists 0 < 1 such that, for all 0 <  < 0 : PJ ( ) ; | |

J ess supH∂B (x) ≤ 1 x∈B1

which, by (5.13), is equivalent to prove that there exists 0 < 1 such that, for all 0 <  < 0 , 1 |B1 | + |B2, |











J (x − y)dydx + B1

B1



B2,

J (x − y)dy.

≤ 2 ess inf x∈B1

J (x − y)dydx B2,

B1

(5.14)

74

5 Nonlocal Cheeger and Calibrable Sets

Now, if (5.14) is not true, then there exists n → 0 such that 1 |B1 | + |B2,n |











J (x − y)dydx + B1

B1



B2,n

(5.15)

J (x − y)dy.

≥ 2 ess inf x∈B1

J (x − y)dydx B2,n

B1

Hence, passing to the limit in (5.15), we get 1 |B1 |





 J (x − y)dydx ≥ 2 ess inf

B1

x∈B1

B1

J (x − y)dy. B1

But, as we see below, the inequality in (5.13) is strict and hence the above inequality gives a contradiction. Let us show that    1 J (x − y)dydx < 2 ess inf J (x − y)dy. x∈B1 B1 |B1 | B1 B1 Since J =

1 χ |B1 | B1 ,

it holds that

 J (x − y)dy = B1

1 |B1 |



χ B1 (x − y)dy = 1 |B1 ∩ B1 (x)|, |B1 | B1

and then the previous inequality turns out to be 1 |B1 |

 |B1 ∩ B1 (x)|dx < 2ess inf|B1 ∩ B1 (x)|. x∈B1

B1

Now, since |B1 ∩ B1 (x)| =

π (|x| + 4)(2 − |x|)2 , 12

we have that 1 |B1 |



 1 |B1 ∩ B1 (x)|dx = (|x| + 4)(2 − |x|)2 dx 16 B1 B1  π 1 = r(r + 4)(2 − r)2 dr 8 0 =

21π . 40

(5.16)

5.2 Nonlocal Calibrable Sets

75

On the other hand, 2ess inf|B1 ∩ B1 (x)| = 2|B1 ∩ B1 (0, 0, 1)| = x∈B1

5π , 6

and we conclude that (5.16) holds. Note that in this example  is not connected. However, this fact is not relevant. We have that the nonlocal perimeter and the nonlocal curvature are continuous with respect to the set in terms of convergence in measure (if En → E in the sense that |En E| → 0, J (x) → H J (x)). Then, we only have to connect the two then PJ (En ) → PJ (E) and H∂E ∂E n balls with a thin bridge to obtain an example of a connected domain such that J ess supH∂ (x) ≤ x∈

PJ () but  is not J -calibrable. ||

Example 5.17 Let  ⊂ RN be a bounded set and assume x ∈ . One can ask if J (x) ≤ sup H∂

x∈∂



 J (x

− y)dy > 0 for all

PJ () . ||

(5.17)

implies J -calibrability. The following example shows that, in general, this is not the case. For N = 1, and for J (z) = 12 χ [−1,1] (z) and   = ] − 3, 3[ \ [−1, 1] ∪] − , [,

0 <  < 1/2,

one can check that J sup H∂ (x) = (1 − 3)+ 

x∈∂

J < H∂ (0) = 1 − 2  J = sup H∂ (x),  x∈

and that 1 + 2 − 3 2 PJ ( ) = . | | 4 + 2 Then, for 


PJ ( ) | |

76

5 Nonlocal Cheeger and Calibrable Sets

and consequently  is not calibrable. Now, for √

0.236

5−2≤
0, is calibrable. Indeed, ess supH∂Lr (x) = x∈∂L r

2πr + 2L Per() 1 < = , r πr 2 + 2rL ||

and Theorem 5.10 gives the result. But, L r is not J -calibrable for the uniform J = 1 χ B1 (0,0)), r > 1 and L > 2 large enough. Moreover, we have that this non-J |B1 (0,0)| calibrable set does not contain a J -Cheeger set. Let us prove these two facts. For the first one, by Theorem 5.14 and Remark 5.15, it is enough to show that, for L large enough, 1 |L r|



 L r

 L r

J (x − y)dydx > 2 ess inf x∈L r

L r

J (x − y)dy.

5.2 Nonlocal Calibrable Sets

77

Now, this condition reads as follows: 1 |L r|



   L L   B + r, 0) ∩ B , 0) |L ∩ B (x)|dx > 2 ( ( 1 r r ,  1 2 L 2

(5.18)

r

since 

 ess inf x∈L r

L r

J (x − y)dy =

L r

J (x0 − y)dy

  for x0 = ( L2 + r, 0). Let us call a(r) = 2B1 ( L2 + r, 0) ∩ Br ( L2 , 0). Now, if CrL =] − L L L L L 2 , 2 [×]0, r − 1[ and Dr =]1 − 2 , 2 − 1[×]r − 1, r[, we have 1 |L r|

 L r

|L r

   2 L πL(r − 1) + ∩ B1 (x)|dx ≥ |r ∩ B1 (x)|dx πr 2 + 2rL DrL   2 2 ) , ≥ πL(r − 1) + (L − 2)(π − πr 2 + 2rL 3

and this last quantity is larger than a(r) if L>

1 2 2 a(r)πr  2 r π − 3r

+π − − a(r)

2

3 ,

2 since π − 3r > a(r). That is, we get that (5.18) holds true for L large enough. Consequently, L r is not J -calibrable. Let us now see that L r does not contain a J -Cheeger set. Arguing by contradiction, assume that there exists E a J -Cheeger set of L r . Then, since E is J -calibrable and having in mind the calculation made above, we have

  2  L L  a(r) = B1 ( + r, 0) ∩ Br ( , 0) π 2 2   1 < L J (x − y)dydx |r | Lr Lr   1 < J (x − y)dydx |E| E E  ≤ 2 inf J (x − y)dy x∈E E

=

2 inf |B1 (x) ∩ E|. π x∈E

78

5 Nonlocal Cheeger and Calibrable Sets

On the other hand, consider a ball Bs such that |Bs | = |E|. By the Isoperimetric Inequality and since Bs is J -calibrable, we have a(r) < ≤

1 |E| 1 |Bs |

  J (x − y)dydx E

E





J (x − y)dydx Bs

Bs

≤ a(s), from where it follows that s > r, and consequently E  Br (− L2 , 0). Hence, if we define 

 L L l + = sup l ∈ − , : |Br (l, 0) ∩ E| > 0 , 2 2 we have l + > − L2 . Therefore, for − L2 < l < l + , we have a(r) < ≤

2 ess inf|B1 (x) ∩ E| π x∈E

 2 ess inf |B1 (x) ∩ E| : x ∈ E ∩ (Br (l + , 0) \ Br (l, 0)) π

≤ a(r) + o(l + − l). Then, letting l → l + we arrive to a contradiction.

5.3

Rescaling

In this section, we relate local and nonlocal Cheeger constants and local and nonlocal calibrable sets under rescaling. Proposition 5.19 Let N ≥ 2. Assume J satisfies (1.19). Let  be an open bounded set of RN , then lim CJ hJ1 () = h1 (). ↓0

Proof Given δ > 0, there exists Eδ ⊂  such that h1 () + δ ≥

Per(Eδ ) . |Eδ |

5.3 Rescaling

79

Then, by Theorem 1.11, we have h1 () + δ ≥ lim ↓0

CJ PJ (Eδ ) CJ J ≥ lim sup h ().  |Eδ |  1 ↓0

By the arbitrariness of δ, we get lim sup ↓0

CJ J h () ≤ h1 ().  1

Let us now suppose that lim inf ↓0

CJ J h () < h1 ().  1

(5.19)

By (5.4), given  > 0, there exists u ∈ BVJ (), u = 0 in RN \ , u  = 1, such that hJ1 () ≤ T V J (u ) ≤ hJ1 () +  2 . Then, by (5.19), lim inf ↓0

CJ T V J (u ) < h1 (). 

Therefore, there exists a sequence n decreasing to 0 such that CJ T V Jn (un ) < h1 (). n Then, for a large ball B containing , 1 CJ T V Jn (un ) = n K1,N

  B

B

|un (x) − un (y)| ρn (x − y)dxdy < h1 (), |x − y|

where ρ (z) =

|z| 1 CJ K1,N J (z). 2 

Consequently, by Theorem 1.9 (observe that ρ satisfies (1.26)), we have that there exists a subsequence of n , denoted equal, such that u n → u

in L1 (B),

80

5 Nonlocal Cheeger and Calibrable Sets

u ∈ BV (B) and, since moreover u = uχ  , 





|Du| =

|Du| +

B



|u|dHN−1 < h1 (). ∂

But, we also get that uL1 () = 1 and consequently, from the above inequality, we get λ1 () < h1 (), which is a contradiction. Therefore, what we supposed in (5.19) is false and then lim inf ↓0

CJ J h () ≥ h1 (),  1  

and the proof concludes.

Note that there are sets which are J -calibrable for every  > 0 small, for example a fixed ball. Our next result says that such sets are calibrable. Corollary 5.20 Let N ≥ 2. Assume J satisfies (1.19). Let  be an open bounded set of RN . If  is Jn -calibrable for a sequence n → 0 as n → +∞, then  is calibrable. Proof Since  is Jn -calibrable, we have CJ Jn CJ PJn () . h1 () = n n || Hence, by Theorem 1.11, CJ Jn Per() h () → n 1 ||

as n → +∞.

Then, by Proposition 5.19, we conclude that Per() = h1 (), || and consequently  is calibrable.

 

6

Nonlocal Heat Content

6.1

The Classical Heat Content

The heat content of a Borel measurable set D ⊂ RN at time t is defined by M. van der Berg in [69] (see also [70]) as:  HD (t) =

T (t)χ D (x)dx, D

with (T (t))t ≥0 being the heat semigroup in L2 (RN ). Therefore, the heat content represents the amount of heat in D at time t if in D the initial temperature is 1 and in RN \ D the initial temperature is 0. The following characterization for the perimeter of a set D ⊂ RN with finite perimeter was given in [61, Theorem 3.3]:  lim

t →0+

π t

 RN \D

T (t)χ D (x)dx = Per(D),

(6.1)

As a consequence, the following result was presented in [69] for the heat content: For an open subset D in RN with finite Lebesgue measure and finite perimeter, there holds  HD (t) = |D| −

√ t Per(D) + o( t) π

as t ↓ 0.

(6.2)

In [3] and [2], the concept of heat content has been extended to more general diffusion (α) processes (see also [35]). More precisely, for 0 < α ≤ 2, let pt : RN → [0, ∞) be the © Springer Nature Switzerland AG 2019 J. M. Mazón et al., Nonlocal Perimeter, Curvature and Minimal Surfaces for Measurable Sets, Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-06243-9_6

81

82

6 Nonlocal Heat Content

probability density such that its Fourier transform verifies α ) pt(α) (x) = e−t |x| .

If one considers (α)

Tt

 (f )(x) =

(α)

RN

f (y)pt (x − y)dy,

then u(x, t) = Tt(α) (f )(x) is the unique weak solution of the initial valued problem: ⎧ α N ⎪ ⎨ ut (x, t) = −(−) 2 u(x, t), (x, t) ∈ R × [0, ∞), ⎪ ⎩

u(x, 0) = f (x)

x ∈ RN .

And, in this context, the heat content of a Borel measurable set D ⊂ RN at time t is defined as:  (α) Tt(α) χ D (x)dx. HD (t) = D

Note that for α = 2, (2) pt

= (4πt)

− N2

  |x|2 exp − , 4t

(2)

is the Gaussian kernel and consequently HD (t) = HD (t); while, for α = 1,  pt(1) (x) =

 π

N+1 2 N+1 2

 t (t + |x|)

N+1 2

is the Poisson heat kernel. In [3] and [2], it is proved that, for bounded sets D in RN of finite perimeter, N ≥ 2, lim

t →0+

 1 |D| − H(α) D (t) = Aα,N Perα (D) t

for 0 < α < 1, where Aα,N is a determined positive constant, lim

t →0+

 1 1  (α) |D| − H (t) = (1 − 1/α)Per(D) D 1/α t π

6.2 The Heat Content for Nonlocal Diffusion with Nonsingular Kernels

83

for 1 < α < 2 and lim

t →0+

  1 1 |D| − H(1) (t) = Per(D) D t ln(1/t) π

for D smooth.

6.2

The Heat Content for Nonlocal Diffusion with Nonsingular Kernels

6.2.1

Definition and a Characterization

Assume J to be continuous with J (0) > 0 and compactly supported. The following nonlocal Cauchy problem has been studied in [12]: ⎧  ⎪ ⎪ ⎨ ut (x, t) =

RN

J (x − y)(u(y, t) − u(x, t))dy, (x, t) ∈ RN × [0, +∞), (6.3)

⎪ ⎪ ⎩ u(x, 0) = u (x), 0

x ∈ RN .

This equation appears naturally from the following considerations: if u(x, t) is thought of as a density at a point x at time t, and J (x − y) is thought of as the probability distribution of jumping from location y to location x, then  RN

J (y − x)u(y, t) dy = (J ∗ u)(x, t)

is the rate at which individuals are arriving at position x from all other places, and  −u(x, t) = −

RN

J (y − x)u(x, t) dy

is the rate at which they are leaving location x to travel to all other sites. This consideration, in the absence of external or internal sources, leads immediately to the fact that the density u satisfies Eq. (6.3). In [12], it is defined a solution of problem (6.3) in the time interval [0, T ] as a function u ∈ W 1,1 (0, T ; L2 (RN )) which satisfies u(x, 0) = u0 and  ut (x, t) =

RN

J (x − y)(u(y, t) − u(x, t)) dy

a.e. in (x, t) ∈ RN × (0, T ).

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6 Nonlocal Heat Content

A simple integration of the equation in (6.3) in space gives that the total mass is preserved, that is: 

 RN

u(x, t)dx =

RN

u0 (x)dx

∀ t ≥ 0.

(6.4)

Our aim in this chapter is to study the heat content associated with the above nonlocal diffusion process and to relate it with the (local) heat content. We give the following definition. Definition 6.1 Given a Lebesgue measurable set D ⊂ RN with finite measure, we define the J-heat content of D in RN at time t by:  HJD (t)

=

u(x, t)dx, D

with u being the solution of (6.3) with the datum u0 = χ D . Note that, from (6.4), we have HJD (0) = |D|, and that no further regularity is required for D besides having finite measure. We have the following interpretation of the J -heat content HJD (t): since u(x, t) represents the density of a population at a point x ∈ RN at time t with initial condition u(x, 0) = χ D (x), then the J -heat content of D at time t represents the size of the population that remains inside D at that time when in D the initial density of the population is 1 and in RN \ D the initial density of population is 0. Our first result relates the J -heat content with the L2 -norm of the solution of (6.3). Let B be the operator defined in L2 (RN ) as: B(u) = v ⇐⇒ v(x) = −J u(x) ∀ x ∈ , where  J u(x) =

RN

J (x − y)(u(y) − u(x))dy.

The domain of B is L2 (RN ) and, in [12], it is showed that B is m-completely accretive in L2 (). Then, this operator B generates a C0 -semigroup (TJ (t))t ≥0 in L2 () which solves

6.2 The Heat Content for Nonlocal Diffusion with Nonsingular Kernels

85

the Cauchy problem (6.3). With this operator, we can describe the J -heat content of a finite measurable set D ⊂ RN as:  HJD (t) = TJ (t)χ D (x)dx, D

and we have the following result. Proposition 6.2 Given a Lebesgue measurable set D ⊂ RN with finite measure, we have *2 *   * * t * χ T HJD (t) = * D* . * J 2 L2 Proof It is enough to prove that the operators TJ (t) are self-adjoint, since then *   *2   , +   * * *TJ t χ D * = TJ t χ D , TJ t χ D * * 2 2 2 2 L = TJ (t)χ D , χ D  = HJD (t). So, let us see that TJ (t) is self-adjoint. Let F : L2 (RN ) → L2 (RN ) be the Fourier–Plancherel transform. We will also write ˆ f = F (f ). If u(t)(x) = u(x, t) := TJ (t)f (x), since ut (t) = J ∗ u(t) − u(t),

(6.5)

applying the Fourier–Plancherel transform, we have uˆ t (ξ, t) = (Jˆ(ξ ) − 1)u(ξ, ˆ t), from which it follows that ˆ

u(ξ, ˆ t) = e(J (ξ )−1)t f-(ξ ).

(6.6)

86

6 Nonlocal Heat Content

Therefore, given f, g ∈ L2 (RN ), we have TJ (t)f, g = F (TJ (t)f ), F (g) ˆ

= e(J (ξ )−1)t f-, g ˆ = f-, e(J (ξ )−1)t g

= f, TJ (t)g,  

as we wanted to show.

6.2.2

The Asymptotic Expansion of the J -heat Content

In this section, we give an asymptotic expansion of the J -heat content, from which a result similar to (6.2) follows. This expansion, obtained from a classical Taylor’s expansion, is given for any time and for any order and with the main fact that all the terms involved in the expansion can be expressed using nonlocal perimeters of the set D with different kernels. We will use the following notation: J = (J ∗)1 , J ∗ J = (J ∗)2 , and for the convolution of n kernels: J ∗ J ∗ . . . ∗ J = (J ∗)n . Observe that, for all n, the kernel (J ∗)n satisfies the same structural conditions than J , in particular  RN

(J ∗)n (x)dx = 1.

For these kernels, we can define, as in (1.9), the (J ∗)k -nonlocal perimeter of a measurable set D:   (J ∗)k (x − y) dy dx, P(J ∗)k (D) = D

RN \D

6.2 The Heat Content for Nonlocal Diffusion with Nonsingular Kernels

87

and the (J ∗)k -nonlocal curvature of D at x: (J ∗) H∂D (x)

 =−

RN

(J ∗)(x − y)(χ D (y) − χ RN \D (y))dy.

We will also use the following relations (see the previous chapters). For a finite measurable set D,   |D| = D

D

(J ∗)k (x − y)dydx + P(J ∗)k (D),

(J ∗)k (x) H∂D

(6.7)

 =1−2

(J ∗)k (x − y)dy

(6.8)

D

and hence 

(J ∗) H∂D (x) dx = k

D

    1 − 2 (J ∗)k (x − y)dy dx D

 

D

(J ∗)k (x − y) dy dx.

= |D| − 2 D

D

that is, 

(J ∗) H∂D (x)dx = 2P(J ∗)k (D) − |D|. k

D

(6.9)

Proposition 6.3 Let D be a finite measurable subset of RN and let u(·, t) = TJ (t)χ D . Then, the following hold: 1. The Fourier transform of u verifies u(ξ, ˆ t) =

L  n=0



   tn n n−k k χ. (−1) Jˆ(ξ ) χ. D (ξ ) + D (ξ ) n! k k=1 n 

ˆ s) +(Jˆ(ξ ) − 1)L+1 u(ξ,

t L+1 (L + 1)!

(6.10)

,

for 0 < s < t. 2. The following expansion holds for u: u(x, t) = e−t

∞   n=0 D

(J ∗)n (x − y)dy

tn , n!

(6.11)

88

6 Nonlocal Heat Content

where

 (J ∗)0 (x − y)dy = χ D (x). D

Proof From (6.6), the Fourier transform of u verifies the evolution problem: ˆ t), uˆ t (ξ, t) = (Jˆ(ξ ) − 1)u(ξ, with the initial condition u(ξ, ˆ 0) = χ. D (ξ ). Hence, ˆ

u(ξ, ˆ t) = e(J (ξ )−1)t χ. D (ξ ).

(6.12)

Using Taylor expansion of the exponential, we deduce L  tn t L+1 ˆ u(ξ, ˆ t) = + (Jˆ(ξ ) − 1)L+1 e(J (ξ )−1)s χ. (Jˆ(ξ ) − 1)n χ. D (ξ ) D (ξ ) n! (L + 1)! n=0     L n   tn n n−k k χ. = (−1) Jˆ(ξ ) χ. D (ξ ) + D (ξ ) n! k n=0 k=1

t L+1 (L + 1)!     L n   tn n n−k ˆ kχ χ . . = (−1) J (ξ ) D (ξ ) D (ξ ) + n! k ˆ

+(Jˆ(ξ ) − 1)L+1 e(J (ξ )−1)s χ. D (ξ )

n=0

k=1

t L+1 , +(Jˆ(ξ ) − 1)L+1 u(ξ, ˆ s) (L + 1)! for 0 < s < t. From (6.12), we also have ˆ

t J (ξ ) u(ξ, ˆ t) = e−t χ. = e−t D (ξ )e

∞  tn (Jˆ(ξ ))n χ. D (ξ ) . n! n=0

Taking here the inverse Fourier transform, we get that u(x, t) = e−t

∞  n=0

=e

−t

  tn F −1 (Jˆ(ξ ))n χ. D (ξ ) (x, t) n!

∞   n=0 D

(J ∗)n (x − y)dy

tn , n!

6.2 The Heat Content for Nonlocal Diffusion with Nonsingular Kernels

89

where  (J ∗)0 (x − y)dy = χ D (x). D

  The main result of this section is the following. Theorem 6.4 Let D be a finite measurable set. Then, ⎛

HJD (t)

= |D| −

n=1

⎞

 

+∞ n   n ⎝

k

k=1

(−1)n−k P(J ∗)k (D)⎠

tn n!

∀t > 0.

(6.13)

Moreover, for L ≥ 1 we have ⎛ ⎞     L  n   |D| 2L L+1 t n  n  J ⎝ t (−1)n−k P(J ∗)k (D)⎠  ≤ HD (t) − |D| +  k n!  (L + 1)! k=1

∀t > 0.

n=1

(6.14) We also have the following expression: HJD (t)

=

+∞    n=0

D D

 (J ∗)n (x −y)dydx

e−t t n n!

∀t > 0,

(6.15)

where   (J ∗)0 (x −y)dydx = |D|. D D

Proof Taking the inverse Fourier–Plancherel transform in (6.10) and integrating over D, we get HJD (t) =

L  n=0



     tn n (J ∗)k (x − y)dydx (−1)n |D| + (−1)n−k n! k D D k=1    t L+1 F −1 (Jˆ(ξ ) − 1)L+1 u(ξ, ˆ s) (x)dx + (L + 1)! D n 

90

6 Nonlocal Heat Content

for n ≥ 1. Now, for n ≥ 1, using (6.7) yields    n  n (J ∗)k (x − y)dydx (−1) |D| + (−1)n−k k D D k=1   n   n = (−1)n |D| + (−1)n−k |D| − P(J ∗)k (D) k k=1   n  n =− (−1)n−k P(J ∗)k (D). k k=1 n

Also, 

  F −1 (Jˆ(ξ ) − 1)L+1 u(ξ, ˆ s) (x)dx D

= (−1)

L+1

|D| +

 L + 1 (−1)L+1−k (J ∗)k ∗ u(x, s)dx. k D

L+1  k=1

(6.16)

Using the fact that for this problem, thanks to the maximum principle, we have 0 ≤ u ≤ 1, we get 

 (J ∗)k ∗ u (x, t)dx ≤ |D|

0≤

∀k ∈ N.

D

Hence, from (6.16) we have that      L + 1   1 L+1  F −1 (Jˆ(ξ ) − 1)L+1 u(ξ, ≤ |D| = 2L |D|. ˆ s) (x)dx   2 k D k=0

From which (6.13) and (6.14) follow. For the second statement in the thesis, let us integrate over D in (6.11), then we get HJD (t) and thus (6.15) is proved.

=e

−t

∞    n=0 D

D

(J ∗)n (x − y)dydx

tn , n!  

6.2 The Heat Content for Nonlocal Diffusion with Nonsingular Kernels

91

We can also proceed in the following way. Integrating in (6.3) over D, we get  

(HJD ) (t) =

 J (x − y)u(y, t)dydx −

RN

D

  = D

u(x, t)dx D

RN

J (x − y)u(y, t)dydx − HJD (t),

and hence, HJD (t) + (HJD ) (t) =

  D

RN

J (x − y)u(y, t)dydx.

(6.17)

Taking the time derivative in (6.17) and using again (6.17), we get (HJD ) (t) + (HJD ) (t) =

  D

  =

RN

D

RN

D

RN

  =

J (x − y)ut (y, t)dydx  J (x − y)  RN

D

= D

RN

(J (y − z)u(z, t) − u(y, t))dx dydx

J (x − y)J (y − z)u(z, t)dzdydx

 

− 



RN

J (x − y)u(y, t)dydx

 (J ∗)2 ∗ u (x, t)dx − HJD (t) − (HJD ) (t),

hence, HJD (t) + 2(HJD ) (t) + (HJD ) (t) =



 (J ∗)2 ∗ u (x, t)dx. D

By induction, it is easy to see that  n     n (HJD )(k) (t) = (J ∗)n ∗ u (x, t)dx, k D k=0 for any n ≥ 1, which is equivalent to what was obtained before. As consequence of the above theorem, we obtain the following nonlocal version of (6.1) and (6.2).

92

6 Nonlocal Heat Content

Corollary 6.5 Let u be the solution of (6.3) for the datum u0 = χ D , with a finite Lebesgue measurable set D in RN , then  1 lim u(x, t)dx = −(HJD ) (0) = PJ (D), (6.18) t →0+ t RN \D or equivalently, HJD (t) = |D| − PJ (D)t + o(t)

as t ↓ 0.

(6.19)

Remark 6.6 Let us make a comment on the relation of the semigroup TJ (t) and the operator J χ D . Observe that for φ(x, t) = TJ (t)χ D (x) − χ D (x), by (6.4) we have  φ(x, t)dx = 0 ∀ t ≥ 0, RN

hence

 RN

φ + (x, t)dx =

 RN

φ − (x, t)dx

∀ t ≥ 0.

Therefore, since φ + (x, t) = (TJ (t)χ D (x) − χ D (x))χ RN \D (x), we have  TJ (t)χ D − χ D L1 = 2

+

RN

φ (x, t)dx = 2

 RN \D

TJ (t)χ D (x)dx.

Then, by (6.18), we get lim

t →0+

1 TJ (t)χ D − χ D L1 = PJ (D). 2t

(6.20)

Note that this formula is similar to the one that holds for the classical heat content; see [61]. Now, since 1 PJ (D) = 2

 

RN

 RN

   

    J (x − y)χ D (y) − χ D (x)dydx   J (x − y)(χ D (y) − χ D (x))dy  dx

=

1 2

=

1 J χ D L1 , 2

RN

RN

6.2 The Heat Content for Nonlocal Diffusion with Nonsingular Kernels

93

we can write (6.20) as: lim

t →0+

1 TJ (t)χ D − χ D L1 = J χ D L1 . t

(6.21)

Moreover, since the operator B = J is the infinitesimal generator of the C0 -semigroup (TJ (t))t ≥0 in L2 (RN ), we have J χ D = lim

t →0+

1 (TJ (t)χ D − χ D ) t

in L2 (RN ).

(6.22)

Hence, (6.21) and (6.22) imply that also J χ D = lim

t →0+

6.2.3

1 (TJ (t)χ D − χ D ) t

in L1 (RN ).

Nonlocal Curvature in the Expansion

We have proved that HJD (0) + 2(HJD ) (0) + (HJD ) (0) =

  D

 RN

J (x − y)J (y − z)dzdydx.

(6.23)

D

Then, having in mind (6.9), we get (HJD ) (0) =

  D



 RN

J (x − y)J (y − z)dzdydx + D

D

J H∂D (x)dx.

(6.24)

Therefore, the Taylor’s expansion gives HJD (t)

 1 = |D| − PJ (D) t + HJ (x)dx t 2 2 D ∂D    1 + J (x − y)J (y − z)dzdydx t 2 + O(t 3 ) as t ↓ 0. 2 D RN D

Observe that, for n ≥ 2, we can express the coefficients (HJD )(n) (0) by using the nonlocal curvature as follows: (HJD )(n) (0) = (−1)n PJ (D)+ 1   n−1  (−1)n−1−k + 2 k n−1 k=1

 D

(J ∗)k H∂D (x)dx

 − D

(J ∗)(k+1) H∂D (x)dx



(6.25) .

94

6 Nonlocal Heat Content

Indeed,   n =− (−1)n−k P(J ∗)k (D) k k=1     n−1  n−1 n−1 n−1−k = (−1) + P(J ∗)k (D) − P(J ∗)n (D) k−1 k k=1

(HJD )(n) (0)

n 

= (−1)n PJ (D) +

n−1     n−1  (−1)n−1−k P(J ∗)k (D) − P(J ∗)(k+1) (D) , k k=1

and hence, using (6.9), we can write 1 P(J ∗)k (D) − P(J ∗)(k+1) (D) = 2

 D

(J ∗)k H∂D (x)dx

 − D

(J ∗)(k+1) H∂D (x)dx



to get (6.25).

6.2.4

A Probabilistic Interpretation

Let us show that the formula given in (6.15), that is, HJD (t) =

+∞    k=0

D D

(J ∗)k (x −y)dydx

 −t k e t

k!

,

has a probabilistic interpretation. As mentioned after giving Definition 6.1 of the J -heat content, if J (x − y) is thought of as the probability distribution of jumping from location y to location x, then  RN

J (x − s)J (s − y)ds

is the probability of jumping from location y to location x in two jumps (the probability of passing through a point s is J (x − s)J (s − y) and we integrate for s ∈ RN ). Further,  RN

 RN

J (x − s)J (s − w)J (w − y)dsdw

is the probability of jumping from location y to location x in three jumps. Let us call these jumps as J -jumps.

6.2 The Heat Content for Nonlocal Diffusion with Nonsingular Kernels

95

Then, from J (x − y) we obtain the probability of a transition from x to D in k steps or J -jumps as:  F

(k)

(x, D) =

(J ∗)k (x −y)dy, D

and we can also set F (0) (x, D) = χ D (x). Observe that F (k) (x, D) also determines how many matter of D goes to x after k jumps, even for k = 0. Also, if we define   a(0, D) =

(J ∗)0 (x −y)dydx = |D|, D D

and   (J ∗)k (x −y)dydx,

a(k, D) =

k = 1, 2, . . . ,

D D

we have that a(k, D) is the amount of matter of D remaining in D after k jumps for any k ≥ 0. From (6.11), we have that u(x, t) is the expected value of the amount of matter of D that goes to x when this matter moves by J -jumps, and the number of J -jumps up to time t, Nt , follows a Poisson distribution with rate t: u(x, t) =

+∞ 

F (k) (x, D)

k=0

e−t t k . k!

It is well known that this function is the transition probability of a pseudo-Poisson process of intensity 1 (see [44, Ch. X]). From (6.15), we have that HJD (t) =

+∞  k=0

a(k, D)

e−t t k k!

is the expected value of the amount of matter of D that remains in D when this matter moves by J -jumps and the number of J -jumps up to time t follows a Poisson distribution with rate t. We can write it as: HJD (t) = E(a(Nt , D)).

96

6 Nonlocal Heat Content

Let us decompose f (k) for k ≥ 2 as: a(k, D) = b(k, D) + c(k, D) where   b(k, D) = D

 D k−1

J (x − x2 )J (x2 − x3 ) · · · J (xk − y)dydxk · · · dx3 dx2 dx D

and   c(k, D) =



D RN−1 \D N−1 D

J (x − x2 )J (x2 − x3 ) · · · J (xn − y)dydxn · · · dx3 dx2dx,

being D n−1 = D × D × · · · × D (n − 1)-times. Let us also define b(0, D) = |D| and   b(1, D) =

J (x −y)dydx. D D

We have that b(k, D) represents the amount of matter of D remaining in D after k jumps all inside D. On the other hand, c(k, D) represents the amount of matter of D remaining in D after k jumps with at least one outside D, of course this has only sense for k ≥ 2. Set accordingly c(0, D) = c(1, D) = 0. Then, we can decompose the J -heat content as: HJD (t) = QJD (t) + RJD (t), where QJD (t) =

+∞ 

b(k, D)

k=0

e−t t k k!

and RJD (t) =

+∞  k=0

c(k, D)

e−t t k . k!

6.2 The Heat Content for Nonlocal Diffusion with Nonsingular Kernels

97

The function QJD (t) is the spectral J -heat content of D and will be studied in Sect. 6.4. We have that QJD (t) < HJD (t)

6.2.5

for all t > 0.

The J -heat Loss of D

Using (6.7), from (6.15) we can get the following expansion for the nonlocal J -heat loss of D in RN at t (see [70]): |D| − HJD (t) =

+∞  k=0

P(J ∗)k (D)

e−t t k , k!

where we set P(J ∗)0 (D) = 0. Using the above notation, we have |D| − HJD (t) = E(g(Nt )), where g(k) = P(J ∗)k (D).

6.2.6

The J -heat Content and the Nonlocal Isoperimetric Inequality

From the isoperimetric inequality given in Theorem 2.1, we have the following result. Corollary 6.7 Let J be radially nonincreasing and having compact support Bδ (0). For any bounded subset D ⊂ RN with |D| > 2δ , we have HJD (t) ≤ HJBr (t)

for small t > 0,

where Br is a ball such that |Br | = |D|. Proof The result is true when D is a ball of radius r, so let us suppose that this is not the case. Then, on account of point 2. of Remark 3.12, we have HJBr (0) = |D| = HJD (0) and (HJBr ) (0) = −PJ (Br ) > −PJ (D) = (HJD ) (0).

98

6 Nonlocal Heat Content

Then, by (6.19), we get HJD (t) < HJBr (t)

for small t > 0,  

and the result follows.

As consequence of Corollary 6.7 and Proposition 6.2, we have the following characterization. Corollary 6.8 Let J be radially nonincreasing and having compact support Bδ (0). The Isoperimetric Inequality (2.2) is equivalent to the inequality: TJ (t)χ D L2 ≤ TJ (t)χ Br L2

for small t > 0,

with Br being a ball such that |Br | = |D| when r > 2δ . A similar result for the local case was proved in [64] (see also [15] and [55]).

6.3

Convergence to the Heat Content when Rescaling the Kernel

For a subset D in RN with finite Lebesgue measure, we will call  HJ,α D (t) =

u(x, t)dx D

the J -heat content of α-intensity of D, where u is the solution of ⎧  ⎪ ⎪ u (x, t) = α ⎨ t

RN

J (x − y)(u(y, t) − u(x, t))dy, x ∈ RN , t ∈ [0, ∞),

⎪ ⎪ ⎩ u(x, 0) = χ (x), D

x ∈ RN .

Observe that HJD (t) is the J -heat content of 1-intensity of D; also, J HJ,α D (t) = HD (αt).

Let v be the solution of ⎧ 1 ⎪ ⎪ ⎨ (v )t (x, t) = 2 [J ∗ v (x, t) − v (x, t)] , x ∈ RN , t ∈ [0, ∞),  ⎪ ⎪ ⎩ v (x, 0) = χ (x), x ∈ RN .  D

6.3 Convergence to the Heat Content when Rescaling the Kernel

99

By Andreu-Vaillo et al. [12, Theorem 1.30], for J radially nonincreasing and compactly supported, we have lim v − vL∞ (RN ×(0,T )) = 0

→0+

for every T > 0, with v being the solution of the heat equation: ⎧ ⎪ ⎪ ⎨ vt (x, t) =

1 v(x, t), x ∈ RN , t ∈ [0, ∞), CJ,2

⎪ ⎪ ⎩ v(x, 0) = χ (x), D

x ∈ RN ,

with 2

CJ,2 =  RN

(6.26)

.

J (x)|xN |2 dx

Set now u(x, t) = v(x, CJ,2 t). Then, u verifies ⎧ N ⎪ ⎨ ut (x, t) = u(x, t), x ∈ R , t ∈ [0, ∞), ⎪ ⎩

u(x, 0) = χ D (x),

x ∈ RN .

Hence, for the solution u of the problem: ⎧ ⎪ ⎨ (u )t (x, t) = ⎪ ⎩

CJ,2 2

(J ∗ u (x, t) − u (x, t)) , x ∈ RN , t ∈ [0, ∞),

u (x, 0) = χ D (x),

x ∈ RN ,

we have that J ,

HD

CJ,2 2





(t) =

 v x, CJ,2 t dx,

u (x, t) dx = D

D

and  lim

→0 D

 v x, CJ,2 t dx =



 v x, CJ,2 t dx =

D

Consequently, we have proved the following result:

 u (x, t) dx = HD (t). D

100

6 Nonlocal Heat Content

Theorem 6.9 Assume J is radially nonincreasing and compactly supported. For a subset D in RN with finite Lebesgue measure, we have J ,

lim HD

CJ,2 2

→0+

(t) = HD (t)

for all t > 0.

That is, if the jumps are rescaled to occur in a ball of radius  and the intensity of the Poisson process that controls the intensity of the jumps is rescaled to the size C 2J , then we are approaching, for  small, the Gaussian heat content. Remark 6.10 In Theorem 1.11, it is shown that, under hypothesis (1.19) on J , lim CJ PJ (E) = P (E), ↓0

for a bounded set E ⊂ RN of finite perimeter. Observe that (J ∗)k = [(J ∗)k ] . Then, by Theorem 6.4, for n ≥ 1, we have (HJD )(n) (0) = −

n    n (−1)n−k P[(J ∗)k ] (D), k k=1

and consequently, by (6.27), there holds 1 J (n) (HD ) (0) = cn Per(D), →0  lim

where  n    1 n ck = (J ∗)n (x)|xN |dx , 2 RN k k=1 that is, 1 n cn = − (−1)n−k 2 k n

k=1

 RN

(J ∗)k (x)|xN |dx .

(6.27)

6.4 The Spectral Heat Content

6.4

101

The Spectral Heat Content

The spectral heat content is given by:  QD (t) =

u(x, t)dx D

for the solution u of the Dirichlet problem: ⎧ ⎪ u (x, t) = u(x, t), (x, t) ∈ D × [0, ∞), ⎪ ⎪ t ⎪ ⎪ ⎪ ⎨ u(x, t) = 0, (x, t) ∈ ∂D × (0, ∞), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u(x, 0) = χ D (x) x ∈ D. The following result was given in [71] for smooth bounded domains: √ 2 1 QD (t) = |D| − √ Per(D) t + (N − 1) 2 π

 H∂D (x)dHN−1 (x) t + O(t 3/2),

t ↓ 0,

∂D

where H∂D is the mean curvature of ∂D. In [12], the following nonlocal Dirichlet problem was also studied: ⎧  ⎪ ⎪ u (x, t) = J (x − y)(u(y, t) − u(x, t))dy, (x, t) ∈ D × [0, ∞), ⎪ t ⎪ ⎪ RN ⎪ ⎪ ⎨ u(x, t) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u(x, 0) = u (x) 0

(x, t) ∈ (RN \ D) × [0, ∞), x ∈ RN . (6.28)

Therefore, we can also define the spectral J -heat content as:  QJD (t) =

u(x, t)dx,

(6.29)

D

with u being the solution of (6.28) for the datum u0 = χ D . Observe that also QJD (0) = |D|.

(6.30)

102

6 Nonlocal Heat Content

We will describe in this section which is the asymptotic expansion of QJD (t). The first term in the expansion is given in (6.30). For the second term, it is easy to get again, from the derivative of (6.29), that QJD (t) + (QJD ) (t) =

  J (x − y)u(y, t)dydx, D

(6.31)

D

consequently,  

QJD (0) + (QJD ) (0) =

J (x − y)dydx, D

D

and hence, (QJD ) (0) = −PJ (D). Note that the first two terms in the expansion of QJD (t) and of HJD (t) coincide. Now, the expression for the next terms differs from that of the J -heat content. Taking derivatives in (6.31), we get QJD (t) + 2(QJD ) (t) + (QJD ) (t) =

   J (x − y)J (y − z)u(z, t)dzdydx. D

D

D

Then, for example, now, instead of (6.23) we have QJD (0) + 2(QJD ) (0) + (QJD ) (0) =

   J (x − y)J (y − z)dzdydx, D

D

D

Hence, (QJD ) (0)

   =

J (x − y)J (y − z)dzdydx + 2PJ (D) − |D|, D

D

D

that is, (QJD ) (0) =

  

 J (x − y)J (y − z)dzdydx +

D

D

D

D

J H∂D (x)dx,

and this term is different from (6.24) in the term with three integrals. Gathering this information, we have the following result.

(6.32)

6.4 The Spectral Heat Content

103

Theorem 6.11 For QJD (t) the J-spectral heat content of D, it holds that  1 QJD (t) = |D| − PJ (D) t + HJ (x)dx t 2 2 D ∂D    1 + J (x − y)J (y − z)dzdydx t 2 + O(t 3 )as t ↓ 0, 2 D D D J (x) is the J-mean curvature at x. where H∂D

Now, similarly to the J -heat content we also have the following complete expansion for the spectral J -heat content. Theorem 6.12 Let D be a finite measurable set. Then, QJD (t) =

+∞ 

b(D, n)

n=0

e−t t n n!

∀t > 0,

where b(0, D) = |D| and, for n ≥ 1,  b(n, D) = J (x1 − x2 )J (x2 − x3 ) · · · J (xn − xn+1 )dxn+1 dxn · · · dx3 dx2 dx1 , D n+1

being D n+1 = D × D × · · · × D, (n + 1)-times. Proof Taking into account the procedure used to get (6.31) and (6.32), by induction, we obtain n    n k=0

k

(QJD )(k) (t) = b(n, D)

for any n ≥ 1. And from here, n    n (QJD )(k) (0) = b(n, D) k k=0

for any n ≥ 0. Therefore using the Cauchy product of series and (6.33), we get that e

t

QJD (t)

=

 n +∞   n=0

=

+∞  n=0

k=0

 (QJD )(k) (0) n 1 t (n − k)! k!

b(D, n)

tn n!

(6.33)

104

6 Nonlocal Heat Content

 

for all t > 0. And, the proof is finished. Set J QJ,α D (t) = QD (αt).

Following the same steps as in Sect. 6.3 and using now [12, Theorem 2.13], we obtain the following rescaling result between the spectral J -heat content and the spectral heat content. Theorem 6.13 Assume J is radially nonincreasing and compactly supported. For a bounded smooth domain D in RN , we have J ,

lim QD

CJ,2 2

→0+

(t) = QD (t)

for all t > 0,

where CJ,2 is given in (6.26).

6.5

Lack of Regularity for the Nonlocal Heat Diffusion

It is well known that there is no regularizing effect on the solutions T (t)u0 of ⎧  ⎪ ⎪ ⎨ ut (x, t) =

RN

J (x − y)(u(y, t) − u(x, t))dy, (x, t) ∈ RN × [0, +∞),

⎪ ⎪ ⎩ u(x, 0) = u (x), 0

x ∈ RN .

We will see in this section some facts on this question. Let D be a finite measurable set. Let u(·, t) = TJ (t)χ D . Using (6.8), we can express (6.11) as follows: ∞

u(x, t) =

tn 1 1 −t  (J ∗)n − e H∂D (x) , 2 2 n!

(6.34)

n=0

(J ∗)0

where H∂D (x) = 1 − 2χ D (x). Proposition 6.14 Let D be a finite measurable set. Let u(·, t) = TJ (t)χ D . Given any level α between 0 and 1, then 

 x ∈ RN : u(x, t) > α = D

for all 0 ≤ t < min{ln(1/α), ln(1/(1 − α))}.

6.5 Lack of Regularity for the Nonlocal Heat Diffusion

105

Proof From (6.34), we have ∞   

 tn (J ∗)n H∂D (x) < (1 − 2α)et x ∈ RN : u(x, t) > α = x ∈ RN : n! n=0

∞ 

 tn (J ∗)n = x∈D : H∂D (x) < 1 + (1 − 2α)et n! n=1

∞

  tn (J ∗)n ∪ x ∈ RN \ D : H∂D (x) < −1 + (1 − 2α)et . n! n=1

(J ∗) Now, since −1 ≤ H∂D ≤ 1, we have that n

−(et − 1) ≤

∞ 

(J ∗)n

H∂D (x)

n=1

tn ≤ et − 1. n!

Then, on one hand, et − 1 < 1 + (1 − 2α)et

⇔

t < ln(1/α)

Hence, for t < ln(1/α), ∞ 

(J ∗)n

H∂D (x)

n=1

tn < 1 + (1 − 2α)et . n!

On the other hand, −(et − 1) > −1 + (1 − 2α)et

⇔

t < ln(1/(1 − α)).

Hence, for t ≤ ln(1/(1 − α)), ∞ 

(J ∗) H∂D (x) n

n=1

tn > −1 + (1 − 2α)et . n!

Consequently, for t < min{ln(1/α), ln(1/(1 − α))}, we have that 

 x ∈ RN : u(x, t) > α = D.  

106

6 Nonlocal Heat Content

Theorem 6.15 Let D be a bounded open set and u(·, t) = TJ (t)χ D . For any x ∈ ∂D, u(x − , t) − u(x + , t) = e−t , where u(x − , t) =

lim

u(y, t)

lim

u(y, t).

y→x, y∈D

and u(x + , t) =

y→x, y ∈ /D

That is, for any time t there is a jump on the boundary ∂D in the solution of (6.3). This jump has the same height e−t at every point. (J ∗)0

Proof From (6.34), since H∂D (x) = 1 − 2χ D (x), and the continuity of the curvature, we have u(x − , t) =

∞

1 1 −t 1 −t  (J ∗)n tn + e − e H∂D (x) 2 2 2 n! n=1

and u(x + , t) =

∞

1 1 −t 1 −t  (J ∗)n tn H∂D (x) . − e − e 2 2 2 n! n=1

 

Hence, the result is straightforward.

The initial velocities on the solution depend on the nonlocal J -curvature at each point. This follows directly from (6.5). Proposition 6.16 Let D be a finite measurable set and u(·, t) = TJ (t)χ D . Then, 1 J ut (x, 0) = − (1 + H∂D (x)) ≤ 0 2 ut (x, 0) =

1 J (1 − H∂D (x)) ≥ 0 2

for x ∈ D, for x ∈ / D.

This implies that  

1 J . x : H∂D (x) < 0 = x : |ut (x, 0)| > 2

7

A Nonlocal Mean Curvature Flow

Consider a family {t }t ≥0 of hypersurfaces embedded in RN parametrized by time t. Assume that each t = ∂Et , the boundary of a bounded open set Et in RN . A geometric flow is an evolution for the sets t → Et governed by a law of the form: V (x, t) = −K(x, Et ),

(7.1)

called surface evolution equation, where V (x, t) stands for the (outer) normal velocity of ∂Et at x and K(·, E) is some generalized curvature of ∂E. If the function K(x, E) depends only on how ∂E looks around x, the flow is called local, this is the case of the classical mean curvature flow, where K(x, E) = H∂E (x) is the mean curvature of ∂E at x. Now, for some relevant geometric flows, K(x, E) is truly nonlocal and depends on the global shape of the evolving set Et itself. It happens for instance for fractional mean curvature flows, where  K(x, E) = lim s(1 − s) δ↓0

χ RN \E (y) − χ E (y)− RN \Bδ (x)

|x − y|N+s

dy,

that has been recently studied by Saez and Valdinoci [65]. In [32], Chambolle, Morini and Ponsiglione study nonlocal curvature flows of the form (7.1) for some general class of generalized curvatures. Let J : RN → [0, +∞[ be a measurable, nonnegative and radially nonincreasing symmetric function verifying  RN

J (z)dz = 1.

© Springer Nature Switzerland AG 2019 J. M. Mazón et al., Nonlocal Perimeter, Curvature and Minimal Surfaces for Measurable Sets, Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-06243-9_7

107

108

7 A Nonlocal Mean Curvature Flow

In Chap. 3, associated with J , it has been studied the J -curvature of a measurable set E ⊂ RN :   J H∂E (x) = − J (x − y)(χ E (y) − χ RN \E (y))dy = 1 − 2 J (x − y)dy. RN

E

J In the case that J is continuous, K(x, E) = H∂E (x) is one of the generalized curvatures considered in [32] and therefore the nonlocal mean curvature flow: J (x). V (x, t) = −H∂E t

(7.2)

As consequence of the results in [32], it is well known the level set formulation of (7.2) and also it is studied a generalized Almgren–Taylor–Wang minimizing movements scheme to approximate the geometric motion associated with (7.2). It is not our aim here to continue with the study of problem (7.2) but to state a similar problem in which also the normal velocity is nonlocal and has a probabilistic interpretation. Note that in the general problem (7.1), the normal velocity V (x, t) is given by the velocity in the direction of the (outer) unitary normal vector of t at x, that we will denote as ν∂E (x), which implies to assume some smoothness of t . To define the nonlocal normal vector, we do not need any regularity of the set.

7.1

Nonlocal Normal Vector

We define the nonlocal normal vector in the following way. Definition 7.1 Given a measurable set E ⊂ RN , we define the J-normal (inward) vector as: 

 νEJ (x)

=

i=N

J (x − y)(y − x) dy =

J (x − y)(yi − xi ) dy

E

E

for x ∈ RN . i=1

Observe that this vector, as the J -curvature, is defined for any point, not necessarily for points on the boundary. If we assume that we are working with a homogeneous population with density 1, and that J (x − y) is the probability that an individual at y jumps to x, then  J (x − y)dy E

7.1 Nonlocal Normal Vector

109

gives the rate of individuals leaving x for going to any other site in E (or the rate of individuals going from E onto x), and 1 J (x − y)dy E



 J (x − y)ydy E

gives the expected place where individuals move from x onto E. As an expected value, it can be not a value in E. Then, the J -normal vector that can be written as: 

 νEJ (x) =

J (x − y)dy

1 E J (x − y)dy



E

 J (x − y)y dy − x ,

 E

is the expected direction that individuals take when they move from x onto E following the law J , modulated in such a way that νEJ (x) = −νRJ N \E (x). Definition 7.2 We say that a measurable set E ⊂ RN is J-regular if νEJ (x) = 0 for all x ∈ ∂E. Definition 7.3 For a J -regular set E, we define the unit (inward) J-normal vector as: J (x) = ν∂E

νEJ (x) |νEJ (x)|

,

x ∈ ∂E.

The notation is similar but the unit normal has ∂E as subindex. Observe that J ν∂E (x) = −ν∂J (RN \E) (x).

Example 7.4 We claim that νBJ r (0) (x) = −C(J, r)

x r

for x ∈ ∂Br (0),

where C(J, r) is the following positive constant (observe that w1 > 0 in Br (re1 )):  C(J, r) :=

J (w)w1 dw. Br (re1 )

In fact,  νBJ r (0) (x)



=

J (x − y)(y − x) dy = − Br (0)

J (z)z dz. Br (x)

(7.3)

110

7 A Nonlocal Mean Curvature Flow

Let Rx be the rotation in RN such that Rx (re1 ) = x, and assume that (ai,j ) is the matrix associated to the rotation Rx . Then, changing variables, using that J is radial, we obtain 





J (z)zi dz =

J (Rx (w))Rx (w)i dw =

Br (x)

Br (re1 )

=

N 

J (w)Rx (w)i dw Br (re1 )



 J (w)wj dw = ai,1

ai,j Br (re1 )

j =1

J (w)w1 dw. Br (re1 )

Now, since Rx (re1 ) = x, we have x = r(a1,1 , a2,1 , . . . , aN,1 ). Therefore,  J (z)zi dz = Br (x)

xi r

 J (w)w1 dw Br (re1 )

and we conclude the proof of (7.3). Example 7.5 Let us now compute this nonlocal normal vector for a set with nonsmooth boundary. Consider the square E = conv{(0, 0), (2, 0), (2, 2), (0, 2)}, and J = 1 χ |B1 (0)| B1 (0) . Then, for 0 ≤ s ≤ 1, we have  νEJ (s, 0) =

i=2

:=

1 π

J ((s, 0) − (y1 , y2 ))(yi − xi ) dy E



i=1

 E

χ B1 ((s,0))(y1 , y2 )(y1 − s) dy,

E



χ B1 ((s,0))(y1 , y2 )y2 dy

 1  2(1 − s 2 )3/2 , 2 + 3s − s 3 . = 6π Note that the classical normal vector is not defined on (0, 0), but  J ν∂E (0, 0) =

 1 1 √ ,√ . 2 2

Also, J ν∂E (s, 0) = (0, 1) = −ν∂E (s, 0),

if 0 < s < 1.

Observe that for this set, J H∂E (s, 0) = 1 −

2 π

 E

χ B1 ((s,0))(y)dy =

3 π − arcsin(s) − s 1 − s 2 . 2

7.2 Nonlocal Mean Curvature Flow

7.2

111

Nonlocal Mean Curvature Flow

We consider a J -regular set E0 and we are interested in a family of J -regular sets Et that satisfies, for every point x ∈ ∂E0 , the law of motion: ⎧ ∂X ⎪ J J ⎨ (x, t) = CJ H∂E (X(x, t))ν∂E (X(x, t)), t t ∂t ⎪ ⎩ X(x, 0) = x ∈ ∂E ,

t > 0, (7.4)

0

where X(x, t) parameterizes the point of the boundary ∂Et where x has moved after a time  −1 has been introduced in previous chapters, it t. The constant CJ = 2 RN J (z)|zN |dz is a rescaling constant whose role appears later. Problem (7.4) can be stated for any other positive constant. In terms of the probabilistic description given above, in (7.4) we are describing the movement of X(x, t) following the nonlocal mean curvature flow given by the J J (X(x, t)) times the push of the curvature H∂E (X(x, t)), which expected direction ν∂E t t is proportional to the difference between the rate of individuals arriving to X(x, t) from Et and the rate of individuals moving to X(x, t) from RN \ Et under the law J . Therefore, when the curvature is positive on the point X(x, t), the movement will point toward Et , in an expectation sense, and the contrary if the curvature is negative; in this last case, we can look at the flow as: J J J J H∂E (X(x, t))ν∂E (X(x, t)) = −H∂E (X(x, t))ν∂(R N \E ) (X(x, t)). t t t t

Note that if Et is a solution of (7.4), then Et is also a solution of ⎧ J J ⎨ ∂X ∂t (x, t) · ν∂Et (X(x, t)) = CJ H∂Et (X(x, t)), ⎩

t > 0, (7.5)

X(x, 0) = x ∈ ∂E0 ,

If we write VJ (x, t) =

∂X J (x, t) · ν∂E (X(x, t)) t ∂t

J for the J-normal velocity of Et at X(x, t) and at time t in the direction of ν∂E (X(x, t)), t then the geometrical evolution problem (7.5) can be written as:

VJ = CJ H∂J ν∂J . Let us see that a circumference that evolves by nonlocal mean curvature collapses in finite time as it does when it evolves by (local) mean curvature.

112

7 A Nonlocal Mean Curvature Flow

Example 7.6 Suppose that N = 2 and J = |Br1(0)| χ Br (0) . Let E0 = BR (0) and assume that the boundary of E0 evolves to the boundary of Et = BR(t )(0) by nonlocal mean curvature following (7.4). By Example 3.6, we have J H∂E (X(x, t)) = 1 − t

2|Br (X(x, t)) ∩ BR(t ) (0)| , |Br (0)|

which is always positive. On the other hand, by Example 7.4, we have  νEJ t (X(x, t)) =−

1 πr 2

=−

J (w)w1 dw BR(t) (R(t )e1)

 w1 dw BR(t) (R(t )e1 )∩Br (0)

X(x, t) R(t)

X(x, t) . R(t)

Therefore, in this case Problem (7.4) is ⎧ X(x,t ) ∂X ⎪ ⎨ ∂t (x, t) = −F (R(t)) R(t ) , ⎪ ⎩

t > 0,

X(x, 0) = x ∈ ∂E0 ,

with F (R(t)) := CJ 3π = 2r

  2|Br ((R(t), 0)) ∩ BR(t ) (0)| 1− |Br (0)|

  2|Br ((R(t), 0)) ∩ BR(t ) (0)| 1− . |Br (0)|

Hence,   t  X(x, t) = x exp − F (R(s))ds , 0

and   t  R(t) = |X(x, t)| = R exp − F (R(s))ds . 0

7.2 Nonlocal Mean Curvature Flow

113

Consequently, R(t) satisfies ⎧  ⎪ ⎨ R (t) = −F (R(t)), ⎪ ⎩

t > 0,

R(0) = R,

and it is a nonincreasing function of t. Assume r ≤ 2R, and let Tr = inf{t ≥ 0 : 2R(t) = r}. Then, r ≤ 2R(t) for all 0 ≤ t ≤ Tr . Let Q(s) = R(s)−1 . Then, for 0 ≤ t ≤ Tr , we have 

R

t = Q(R(t)) − Q(R) = −





R

Q (R(s))ds =

R(t )

R(t )

1 ds, F (R(s))

hence,  Tr =

R r 2

1 ds. F (R(s))

Now, for 0 ≤ t ≤ Tr , we have |Br ((R(t), 0)) ∩ BR(t ) (0)| −1

= r cos 2



r 2R(t)



−1

+ R(t) cos 2

 1−

r2 2R(t)2

 −

13 3 r (2R(t) + r), 2

and then, F (R(t)) = 1 1− πr 2

  2 −1 r cos

r 2R(t)



On the other hand, since R(t) ≤

−1

+ R(t) cos 2

r 2

 1−

r2 2R(t)2



 13 3 − r (2R(t) + r) . 2

for all t > Tr , we have

3π F (R(t)) = 2r

  2R(t)2 1− r2

for all t > Tr .

114

7 A Nonlocal Mean Curvature Flow

Consequently, R(t) satisfies ⎧  ⎪ ⎨ R (t) = ⎪ ⎩

3R(t )2 r3

−

3π 2r ,

for all t > Tr ,

R(Tr ) = 2r .

whose general solution is given by: √  √  r π − R(t) 6 π log √ = 2 t +C r r π + R(t)

for all t ≥ Tr .

Since R(Tr ) = 2r , we get √  √  2 π −1 6 π C = log √ − 2 Tr , 2 π +1 r and therefore, √  √  √  r π − R(t) 2 π +1 6 π log √ √ = 2 (t − Tr ) r r π + R(t) 2 π −1

for all t ≥ Tr .

From here,   √   √ 2 π −1 6 π (t − T ) = √ exp r r2 2 π +1

√ r π − R(t) √ r π + R(t)

and this collapses in finite time Tre

 √  r2 2 π +1 = Tr + √ log √ . 6 π 2 π −1

(7.6)

For the (local) √ mean curvature flow, the circle collapses at time R 2 /2 since in this case R(t) is given by R 2 − 2t. Observe that if we take r ↓ 0 in (7.6) we obtain also that collapsing time:  lim Tre = lim r↓0

r↓0

R r 2

1 ds = F (R(s))

Consider the rescaled kernel J (x) = 1N J chapters. In Example 7.4, we have seen that νBJ r (0) (x) = −C(J, r)

x

x r





R 0

sds =

R2 . 2

for  > 0, yet introduced in previous

for x ∈ ∂Br (0),

7.2 Nonlocal Mean Curvature Flow

where C(J, r) =

115



Br (re1 ) J (w)w1 dw.

1 1 CJ C(J , r) = CJ N  

 J

Now,

w 

Br (re1 )





2

w1 dw = 

J (z)|zN |dz

RN

B r ( r e1 )

(z)z1 dz,



and consequently, CJ C(J , r) → 1 as  → 0. Therefore, lim CJ νBJr (0) (x) = − ↓0

x = −νBr (0) (x). r

Let us see that this convergence result is true for smooth domains. Theorem 7.7 Let E ⊂ RN be a smooth set such that ∂E is of class C 2 . Then, for every x ∈ ∂E, we have lim CJ νEJ (x) = −ν∂E (x),

(7.7)

↓0

where ν∂E (x) is the outward unit normal vector at x ∈ ∂E. And, for the unit J -normal vector, we have J (x) = −ν∂E (x), lim ν∂E ↓0

Proof We can assume that x = 0 ∈ ∂E. Namely, suppose ∂E is described as a graph in normal coordinates, meaning that, in an open ball Br0 , ∂E coincides with the graph of a C 2 function ϕ : Br0 ∩ RN−1 → R with ϕ(0) = 0 and ∇ϕ(0) = 0 such that E ∩ Br0 = {(y1 , . . . , yN ) : yN < ϕ(y1 , . . . , yN−1 )}. Then, to prove (7.7), we show that  lim CJ J (y)y dy = (0, 0, . . . , 0, −1) = −ν∂E (0). ↓0

E

For  > 0 such that r ≤ r0 , we have  CJ =

CJ J (y)y dy = N  E

CJ  N+1

 J E∩Br (0)

 {yN