Elsevier

Nonlinear Analysis

Volume 211, October 2021, 112395
Nonlinear Analysis

Large mass minimizers for isoperimetric problems with integrable nonlocal potentials

https://doi.org/10.1016/j.na.2021.112395Get rights and content

Abstract

This paper is concerned with volume-constrained minimization problems derived from Gamow’s liquid drop model for the atomic nucleus, involving the competition of a perimeter term and repulsive nonlocal potentials. We consider a large class of potentials, given by general radial nonnegative kernels which are integrable on Rn, such as Bessel potentials, and study the behavior of the problem for large masses (i.e., volumes). Contrary to the small mass case, where the nonlocal term becomes negligible compared to the perimeter, here the nonlocal term explodes compared to it. However, using the integrability of those kernels, we rewrite the problem as the minimization of the difference between the classical perimeter and a nonlocal perimeter, which converges to a multiple of the classical perimeter as the mass goes to infinity. Renormalizing to a fixed volume, we show that, if the first moment of the kernels is smaller than an explicit threshold, the problem admits minimizers of arbitrarily large mass, which contrasts with the usual case of Riesz potentials. In addition, we prove that, any sequence of minimizers converges to the ball as the mass goes to infinity. Finally, we study the stability of the ball, and show that our threshold on the first moment of the kernels is sharp in the sense that large balls go from stable to unstable. A direct consequence of the instability of large balls above this threshold is that there exist nontrivial compactly supported kernels for which the problems admit minimizers which are not balls, that is, symmetry breaking occurs.

Introduction

We study large mass minimizers for a variant of Gamow’s liquid drop model for the atomic nucleus, in which the repulsive term is given by a general nonnegative, integrable, radial kernel. More precisely, given G:Rn{0}[0,+) a measurable nonnegative radial function with GL1(Rn) and n2, we originally consider the minimization problem minP(E)+E×EG(xy)dxdy:|E|=m,where the minimum is taken over all sets of finite perimeter of volume |E|=m – which we call the mass – and P(E) denotes the perimeter of E. Observe that this problem exhibits a competition between two terms and is thus nontrivial: the local perimeter term constrains the set E to concentrate as much as possible, while the nonlocal term acts as a repulsive term, forcing E to spread. Indeed, it is known that the perimeter is minimized by balls under volume constraint, while the nonlocal term is maximized by balls if G is in addition radially nonincreasing1 (by Riesz’s symmetric rearrangement, using e.g. [25, Chapter 3.7] and the fact that G is equal to its symmetric rearrangement in that case).

As we show just below, the integrability assumption on G allows us to reformulate this problem as min{P(E)γPerGλ(E):|E|=|B1|},where B1 stands for the open unit ball of Rn centered at the origin, λ and γ are positive constants, the kernel Gλ is given by Gλ()λn+1G(λ),and the functional PerG is defined by PerG(E)E×EcG(xy)dxdy.The parameter λ>0 will represent the mass (to the power 1n), and γ>0 will be chosen to adjust the first moment of G (that is, its integral against the measure |x|dx) to a particular universal constant. As will be justified later on, PerG should be considered as a “nonlocal perimeter”, which behaves in several ways as a standard perimeter term rather than as a volume term.

Before elaborating on the reformulation, let us say a few words on the original problem (). This problem has been studied extensively in the literature when G is a Riesz kernel (whose definition is recalled in Section 2.2) or a general integrable kernel with compact support. In the Riesz case, it is known that the problem admits the ball as unique minimizer below a critical mass, up to translations, and it is conjectured that there is no minimizer above a (possibly different) critical mass. This conjecture has already been proven in a few cases. We will discuss the Riesz and compact support cases further, and provide references, in the next section.

In this paper, we are interested in kernels decreasing faster than Riesz kernels at infinity, enough to make them integrable, but not necessarily compactly supported, and we focus exclusively on the case of large masses. Let us remark that, even if the kernel decreases rapidly at infinity, the asymptotic behavior of the problem for large masses is very different than that of small masses: indeed, in the case of Riesz (or Bessel) kernels, the nonlocal term is negligible compared to the perimeter as the mass vanishes, so that the problem consists of minimizing the perimeter plus a vanishing perturbation; here, the nonlocal term explodes compared to the perimeter as the mass goes to infinity, as can be seen by writing P(E)+E×EG(xy)dxdy=λn1P(F)+λn+1F×FG(λ(xy))dxdy=λn1(P(F)+λGL1(Rn)PerGλ(F)),where EλF with |F|=|B1|, since PerGλ(F) is of the same order as P(F), as we will see.

Rewriting the nonlocal repulsive term as E×EG(xy)dxdy=mGL1(Rn)E×EcG(xy)dxdy,we see that () is in fact strictly equivalent to min{P(E)PerG(E):|E|=m}.Now, to further normalize our problem, for k{0,1}, we define the quantity IGkRn|x|kG(x)dx,and for every positive natural numbers p and n, we denote by Kp,n the constant defined by Kp,nSn1|ex|pdn1(x),which does not depend on eSn1 by symmetry. Here Sn1 denotes the unit sphere in Rn and n1 the (n1)-dimensional Hausdorff measure. Now, up to dividing G by the constant γ=(IG1K1,n)2, without renaming G, we may assume that IG1=2K1,n,and may look instead at the problem min{P(E)γPerG(E):|E|=m},where γ>0. The choice of the constant in (1.3) will be justified in Section 3.1. Keep in mind that the parameter γ plays the role of a constant times IG1, the first moment of G. As a last simplification step, in order to study the asymptotic behavior when the mass goes to infinity, it is more convenient to look at the rescaled problem with fixed mass equal to the volume of the unit ball in Rn. Given m>0, setting λm|B1|1n and Fλ1E, it is then easy to see that |F|=|B1|, IGλ1=IG1, and by a change of variables P(E)γPerG(E)=λn1(P(F)γPerGλ(F)),so that the set E is a minimizer of (1.4) if and only if F is a minimizer of the rescaled problem (Pγ,λ). Let us emphasize that even in the reformulation (Pγ,λ), the nonlocal perturbation still does not vanish as λ goes to infinity, contrary to the Riesz case for small masses, and our results hold for any γ(0,1). In the rest of the paper, we shall always work with this equivalent formulation.

Except in Section 2, we shall always assume that G satisfies the two following general hypotheses:

  • (H1)

    G is radial, that is, there exists a nonnegative function g:(0,+)R such that G(x)=g(|x|) for na.e.xRn;

  • (H2)

    GL1(Rn), and IG1=2K1,n.

Starting from Section 5, dedicated to the study of the stability of the ball, we may add the extra assumptions

  • (H3)

    G(x)=o(|x|αn) near the origin, for some α>0;

  • (H4)

    G(x)=o(|x|(n+β)) at infinity for some β>0 when n3, and G(x)=o(|x|3) at infinity when n=2;

  • (H5)

    GC1(Rn{0}).

These extra assumptions are required essentially in order to be able to use directly computations from [14] for the second variation of the nonlocal perimeter. The stronger assumption (H4) in dimension 2 is due to the presence of a Jacobian determinant when integrating on the sphere, which appears to be singular only in dimension 2 (see Lemma 5.9). As we will see, Bessel kernels satisfy those general assumptions (see Sections 2.4 Intermediate case: Bessel kernels, 3.2 Bessel kernels).

We are interested in the asymptotic behavior of the minimization problem (Pγ,λ) as λ (that is, the mass) goes to infinity, and give answers to several natural questions: does (Pγ,λ) admit a minimizer? If so, what do minimizers look like, are they regular? Can the unit ball be a minimizer? We decided to state in a concise manner just below three of the main results obtained in the paper, and an application to Bessel kernels, however these results are not necessarily arranged in the same way in the paper.

Theorem A Consequence of Theorem 4.5, Theorem 4.14

Assume γ<1. Then there exists λe=λe(n,γ,G) such that, for any λ>λe, (Pγ,λ) admits a minimizer, and in addition, minimizers have a C1,12 reduced boundary and are essentially connected.  The dependency of me in G is not known. It depends on the speed of convergence of η(λ) to zero.

Let us point out that the regularity of the reduced boundary is actually true for any minimizer, if one exists, no matter the value of γ and λ. Connectedness of minimizers also holds in any case, provided that G is strictly positive (see Theorem 4.14). In terms of the original problem (), Theorem A means that if IG1<2K1,n, then there exists a critical mass me=me(n,G), such that, above this mass, the problem admits a minimizer. To our knowledge, this is the first time existence of minimizers of arbitrarily large mass is obtained in a Gamow-type problem on the whole space for non-compactly supported kernels without the presence of an external attractive background potential, as is often the case (see e.g. [1], [2], [17]). Let us also mention [3], where it is shown that, if the perimeter is weighted by a power-law density growing sufficiently fast at infinity, then minimizers always exist, and are balls in the large mass regime.

The main obstacle for proving existence with the direct method in the calculus of variations is the possibility for a minimizing sequence to have some mass escape at infinity. We solve this problem by showing that, for large values of λ, a minimizing sequence may be constrained inside a ball via a truncation lemma. This relies heavily on the fact that the nonlocal perimeter behaves to some extent as the classical perimeter and converges to it as λ goes to infinity. Note that the general kernels we consider (and in particular Bessel kernels) do not behave as nicely as Riesz kernels under scaling, which are homogeneous. The C1,12-regularity of the reduced boundary of minimizers follows by results in [29] on quasi-minimizers for the perimeter.

For any γ and λ, let us define the functional to be minimized Fγ,Gλ(E)P(E)γPerGλ(E).When γ[0,1), we are able to compute the Γ-limit in L1 of these functionals as λ goes to infinity, and we show independently that, up to translations, any sequence of minimizers converges to the unit ball. More precisely, we prove that minimizers are included in the set difference between two balls whose radii converge to 1 as λ goes to infinity, which implies in particular Hausdorff convergence of the boundaries of minimizers to the unit sphere, up to translations.

Theorem B See Theorem 4.5, Theorem 4.12 for More Precise Statements

Assume γ<1. For any minimizer E of (Pγ,λ) with λ>λe, up to a translation, we have B¯1ηγ,G(λ)EB1+ηγ,G(λ),where ηγ,G is a function depending only on n, γ and G which vanishes at infinity. In addition, the family of functionals Fγ,Gλ (with the added constraint of being the indicator function of a set of finite perimeter with volume |B1|) Γ-converges in L1 to 1γP as λ goes to infinity.

Of course, the Γ-convergence of the functional to a positive multiple of the perimeter implies that any converging sequence of minimizers of (Pγ,λ) with λ converges to the unit ball, but (1.6) is stronger and in fact a direct consequence of the proof of existence above λe.

Then we recall a well-suited notion of stability for functionals on sets under volume constraint (see Definition 5.4), and show that the threshold γ=1 is a stability threshold of the unit ball for (Pγ,λ) for large values of λ.

Theorem C Consequence of Theorem 5.8, Theorem 5.14

Assume that G satisfies all the hypotheses (H1) to (H5). Then the following hold:

  • (i)

    if γ<1, then there exists λs=λs(n,γ,G)>0 such that, for any λ>λs, B1 is a (critical) stable set for Fγ,Gλ;

  • (ii)

    if γ>1, then there exists λu=λu(n,γ,G) such that, for any λ>λu, B1 is a (critical) unstable set for Fγ,Gλ: in particular, it cannot be a minimizer, i.e., symmetry-breaking occurs.

In terms of the original problem (), this means that the threshold 2K1,n for IG1 is a threshold for which large balls go from stable to unstable.

The proofs for the stability and instability of the ball rely essentially on the two following ingredients:

  • (i)

    the decomposition in spherical harmonics of the Jacobi operator associated with the second variation of the perimeter and of the nonlocal term (given by the so-called Funk–Hecke formula for the latter);

  • (ii)

    results analogous to the one by J. Bourgain, H. Brezis, and P. Mironescu in [9] for Sobolev spaces on spheres, that is, computation of the limit and of a sharp “asymptotic” upper bound for the quantity Sn1×Sn1|f(x)f(y)|2|xy|2ηε(xy)dxn1dyn1,where (ηε)ε>0 is a (n1)-dimensional approximation of identity, and f belongs to H1(Sn1).

A particularly interesting consequence of Theorem C is that there exist kernels for which (Pγ,λ) admits nontrivial minimizers, that is, minimizers which are not balls. Indeed, working from the formulation (), S. Rigot proved in [29] that (Pγ,λ) always admits a minimizer whenever G is compactly supported. Hence, taking γ>1, minimizers still exist but cannot be the unit ball when λ is large enough, since it is unstable.

Theorem A, Theorem B, Theorem C directly apply when G is a so-called Bessel kernel: for every α,κ>0, we denote by Bκ,α the Bessel kernel of order α defined as the fundamental solution of the operator (IκΔ)α2, that is, (IκΔ)α2Bκ,α=δ0 in D(Rn),where δ0 is the Dirac distribution at the origin. We then have:

Application D Corollary 3.12 and Lemma 3.13

For every κ,α(0,+), we consider the problem () with G=Bκ,α. Let us define καπ(n+1)Γα22Γ1+α22.Then we have:

  • (i)

    if κ<κα, there exist me=me(α,κ)>0 and ms=ms(α,κ)>0 such that, for every m>me, () admits a minimizer, and for every m>ms, the ball of volume m centered at the origin, denoted by [B]m, is a stable critical point for the functional of (). In addition, rescaling minimizers so that they are of volume |B1|, and translating them, they converge to the unit ball as m goes to infinity;

  • (ii)

    if κ>κα, there exists mu=mu(α,κ) such that for every m>mu, [B]m is an unstable critical point of the functional of (). In particular, [B]m cannot be a minimizer.

In view of Theorem A, Theorem B, Theorem C, we conjecture that for γ<1, there should be a critical value λB such that, for λ>λB, the unique minimizer of (Pγ,λ) is the unit ball, up to translations. This conjecture will be the subject of a future work.

This paper is organized as follows. In Section 2 we discuss a few variants of Gamow’s liquid drop model which have already been studied in the literature, and we motivate the choice of our assumptions (H1), (H2). We also recall some well-known results on isoperimetric inequalities. In Section 3 we establish basic prerequisites on nonlocal perimeters and on Bessel kernels, which justify Application D. Section 4 is devoted to the proofs of Theorem A, Theorem B. First, we prove existence of minimizers for γ<1 and λ large enough, as well as convergence to the ball as λ goes to infinity in Theorem 4.5. Then, we compute the Γ-limit of the functionals Fγ,Gλ as λ goes to infinity, and conclude this section by establishing C1,12-regularity (applying directly results from [29]) and connectedness of minimizers. In Section 5, we focus on the stability of the unit ball for large λ, and show that γ=1 is a threshold for which the unit ball goes from stable to unstable, i.e., Theorem C. To conclude on the stability issue, we need to study asymptotics for some nonlocal seminorms on the sphere, which is done in Appendix A: here we compute the limit of these seminorms as the kernels concentrate to the Dirac distribution, and obtain a uniform upper bound which is asymptotically sharp.

For any set ERn, we define EcRnE, and we write |E| for its volume (that is, its Lebesgue measure) whenever E is measurable. We write EF for the union of two sets which are disjoint. Given two sets E and F, we denote by EF(EF)(FE) their symmetric difference. We say that two sets E and F in Rn are equivalent if |EF|=0.

We denote by k the k-dimensional Hausdorff measure in Rn, and by dim(E) the Hausdorff dimension of a set ERn. When integrating w.r.t. the measure k in a variable x, we use the notation dxk instead of the more standard but less compact dk(x).

We denote by Br(x) the open ball in Rn of radius r centered at x. For simplicity we write Br when x is the origin. The volume of B1 is ωn|B1|=πn2Γ1+n2, and the area of the unit sphere Sn1 is n1(Sn1)=nωn. More generally we denote by Sk the k-dimensional unit sphere, and for simplicity we write |Sk|=k(Sk) for its surface area. For any m>0 and xRn we let [B]x,m be the open ball of volume m centered at x, or simply [B]m if x=0.

For any nontrivial open set ΩRn, we denote by BV(Ω) the space of functions with bounded variation in Ω, and for any fBV(Ω) we let |Df| be its total variation measure, and set [f]BV(Ω)Ω|Df|. For a set of finite perimeter E in Ω, we let 1EBV(Ω) be its characteristic function (i.e., 1E(x)=1 if xE and 0 otherwise), and define its perimeter in Ω by P(E;Ω)Ω|D1E|. If Ω=Rn we simply write P(E)P(E;Rn). We denote by μED1E the Gauss–Green measure associated with the set of finite perimeter E and νE(x) the outer unit normal of E at x, where E stands for the reduced boundary of E. We refer e.g. to [13, Chapter 5] or [27] for further details on functions of bounded variations and sets of finite perimeter.

Section snippets

No repulsion: the classical isoperimetric problem

First let us say a few words about the simplest case for (), that is, when G0. In that case, () is the classical isoperimetric problem which consists in minimizing the perimeter under a volume constraint. It is known that the unique minimizer is the ball, up to translations (see e.g. [12]), which gives the classical isoperimetric inequality P(E)P([B]m),for any set of finite perimeter E with volume m, and can be rewritten P(E)nωn1n|E|11n.Knowing that balls are solutions to the classical

Preliminaries

For the rest of the paper, we shall always assume that the kernel G satisfies assumptions (H1) to (H2).

Existence and convergence as m

In order to prove existence of minimizers for large masses, we want to use the direct method in the calculus of variations, starting from a minimizing sequence. When γ<1, we will see that any minimizing sequence is bounded in BV(Rn), but in order to get compactness in L1(Rn) and pass to the limit, we need to show that no mass escapes at infinity. To do so, we will need to establish a few lemmas. First we show that for large masses, if the energy Fγ,Gλ of some set E, (where Fγ,Gλ is defined by

Stability of the ball

As in the previous section, we always assume that the kernel G satisfies assumptions (H1), (H2).

Acknowledgments

The author is very grateful to his PhD advisor V. Millot for introducing him to this problem, and to B. Merlet for his suggestions to improve the clarity of the paper. M. Pegon is supported by the Labex CEMPI (ANR-11-LABX-0007-01).

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