Large mass minimizers for isoperimetric problems with integrable nonlocal potentials
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 a measurable nonnegative radial function with and , we originally consider the minimization problem where the minimum is taken over all sets of finite perimeter of volume – which we call the mass – and denotes the perimeter of . Observe that this problem exhibits a competition between two terms and is thus nontrivial: the local perimeter term constrains the set to concentrate as much as possible, while the nonlocal term acts as a repulsive term, forcing 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 is in addition radially nonincreasing1 (by Riesz’s symmetric rearrangement, using e.g. [25, Chapter 3.7] and the fact that is equal to its symmetric rearrangement in that case).
As we show just below, the integrability assumption on allows us to reformulate this problem as where stands for the open unit ball of centered at the origin, and are positive constants, the kernel is given by and the functional is defined by The parameter will represent the mass (to the power ), and will be chosen to adjust the first moment of (that is, its integral against the measure ) to a particular universal constant. As will be justified later on, 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 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 where with , since is of the same order as , as we will see.
Rewriting the nonlocal repulsive term as we see that () is in fact strictly equivalent to Now, to further normalize our problem, for , we define the quantity and for every positive natural numbers and , we denote by the constant defined by which does not depend on by symmetry. Here denotes the unit sphere in and the -dimensional Hausdorff measure. Now, up to dividing by the constant , without renaming , we may assume that and may look instead at the problem where . The choice of the constant in (1.3) will be justified in Section 3.1. Keep in mind that the parameter plays the role of a constant times , the first moment of . 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 . Given , setting and , it is then easy to see that , , and by a change of variables so that the set is a minimizer of (1.4) if and only if is a minimizer of the rescaled problem (). Let us emphasize that even in the reformulation (), the nonlocal perturbation still does not vanish as goes to infinity, contrary to the Riesz case for small masses, and our results hold for any . In the rest of the paper, we shall always work with this equivalent formulation.
Except in Section 2, we shall always assume that satisfies the two following general hypotheses:
- (H1)
is radial, that is, there exists a nonnegative function such that for ;
- (H2)
, and
Starting from Section 5, dedicated to the study of the stability of the ball, we may add the extra assumptions
- (H3)
near the origin, for some ;
- (H4)
at infinity for some when , and at infinity when ;
- (H5)
.
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 is due to the presence of a Jacobian determinant when integrating on the sphere, which appears to be singular only in dimension (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 () as (that is, the mass) goes to infinity, and give answers to several natural questions: does () 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 . Then there exists such that, for any , () admits a minimizer, and in addition, minimizers have a reduced boundary and are essentially connected. The dependency of in is not known. It depends on the speed of convergence of 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 and . Connectedness of minimizers also holds in any case, provided that is strictly positive (see Theorem 4.14). In terms of the original problem (), Theorem A means that if , then there exists a critical mass , 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 , 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 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 -regularity of the reduced boundary of minimizers follows by results in [29] on quasi-minimizers for the perimeter.
For any and , let us define the functional to be minimized When , we are able to compute the -limit in of these functionals as 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 as 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 . For any minimizer of () with , up to a translation, we have where is a function depending only on , and which vanishes at infinity. In addition, the family of functionals (with the added constraint of being the indicator function of a set of finite perimeter with volume ) -converges in to as goes to infinity.
Of course, the -convergence of the functional to a positive multiple of the perimeter implies that any converging sequence of minimizers of () with converges to the unit ball, but (1.6) is stronger and in fact a direct consequence of the proof of existence above .
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 is a stability threshold of the unit ball for () for large values of .
Theorem C Consequence of Theorem 5.8, Theorem 5.14 Assume that satisfies all the hypotheses (H1) to (H5). Then the following hold: if , then there exists such that, for any , is a (critical) stable set for ; if , then there exists such that, for any , is a (critical) unstable set for : in particular, it cannot be a minimizer, i.e., symmetry-breaking occurs.
In terms of the original problem (), this means that the threshold for 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 where is a -dimensional approximation of identity, and belongs to .
A particularly interesting consequence of Theorem C is that there exist kernels for which () admits nontrivial minimizers, that is, minimizers which are not balls. Indeed, working from the formulation (), S. Rigot proved in [29] that () always admits a minimizer whenever is compactly supported. Hence, taking , minimizers still exist but cannot be the unit ball when is large enough, since it is unstable.
Theorem A, Theorem B, Theorem C directly apply when is a so-called Bessel kernel: for every , we denote by the Bessel kernel of order defined as the fundamental solution of the operator , that is, where is the Dirac distribution at the origin. We then have:
Application D Corollary 3.12 and Lemma 3.13 For every , we consider the problem () with . Let us define Then we have: if , there exist and such that, for every , () admits a minimizer, and for every , the ball of volume centered at the origin, denoted by , is a stable critical point for the functional of (). In addition, rescaling minimizers so that they are of volume , and translating them, they converge to the unit ball as goes to infinity; if , there exists such that for every , is an unstable critical point of the functional of (). In particular, cannot be a minimizer.
In view of Theorem A, Theorem B, Theorem C, we conjecture that for , there should be a critical value such that, for , the unique minimizer of () 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 and large enough, as well as convergence to the ball as goes to infinity in Theorem 4.5. Then, we compute the -limit of the functionals as goes to infinity, and conclude this section by establishing -regularity (applying directly results from [29]) and connectedness of minimizers. In Section 5, we focus on the stability of the unit ball for large , and show that 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 , we define , and we write for its volume (that is, its Lebesgue measure) whenever is measurable. We write for the union of two sets which are disjoint. Given two sets and , we denote by their symmetric difference. We say that two sets and in are equivalent if .
We denote by the -dimensional Hausdorff measure in , and by the Hausdorff dimension of a set . When integrating w.r.t. the measure in a variable , we use the notation instead of the more standard but less compact .
We denote by the open ball in of radius centered at . For simplicity we write when is the origin. The volume of is , and the area of the unit sphere is . More generally we denote by the -dimensional unit sphere, and for simplicity we write for its surface area. For any and we let be the open ball of volume centered at , or simply if .
For any nontrivial open set , we denote by the space of functions with bounded variation in , and for any we let be its total variation measure, and set . For a set of finite perimeter in , we let be its characteristic function (i.e., if and otherwise), and define its perimeter in by . If we simply write . We denote by the Gauss–Green measure associated with the set of finite perimeter and the outer unit normal of at , where stands for the reduced boundary of . 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 . 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 for any set of finite perimeter with volume , and can be rewritten Knowing that balls are solutions to the classical
Preliminaries
For the rest of the paper, we shall always assume that the kernel satisfies assumptions (H1) to (H2).
Existence and convergence as
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 , we will see that any minimizing sequence is bounded in , but in order to get compactness in 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 of some set , (where is defined by
Stability of the ball
As in the previous section, we always assume that the kernel 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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