Nonlocal Shape Optimization via Interactions of Attractive and Repulsive Potentials

Abstract

We consider a class of nonlocal shape optimization problems for sets of fixed mass where the energy functional is given by an attractive/repulsive interaction potential in powerlaw form. We find that the existence of minimizers of this shape optimization problem depends crucially on the value of the mass. Our results include existence theorems for large mass and nonexistence theorems for small mass in the class where the attractive part of the potential is quadratic. In particular, for the case where the repulsion is given by the Newtonian potential, we prove that there is a critical value for the mass, above which balls are the unique minimizers, and below which minimizers fail to exist. The proofs rely on a relaxation of the variational problem to bounded densities, and recent progress on nonlocal obstacle problems.

Uniqueness and radial symmetry of minimizers for a nonlocal variational problem

“…We conclude that if the ratio m/M is less or equal to 4π 3 , then u(r) is the minimizer. Since there is no gap between the critical ratio for phase 1 and critical ratio for phase 3, we conclude, as it is well known (see [1]), that there no phase 2 minimizers. However, as we will see, things are different if q = 3 or q = 4.…”

Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits

“…We begin in section 5.1 by describing the details of our numerical implementation, which include various refinements over previous works, such as regridding to reduce the number of particles required for convergence [55,82]. In section 5.2, we provide several numerical examples of the slow diffusion limit and properties of the constrained aggregation equation, particularly critical mass behavior relating to open problems in geometric shape optimization [34,82,91]. In section 5.3, we give numerical examples illustrating the relationship between singular limits and metastability behavior, both as aggregation becomes localized and as diffusion vanishes.…”

Aggregation-diffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit

“…If K satisfies (ATT)(ii), then the interaction potential K is strictly increasing in every coordinate outside of some fixed set, and the existence of a minimizer in P(R d ) for the energy E ∞ over P(R d ) follows simply by [20,Proposition 4.1]. The minimizer in P(R d ) is in fact compactly supported by [20,Lemma 4.4], and therefore is in P 2 (R d ).…”

“…More recently, several works have also considered the constrained aggregation equation and minimizers of the constrained interaction energy E ∞ . Minimizers of E ∞ are directly related to a shape optimization problem introduced by Burchard, Choksi, and the second author [20]: given a repulsive-attractive power-law interaction potential K, as in equation (1.3), minimize E(Ω) = 1 2 Ω Ω K(x − y) dxdy over sets Ω ⊆ R d of volume M. (1.4) Competition between the attraction parameter q and the repulsion parameter p in the definition of K determines existence, nonexistence, and qualitative properties of minimizers, providing a counterpoint to the well-studied nonlocal isoperimetric problem. (See [41] for a survey.)…”