Existence and Regularity in the Small-Mass Regime for a Hartree--Ohta-Kawasaki Shape Optimization Problem

TL;DR

This paper proves the existence and regularity of near-spherical minimizers for a shape optimization problem involving local and nonlocal energies in the small-mass regime.

Contribution

It establishes existence and smoothness of minimizers close to a ball, introducing novel analysis of Coulombic repulsion without sign constraints.

Findings

Minimizers exist and are $C^{2,\alpha}$ perturbations of a ball in the small mass regime.

The nonlocal Coulomb term acts as both a scattering and homogenizing force.

The proof combines surgery techniques, $\Gamma$-convergence, PDE theory, and free boundary regularity.

Abstract

We consider a shape optimization problem for a hybrid energy combining local confinement and nonlocal Coulomb repulsion. Specifically, for any open set $\Omega \subseteq \mathbb{R}^3$ of prescribed volume, we consider the ground state energy of an $L^2$-normalized function supported in $\Omega$, defined as a linear combination of its homogeneous $\dot{H}^1$ and $\dot{H}^{-1}$ seminorms. We show that in the small mass regime, volume-constrained minimizers of this geometric functional exist and are $C^{2,\alpha}$ perturbations of a ball. The proof relies on a combination of surgery techniques, $\Gamma$-convergence, elliptic PDE theory, and one-phase free boundary regularity. A key novelty of this paper lies in the treatment of the Coulombic repulsive term: unlike standard competitive models, the lack of (a priori) sign constraints on the optimal functions forces the nonlocal term to exhibit two natures: it acts both as a scattering and an homogenizing force.