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

We consider a shape optimization problem for a hybrid energy combining local confinement and nonlocal Coulomb repulsion. Specifically, for any open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ of prescribed volume, we consider the ground state energy of an $L^2$-normalized function supported in $\mathrm{\Omega }$, 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, $\mathrm{\Gamma }$-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.