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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.13895 |
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| _version_ | 1866917410681913344 |
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| author | Mazzoleni, Dario Moraschi, Riccardo Ruffini, Berardo |
| author_facet | Mazzoleni, Dario Moraschi, Riccardo Ruffini, Berardo |
| contents | We consider a shape optimization problem for a hybrid energy combining local confinement and nonlocal Coulomb repulsion. Specifically, for any open set $Ω\subseteq \mathbb{R}^3$ of prescribed volume, we consider the ground state energy of an $L^2$-normalized function supported in $Ω$, 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 $C^{2,α}$ perturbations of a ball. The proof relies on a combination of surgery techniques, $Γ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13895 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence and Regularity in the Small-Mass Regime for a Hartree--Ohta-Kawasaki Shape Optimization Problem Mazzoleni, Dario Moraschi, Riccardo Ruffini, Berardo Analysis of PDEs We consider a shape optimization problem for a hybrid energy combining local confinement and nonlocal Coulomb repulsion. Specifically, for any open set $Ω\subseteq \mathbb{R}^3$ of prescribed volume, we consider the ground state energy of an $L^2$-normalized function supported in $Ω$, 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 $C^{2,α}$ perturbations of a ball. The proof relies on a combination of surgery techniques, $Γ$-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. |
| title | Existence and Regularity in the Small-Mass Regime for a Hartree--Ohta-Kawasaki Shape Optimization Problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.13895 |