Uniform $C^{1,\alpha}$-regularity for almost-minimizers of some nonlocal perturbations of the perimeter

Michael Goldman (1) , Benoît Merlet (2, 3) , Marc Pegon (2, 3)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 RAPSODI - Reliable numerical approximations of dissipative systems
3 LPP - Laboratoire Paul Painlevé - UMR 8524

In this paper, we establish a $C^{1,\alpha}$-regularity theorem for almost-minimizers of a functional $\mathcal{F}_{\varepsilon,\gamma}=P-\gamma P_{\varepsilon}$, where $\gamma\in (0,1)$ and $P_{\varepsilon}$ is a nonlocal energy converging to the perimeter as $\varepsilon$ vanishes. Our theorem provides a criterion for $C^{1,\alpha}$-regularity at a point of the boundary, which is uniform as the parameter $\varepsilon$ goes to $0$. Building upon previous work by the last two authors, as an application of this theorem we obtain that volume-constrained global minimizers of $\mathcal{F}_{\varepsilon,\gamma}$ are balls for any $\varepsilon$ small enough. For small $\varepsilon$, this minimization problem corresponds to the large mass regime for a Gamow-type problem where the nonlocal repulsive term is given by an integrable kernel $G$ with sufficiently fast decay at infinity.

Mots clés: geometric variational problems; nonlocal isoperimetric problems; nonlocal perimeters; regularity; liquid drop model

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]; Physique mathématique [math-ph]

Michael Goldman, Benoît Merlet, Marc Pegon. Uniform $C^{1,\alpha}$-regularity for almost-minimizers of some nonlocal perturbations of the perimeter. 2022. ⟨hal-03623418v1⟩