Uniform $C^{1,\alpha}$-regularity for almost-minimizers of some nonlocal perturbations of the perimeter
Résumé
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.
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Origine | Fichiers produits par l'(les) auteur(s) |
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Format | Figure, Image |
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Format | Figure, Image |
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Format | Figure, Image |
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Origine | Fichiers produits par l'(les) auteur(s) |