Uniform $C^{1,α}$-regularity for almost-minimizers of some nonlocal perturbations of the perimeter
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| Format: | Preprint |
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2022
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| _version_ | 1866916392123498496 |
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| author | Goldman, Michael Merlet, Benoît Pegon, Marc |
| author_facet | Goldman, Michael Merlet, Benoît Pegon, Marc |
| contents | In this paper, we establish a $C^{1,α}$-regularity theorem for almost-minimizers of the functional $\mathcal{F}_{\varepsilon,γ}=P-γP_{\varepsilon}$, where $γ\in(0,1)$ and $P_{\varepsilon}$ is a nonlocal energy converging to the perimeter as $\varepsilon$ vanishes. Our theorem provides a criterion for $C^{1,α}$-regularity at a point of the boundary which is uniform as the parameter $\varepsilon$ goes to $0$. Since the two terms in the energy are of the same order when $\varepsilon$ is small, we are considering here much stronger nonlocal interactions than those considered in most related works. As a consequence of our regularity result, we obtain that, for $\varepsilon$ small enough, volume-constrained minimizers of $\mathcal{F}_{\varepsilon,γ}$ are balls. 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 $G$ with sufficiently fast decay at infinity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_11006 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Uniform $C^{1,α}$-regularity for almost-minimizers of some nonlocal perturbations of the perimeter Goldman, Michael Merlet, Benoît Pegon, Marc Analysis of PDEs Mathematical Physics Optimization and Control 28A75, 49Q05, 49Q10, 49Q20 In this paper, we establish a $C^{1,α}$-regularity theorem for almost-minimizers of the functional $\mathcal{F}_{\varepsilon,γ}=P-γP_{\varepsilon}$, where $γ\in(0,1)$ and $P_{\varepsilon}$ is a nonlocal energy converging to the perimeter as $\varepsilon$ vanishes. Our theorem provides a criterion for $C^{1,α}$-regularity at a point of the boundary which is uniform as the parameter $\varepsilon$ goes to $0$. Since the two terms in the energy are of the same order when $\varepsilon$ is small, we are considering here much stronger nonlocal interactions than those considered in most related works. As a consequence of our regularity result, we obtain that, for $\varepsilon$ small enough, volume-constrained minimizers of $\mathcal{F}_{\varepsilon,γ}$ are balls. 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 $G$ with sufficiently fast decay at infinity. |
| title | Uniform $C^{1,α}$-regularity for almost-minimizers of some nonlocal perturbations of the perimeter |
| topic | Analysis of PDEs Mathematical Physics Optimization and Control 28A75, 49Q05, 49Q10, 49Q20 |
| url | https://arxiv.org/abs/2209.11006 |