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arXiv · 1803.04911

Some overdetermined problems related to the anisotropic capacity

Chiara Bianchini·Giulio Ciraolo·Paolo Salani
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Abstract

We characterize the Wulff shape of an anisotropic norm in terms of solutions to overdetermined problems for the Finsler $p$-capacity of a convex set $\Omega \subset \mathbb{R}^N$, with $1<p<N$. In particular we show that if the Finsler $p$-capacitary potential $u$ associated to $\Omega$ has two homothetic level sets then $\Omega$ is Wulff shape. Moreover, we show that the concavity exponent of $u$ is $q=-(p-1)/(N-p)$ if and only if $\Omega$ is Wulff shape.

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Chiara Bianchini, Giulio Ciraolo, Paolo Salani. 2018-03-13. Some overdetermined problems related to the anisotropic capacity. https://arxiv.org/abs/1803.04911

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