An overdetermined problem associated to the Finsler Laplacian

Giulio Ciraolo; Antonio Greco

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arXiv · 2010.03042

An overdetermined problem associated to the Finsler Laplacian

Giulio Ciraolo·Antonio Greco

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Abstract

We prove a rigidity result for the anisotropic Laplacian. More precisely, the domain of the problem is bounded by an unknown surface supporting a Dirichlet condition together with a Neumann-type condition which is not translation-invariant. Using a comparison argument, we show that the domain is in fact a Wulff shape. We also consider the more general case when the unknown surface is required to have its boundary on a given conical surface: in such a case, the domain of the problem is bounded by the unknown surface and by a portion of the given conical surface, which supports a homogeneous Neumann condition. We prove that the unknown surface lies on the boundary of a Wulff shape.

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Giulio Ciraolo, Antonio Greco. 2020-10-06. An overdetermined problem associated to the Finsler Laplacian. https://arxiv.org/abs/2010.03042

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