An overdetermined problem associated to the Finsler Laplacian
Analysis of PDEs
2020-10-08 v1
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.
Keywords
Cite
@article{arxiv.2010.03042,
title = {An overdetermined problem associated to the Finsler Laplacian},
author = {Giulio Ciraolo and Antonio Greco},
journal= {arXiv preprint arXiv:2010.03042},
year = {2020}
}