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We prove a rigidity result for the anisotropic Laplacian. More precisely, thedomain of the problem is bounded by an unknown surface supporting a Dirichletcondition together with a Neumann-type condition which is nottranslation-invariant. Using a comparison argument, we show that the domain isin fact a Wulff shape. We also consider the more general case when the unknownsurface is required to have its boundary on a given conical surface: in such acase, the domain of the problem is bounded by the unknown surface and by aportion of the given conical surface, which supports a homogeneous Neumanncondition. We prove that the unknown surface lies on the boundary of a Wulffshape.

An overdetermined problem associated to the Finsler Laplacian