  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.
