Wulff Rigidity for Overdetermined Anisotropic $k$-Hessian Equations

Abstract

We establish Wulff rigidity for overdetermined anisotropic $k$-Hessian equations associated with a smooth and strongly convex Finsler gauge. The solutions are subject to homogeneous Dirichlet boundary and a prescribed anisotropic gradient condition on a smooth domain. We prove that the domain must be a Wulff shape. The same rigidity holds under a prescribed positive weighted Neumann-type condition. These results extend the Wulff rigidity for the endpoint cases $k=1$ (Cianchi–Salani, Math. Ann., 2009; Wang–Xia Arch. Ration. Mech. Anal., 2011) and $k=n$ ( Cianchi–Salani, J. Funct. Anal., 2023) to the intermediate cases $2\le k<n$.