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arXiv · 2610.39756

Wulff Rigidity for Overdetermined Anisotropic $k$-Hessian Equations

Lichun Liang·Yanjun Liu
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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, \emph{Math. Ann.}, 2009; Wang--Xia \emph{Arch. Ration. Mech. Anal.}, 2011) and $k=n$ ( Cianchi--Salani, \emph{J. Funct. Anal.}, 2023) to the intermediate cases $2\le k<n$.

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Lichun Liang, Yanjun Liu. 2026-10-29. Wulff Rigidity for Overdetermined Anisotropic $k$-Hessian Equations. https://arxiv.org/abs/2610.39756

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