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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, \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≤ k<n.
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BibTeX
@misc{liang2026wulff,
title = {{Wulff Rigidity for Overdetermined Anisotropic $k$-Hessian Equations}},
author = {Lichun Liang and Yanjun Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39756},
eprint = {2610.39756},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39756},
}
RIS
TY - UNPB
TI - Wulff Rigidity for Overdetermined Anisotropic k-Hessian Equations
AU - Lichun Liang
AU - Yanjun Liu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39756
UR - https://arxiv.org/abs/2610.39756
AB - 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≤ k<n.
ER -