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Overdetermined problems for Weingarten hypersurfaces

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Abstract

Given a bounded domain of \({{\mathbb {R}}}^{n}\) of class \( C^{2}\), we prove the symmetry of solutions to overdetermined problems obtained by adding both zero Dirichlet and constant Neumann boundary conditions to a class of fully nonlinear equations \(\sigma _{k}(\lambda )=C_{n}^{k}\), where \(\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _n)\) are the principal curvatures of a graph. Our method of proof relies on the maximum principle for a suitable P-function and associated Pohozaev type identities.

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Acknowledgements

The author would like to express her gratitude to Prof. Xinan Ma for his discussions and advice in this research. The research is supported by National Science Foundation of China No. 11721101 and No. 11871255.

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  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Xiaohan Jia

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  1. Xiaohan Jia
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Correspondence to Xiaohan Jia.

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Communicated by F. H. Lin.

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Jia, X. Overdetermined problems for Weingarten hypersurfaces. Calc. Var. 59, 78 (2020). https://doi.org/10.1007/s00526-020-01737-1

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