General monotone formula for homogeneous $k$-Hessian equation in the exterior domain and its applications
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918102429597696 |
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| author | Yin, Jiabin Zhou, Xingjian |
| author_facet | Yin, Jiabin Zhou, Xingjian |
| contents | In this paper, we deal with an overdetermined problem for the $k$-Hessian equation ($1\leq k<\frac n2$) in the exterior domain and prove the corresponding ball characterizations. Since that Weinberger type approach seems to fail to solve the problem, we give a new perspective to solve exterior overdetermined problem by combining two integral identities and geometric inequalities inspired by Brandolini-Nitsch-Salani's results \cite{BNS}. Meanwhile, we establish general monotone formulas to derive geometric inequalities related to $k$-admissible solution $u$ in $\mathbb R^n\setminusΩ$, where $Ω$ is smooth, $k$-convex and star-shaped domain, which constructed by Ma-Zhang\cite{MZ} and Xiao\cite{xiao}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01434 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | General monotone formula for homogeneous $k$-Hessian equation in the exterior domain and its applications Yin, Jiabin Zhou, Xingjian Analysis of PDEs Differential Geometry In this paper, we deal with an overdetermined problem for the $k$-Hessian equation ($1\leq k<\frac n2$) in the exterior domain and prove the corresponding ball characterizations. Since that Weinberger type approach seems to fail to solve the problem, we give a new perspective to solve exterior overdetermined problem by combining two integral identities and geometric inequalities inspired by Brandolini-Nitsch-Salani's results \cite{BNS}. Meanwhile, we establish general monotone formulas to derive geometric inequalities related to $k$-admissible solution $u$ in $\mathbb R^n\setminusΩ$, where $Ω$ is smooth, $k$-convex and star-shaped domain, which constructed by Ma-Zhang\cite{MZ} and Xiao\cite{xiao}. |
| title | General monotone formula for homogeneous $k$-Hessian equation in the exterior domain and its applications |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2506.01434 |