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Volume 36, Issue 1
Serrin-Type Overdetermined Problem in $\mathbb H^n$

Zhenghuan Gao, Xiaohan Jia & Jin Yan

DOI: 10.4208/jpde.v36.n1.7

J. Part. Diff. Eq., 36 (2023), pp. 102-118.

Published online: 2022-12

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  • Abstract

In this paper, we prove the symmetry of the solution to overdetermined problem for the equation $\sigma_k(D^2u-uI)=C_n^k$ in hyperbolic space. Our approach is based on establishing a Rellich-Pohozaev type identity and using a $P$ function. Our result generalizes the overdetermined problem for Hessian equation in Euclidean space.

  • Keywords

Overdetermined problems hyperbolic space P functions Rellich-Pohozaev identity.

  • AMS Subject Headings

35N25, 35B06, 35B50, 53C24

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

gzh@shu.edu.cn (Zhenghuan Gao)

jiaxiaohan@xmu.edu.cn (Xiaohan Jia)

yjoracle@mail.ustc.edu.cn (Jin Yan)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-36-102, author = {Gao , ZhenghuanJia , Xiaohan and Yan , Jin}, title = {Serrin-Type Overdetermined Problem in $\mathbb H^n$}, journal = {Journal of Partial Differential Equations}, year = {2022}, volume = {36}, number = {1}, pages = {102--118}, abstract = {

In this paper, we prove the symmetry of the solution to overdetermined problem for the equation $\sigma_k(D^2u-uI)=C_n^k$ in hyperbolic space. Our approach is based on establishing a Rellich-Pohozaev type identity and using a $P$ function. Our result generalizes the overdetermined problem for Hessian equation in Euclidean space.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v36.n1.7}, url = {http://global-sci.org/intro/article_detail/jpde/21296.html} }
TY - JOUR T1 - Serrin-Type Overdetermined Problem in $\mathbb H^n$ AU - Gao , Zhenghuan AU - Jia , Xiaohan AU - Yan , Jin JO - Journal of Partial Differential Equations VL - 1 SP - 102 EP - 118 PY - 2022 DA - 2022/12 SN - 36 DO - http://doi.org/10.4208/jpde.v36.n1.7 UR - https://global-sci.org/intro/article_detail/jpde/21296.html KW - Overdetermined problems KW - hyperbolic space KW - P functions KW - Rellich-Pohozaev identity. AB -

In this paper, we prove the symmetry of the solution to overdetermined problem for the equation $\sigma_k(D^2u-uI)=C_n^k$ in hyperbolic space. Our approach is based on establishing a Rellich-Pohozaev type identity and using a $P$ function. Our result generalizes the overdetermined problem for Hessian equation in Euclidean space.

Gao , ZhenghuanJia , Xiaohan and Yan , Jin. (2022). Serrin-Type Overdetermined Problem in $\mathbb H^n$. Journal of Partial Differential Equations. 36 (1). 102-118. doi:10.4208/jpde.v36.n1.7
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