The Obata equation with Robin boundary condition
The Obata equation with Robin boundary condition
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DOI：
10.4171/rmi/1212
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发表时间：
2019-01
期刊：
影响因子：
--
通讯作者：
Xuezhang Chen;Mijia Lai;Fang Wang
中科院分区：
文献类型：
--
作者：
Xuezhang Chen;Mijia Lai;Fang Wang
研究了带边界流形上具有Robin边界条件$\frac{\partial f}{\partial \nu}+af=0$的Obata方程，其中$a \in \mathbb{R}\setminus\{0\}$. Dirichlet和Neumann边界条件以前由Reilly \cite{R}、Escobar \cite{Es}和Xia \cite{X}研究过。与他们的结果相比，符号$a$在这里起着重要的作用。这一新发现表明，除了球面域之外，对于$a>0$和$a<0$，还存在其他流形。我们还考虑了具有非零Neumann条件$\frac{\partial f}{\partial \nu}=1$的Obata方程。
We study the Obata equation with Robin boundary condition $\frac{\partial f}{\partial \nu}+af=0$ on manifolds with boundary, where $a \in \mathbb{R}\setminus\{0\}$. Dirichlet and Neumann boundary conditions were previously studied by Reilly \cite{R}, Escobar \cite{Es} and Xia \cite{X}. Compared with their results, the sign of $a$ plays an important role here. The new discovery shows besides spherical domains, there are other manifolds for both $a>0$ and $a<0$. We also consider the Obata equation with non-vanishing Neumann condition $\frac{\partial f}{\partial \nu}=1$.