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arXiv · 1901.02206

The Obata equation with Robin boundary condition

Xuezhang Chen·Mijia Lai·Fang Wang
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Abstract

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 $\frac{\partial f}{\partial \nu}=1$.

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Xuezhang Chen, Mijia Lai, Fang Wang. 2019-01-08. The Obata equation with Robin boundary condition. https://arxiv.org/abs/1901.02206

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