An Overdetermined Steklov Eigenvalue Problem on Riemannian Manifolds with Nonnegative Ricci Curvature

초록

We consider an overdetermined Steklov eigenvalue problem on a domain in a Riemannian manifold with nonnegative Ricci curvature. We prove that, given a compact connected domain Omega$\Omega$ with nonnegative Gaussian curvature with C2$C{<}^{>}2$ boundary, if the first Steklov eigenfunction is a solution to the overdetermined problem, then the domain Omega$\Omega$ is flat and the boundary partial derivative Omega$\partial \Omega$ consists of geodesics or geodesic circles. This can be regarded as a generalization of the result by Payne-Philippin [21], where they assumed that Omega$\Omega$ is a simply-connected domain in R2$\mathbb{R}{<}^{>}2$. We also obtain a similar result for the same overdetermined problem on a higher-dimensional compact connected manifold with C2$C{<}^{>}2$ boundary with nonnegative Ricci curvature.

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