An overdetermined problem of the biharmonic operator on Riemannian manifolds

Abstract

AbstractLet $(M,g)$ ( M , g ) be an n-dimensional complete Riemannian manifold with nonnegative Ricci curvature. In this paper, we consider an overdetermined problem of the biharmonic operator on a bounded smooth domain Ω in M. We deduce that the overdetermined problem has a solution only if Ω is isometric to a ball in $\mathbb{R}^{n}$ R n . Our method is based on using a P-function and the maximum principle argument. This result is a generalization of the overdetermined problem for the biharmonic equation in Euclidean space.