Gradient estimate of a Dirichlet eigenfunction on a compact manifold with boundary
Yiqian Shi, Bin Xu · Forum Mathematicum · 2011
Abstract. Let be an eigenfunction with respect to the Dirichlet Laplacian on a compact Riemannian manifold N with boundary: in the interior of N and on the boundary of N . We show the following gradient estimate on : for every , there holds , where C is a positive constant depending only on N . In the proof, we use a basic geometrical property of nodal sets of eigenfunctions and elliptic a priori estimates.