Estimates for eigenvalues of the Neumann and Steklov problems

Abstract

We prove Li-Yau-Kröger type bounds for Neumann-type eigenvalues of the poly-harmonic operator and of the biharmonic operator on bounded domains in a Euclidean space. We also prove sharp estimates for lower order eigenvalues of a biharmonic Steklov problem and of the Laplacian, which directly implies two sharp Reilly-type inequalities for the corresponding first nonzero eigenvalue.

18 pages. We have made revisions to v1 based on some useful suggestions from colleagues. Comments are welcome