# Weyl-type bounds for Steklov eigenvalues

Research paper by Luigi Provenzano, Joachim Stubbe

Indexed on: 03 Nov '16Published on: 03 Nov '16Published in: arXiv - Mathematics - Spectral Theory

#### Abstract

We present upper and lower bounds for Steklov eigenvalues for domains in \$\mathbb{R}^{N+1}\$ with \$C^2\$ boundary compatible with the Weyl asymptotics. In particular, we obtain sharp upper bounds on Riesz-means and the trace of corresponding Steklov heat kernel. The key result is a comparison of Steklov eigenvalues and Laplacian eigenvalues on the boundary of the domain by applying Pohozaev-type identities on an appropriate tubular neigborhood of the boundary and the min-max principle. Asymptotically sharp bounds then follow from bounds for Riesz-means of Laplacian eigenvalues.