Weyl-type bounds for Steklov eigenvalues

2016/11/03 by Provenzano, Luigi, Stubbe, Joachim
#35P20 #58C40 #FOS: Mathematics #Primary: 35P15 #Secondary: 35J25 #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1611.00929

Abstract

We present upper and lower bounds for Steklov eigenvalues for domains in ℝ^N+1 with C² 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.

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