Spectral asymptotics for Robin Laplacians on Lipschitz sets

TL;DR

This paper establishes precise two-term spectral asymptotics for Robin Laplacians on Lipschitz domains, extending known results for Neumann conditions and analyzing differences in eigenvalue means.

Contribution

It provides the first two-term asymptotic formulas for Riesz means of Robin Laplacian eigenvalues on Lipschitz sets, including the difference from Neumann eigenvalues.

Findings

Two-term asymptotics for Riesz means of Robin eigenvalues.

The second term matches the Neumann case.

Leading order difference between Robin and Neumann eigenvalues derived.

Abstract

We prove two-term spectral asymptotics for the Riesz means of the eigenvalues of the Laplacian on a Lipschitz domain with Robin boundary conditions. The second term is the same as in the case of Neumann boundary conditions. This is valid for Riesz means of arbitrary positive order. For orders at least one and under additional assumptions on the function determining the boundary conditions we derive leading order asymptotics for the difference between Riesz means of Robin and Neumann eigenvalues.