Eigenvalue asymptotic of Robin Laplace operators on two-dimensional domains with cusps

TL;DR

This paper derives the asymptotic distribution of eigenvalues for Robin Laplace operators on 2D domains with cusps, revealing how geometry and boundary conditions influence spectral properties.

Contribution

It provides a new formula for eigenvalue asymptotics of Robin Laplacians on cusp domains, linking spectral behavior to geometric and boundary features.

Findings

Eigenvalue distribution formula depends on cusp geometry

Eigenvalue asymptotics are influenced by boundary conditions

Results extend understanding of spectral theory on singular domains

Abstract

We consider Robin Laplace operators on a class of two-dimensional domains with cusps. Our main results include the formula for the asymptotic distribution of the eigenvalues of such operators. In particular, we show how the eigenvalue asymptotic depends on the geometry of the cusp and on the boundary conditions.