Laplacian eigenvalues with a large negative Robin parameter on a part of the boundary

TL;DR

This paper analyzes the asymptotic behavior of Laplacian eigenvalues with strongly attractive Robin boundary conditions on part of the boundary of smooth planar domains, revealing their dependence on boundary geometry.

Contribution

It provides new asymptotic formulas for eigenvalues under Robin conditions with large negative parameters, including effects of boundary curvature.

Findings

Eigenvalues asymptotics are described by an effective boundary operator.

Boundary curvature influences eigenvalue asymptotics.

Results apply to various typical geometries.

Abstract

We consider the Laplacian eigenvalues for smooth planar domains with strongly attractive Robin conditions imposed on a part of the boundary and Neumann condition on the remaining boundary. The asymptotics of individual eigenvalues is described in terms of an effective operator on an interval with boundary conditions at the endpoints. For several typical geometries a more precise asymptotics in terms of the boundary curvature is obtained.