Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions

TL;DR

This paper analyzes the spectral behavior of the Laplacian in a narrow curved strip with mixed boundary conditions, deriving uniform two-term asymptotics for eigenvalues as the strip width shrinks to zero.

Contribution

It provides the first detailed asymptotic analysis of Laplacian eigenvalues in curved strips with combined boundary conditions, highlighting local geometric influences.

Findings

Two-term eigenvalue asymptotics depend only on extremal curvature points.

Asymptotics are uniform and local, depending on curvature ratios.

Results connect asymptotics with norm-resolvent convergence.

Abstract

We consider the Laplacian in a domain squeezed between two parallel curves in the plane, subject to Dirichlet boundary conditions on one of the curves and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the curves tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the curvature radii of the Neumann boundary to the Dirichlet one is the biggest. We also show that the asymptotics can be obtained from a form of norm-resolvent convergence which takes into account the width-dependence of the domain of definition of the operators involved.