On the eigenvalues of the biharmonic operator with Neumann boundary conditions on a thin set

TL;DR

This paper investigates how the eigenvalues of the biharmonic operator with Neumann boundary conditions behave on a thin boundary layer of a domain, showing they converge to a limit described by a boundary system of differential equations.

Contribution

It establishes the convergence of eigenvalues for the biharmonic operator on thin boundary regions to a boundary differential system, providing new insights into boundary layer spectral analysis.

Findings

Eigenvalues converge to a boundary system of differential equations.

The convergence result applies to the biharmonic operator with Neumann conditions.

Provides a rigorous mathematical framework for spectral limits on thin sets.

Abstract

Let $\Omega$ be a bounded domain in $\mathbb R^2$ with smooth boundary $\partial\Omega$, and let $\omega_h$ be the set of points in $\Omega$ whose distance from the boundary is smaller than $h$. We prove that the eigenvalues of the biharmonic operator on $\omega_h$ with Neumann boundary conditions converge to the eigenvalues of a limiting problem in the form of system of differential equations on $\partial\Omega$.