Spectral analysis of the biharmonic operator subject to Neumann boundary conditions on dumbbell domains

TL;DR

This paper investigates how the eigenvalues and eigenprojections of the biharmonic operator with Neumann boundary conditions behave on dumbbell-shaped domains as the connecting channel becomes very thin, revealing unique limit distortions influenced by material properties.

Contribution

It provides a detailed spectral analysis of the biharmonic operator on dumbbell domains, highlighting the effects of a shrinking channel on eigenvalues and eigenprojections, and introduces a novel limit equation affected by a Poisson-like parameter.

Findings

Eigenvalues converge to a limit depending on the domain's disconnected parts.

Eigenprojections exhibit specific asymptotic behavior as the channel narrows.

The limiting equation is distorted by a factor related to the Poisson coefficient.

Abstract

We consider the biharmonic operator subject to homogeneous boundary conditions of Neumann type on a planar dumbbell domain which consists of two disjoint domains connected by a thin channel. We analyse the spectral behaviour of the operator, characterizing the limit of the eigenvalues and of the eigenprojections as the thickness of the channel goes to zero. In applications to linear elasticity, the fourth order operator under consideration is related to the deformation of a free elastic plate, a part of which shrinks to a segment. In contrast to what happens with the classical second order case, it turns out that the limiting equation is here distorted by a strange factor depending on a parameter which plays the role of the Poisson coefficient of the represented plate.