Asymptotic behaviour of the Steklov problem on dumbbell domains

TL;DR

This paper investigates how the eigenvalues and eigenvectors of the Steklov problem behave asymptotically in dumbbell-shaped domains with thin connecting tubes, revealing dimension-dependent phenomena and limit behaviors.

Contribution

It provides a detailed analysis of the asymptotic eigenvalue behavior in dumbbell domains, highlighting differences between two and higher dimensions and deriving associated limit problems.

Findings

Eigenvalues collapse to zero at a rate depending on the tube width

Dimension two exhibits fundamentally different asymptotic behavior

Limit problems vary significantly between two and higher dimensions

Abstract

We analyse the asymptotic behaviour of the eigenvalues and eigenvectors of a Steklov problem in a dumbbell domain consisting of two Lipschitz sets connected by a thin tube with vanishing width. All the eigenvalues are collapsing to zero, the speed being driven by some power of the width which multiplies the eigenvalues of a one dimensional problem. In two dimensions of the space, the behaviour is fundamentally different from the third or higher dimensions and the limit problems are of different nature. This phenomenon is due to the fact that only in dimension two the boundary of the tube has not vanishing surface measure.