Singularity of eigenfunctions at the junction of shrinking tubes. Part I

TL;DR

This paper analyzes the asymptotic behavior of eigenfunctions of the Dirichlet Laplacian in domains connected by shrinking tubes, revealing a singularity of order N-1 and the sign behavior near the junction.

Contribution

It provides a precise description of the singularity and asymptotic profile of eigenfunctions at the junction of shrinking tubes, under a nondegeneracy condition.

Findings

Eigenfunctions concentrate in one domain as the tube shrinks.

Normalized eigenfunctions develop a singularity of order N-1 at the junction.

Eigenfunctions remain one-sign near the junction, with nodal sets not entering the channel.

Abstract

Consider two domains connected by a thin tube: it can be shown that, generically, the mass of a given eigenfunction of the Dirichlet Laplacian concentrates in only one of them. The restriction to the other domain, when suitably normalized, develops a singularity at the junction of the tube, as the channel section tends to zero. Our main result states that, under a nondegeneracy condition, the normalized limiting profile has a singularity of order N-1, where N is the space dimension. We give a precise description of the asymptotic behavior of eigenfunctions at the singular junction, which provides us with some important information about its sign near the tunnel entrance. More precisely, the solution is shown to be one-sign in a neighborhood of the singular junction. In other words, we prove that the nodal set does not enter inside the channel.