On the splitting of Neumann eigenvalues in perforated domains

TL;DR

This paper studies how small spherical holes in a domain cause multiple Neumann Laplacian eigenvalues to split into simpler eigenvalues, showing this is a generic phenomenon under certain conditions.

Contribution

It proves that eigenvalue splitting is a generic property when perturbing domains with small holes, extending asymptotic formulas to arbitrary-shaped holes in higher dimensions.

Findings

Eigenvalues split into lower multiplicity branches when small holes are introduced.

The splitting occurs generically outside a set of Hausdorff dimension N-1.

An asymptotic expansion for eigenvalues is established, valid for arbitrary-shaped holes in dimensions N≥3.

Abstract

We address the problem of splitting of eigenvalues of the Neumann Laplacian under singular domain perturbations. We consider a domain perturbed by the excision of a small spherical hole shrinking to an interior point. Our main result establishes that the splitting of multiple eigenvalues is a generic property: if the center of the hole is located outside a set of Hausdorff dimension $N-1$ and the radius is sufficiently small, multiple eigenvalues split into branches of lower multiplicity. The proof relies on the validity of an asymptotic expansion for the perturbed eigenvalues in terms of the scaling parameter. Such an asymptotic formula is of independent interest and generalizes previous results; notably, in dimension $N\geq 3$, it is valid for holes of arbitrary shape.