Nodal sets of thin curved layers

TL;DR

This paper investigates the behavior of eigenfunction nodal sets in thin curved layers, showing convergence to an effective operator and that nodal sets intersect the boundary, advancing understanding of spectral properties in geometric domains.

Contribution

It strengthens existing perturbation results by proving eigenfunction convergence in H"older spaces and demonstrates that nodal sets always meet the boundary in thin tubular neighborhoods.

Findings

Eigenfunctions converge in H"older spaces as the layer thickness tends to zero.

Nodal sets of eigenfunctions intersect the boundary of the tubular neighborhood.

Spectral properties are approximated by an effective Schrödinger operator on the hypersurface.

Abstract

This paper is concerned with the location of nodal sets of eigenfunctions of the Dirichlet Laplacian in thin tubular neighbourhoods of hypersurfaces of the Euclidean space of arbitrary dimension. In the limit when the radius of the neighbourhood tends to zero, it is known that spectral properties of the Laplacian are approximated well by an effective Schr\"odinger operator on the hypersurface with a potential expressed solely in terms of principal curvatures. By applying techniques of elliptic partial differential equations, we strengthen the known perturbation results to get a convergence of eigenfunctions in H\"older spaces. This enables us in particular to conclude that every nodal set has a non-empty intersection with the boundary of the tubular neighbourhood.