Overdetermined problems with sign-changing eigenfunctions in unbounded periodic domains

TL;DR

This paper constructs specific unbounded periodic domains where the Laplacian's eigenfunctions change sign and satisfy overdetermined boundary conditions, revealing new geometric configurations for such spectral problems.

Contribution

It demonstrates the existence of unbounded, periodic, and radially symmetric domains with sign-changing eigenfunctions under overdetermined boundary conditions, expanding the class of known spectral domain configurations.

Findings

Existence of nontrivial unbounded domains with sign-changing eigenfunctions.

Construction of domains bifurcating from cylinders or slabs.

Domains exhibit periodicity and radial symmetry in certain variables.

Abstract

We prove the existence of nontrivial unbounded domains $\O$ in the Euclidean space $\R^d$ for which the Dirichlet eigenvalue problem for the Laplacian on $\Omega$ admits sign-changing eigenfunctions with constant Neumann values on $\partial \Omega$. We also establish a similar result by studying a partially overdetermined problem on domains with two boundary components and opposite Neumann boundary values. The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from straight (generalized) cylinder or slab.