Computing eigenvalues of the Laplacian on rough domains

TL;DR

This paper establishes a framework for approximating eigenvalues of the Dirichlet Laplacian on rough, fractal-boundary domains using a universal algorithm, with convergence guarantees under mild geometric conditions.

Contribution

It introduces a Mosco convergence theorem for rough domains, develops a novel Poincaré inequality, and constructs a universal algorithm for spectral computation on complex geometries.

Findings

Spectral convergence is achieved for domains satisfying geometric hypotheses.

A universal algorithm can compute eigenvalues on many fractal-boundary domains.

Counterexample shows no universal algorithm exists for all bounded domains.

Abstract

We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincar\'e-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counter example showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.