Computing eigenvalues of the Laplacian on rough domains

by   Frank Rösler, et al.

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 Poincaré-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.


page 33

page 35


Hardy-Muckenhoupt Bounds for Laplacian Eigenvalues

We present two graph quantities Psi(G,S) and Psi_2(G) which give constan...

Integer Laplacian eigenvalues of strictly chordal graphs

In this paper, we establish the relation between classic invariants of g...

Dirichlet spectral-Galerkin approximation method for the simply supported vibrating plate eigenvalues

In this paper, we analyze and implement the Dirichlet spectral-Galerkin ...

Asymptotic analysis and numerical computation of the Laplacian eigenvalues using the conformal mapping

We consider the Dirichlet and Neumann eigenvalues of the Laplacian for a...

Discrete approximations to Dirichlet and Neumann Laplacians on a half-space and norm resolvent convergence

We extend recent results on discrete approximations of the Laplacian in ...

On the semiclassical spectrum of the Dirichlet-Pauli operator

This paper is devoted to semiclassical estimates of the eigenvalues of t...

An efficient unconditionally stable method for Dirichlet partitions in arbitrary domains

A Dirichlet k-partition of a domain is a collection of k pairwise disjoi...

Please sign up or login with your details

Forgot password? Click here to reset