Laplacian Eigenproblems on Product Regions and Tensor Products of Sobolev Spaces

TL;DR

This paper characterizes eigenvalues and eigenfunctions of the Laplacian on product domains, showing they are tensor products of eigenfunctions on individual factors, and explores related Sobolev space structures.

Contribution

It provides new characterizations of Laplacian eigenproblems on product regions and connects these to tensor product structures of Sobolev spaces.

Findings

Eigenfunctions on product domains are products of eigenfunctions on factors.

Characterization of Hilbert-Sobolev spaces as tensor products.

New description of the trace space.

Abstract

Characterizations of eigenvalues and eigenfunctions of the Laplacian on a product domain are obtained. When zero Dirichlet, Robin or Neumann boundary conditions are specified on each factor, then the eigenfunctions on the product domain are precisely the products of the eigenfunctions on the individual factors. There is a related result when Steklov boundary conditions are specified on the second factor. These results enable the characterization of certain Hilbert-Sobolev spaces as tensor products and descriptions of some orthogonal bases of the spaces. A different characterization of the trace space is also found.