Laplacian eigenproblems on product regions and tensor products of Sobolev spaces

摘要

Characterizations of eigenvalues and eigenfunctions of the Laplacian on a product domain Omega(p) := Omega(1) x Omega(2) are obtained. When zero Dirichlet, Robin or Neumann conditions are specified on each factor, then the eigenfunctions on Omega(p) are precisely the products of the eigenfunctions on the sets Omega(1), Omega(2) separately. There is a related result when Steklov boundary conditions are specified on Omega(2). These results enable the characterization of H-1(Omega(p)) and H-0(1)(Omega(p)) as tensor products and descriptions of some orthogonal bases of the spaces. A different characterization of the trace space of H-1(Omega(p)) is found. (C) 2015 Elsevier Inc. All rights reserved.