Laplacian eigenproblems on product regions and tensor products of Sobolev spaces

Characterizations of eigenvalues and eigenfunctions of the Laplacian on a product domain Ωp:=Ω1×Ω2Ωp:=Ω1×Ω2 are obtained. When zero Dirichlet, Robin or Neumann conditions are specified on each factor, then the eigenfunctions on ΩpΩp are precisely the products of the eigenfunctions on the sets Ω1Ω1, Ω2Ω2 separately. There is a related result when Steklov boundary conditions are specified on Ω2Ω2. These results enable the characterization of H1(Ωp)H1(Ωp) and H01(Ωp) as tensor products and descriptions of some orthogonal bases of the spaces. A different characterization of the trace space of H1(Ωp)H1(Ωp) is found.