Eigenfunction expansions on arbitrary domains

Abstract

Consider the boundary-value problem for the field ψ ( x ) which satisfies the linear partial differential equation in an arbitrary domain with data given on the boundary . It is generally believed that, unless is the union of constant coordinate lines in a separable coordinate system for the operator , the problem cannot be solved by the classical method of eigenfunction expansions. We show how this apparent limitation can be overcome. The key idea is to embed in a larger embedding domain , which is endowed with a complete set of eigenfunctions of the operator , where the λ n are the eigenvalues. We can now expand ψ ( x ) in terms of this set, i.e. . Although the unknown scalars { a n } can no longer be determined by the use of an inner product, a least-squares procedure which minimizes the error in the boundary data yields the scalars to as high a precision, in principle, as needed. Examples are given of steady heat conduction in two and three dimensions, governed by Laplace's equation, and of Stokes flow in a container, governed by the biharmonic equation, all in non-simple domains. The scope of a powerful classical method has, by this extension, been enlarged very considerably. It is believed that it will be of great use in solving practical, linear boundary-value problems, which until now had to be solved by brute force numerical methods.