Steklov eigenproblems and the representation of solutions of elliptic boundary value problems

This paper describes some properties and applications of Steklov eigenproblems for prototypical second-order elliptic operators on bounded regions in R-n. Results are described for Schroedinger and weighted harmonic equations. A variational description of the least eigenvalue leads to optimal L-2-trace inequalities. It is shown that the eigenfunctions provide complete orthonormal bases of certain closed subspaces of H-1(Omega) and also of L-2(partial derivativeOmega, dsigma). This allows the description, and representation, of solution operators for homogeneous elliptic equations subject to inhomogeneous Dirichlet, Neumann or Robin boundary data. They are also used to describe Robin to Dirichlet and Neumann to Dirichlet operators for these equations, and to describe the spectrum of these operators. The allowable regions are quite general; in particular classes of bounded regions with a finite number of disjoint Lipschitz components for the boundary are allowed.