Unconstrained Variational Principles for Linear Elliptic Eigenproblems

This paper introduces and studies some unconstrained variational principles for finding eigenvalues, and associated eigenvectors, of a pair of bilinear forms (a, m) on a Hilbert space V. The functionals involve a parameter mu and are smooth with well-defined second variations. Their non-zero critical points are eigenvectors of (a, m) with associated eigenvalues given by specific formulae. There is an associated Morse-index theory that characterizes the eigenvector as being associated with the jth eigenvalue. The requirements imposed on the forms (a, m) are appropriate for studying elliptic eigenproblems in Hilbert-Sobolev spaces, including problems with indefinite weights. The general results are illustrated by analyses of specific eigenproblems for second order elliptic Robin, Steklov and general eigenproblems.