Unconstrained Variational Principles and Morse Indices for Linear Elliptic Eigenproblems

Variational principles for finding eigenvalues, and associated eigenvectors, for symmetric matrices and compact self-adjoint linear operators have been studied by many authors for some time now. Here we shall introduce and study unconstrained variational principles for the eigenproblem of a pair of bilinear forms (a, m) on a Hilbert space. Each functional in the one-parameter family of functionals has well-defined first and second variations.

First variations characterize the critical points as eigenvectors of (a, m) with associated eigenvalues given by specific formulae. Properties of the set of critical points, that depend on the parameter value of the family of functionals, are given and summarized by a bifurcation diagram. Second variations enable a Morse index theory that characterizes the critical point as being associated with the j-th eigenvalue.

The framework is quite general, but the assumption on (a, m) are appropriate for the study of second-order divergence form elliptic problems in Hilbert-Sobolev spaces, including problems with non-zero boundary data and indefinite weights. These problems include Robin, Steklov and general eigenvalue problems.