Unconstrained Variational Principles for Linear Elliptic Eigenproblems^∗

Department of Mathematics, University of Houston, Houston, Tx 77204-3008, USA
auchmuty@uh.edu; mauricio@math.uh.edu

Received: 3 October 2013

Abstract

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 μ 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.