Unconstrained Variational Principles for Linear Elliptic Eigenproblems

Auchmuty, G.; Rivas, M.A.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 165-189.

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 $j$th 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.

Classification : 35P15, 49R05, 58E05
Mots clés : Robin eigenproblems, Steklov eigenproblems, Morse indices, unconstrained variational problems

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