Geodesic
Unconstrained Variational Principles for Linear Elliptic Eigenproblems
Auchmuty, G.1 ; Rivas, M.A.1
1 Department of Mathematics, University of Houston, Houston, Tx 77204-3008, USA
ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 165-189

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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.

Reçu le :
DOI : 10.1051/cocv/2014021
Classification : 35P15, 49R05, 58E05
Keywords: Robin eigenproblems, Steklov eigenproblems, Morse indices, unconstrained variational problems

Auchmuty, G. 1 ; Rivas, M.A. 1

1 Department of Mathematics, University of Houston, Houston, Tx 77204-3008, USA 
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     title = {Unconstrained {Variational} {Principles} for {Linear} {Elliptic} {Eigenproblems}},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {165--189},
     publisher = {EDP-Sciences},
     volume = {21},
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Auchmuty, G.; Rivas, M.A. Unconstrained Variational Principles for Linear Elliptic Eigenproblems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 165-189. doi: 10.1051/cocv/2014021

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