Unconstrained Variational Principles for Linear Elliptic Eigenproblems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 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 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.

Received:
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
@article{COCV_2015__21_1_165_0,
     author = {Auchmuty, G. and Rivas, M.A.},
     title = {Unconstrained {Variational} {Principles} for {Linear} {Elliptic} {Eigenproblems}},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {165--189},
     publisher = {EDP-Sciences},
     volume = {21},
     number = {1},
     year = {2015},
     doi = {10.1051/cocv/2014021},
     mrnumber = {3348419},
     zbl = {1327.35267},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2014021/}
}
TY  - JOUR
AU  - Auchmuty, G.
AU  - Rivas, M.A.
TI  - Unconstrained Variational Principles for Linear Elliptic Eigenproblems
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2015
SP  - 165
EP  - 189
VL  - 21
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2014021/
DO  - 10.1051/cocv/2014021
LA  - en
ID  - COCV_2015__21_1_165_0
ER  - 
%0 Journal Article
%A Auchmuty, G.
%A Rivas, M.A.
%T Unconstrained Variational Principles for Linear Elliptic Eigenproblems
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2015
%P 165-189
%V 21
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2014021/
%R 10.1051/cocv/2014021
%G en
%F COCV_2015__21_1_165_0
Auchmuty, G.; Rivas, M.A. Unconstrained Variational Principles for Linear Elliptic Eigenproblems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 1, pp. 165-189. doi : 10.1051/cocv/2014021. http://www.numdam.org/articles/10.1051/cocv/2014021/

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. MPS-SIAM, Philadelphia (2005). | MR | Zbl

G. Auchmuty, Dual Variational Principles for Eigenvalue Problems, in Nonlinear Functional Analysis and its Applications Part 1, edited by F.E. Browder. Vol. 45 of Proc. Symp. Pure Math. AMS (1986) 55–72. | MR | Zbl

G. Auchmuty, Unconstrained Variational Principles for Eigenvalues of Real Symmetric Matrices. SIAM J. Appl. Math 20 (1989) 1186–1207. | DOI | MR | Zbl

G. Auchmuty, Variational Principles for Eigenvalues of Compact Operators, SIAM J. Appl. Math. 20 (1989) 1321–1335. | DOI | MR | Zbl

G. Auchmuty, Variational Principles for Self-Adjoint Elliptic Eigenproblems, in Nonsmooth/Nonconvex Mechanics. Edited by Gao, Ogden and Stavroulakis. Kluwer, Dordrecht (2001) 15–42. | MR | Zbl

G. Auchmuty, Bases and Comparison Results for Linear Elliptic Eigenproblems. J. Math. Anal. Appl. 390 (2012) 394–406. | DOI | MR | Zbl

F. Belgacem, Elliptic Boundary Value Problems with indefinite weights. Vol. 368 of Research Notes Math. Pitman (1997). | MR | Zbl

P. Blanchard and E. Brüning, Variational Methods in Mathematical Physics. Springer-Verlag, Berlin (1992). | MR | Zbl

B. Belinskiy, Eigenvalue problems for Elliptic type partial differential operators with spectral parameter contained linearly in boundary conditions, in Proc. of Partial Differential and Integral Equations. Edited by Begehr. Kluwer (1999). | MR | Zbl

D.E. Edmunds and W.D. Evans, Spectral Theory and Differential Operators. Oxford University Press, Oxford (1987). | MR | Zbl

E. Zeidler, Nonlinear Functional Analysis and its Applications, II/A, Linear Monotone Operators. Springer Verlag, New York (1990). | Zbl

E. Zeidler, Nonlinear Functional Analysis and its Applications, III, Variational Methods Operators. Springer Verlag, New York (1985). | MR | Zbl

Cited by Sources: