Abstract
This paper describes some families of unconstrained variational principles for finding eigenvalues and eigenfunctions of symmetric closed linear operators on a Hilbert space. The functionals involved are smooth, with well-defined second derivatives and Morse-type indices associated with nondegenerate critical points. This leads to an analog of the Courant-Fischer-Weyl minimax theory, where an analysis of the second derivative at a critical point leads to the determination of which eigenvalue of the operator is associated with this eigenfunction. An extension to weighted eigenproblems is also described. The case of linear second order elliptic operators on a nice bounded set and subject to zero Dirichlet conditions is treated in some detail.
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© 2001 Kluwer Academic Publishers
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Auchmuty, G. (2001). Variational Principles for Self-Adjoint Elliptic Eigenproblems. In: Gao, D.Y., Ogden, R.W., Stavroulakis, G.E. (eds) Nonsmooth/Nonconvex Mechanics. Nonconvex Optimization and Its Applications, vol 50. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0275-9_2
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DOI: https://doi.org/10.1007/978-1-4613-0275-9_2
Publisher Name: Springer, Boston, MA
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