Title:Variational Principles for Self-adjoint Eigenvalue Problems
Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle
Time and Date:4:00 pm, Thu, 9th Dec 2010
This talk will describe a number of dierent variational principles for self-adjoint
eigenvalue problems that arose from considerations of convex and nonlinear analysis.
First some unconstrained variational principles that are smooth analogues of the
classical Rayleigh principles for eigenvalues of symmetric matrices will be described. In
particular the critical points are eigenvectors and their norms are related to the eigenvalues
of the matrix. Moreover the functions have a nice Morse theory with the Morse indices
describing the ordering of the eigenvector.
Next an unconstrained variational principle for eigenfunctions of elliptic operators
will be illustrated for the classical Dirichlet Laplacian eigenproblem. The critical points
of this problems have a Morse theory that plays a similar role to the classical Courant-
Fischer-Weyl minimax theory.
Finally I will describe certain Steklov eigenproblems and indicate how they are used
to develop a spectral characterization of trace spaces of Sobolev fundtions.