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.