Spectral theory for linear operators of mixed type and applications to nonlinear Dirichlet problems

For a class of linear partial differential operators L of mixed elliptic-hyperbolic type in divergence form with homogeneous Dirichlet data on the entire boundary of suitable planar domains, we exploit the recent weak well-posedness result of (Lupo, Morawetz, Payne 2007) and minimax methods to establish a complete spectral theory in the context of weighted Lebesgue and Sobolev spaces. The results represent the first robust spectral theory for mixed type equations. In particular, we find a basis for a weighted version of the space H^1_0(Omega) comprised of weak eigenfunctions which are orthogonal with respect to a natural bilinear form associated to L. The associated eigenvalues {lambda_k} are all non zero, have finite multiplicity and yield a doubly infinite sequence tending to plus and minus infinity. The solvability and spectral theory are then combined with topological methods of nonlinear analysis to establish the first results on existence, existence with uniqueness and bifurcation from (lambda_k, 0) for associated semilinear Dirichlet problems.