Spectral Theory for Linear Operators of Mixed Type and Applications to Nonlinear Dirichlet Problems

Abstract

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 [8] 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 comprised of weak eigenfunctions which are orthogonal with respect to a natural bilinear form associated to L. The associated eigenvalues {λ k } k∈ℕ are all non-zero, have finite multiplicity and yield a doubly infinite sequence tending to ± ∞. 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 (λ k , 0) for associated semilinear Dirichlet problems.