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.

Keywords:

• Bifurcation
• Mixed type PDE
• Nonlinear boundary value problems
• Spectral theory
• Topological methods

Mathematics Subject Classification:

• 35M10
• 35P30
• 35A16
• 35B32

Acknowledgments

Lupo and Payne supported by MIUR project “Equazioni alle Derivate Parziali e Disuguaglianze Funzionali: Aspetti Geometrici, Proprietà Qualitative e Applicazioni”.

Monticelli supported by MIUR project “Metodi Variazionali e Topologici Nello Studio di Fenomeni Nonlineari”.