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

• Published in: Communications in partial differential equations Vol. 37; no. 9; pp. 1495 - 1516
• Main Authors: Lupo, Daniela, Monticelli, Dario D., Payne, Kevin R.
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis Group 01.09.2012; Taylor & Francis Ltd
• Subjects: Bifurcation; Markov analysis; Mixed type PDE; Nonlinear boundary value problems; Spectral theory; Studies; Theory; Topological methods
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2012.686549

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.