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 [ 8 ] and minimax methods to establish a complete spectra...
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| Published in: | Communications in partial differential equations Vol. 37; no. 9; pp. 1495 - 1516 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Philadelphia
Taylor & Francis Group
01.09.2012
Taylor & Francis Ltd |
| Subjects: | |
| ISSN: | 0360-5302, 1532-4133 |
| Online Access: | Get full text |
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| Summary: | 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. |
|---|---|
| Bibliography: | SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 |
| ISSN: | 0360-5302 1532-4133 |
| DOI: | 10.1080/03605302.2012.686549 |