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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Vydáno v:Communications in partial differential equations Ročník 37; číslo 9; s. 1495 - 1516
Hlavní autoři: Lupo, Daniela, Monticelli, Dario D., Payne, Kevin R.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Philadelphia Taylor & Francis Group 01.09.2012
Taylor & Francis Ltd
Témata:
Bifurcation
Markov analysis
Mixed type PDE
Nonlinear boundary value problems
Spectral theory
Studies
Theory
Topological methods
ISSN:0360-5302, 1532-4133
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Popis
Shrnutí: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.
Bibliografie:SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 14
ISSN:0360-5302
1532-4133
DOI:10.1080/03605302.2012.686549