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An abstract Nash-Moser theorem with parameters and applications to PDEs

TitleAn abstract Nash-Moser theorem with parameters and applications to PDEs
Publication TypeJournal Article
Year of Publication2010
AuthorsBerti, M, Bolle, P, Procesi, M
JournalAnnales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis
KeywordsAbstracting; Aircraft engines; Finite dimensional; Hamiltonian PDEs; Implicit function theorem; Invariant tori; Iterative schemes; Linearized operators; Mathematical operators; Moser theorem; Non-Linearity; Nonlinear equations; Nonlinear wave equation; Periodic solution; Point of interest; Resonance phenomena; Small divisors; Sobolev; Wave equations

We prove an abstract Nash-Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the "tame" estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large "clusters of small divisors", due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity. © 2009 Elsevier Masson SAS. All rights reserved.


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