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Variational methods for Hamiltonian PDEs

TitleVariational methods for Hamiltonian PDEs
Publication TypeJournal Article
Year of Publication2008
AuthorsBerti, M
JournalNATO Science for Peace and Security Series B: Physics and Biophysics
ISBN Number9781402069628

We present recent existence results of periodic solutions for completely resonant nonlinear wave equations in which both "small divisor" difficulties and infinite dimensional bifurcation phenomena occur. These results can be seen as generalizations of the classical finite-dimensional resonant center theorems of Weinstein-Moser and Fadell-Rabinowitz. The proofs are based on variational bifurcation theory: after a Lyapunov-Schmidt reduction, the small divisor problem in the range equation is overcome with a Nash-Moser implicit function theorem for a Cantor set of non-resonant parameters. Next, the infinite dimensional bifurcation equation, variational in nature, possesses minimax mountain-pass critical points. The big difficulty is to ensure that they are not in the "Cantor gaps". This is proved under weak non-degeneracy conditions. Finally, we also discuss the existence of forced vibrations with rational frequency. This problem requires variational methods of a completely different nature, such as constrained minimization and a priori estimates derivable from variational inequalities. © 2008 Springer Science + Business Media B.V.


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