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Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces

TitleNonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces
Publication TypeJournal Article
Year of Publication2011
AuthorsBerti, M, Procesi, M
JournalDuke Mathematical Journal
Volume159
Issue3
Start Page479
Date Published2011
ISSN0012-7094
Abstract

We develop linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces relevant for the theory of evolutionary Hamiltonian PDEs. A basic tool is the theory of the highest weight for irreducible representations of compact Lie groups. This theory provides an accurate description of the eigenvalues of the Laplace-Beltrami operator as well as the multiplication rules of its eigenfunctions. As an application, we prove the existence of Cantor families of small amplitude time-periodic solutions for wave and Schr¨odinger equations with differentiable nonlinearities. We apply an abstract Nash-Moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the Laplace operator. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on Banach scales of Sobolev functions.

DOI10.1215/00127094-1433403

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