%0 Journal Article
%J Duke Mathematical Journal
%D 2011
%T Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces
%A Massimiliano Berti
%A Michela Procesi
%X 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.
%B Duke Mathematical Journal
%V 159
%8 2011
%G eng
%N 3
%& 479
%R 10.1215/00127094-1433403