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 Schrödinger 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.

Nonlinear wave and Schroedinger equations on compact Lie groups and homogeneous spaces / Berti, M.; Procesi, M.. - In: DUKE MATHEMATICAL JOURNAL. - ISSN 0012-7094. - 159:3(2011), pp. 479-538. [10.1215/00127094-1433403]

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

Berti, M.;
2011-01-01

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 Schrödinger 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.
2011
159
3
479
538
https://projecteuclid.org/euclid.dmj/1314648277#info
Berti, M.; Procesi, M.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/14832
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