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

Publication Type | Journal Article |

Year of Publication | 2011 |

Authors | Berti, M, Procesi, M |

Journal | Duke Mathematical Journal |

Volume | 159 |

Issue | 3 |

Start Page | 479 |

Date Published | 2011 |

ISSN | 0012-7094 |

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. |

DOI | 10.1215/00127094-1433403 |

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

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