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15 August 2006 Cantor families of periodic solutions for completely resonant nonlinear wave equations
Massimiliano Berti, Philippe Bolle
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Duke Math. J. 134(2): 359-419 (15 August 2006). DOI: 10.1215/S0012-7094-06-13424-5

Abstract

We prove the existence of small amplitude, (2π/ω)-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency ω belonging to a Cantor-like set of asymptotically full measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser implicit function theorem. In spite of the complete resonance of the equation, we show that we can still reduce the problem to a finite-dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows us to deal also with nonlinearities that are not odd and with finite spatial regularity

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Massimiliano Berti. Philippe Bolle. "Cantor families of periodic solutions for completely resonant nonlinear wave equations." Duke Math. J. 134 (2) 359 - 419, 15 August 2006. https://doi.org/10.1215/S0012-7094-06-13424-5

Information

Published: 15 August 2006
First available in Project Euclid: 8 August 2006

zbMATH: 1103.35077
MathSciNet: MR2248834
Digital Object Identifier: 10.1215/S0012-7094-06-13424-5

Subjects:
Primary: 35L05, 37K50
Secondary: 58E05

Rights: Copyright © 2006 Duke University Press

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Vol.134 • No. 2 • 15 August 2006
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