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Sobolev Periodic Solutions of Nonlinear Wave Equations in Higher Spatial Dimensions

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Abstract

We prove the existence of Cantor families of periodic solutions for nonlinear wave equations in higher spatial dimensions with periodic boundary conditions. We study both forced and autonomous PDEs. In the latter case our theorems generalize previous results of Bourgain to more general nonlinearities of class Ck and assuming weaker non-resonance conditions. Our solutions have Sobolev regularity both in time and space. The proofs are based on a differentiable Nash–Moser iteration scheme, where it is sufficient to get estimates of interpolation-type for the inverse linearized operators. Our approach works also in presence of very large “clusters of small divisors”.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni R. Caccioppoli, Università degli Studi Napoli Federico II, Via Cintia Monte S. Angelo, 80126, Naples, Italy

    Massimiliano Berti

  2. Université d’ Avignon et des Pays de Vaucluse, Laboratoire d’ Analyse non Linéaire et Géométrie (EA 2151), 84018, Avignon, France

    Philippe Bolle

Authors
  1. Massimiliano Berti
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  2. Philippe Bolle
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Correspondence to Massimiliano Berti.

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Communicated by P. Rabinowitz

Supported by MIUR “Variational Methods and Nonlinear Differential Equations” and by the European Research Council under FP7.

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Berti, M., Bolle, P. Sobolev Periodic Solutions of Nonlinear Wave Equations in Higher Spatial Dimensions. Arch Rational Mech Anal 195, 609–642 (2010). https://doi.org/10.1007/s00205-008-0211-8

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