Proof of Some Auxiliary Results

  • Massimiliano Berti
  • Jean-Marc Delort
Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 24)


The goal of this Chapter is to prove that the non-resonance condition required for the normal form construction is fulfilled for all the values of the gravity-capillary parameters except a zero measure set. A key property of the proof is that the function which describes the linear dispersive relation of the gravity-capillary water waves equations is sub-analytic. In addition in the last section we prove an expansion of the Dirichlet–Neumann operator that has been used in the preceding chapters.


  1. 31.
    Delort, J.-M., Szeftel, J.: Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres. Int. Math. Res. Not. 37, 1897–1966 (2004). MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  • Massimiliano Berti
    • 1
  • Jean-Marc Delort
    • 2
  1. 1.Department of MathematicsInternational School for Advanced Studies SISSATriesteItaly
  2. 2.LAGASorbonne Paris-Cité/University Paris 13VilletaneuseFrance

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