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Article Dans Une Revue Communications in Partial Differential Equations Année : 2009

Paralinearization of the Dirichlet to Neumann operator

Guy Metivier
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Thomas Alazard

Résumé

This paper is concerned with {\em a priori\/} $C^\infty$ regularity for three-dimensional doubly periodic travelling gravity waves whose fundamental domain is a symmetric diamond. The existence of such waves was a long standing open problem solved recently by Iooss and Plotnikov. The main difficulty is that, unlike conventional free boundary problems, the reduced boundary system is not elliptic for three-dimensional pure gravity waves, which leads to small divisors problems. Our main result asserts that sufficiently smooth diamond waves which satisfy a diophantine condition are automatically~$C^\infty$. In particular, we prove that the solutions defined by Iooss and Plotnikov are $C^\infty$. Two notable technical aspects are that (i) no smallness condition is required and (ii) we obtain an exact paralinearization formula for the Dirichlet to Neumann operator.
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Dates et versions

hal-00777059 , version 1 (16-01-2013)

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  • HAL Id : hal-00777059 , version 1

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Guy Metivier, Thomas Alazard. Paralinearization of the Dirichlet to Neumann operator. Communications in Partial Differential Equations, 2009, 34, pp.1634--1704. ⟨hal-00777059⟩
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