Cahiers du CEREMADE 

Unité
Mixte de Recherche du C.N.R.S. N°7534 

Abstract : By applying the mountainpass lemma to an energy functional, we establish the existence of twodimensional water waves on the surface of an infinitely deep ocean in a constant gravity field. The formulation used, which is due to K. I. Babenko (and later to others, independently), has as its independent variable an amplitude function which gives the surface elevation, its nonlinear term is purely quadratic but it is nonlocal because it involves the Hilbert transform. Moreover the energy functional from which it is derived is rather degenerate and offers an important challenge in the calculus of variations.
In the present treatment the first step is to truncate the integrand, and then to penalize and regularize it. The mountainpass lemma gives the existence of critical points of the resulting problem.
To check that, in the limit of vanishing regularization, the critical points converge to a nontrivial water wave, we need {\em a priori} estimates and information on their Morse index in the spirit of a work by Amann and Zehnder.
The amplitudes of the waves so obtained are compared with those obtained from the bifurcation argument of Babenko, and are found to extend the parameter range where existence is known. We also compare our approach with a minimizationunderconstraint method used by R.~E.~L.~Turner. 





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