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Surface water waves as saddle points of the energy. (English) Zbl 1222.76019
Summary: By applying the mountain-pass lemma to an energy functional, we establish the existence of two-dimensional 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 [Sov. Math., Dokl. 35, No. 3, 647–650 (1987); translation from Dokl. Akad. Nauk SSSR 294, No. 6, 1289–1292 (1987; Zbl 0641.76008); Sov. Math., Dokl. 35, No. 3, 599–603 (1987); translation from Dokl. Akad. Nauk SSSR 294, No. 5, 1033–1037 (1987; Zbl 0641.76007)] (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 mountain-pass 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 non-trivial water wave, we need a priori estimates and information on their Morse index in the spirit of the work by H. Amann and E. Zehnder [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 7, 539–603 (1980; Zbl 0452.47077)]. 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 by analytical methods. We also compare our approach with the minimization-under-constraint method used by R. E. L. Turner [J. Differ. Equations 55, 401–438 (1984; Zbl 0574.76015)].

MSC:
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
47G30 Pseudodifferential operators
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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