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Minimization methods for quasi-linear problems with an application to periodic water waves. (English) Zbl 1077.76058

Summary: Penalization and minimization methods are used to give an abstract ”semiglobal” result on the existence of nontrivial solutions of parameter-dependent quasi-linear differential equations in variational form. A consequence is a proof of existence, by infinite-dimensional variational means, of bifurcation points for quasi-linear equations which have a line of trivial solutions. The approach is to penalize the functional twice. Minimization gives the existence of critical points of the resulting problem, and a priori estimates show that the critical points lie in a region unaffected by the leading penalization. The other penalization contributes to the value of the parameter. As applications, we prove the existence of periodic water waves, with and without surface tension.

MSC:

76M30 Variational methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B45 Capillarity (surface tension) for incompressible inviscid fluids
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