Variational Techniques for Water Waves and Singular Perturbations

This thesis aims to provide a variational framework for the study of two problems that arise from fluid dynamics and continuum mechanics. The first part concerns a free boundary approach for the existence of periodic water waves. This is a notoriously hard problem as the only variational solutions of the unconstrained problem are waves with flat profiles. Nevertheless, it is shown that by
considering an additional Dirichlet condition on part of the lateral boundary, nontrivial solutions can be found among minimizers of the classical Alt-Caffarelli functional. The second part of the thesis focuses on a regularization by singular perturbations of a mixed Dirichlet-Neumann boundary value problem. The asymptotic behavior of the solutions to the perturbed problems is studied by means of
an asymptotic development by Gamma-convergence, recovering classical results in the literature.