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<br>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<br>an asymptotic development by Gamma-convergence, recovering classical results in the literature.<br>