Application aux équations aux dérivées partielles d'une méthode par point fixe et le problème des deux puits

This thesis consists of two parts. The first part is about a variant of Banach's fixed point theorem and its applications to several partial differential equations (PDE's), abstractly of the form $\mathcal Lu + \mathcal Q(u) = f.$ The main result of this first part asserts that an equation having this form admits a solution if the datum $f$ satisfies a certain smallness assumption. This result (we call it \emph{the fixed point method}) is relatively simple to use and can be applied to a large variety of PDE's. The downside is that it guarantees the existence of solutions only for "small" data. The equations we deal with are Jacobian equations, non-linear elliptic PDE's, transport problems and the semi-linear wave equation. The second part of the thesis treats the $\emph{two well problem}$ in two dimensions $\nabla u \in \mathbb{S}_A \cup \mathbb{S}_B\quad \text{almost everywhere in}\ \Omega,$ $u = u_0\quad \text{on}\ \partial\Omega.$ For the non-degenerate case $\det(A),\det(B) \neq 0$, we show a non-existence result for piecewise regular solutions if $A$ and $B$ are non-orthogonal. For the degenerate and semi-degenerate cases, we give a characterisation for the rank-one convex hull of $\mathbb{S}_A \cup \mathbb{S}_B$ and several existence results for Lipschitz and piecewise affine solutions. Finally, for each case, we construct several explicit non-trivial solutions for well-chosen boundary conditions $u_0$.