Infoscience

Thesis

# 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 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 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 $u_0$.

Thèse École polytechnique fédérale de Lausanne EPFL, n° 6693 (2015)
Programme doctoral Mathématiques
Faculté des sciences de base
Institut de mathématiques d'analyse et applications
Chaire d'analyse mathématique et applications
Jury: Prof. Thomas Mountford (président) ; Prof. Bernard Dacorogna (directeur de thèse) ; Prof. Marco Picasso, Dr Olivier Kneuss, Prof. Paolo Marcellini (rapporteurs)

Public defense: 2015-10-9

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Record created on 2015-09-29, modified on 2016-08-09