In this paper, we present a method for the numerical minimization of the Mumford–Shah functional that is based on the idea of topological asymptotic expansions. The basic idea is to cover the expected edge set with balls of radius ε > 0 and use the number of balls, multiplied with 2ε, as an estimate for the length of the edge set. We introduce a functional based on this idea and prove that it c...

For ⌦ varying among open bounded sets in Rn with a Lipschitz boundary @⌦, we consider shape functionals J(⌦) defined as the infimum over a Sobolev space of an integral energy of the kind R ⌦[f(ru) + g(u)], under Dirichlet or Neumann conditions on @⌦. Under fairly weak assumptions on the integrands f and g, we prove that, when a given domain ⌦ is deformed into a one-parameter family of domains ⌦...

We investigate the approximation of the Monge problem (minimizing ∫ Ω |T (x)− x|dμ(x) among the vector-valued maps T with prescribed image measure T#μ) by adding a vanishing Dirichlet energy, namely ε ∫ Ω |DT |. We study the Γ-convergence as ε → 0, proving a density result for Sobolev (or Lipschitz) transport maps in the class of transport plans. In a certain two-dimensional framework that we a...

Cette thèse est consacrée à l’étude de divers problèmes de modélisation en élasticité non linéaire. Les deux premières parties portent sur la justification des modèles bidimensionnels de plaques non linéaires. La motivation est commune, mais les approches, développement asymptotique formel dans un cas, Γ-convergence dans l’autre diffèrent. Le dernier chapitre traite de la modélisation des solid...