We discuss the Γ-convergence, under the appropriate scaling, of the energy functional ‖u‖Hs(Ω) + ∫ Ω W (u) dx, with s ∈ (0, 1), where ‖u‖Hs(Ω) denotes the total contribution from Ω in the Hs norm of u, and W is a double-well potential. When s ∈ [1/2, 1), we show that the energy Γ-converges to the classical minimal surface functional – while, when s ∈ (0, 1/2), it is easy to see that the functio...

We study, through a Γ-convergence procedure, the discrete to continuum limit of Ising type energies of the form Fε(u) = − ∑ i,j ci,juiuj , where u is a spin variable defined on a portion of a cubic lattice εZ ∩ Ω, Ω being a regular bounded open set, and valued in {−1, 1}. If the constants ci,j are non negative and satisfies suitable coercivity and decay assumptions, we show that all possible Γ-...

The distributional k-dimensional Jacobian of a map u in the Sobolev space W 1,k−1 which takes values in the the sphere S can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary M o...

We study the asymptotic behavior of a sequence of Dirichlet problems for second order linear operators in divergence form { −div(σε∇u) = f in Ω, u ∈ H1 0 (Ω), where (σε) ⊂ L∞(Ω;Rn×n) is uniformly elliptic and possibly non-symmetric. On account of the variational principle of Cherkaev and Gibiansky [1], we are able to prove a variational characterization of the H-convergence of (σε) in terms of ...

We review recent results on the homogenization in Sobolev spaces with variable exponents. In particular, we are dealing with the Γ-convergence of variational functionals with rapidly oscillating coefficients, the homogenization of the Dirichlet and Neumann variational problems in strongly perforated domains, as well as double porosity type problems.The growth functions also depend on the small ...

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...