We prove a homogenization theorem for non-convex functionals depending on vector-valued functions, defined on Sobolev spaces with respect to oscillating measures. The proof combines the use of the localization methods of Γ-convergence with a ‘discretization’ argument, which allows to link the oscillating energies to functionals defined on a single Lebesgue space, and to state the hypothesis of ...

Abstract. The Allen–Cahn action functional is related to the probability of rare events in the stochastically perturbed Allen–Cahn equation. Formal calculations suggest a reduced action functional in the sharp interface limit. We prove in two and three space dimensions the corresponding lower bound. One difficulty is that diffuse interfaces may collapse in the limit. We therefore consider the l...

We characterize the functionals which are Mosco-limits, in the L2(Ω) topology, of some sequence of functionals of the kind Fn(u) := ∫ Ω αn(x)|∇u(x)| dx , where Ω is a bounded domain of RN (N ≥ 3). It is known that this family of functionals is included in the closed set of Dirichlet forms. Here, we prove that the set of Dirichlet forms is actually the closure of the set of diffusion functionals...

From elliptic regularity one immediately has that, should such a minimizer u exist, u ∈ C∞(Ω̄). Then u is in fact a smooth (analytic, even) harmonic function obtaining the boundary value g, i.e. a solution to the classical Dirichlet problem with boundary data g. A standard exercise in the Direct Method of the Calculus of Variations provides for the existence of a minimizer, i.e. ∃u ∈ H g (Ω;R) s...

In this paper we analyse the behaviour of a pile-up of vertically periodic walls of edge dislocations at an obstacle, represented by a locked dislocation wall. Starting from a continuum non-local energy Ec modelling the interactions—at a typical length scale of 1/c— of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy by Γ-convergence in the lim...

The rigorous derivation of lower dimensional models for thin structures is a question of great interest in mechanics and its applications. In the early 90’s a rigorous approach to dimension reduction has emerged in the stationary framework and in the context of nonlinear elasticity. This approach is based on Γ-convergence and, starting from the seminal paper [3, 4], has led to establish a hiera...

The wetting and contact angle hysteresis on chemically patterned surfaces in two dimensions are analyzed from a stationary phase-field model for immiscible two phase fluid. We first study the sharp interface limit of the model by the method of matched asymptotic expansions. We then justify the results rigorously by the Γ-convergence theory for the related variational problem and study the prope...

This paper deals with the quasistatic crack growth of a homogeneous elastic brittle thin film. It is shown that the quasistatic evolution of a three-dimensional cylinder converges, as its thickness tends to zero, to a two-dimensional quasistatic evolution associated with the relaxed model. Firstly, a Γ-convergence analysis is performed with a surface energy density which does not provide weak c...

This is the first in a series of two papers in which we derive a Γ-expansion for a two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion, also known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlocal Coulomb type energy. We focus on the regime where one of the phases has very small volume fra...

Abstract. Homogenization of integral functionals is studied under the constraint that admissible maps have to take their values into a given smooth manifold. The notion of tangential homogenization is defined by analogy with the tangential quasiconvexity introduced by Dacorogna, Fonseca, Malý & Trivisa [18]. For energies with superlinear or linear growth, a Γ-convergence result is established i...