We consider a variational problem related to the shape of charged liquid drops at equilibrium. We show that this problem never admits local minimizers with respect to L^1 perturbations preserving the volume. However, we prove that the ball is stable under small C^(1,1) perturbations when the charge is small enough

We study pattern formation for a variational model displaying competition between a local term penalizing interfaces and a non-local term favoring oscillations. By means of a Γ−convergence analysis, we show that as the parameter J converges to a critical value J c , the minimizers converge to periodic one-dimensional stripes. A similar analysis has been previously performed by other authors for related discrete systems. In that context, a central point is that each "angle" comes with a strictly positive contribution to the energy. Since this is not anymore the case in the continuous setting, we need to overcome this difficulty by slicing arguments and a rigidity

This paper is about quantitative linearization results for the Monge-Ampère equation with rough data. We develop a large-scale regularity theory and prove that if a measure µ is close to the Lebesgue measure in Wasserstein distance at all scales, then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding Poisson equation. The main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures. This is used in a Campanato iteration which transfers the information through the scales.

A variational model for the epitaxial deposition of a film on a rigid substrate in the presence of a crystallographic misfit is studied. The scaling behavior of the minimal energy in terms of the volume of the film and the amplitude of the misfit is considered, and reduced models in the various regimes are derived by Γ-convergence methods. Depending on the relation between the thickness of the film and the amplitude of the misfit, the surface or the elastic energy contribution dominate, and in the critical case the two contributions balance. In particular, the formation of islands is proven if the amplitude of the misfit is large compared to the volume of the film.