The Γ-convergence theory shows that under certain conditions the diblock copolymer equation has spot and ring solutions. We determine the asymptotic properties of the critical eigenvalues of these solutions in order to understand their stability. In two dimensions a threshold exists for the stability of the spot solution. It is is stable if the sample size is small and unstable if the sample si...

Abstract. The energy of the Francfort–Marigo model of brittle fracture can be approximated, in the sense of Γ-convergence, by the Ambrosio–Tortorelli functional. In this work we formulate and analyze two adaptive finite element algorithms for the computation of its (local) minimizers. For each algorithm we combine a Newton-type method with residual-driven adaptive mesh refinement. We present tw...

Abstract. Partial localization is the phenomenon of self-aggregation of mass into high-density structures that are thin in one direction and extended in the others. We give a detailed study of an energy functional that arises in a simplified model for lipid bilayer membranes. We demonstrate that this functional, defined on a class of two-dimensional spatial mass densities, exhibits partial loca...

Studying a one-dimensional analogue of the spontaneous curvature model for twocomponent lipid bilayer membranes we consider planar curves that are made of a material with two phases. Each phase induces a different preferred curvature, and these curvatures as well as phase boundaries may cause the curves to develop kinks. We introduce a family of models for smooth curves and phase fields that ap...

In this paper we generalize to arbitrary dimensions a one-dimensional equicoerciveness and Γ-convergence result for a second derivative perturbation of Perona-Malik type functionals. Our proof relies on a new density result in the space of special functions of bounded variation with vanishing diffuse gradient part. This provides a direction of investigation to derive approximation for functiona...

In this paper the space of images is considered as a Riemannian manifold using the metamorphosis approach [24, 34, 35], where the underlying Riemannian metric simultaneously measures the cost of image transport and intensity variation. A robust and effective variational time discretization of geodesics paths is proposed. This requires to minimize a discrete path energy consisting of a sum of co...

We investigate the nonsmooth and nonconvex L1-Potts functional in discrete and continuous time. We show Γ-convergence of discrete L1-Potts functionals toward their continuous counterpart and obtain a convergence statement for the corresponding minimizers as the discretization gets finer. For the discrete L1-Potts problem, we introduce an O(n2) time and O(n) space algorithm to compute an exact m...

We consider a variational model which describes a complex system composed, in its reference configuration, of a periodic distribution of ‘small’ interacting particles immersed in a continuous medium. We describe its macroscopic limit via Gamma-convergence, highlighting different regimes. In particular, we show how the interplay between the particles and the continuum leads, for a critical size ...

Let M‾ be a compact smoothly stratified pseudomanifold with boundary, satisfying the Witt assumption. In this paper we introduce de Rham signature and Hodge of M‾, prove their equality. Next, building also on recent work Albin Gell-Redman, extend Atiyah-Patodi-Singer index theory established in our previous under hypothesis that has stratification depth 1 to general case, establishing particula...