I will first review the classical partitioning problem, discuss the double/triple/K-bubble conjectures, and introduce a new partitioning problem that arises in the study of triblock copolymers in certain limiting regimes. I will present the setting for this new geometrical problem, discuss existence and uniqueness of locally minimizing solutions as well as answers to several conjectures. These...
We discuss the relaxation on $L^1$ of polyconvex functions with linear growth, and recall some old and new results. As prototype of this class of energies, the analysis of the area functional leads to the main example of nonlocality and non-subadditivity, actually confirming a conjecture by De Giorgi (proved by Acerbi and Dal Maso). We discuss what has been recently done to understand the...
When performing a parabolic blowup analysis of singularities in 2D multiphase mean curvature flow, one is led to the notion of self-similar shrinker: Networks whose evolution by mean curvature is given by shrinking homotheties. It can be shown that they are critical points of an entropy given by the interface length functional with a suitable Gaussian weight.
Furthermore, this entropy is...
Taking advantage of monotone quantities along geometric flow to derive functional inequalities is a recurring scheme in geometric analysis.
Recently, we have provided a unified perspective on a broad range of monotonicity formulas in both linear and nonlinear potential theory, as well as along the inverse mean curvature flow. The quantities involved in this study are generalizations and...
We consider a variant of the sticky disk model for N interacting particles in the plane, where distances are evaluated by means of the supremum norm instead of the Euclidean norm. We show crystallization for minima of such an energy (for fixed N) and we prove Gamma-convergence (in the limit as N goes to infinity) of suitably rescaled energies to the anisotropic perimeter with octagonal Wulff...
This talk addresses the question of uniqueness of the minimizers of a convex but not strictly convex integral functional with linear growth in a two-dimensional setting. The integrand - whose precise form derives directly from the theory of perfect plasticity - behaves quadratically close to the origin and grows linearly once a specific threshold is reached. We make use of spatial hyperbolic...
In this talk I present some recent dimension-reduction results for elastic materials with voids. We consider three-dimensional
models with an elastic bulk and an interfacial energy featuring a Willmore-type curvature penalization. By Gamma-convergence we
rigorously derive lower-dimensional models for rods and plates where the effective limit comprises a classical elastic bending
energy and...
We develop the Direct Method in the Calculus of Variations for free-discontinuity energies whose bulk and surface densities exhibit superlinear growth, respectively for large gradients and small jump amplitudes. A distinctive feature of this kind of models is that the functionals are defined on SBV functions whose jump sets may have infinite measure. Establishing general lower semicontinuity...
In this talk I will present recent results obtained with J.F. Babadjian and B. Buet about the regularizing effects of curvature terms for interface models with strong anisotropy. We will consider two main (related) questions for two types of problems. The questions are lower semi-continuity of the energies and phase-field approximations. The models are isoperimetric problems on the one hand...
We present the classification of area-strict limits of planar BV homeomorphisms. This class of mappings allows for cavitations and fractures but fulfils a suitable generalization of the INV condition. As pointed out by J. Ball, these features are expected in limit configurations of elastic deformations. Few years ago, De Philippis and Pratelli introduced the no-crossing condition which...
Mean curvature flow has been a central object in geometric analysis. Weak solutions describe the evolution past singularities, but different solution concepts might lead to different behavior. In this talk, I'll present recent results on the relation between the viscosity solution and distributional solutions. I will also present extensions to the associated obstacle problem, introduce weak...
In the talk I will present some recent results obtained in collaboration with A. Chambolle and M. Morini, concerning some fully discrete (in both space and time) and explicit schemes for the mean curvature flow of boundaries. These schemes are based on an elementary diffusion step and a more costly redistancing operation. We give an elementary convergence proof for the schemes under the...
In a pioneering paper published on JDG in 1993, Leon Simon established a powerful method to demonstrate, among other things, the validity of the following result: if a multiplicity one minimal k-surface (stationary varifold) is sufficiently close, in the unit ball and in a weak measure-theoretic sense, to the stationary cone given by the union of three k-dimensional half-planes meeting along a...
I will present a phase field approximation for sharp interface energies, defined on partitions, as appropriate for modeling grain boundaries in polycrystals. The label takes value in O(d)/G, where G is the point group of a lattice, the phase-field approximation fully respects the symmetry. These functionals can be used for the simulation of grain growth or for image reconstruction of grain...
We prove that minimizers of fractional Gagliardo seminorms, among piecewise affine functions defined on the real line with two given - opposite - slopes (suitably prescribing the length scale of the oscillations) are periodic.
We extend such a result to a less rigid setting that allows to study also the gradient flow of the energy functionals.
Our analysis applies to the van der Merwe...
We will gently review both classical and recent results concerning the variational analysis of energy concentration phenomena in magnetic models defined on lattices in the limit as the lattice spacing tends to zero. The discussion includes results on the classical XY model on the square lattice, when the magnetization takes values in S^1, along with some of its variants. We will then focus on...
Nematic films are thin fluid structures, ideally two dimensional, endowed with an in-plane degenerate nematic order.
Some variational models for nematic films have been introduced by Giomi in 2012 and by Napoli and Vergori in 2018. At equilibrium, the shape of the nematic film results from the competition between surface tension, which favors the minimization of the area, and the nematic...
In recent years, the study of highly non-convex differential
inclusions increased a lot, also motivated by applications to materials
science. Due to the lack of convexity, according to the prescribed
regularity, there may be either many (flexibility) or one (rigidity) class
of solutions.
After introducing and motivating the problem, we try to find information
about the threshold...
The classical capillarity perimeter of a set in a half-space is defined as the sum of its relative perimeter inside the half-space and a constant multiple of the area of the portion of its boundary lying on the boundary of the half-space. The isoperimetric capillarity problem seeks to minimize the capillarity perimeter under a volume constraint. A classical isoperimetric inequality implies...
In this talk, we consider an isoperimetric problem for periodic planar Tilings allowing for unequal repeating cells. We discuss general existence and regularity results and we study classification results for double Tilings, i.e. Tilings with two repeating cells. In this case, we explicitly compute the associated energy profile and we give a complete description of the phase transitions. Based...
I will treat a class of sharp interface models for partial defects, in which partial defects are codimension 2 objects connected by codimension 1 objects. I will present two derivations of such energies. In dimension 2 a discrete model for crystal defects based on nearest neighbours and next to nearest neighbours interaction, via period potentials. In the asymptotic limit as the lattice...
