Federico Luigi Dipasquale : The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics

We consider a variational model for ferronematics - composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes $\Q$-tensor for the liquid crystal component and a magnetisation vector field $\M$, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between $\Q$ and $\M$. We report on some recent results on the asymptotic behaviour of (not necessarily minimising) critical points as a small parameter $\epsilon$ tends to zero. Our main results show that the energy concentrates along distinct singular sets: the (rescaled) energy density for the $\Q$-component concentrates, to leading order, on a finite number of singular points, while the energy density for the $\M$-component concentrates along a one-dimensional rectifiable set. Moreover, we will see that the curvature of the singular set for the $\M$-component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e. the singular set for the $\Q$-component. Joint work with G. Canevari (University of Verona) and B. Stroffolini (University of Naples Federico II). 

Tiziana Giorgi :  Chiral structures in chiral and achiral materials 

We study the formation of chiral structures in chiral and achiral liquid crystal systems in the simplified setting of one-dimension spacial dependence. For chiral liquid crystals, we characterize minimizers when the intrinsic pitch conflicts with the twist imposed by the boundary conditions and it is within a certain range of values. For nematic liquid crystals composed of achiral bent-core molecules, we analyze the twist-bend nematic phase. Numerical simulations based on constrained minimization explore the effects of a constant applied magnetic field. This is joint work with Carlos García-Cervera and Sookyung Joo.

Michal Lasica : SBV-morphisms and image similarity measures 

Perhaps the simplest way to quantitatively compare two images (of the same size) is to sum absolute differences of pixel intensities. To account for possible change of perspective or movement of objects, one may compose one of the images with a domain deformation. Recently, Ball and Horner introduced a variational image comparison model, amounting to minimization over possible deformations of a quantity composed of such integral distance and a nonlinear elasticity term. The latter ensures that deformations of finite energy are homeomorphic, the deformed image is well defined, and a minimizing deformation exists. 

In order to naturally take into account possible cuts or permutation of objects, we propose a variant of this model with a two-sided Mumford-Shah-type energy. The energy and its domain (which now includes discontinuous maps) needs to be carefully defined. On the way we revisit basic questions related to invertibility of measurable maps. Eventually we show lower semicontinuity and existence of minimizers. Our main tool is Hajłasz's change of variables formula. 

This is joint work with Zofia Grochulska and Anastasia Molchanova.  

Benjamin Lledos :  Continuity and Uniqueness for BV minimisers 

In this talk, we study the continuity and uniqueness for the minimisers of multidimensional scalar variational problems in the space of functions of bounded variation. The integrand is assumed to be convex (but not necessarily strictly convex), with linear growth from below (but not necessarily from above), while the domain  and the boundary condition must satisfy suitable geometric and regularity conditions, such as convexity or Lipschitz continuity.

Piotr Rybka :  A few words on convergence of families of gradient flows viewed from the variational point of view 

There is no unique definition of gradient flow and we will take advantage of this fact. When we study the question of stabilization of solutions of a perturbed gradient flow our work would be easier if the pertubation is a gradient flow. We will show when this is possible. When we deal with a family of gradient flows we would like to ensure that the limit is another gradient flow. We will present old and new result to this effect. We will present a few examples.

Yana Teplitskaya :  Steiner trees and maximal distance minimizers. Recent results and open problems

I will talk about two problems: maximal distance minimizers and Steiner trees. Maximal distance minimizers concern finding a connected set of minimal length whose closed $r$-neighborhood covers a given compact set, whereas Steiner trees aim to find a minimum-length set connecting a prescribed set of points. For both problems, I briefly summarize known results and highlight the open questions and recent results. Joint work with M. Basok, D. Cherkashin, A. Gordeev, E. Paolini, N. Rastegaev and E. Stepanov.

 Epifanio Virga :  A variational theory for soft shells

Three independent deformation modes can be identified for shells: these are stretching, drilling and bending. Soft shells are especially sensitive to drilling. I shall present a direct variational theory for soft shells in which each independent mode contributes a separate invariant elastic energy. Quite naturally, this turns out to be a second-grade theory, albeit in a rather indirect (perhaps also unfamiliar) way. Thus, both stress and hyper-stress tensors need to be identified and their role in interpreting the equilibrium equations as balances of forces and torques is then illuminated.

 Kai Xiao :  Sobolev maps to manifolds: the distributional Jacobian and its applications

Starting with the observation that the strong density of smooth maps fails when considering maps that are restricted to take their values on a closed manifold, a natural question is how to characterize the maps that can be approximated with smooth maps? A good candidate to answer this in special case where the target manifold is a sphere is the distributional Jacobian, which detects the topological singularities of the map that thwart to the strong approximation with smooth maps, thus a map can be approximated if and only if its Jacobian is 0. I will introduce, when considering fractional Sobolev maps, a generalization of the Jacobian to a broader class of target manifolds that share some convenient topological properties with the sphere, and use it to characterize the closure of smooth maps in terms of restrictions on generic skeletons.