We consider the dynamics of a system of particles with logarithmic attractive interaction, on the torus, at inverse temperature beta. We show phase transitions on the stability and uniqueness of the uniform distribution. Investigating the mean-field convergence of the system by the modulated free energy method, we deduce that uniform-in-time convergence is not always true. This is joint work...
Smectic liquid crystals are a phase of matter in which the constituent molecules tend to align locally parallel to one another and to arrange themselves in layers. Experimental evidence shows that the configuration of the layers in smectic films may be rather complex, possibly with defects - that is, localised regions of sharp change in the orientation of the layers. Defects may occur at...
A mapping of nonzero topological degree from the boundary of a disk to the circle cannot be extended by a continuous mapping defined on the whole disk to the circle as homotopy theory asserts. However, this extension is possible if one allows the extended mapping to have discontinuous points, also called singularities. Geometrical and physical situations that we will describe motivate the...
Modelling a liquid crystal outside a colloidal particle with the Landau - de Gennes model when a uniform magnetic field is considered leads to a frustrated system. Line and point singularities are likely to appear. We derive, by means of variational convergence, a limiting model of a suitably rescaled energy that can be written in terms of geometric objects.
This is a joint work with D....
In this talk, we consider a variational model which has been introduced in the literature to model the deposition of a thin crystalline film on a rigid substrate, allowing for the formation of dislocations. The energy functional takes into account the surface energy of the film’s free surface, the elastic energy due to the crystallographic misfit between the film and the substrate, and the...
We study a porous medium-type equation whose pressure is given by a nonlocal Lévy operator associated to a symmetric jump Lévy kernel. The class of nonlocal operators under consideration appears as a generalization of the classical fractional Laplace operator. For the class of Lévy-operators, we construct weak solutions using a variational minimizing movement scheme. The lack of interpolation...
We discuss the absence of Lavrentiev gap for minimization problems in the calculus of variations when the functional depends on the space variable, the function and the gradient. Namely, can we approximate a function $u$ of finite energy by a sequence of Lipschitz functions whose energy converges to that of the original function? In general, it is not true, thus, we have to put some...
I will discuss recent results concerning the closability of certain directional derivative and Jacobian-type differential operators and their implications for the structure of flat chains and metric currents. Additionally, I will present a new, elementary proof of Ambrosio and Kirchhiem's flat chain conjecture, in the case of 1-dimensional currents. This conjecture asserts that metric currents...
The modelling of shape-memory alloys displays a striking dichotomy between rigidity and flexibility. On the one hand, without any additional regularity solutions can be highly irregular and non-unique, they are very flexible. On the other hand, often, at higher regularity, which physically can be viewed as augmenting the model by an interfacial energy, the solutions become very rigid and obey...
In the Ginzburg-Landau nodel superconductors are characterized by a parameter $\kappa$ called the Ginzburg-Landau parameter. If $\kappa<\frac{1}{\sqrt{2}}$ the superconductors are classified as type-I, if $\kappa>\frac{1}{\sqrt{2}}$ they are classified as type-II. While in type-II superconductors vortices appear, in type-I superconductors normal and superconducting regions are formed,...
Representation results for Lipschitz (or even absolutely continuous) curves $\mu:[0,T]\to \mathcal{P}_p(\mathbb{R}^d)$, $p>1$, with values in
the Wasserstein space $(\mathcal{P}_p(\mathbb{R}^d),W_p)$ of Borel probability measures in $\mathbb{R}^d$ with finite $p$-moment provide a crucial tool to study evolutionary PDEs and geometric problems in a measure-theoretic setting.
They are...
We consider optimization problems under partial differential equation constraints. It is assumed that the p.d.e. arises from the minimization of a convex non-linear (non-quadratic) energy. We prove that the optimization problem is self-adjoint when the objective function is the dual energy. In other words, the differential of the objective function with respect to the optimization variable...
In this talk we first consider the nonlocal-to-local convergence of exchange energy functionals in Micromagnetics, extending the Bourgain-Brezis-Mironescu formula in order to encompass the scenario where also antisymmetric contributions are encoded.
In a first stage, the nonlocal approximation is given by a pointwise convergence result, obtaining as byproduct a rigorous justification of the...
I will present a recent quantitative result concerning the stochastic homogenization of the
so-called Griffith type model arising in fracture mechanics : the energy $E_\varepsilon(u)$ for $u\in\mathrm{SBV}$ takes the form of
$$
E_\varepsilon(u) = \int_{\Omega\setminus S_u} F\left(\frac{\cdot}{\varepsilon},\nabla u\right)-f\cdot u+\int_{S_u}...
In this talk, we investigate existence and nonexistence of positive and nodal action ground states for the nonlinear Schrödinger equation on metric graphs.
For noncompact graphs with finitely many edges, we detect purely topological sharp conditions preventing the existence of ground states or of nodal ground states. We also investigate analogous conditions of metrical nature. The negative...
We consider shape optimisation problems for sets of prescribed mass, where the driving energy functional is nonlocal and anisotropic. More precisely, we deal with the case of attractive/repulsive interactions in two and three dimensions, where the attraction is quadratic and the repulsion is given by an anisotropic variant of the Coulomb potential.
Under the sole assumption of strict...
Given compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$ and $p \in (1, \infty)$, the question of traces for Sobolev mappings consists in characterising the mappings from $\partial \mathcal{M}$ to $\mathcal{N}$ that can arises of maps in the first-order Sobolev space $\smash{\dot{W}}^{1, p} (\mathcal{M}, \mathcal{N})$.
A direct application of Gagliardo's characterisation of traces...
According to the variational approach to fracture introduced by Francfort and Marigo, the Mumford-Shah energy is commonly used to modelize brittle cracks in an elastic material. Ambrosio and Tortorelli proposed a variational phase field regularization of this functional which, through a Gamma-convergence result, leads to the convergence of global minimizers. This result gives however no...
The Steiner tree problem is a problem of connecting a given compact set by a shortest way. By a full Steiner tree we name a solution of the Steiner tree problem without vertices of degree 2.
I will talk about two solutions to the Steiner problem with given data differ by only one vertex. Every solution for the first data appears to be a full Steiner tree with an infinite number of branching...
In this talk, we discuss the following question: knowing that the first Dirichlet-Laplacian eigenvalue of an open set is close to the one of the ball of same volume (which is the minimizer due to Kaber-Krahn’s inequality), can we say that the other eigenvalues of this set are also close to the ones of the ball? More precisely we seek for quantitative estimates of the...
I will present a convergence result for solutions of Allen-Cahn type systems with a multiple-well potential involving the usual fractional Laplacian in the regime of the so-called nonlocal minimal surfaces.
In the singular limit, solutions converge in a certain sense to stationary points of a nonlocal (or fractional) energy for partitions of the domain with (in general) non homogeneous...
We consider the problem of minimising the (simplest) Landau-de Gennes (LdG) energy in two-dimensional discs, under axial symmetry, a physically relevant pointwise norm-constraint in the interior, and radial anchoring on the boundary. The goal is to study the uniaxial or biaxial character of minimisers. We show that the latter depends crucially on the value of a parameter $\lambda \geq 0$...
The so called Griffith functional has been introduced to model the equilibrium state of a fracture in linearized elasticity. According to this model the equilibrium state of a fracture is defined as a minimizer of the functional
$$
\mathcal{G}(u,K) := \int_{\Omega \setminus K} |e(u)|^2 \, \mathrm{d}x + \mathcal{H}^{N-1}(K),
$$
among pairs $(u,K)$ such that $K$ is a subset of...
I will discuss existence and regularity of periodic tessellations of the Euclidean space, with possibly unequal cells, which minimize a general perimeter functional. In will present some examples in the planar case and some open problems.
The Willmore energy is fundamental in the study of curved surfaces and arise in various context, such as cell biology, optics, general relativity... Since the work of Mondino-Nguyen in 2018, the Willmore energy can also be understood as the only way to merge the study of minimal surfaces and conformal geometry. Despite its first appearance during the 1810s in the work of Germain and Poisson,...
The classical mean curvature flow of regular surfaces may develop singularities in finite time and is not well defined beyond. Various extensions have been proposed which are meaningful for all positive times. However, they are generally not well defined or inconvenient for more general "surface-type" objects such as point clouds. In this talk, I will present a new notion of approximate mean...
