Conference on Calculus of Variations in Lille - 4th edition - June 24-28 2024

Contribution List

1. The attractive log gas: propagation of chaos, stability and uniqueness questions

Sylvia Serfaty (NYU Courant Institute)

6/24/24, 10:30 AM

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...

2. A free-discontinuity problem for smectic liquid crystals

Giacomo Canevari (Università degli studi di Verona)

6/24/24, 11:30 AM

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...

3. Asymptotic behavior of $p$-Harmonic mappings when $p$ goes to $2$

Benoît Van Vaerenbergh (Université catholique de Louvain)

6/24/24, 2:00 PM

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...

4. Variational Convergence of Liquid Crystal Energies to Line and Surface Energies

François Alouges (ENS Paris-Saclay)

6/24/24, 2:30 PM

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....

34. Poster session

6/24/24, 4:00 PM

7. On a variational model for epitaxial growth with dislocations

Barbara Zwicknagl (Humboldt-Universität zu Berlin)

6/25/24, 9:00 AM

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...

8. Gradient flow solutions for porous medium equations with nonlocal Lévy-type pressure

David Padilla-Garza (IST Austria)

6/25/24, 10:00 AM

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...

9. Nonoccurrence of Lavrentiev gap for a general minimization problem

Benjamin Lledos (Université catholique de Louvain)

6/25/24, 11:00 AM

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...

10. Closability of differential operators and structure of currents

Andrea Marchese (Università di Trento)

6/25/24, 11:30 AM

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...

11. Rigidity, Flexibility and Scaling for Models of Shape-Memory Alloys

Angkana Rüland (Universität Bonn)

6/25/24, 2:00 PM

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...

12. A $\Gamma$-convergence result for 2D type-I superconductors

Alessandro Cosenza (Université Paris Cité)

6/25/24, 3:00 PM

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,...

13. BV curves of measures and the continuity equation

Riccarda Rossi (Università degli Studi di Brescia)

6/25/24, 4:00 PM

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...

14. Self-adjoint problems in the optimization of non-linear pde models

Grégoire Allaire (École Polytechnique)

6/26/24, 9:00 AM

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...

15. Nonlocal-to-local analysis of energies in Micromagnetics

Rossella Giorgio (TU Wien)

6/26/24, 10:00 AM

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...

16. Quantitative stochastic homogenization of variational models arising in fracture mechanics

Nicolas Clozeau (IST Austria)

6/26/24, 11:00 AM

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}...

17. Constant sign and sign changing NLS ground states on metric graphs

Colette De Coster (Université Polytechnique Hauts-de-France)

6/26/24, 11:30 AM

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...

31. Shape Optimisation for nonlocal anisotropic energies

Lucia Scardia (Heriot-Watt University)

6/26/24, 2:00 PM

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...

21. The extension of traces for Sobolev mappings between manifolds

Jean Van Schaftingen (Université catholique de Louvain)

6/27/24, 9:00 AM

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...

22. Convergence results for critical points of the one-dimensional Ambrosio-Tortorelli functional with an obstacle condition

Martin Rakovsky (Université Paris-Saclay)

6/27/24, 10:00 AM

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...

23. Two examples of Steiner trees with a countable number of terminals and branching points

Yana Teplitskaya (Université Paris-Saclay)

6/27/24, 11:00 AM

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...

24. Sharp stability for the Dirichlet spectrum near the ball

Jimmy Lamboley (Sorbonne Université)

6/27/24, 11:30 AM

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...

25. Fractional multiphase transitions & nonlocal minimal partitions: closed and open questions

Vincent Millot (Université Paris-Est Créteil)

6/27/24, 2:00 PM

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...

26. Biaxiality vs uniaxiality in Landau-de Gennes minimisers in 2D discs

Federico Luigi Dipasquale (Scuola Superiore Meridionale)

6/27/24, 3:00 PM

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 $\lambda \geq 0$...

27. Blow-up limits for Griffith minimizers

Antoine Lemenant (Université de Lorraine)

6/27/24, 4:00 PM

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...

28. Periodic partitions with minimal perimeter

Matteo Novaga (Università di Pisa)

6/28/24, 9:00 AM

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.

29. Concentration compactness phenomena for Willmore surfaces

Dorian Martino (Université Paris Cité)

6/28/24, 10:00 AM

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,...

18. Approximate mean curvature flows for general data, and their limits

Simon Masnou (Université Claude Bernard Lyon 1)

6/28/24, 11:00 AM

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...