Shape optimization, PDE, and Nonlinear Analysis

Liste des Contributions

9. Convergence of Ambrosio-Tortorelli functionals on varying surfaces

Benjamin Lledos

07/10/2026 14:00

We study Ambrosio-Tortorelli approximations of the Mumford-Shah functional on varying hypersurfaces, formulated in terms of Sobolev functions on rectifiable currents. We prove a compactness result for bounded-energy sequences and establish a lower bound for a generalized Mumford-Shah functional on the limiting current. In dimension two, we obtain the optimal lower bound and construct recovery...

2. About the maximization of the gradient of the torsion function

Ilias Ftouhi

07/10/2026 15:00

In this talk, we address the problem of maximizing the supremum norm of the gradient of the torsion function over planar convex domains with fixed measure (or perimeter). We prove that an optimal shape exists, its boundary possesses $C^1$ regularity, and it features at least one straight segment. The proofs leverage probabilistic methods along with a novel boundary Harnack principle for the...

3. Sharp and quantitative bounds for torsional rigidity with Dirichlet and Robin boundary conditions

Alba Lia Masiello

07/10/2026 16:30

Torsional rigidity is a classical quantity associated with the Poisson problem with Dirichlet boundary conditions and is closely related to the geometry of the underlying domain. In this talk we consider open, bounded and convex sets in R^n and present sharp geometric inequalities relating torsional rigidity to basic geometric quantities such as the perimeter, the measure of the set, and the...

4. Flagfolds: multi-dimensional varifolds to handle discrete surfaces

Blanche Buet

08/10/2026 09:00

We propose a natural framework for the study of surfaces and their different discretizations based on varifolds. Varifolds have been introduced by Almgren to carry out the study of minimal surfaces. Though mainly used in the context of rectifiable sets, they turn out to be well suited to the study of discrete type objects as well. While the structure of varifold is flexible enough to adapt to...

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

Nicolas Clozeau

08/10/2026 10:30

I will present a recent quantitative result concerning the stochastic homogenization of the so-called Griffith type model arising in fracture mechanics: given a body $\Omega\subset \mathbb{R}^3$, the energy $E_\varepsilon(u)$ for deformation $u\in \mathrm{SBV}(\Omega)$ takes the form of

$$E_\varepsilon(u):=\int_{\Omega\backslash S_u} F(\tfrac{\cdot}{\varepsilon},\nabla...

6. On a quasilinear Schrödinger equation: the small frequency limit

Simona Rota-Nodari

08/10/2026 11:30

In this talk, I will present some recent results on a quasilinear Schrödinger equation with a power nonlinearity. After showing the uniqueness and the non-degeneracy of the positive radial solution $u_\omega$ for all $\omega>0$, I will describe its asymptotic behavior in the limit $\omega\to 0$. This gives some important information about the orbital stability of $u_\omega$ and the uniqueness...

11. Minimization of a quasi-linear Schrödinger functional for generalized $\alpha$-non-linearities

Mathis Dauchy

08/10/2026 14:00

In this talk, I will present the minimization of a quasi-linear Schrödinger functional derived from an effective model in quantum physics, with generalized $\alpha$-power non-linearities. We generalize, for $\alpha >0$, the existence result already known for $\alpha = 1$. More precisely, we show the existence of a critical coupling constant such that a minimizer always exists above this...

12. Energy scaling for von Kármán elastic plates with positional constraints

Giacomo Vizzari

08/10/2026 14:40

The studies behind the behavior of elastic plates under different kinds of stress (and the resulting deformations) have often focused on models that study the mid-plane section of the plate, reducing it to a two dimensional problem. One of the most studied models is the von Kármán model, with energy functional
$$ U_h = \int \bigg|\frac{1}{2}\nabla v\, \otimes \nabla v \,+\,\text{sym}\,\nabla...

10. Spectral analysis of the Dirac operator in shrinking domains

Léa Mazzouza

08/10/2026 15:20

I will be presenting a theorem on the spectral analysis of the Dirac operator with infinite mass boundary conditions in thin planar domains. The main objective is to understand how the geometry of the domain shrinking influences the energy levels and localization properties of the eigenmodes. Using tools from pseudo-differential calculus and semiclassical analysis, we derive asymptotic...

14. Shape optimization in thermoelasticity

Marc Dambrine

08/10/2026 16:30

This presentation focuses on shape optimization in thermoelasticity, motivated by the design of engine components subjected to thermal stresses.

First, I will discuss a linear framework in which the objective function—a von Mises norm under volume constraint—is analyzed using shape calculus and an adjoint state. This calculation involves a subtlety related to the boundary condition of the...

1. On the Lavrentiev phenomenon for multidimensional problems in the Calculus of Variations

Pierre Bousquet

09/10/2026 09:00

We present some results on the Lavrentiev phenomenon for some scalar problems in the multidimensional Calculus of Variations. Our main goal is to identify a unifying and natural framework that entitles one to discard this phenomenon and the related Lavrentiev gaps.

13. Conductivity in Random Media: From Nonlinear Diffusion to Dielectric Breakdown

Théo Lavier

09/10/2026 10:30

We present work in progress on quantitative homogenization results for viscosity solutions of $\infty$-Laplacian-type equations of the form$$\nabla\!\left(a\!\left(\tfrac{\cdot}{\varepsilon}\right)\vert{}\nabla u_\varepsilon\vert{}^2\right)\cdot \nabla u_\varepsilon = 0,$$subject to Dirichlet boundary conditions in dimensions $d\leq 3$, where the coefficient field $a : \mathbb{R}^d\rightarrow...

8. Concentration properties for the Aviles-Giga energy

Xavier Lamy

09/10/2026 11:10

The Aviles-Giga energy is a phase transition model related to liquid crystals, micromagnetics and elasticity. Sharp interface limits of bounded energy are weak solutions of the 2D eikonal equation: unit vector fields $m$ with zero divergence (in the sense of distributions). Partial information about the limit energy cost of a given solution $m$ is encoded in a family of signed measures called...