Journées Nîmoises d’optimisation de forme

15 juin 2026, 12:00 → 16 juin 2026, 17:30 Europe/Paris

Amphi A3 (Fort Vauban, Nîmes)

   

Le laboratoire MIPA de Nîmes université organise la conférence « Journées Nîmoises d’optimisation de forme »  les 15 et 16 juin 2026. La conférence aura lieu sur le site Vauban de l'université.    

 

Cette conférence regroupera des spécialistes de l'optimisation de forme et d'autres domaines de l'analyse mathématique. Les intervenants seront : 

Samuel Amstutz (LMA, Avignon)

Pierre Bousquet (IMT, Toulouse)

Nicolas Clozeau (IMATH, Toulon)

Gisella Croce (SAMM, Paris)

Anas El Hachimi (CEA, Aix-en-Provence)

Zakaria Fattah (MIPA, Nîmes)

Jessica Guerand (IMAG, Montpellier)

Jimmy Lamboley (DMA-ENS et IMJ-PRG, Paris)

Thuong Nguyen (LAMPS, Perpignan)

Reza Pakzad (IMATH, Toulon)

 

 

 

Comité d'organisation : Ilias Ftouhi et Benjamin Lledos.

 

Le comité d'organisation remercie la fédération occitane de mathématiques pour sa contribution à l'évènement.

lun. 15 juin

Buffet d'accueil

Exposé

1 Une inégalité isopérimétrique quantitative en $\mathbb{R}^3$

Dans ce séminaire nous allons étudier un problème de stabilité pour l'inégalité isopérimétrique dans $\mathbb{R}^3$, faisant intervenir une asymétrie de type Hausdorff. Pour un corps convexe $F \subset \mathbb{R}^3$ de volume $4\pi/3$ (c'est-à-dire le volume de la boule unité dans l'espace), nous considérons la fonctionnelle
$$ Q^*(F) := \frac{\delta(F)\log\bigl(\delta(F)+\delta(F)^{-1}\bigr)P(F)^3} {\lambda^*(F)^2}, $$ où $\delta(F)$ désigne le déficit isopérimétrique, c'est-à-dire la différence entre le périmètre $P(F)$ et le périmètre de la boule unité, et $$\lambda^*(F):=\inf_{x\in\mathbb{R}^3} d_H(F,B+x)$$ la distance de Hausdorff à la famille des boules unités. Le facteur logarithmique reflète l'échelle capacitaire optimale au voisinage de la boule en dimension trois, tandis que le facteur $P(F)^3$ assure la coercivité. Notre résultat principal est l'existence du minimum de $Q^*$ dans la classe des corps convexes de volume $4\pi/3$.

Ce travail est en collaboration avec S. Bove et G. Pisante.

Orateur: Gisella Croce

Exposé

2 Régularité et unicité des minima $BV$

Les minima de problèmes de calcul des variations scalaires posés dans $BV$ sont généralement discontinus et sont rarement uniques, lorsque l'intégrande associé est supposé convexe (mais pas nécessairement strictement convexe). En l'absence de toute hypothèse de croissance linéaire, on montre comment des conditions géométriques ou de régularité sur le domaine d'intégration ou la condition au bord (convexité, continuité) impliquent la continuité des minima, une propriété clé pour établir leur unicité. Il s'agit d'un travail en collaboration avec Benjamin Lledos (MIPA).

Orateur: Pierre Bousquet

Pause café

Exposé

3 Lemme de Gehring pour les équations cinétiques et ultraparaboliques

Le lemme de Gehring affirme qu’une fonction satisfaisant une inégalité de Hölder inverse sur des sous-domaines possède une meilleure intégrabilité. Introduit à l’origine par Gehring dans le cadre de problèmes ouverts en théorie des applications quasiconformes, ce résultat a depuis été adapté à l’étude de gain d'intégrabilité des gradients de solutions d’équations elliptiques et paraboliques.

Dans cet exposé, je présenterai des résultats obtenus dans le cadre de différentes collaborations : avec Cyril Imbert et Clément Mouhot pour l’équation cinétique de Fokker–Planck, ainsi qu’avec Francesca Anceschi et Teresa Isernia pour des équations ultraparaboliques non linéaires. La première étape clé consiste à établir un lemme de type Gehring sur des sous-domaines de type cylindres cinétiques et ultraparaboliques. La seconde étape à obtenir des inégalités de Hölder inverses des gradients des solutions à l'aide d'inégalités de type Poincaré, énergie, intégrabilité des solutions.

Orateur: Jessica Guerand

Social dinner

mar. 16 juin

Exposé

4 Boundary Regularity of a Fourth Order Alt–Caffarelli Problem

In this talk, we will present recent results obtained with Mickaël Nahon on a free boundary problem involving the bi-laplace operator. This can be seen as a higher-order analogue of the classical Alt–Caffarelli free boundary problem from 1981. This kind of results can be applied to different shape optimization problems, including the minimization of the first buckling eigenvalue under area constraint, which is conjectured to be solved by the ball.

We will focus on regularity properties in dimension two, with particular emphasis on boundary regularity near points of density less than 1. We prove full regularity of the free boundary, which is analytic outside a possible singular set corresponding to angles with opening approximately 1.43π. The analysis relies on the monotonicity formula introduced by Dipierro, Karakhanyan, and Valdinoci, together with a new epiperimetric inequality that strengthens their approach.

Orateur: Jimmy Lamboley

Exposé

5 A Unified Semi-Smooth Newton--PDAS Framework for Finite-Strain Contact with SMA-Inspired Transformation

In this work, we propose a unified computational framework for large-strain dynamic contact in shape-memory-alloy (SMA) structures, with application to stent deployment in an artery-like environment. The model couples persistent unilateral contact and Coulomb friction at interfaces with finite-strain SMA inelastic transformations. A key feature is that both interface constraints and constitutive transformation conditions are written in a common KKT/NCP complementarity form. The resulting nonsmooth problem is solved by a single semi-smooth Newton– PDAS strategy, where Gauss-point active sets for SMA evolution are treated within the same loop as contact active sets. A midpoint time discretization is adopted to preserve robust transient behavior. The framework is assessed on academic superelastic and dynamic plasticity-type benchmarks, including exact-solution-oriented verification settings. It is then validated on a 2D stent–artery contact test under loading–unloading cycles with frictional interaction. Numerical results show competitive iteration counts versus radial-return-type updates.

Orateur: Thuong Nguyen

Pause café

Exposé

6 Topological derivative for Kirchhoff-Love shells

Joint work with Michael Gfrerer (TU Graz)

I will present a recent work dedicated to the topological sensitivity analysis for the linear Kirchhoff-Love shell model in the case of an elastic inclusion. It extends prior works on plates, but significant additional difficulties are due to curvature, in a more or less direct way. In particular, the coupling between the bending and in-plane deformations mixes first and second order derivatives of displacements. As one consequence, the computation of the polarization tensor is highly involved, and we are only able to obtain a closed formula at an umbilical point. I will present elements of asymptotic analysis, theoretical results and numerical validations.

Orateur: Samuel Amstutz

Exposé

7 Reverse Isoperimetric Inequalities

The classical isoperimetric inequality gives a sharp lower bound on the perimeter of a domain with fixed volume, with equality attained by the ball. Reverse isoperimetric inequalities ask the opposite question: how large can the perimeter or surface area be under a volume constraint? Since this problem is generally ill-posed without additional assumptions, one has to impose geometric constraints such as convexity, inclusion, or curvature bounds.

In this talk, we will discuss reverse isoperimetric inequalities under different geometric constraints. Particular attention will be given to the class of convex bodies with prescribed bounds on the radius of curvature

Orateur: Zakaria Fattah

Buffet

Exposé

8 A theorem by Goldstein and Vodopyanov and its generalizations

A theorem by Goldstein and Vodopyanov from 70s states the following: Let $\Omega \subset {\mathbb R}^n$ be a domain and $u : \Omega \to {\mathbb R}^n$  a deformation of $W^{1,n}$ regularity. If the Jacobian determinant ${\rm det}\, Du$ is positive almost everywhere, then $u$ is continuous. We will discuss a few recent generalizations of this theorem to other function spaces (such as fractional Sobolev spaces), and to the case when the codomain of $u$ is a smooth $n$-manifold with non-trivial topology rather than ${\mathbb R}^n$.

Orateur: Reza Pakzad

Exposé

9 Stable full-field simulation of a multiscale elliptic equation by means of Quantized Tensor Trains

This work addresses the numerical approximation of elliptic differential equations in heterogeneous media, with a particular focus on high-resolution discretizations and large computational challenges. Our main motivation is handling multiscale problems with oscillating coefficient. The proposed framework relies on tensor-based techniques, in particular Quantized Tensor Train (QTT) representations, to efficiently encode high-dimensional structures arising from fine discretizations. The originality of our approach lies in the construction of a QTT-based methodology that allows an accurate control of the gradient of the solution, even for very fine discretization, without relaying on an explicit mesh. This approach provides compact representations of both the problem data and the solution, while remaining flexible with respect to complex and heterogeneous coefficient fields. In addition, a comparison with other solvers is provided to demonstrate the efficiency of our approach. The results suggest that tensor-based solvers, and QTT representations in particular, offer a promising direction for large-scale simulations where classical methods may face limitations in terms of memory and computational cost.

Orateur: Anas El Hachimi

Pause café

Exposé

10 Homogenization of singular SPDEs

I will present an ongoing joint research project with Harprit Singh (postdoctoral researcher at the University of Vienna) on the stochastic homogenization of parabolic singularstochastic PDEs (singular SPDEs) in correlated media. Our goal is to understand the large-scale behavior of such equations, that is, to derive an effective equation, for parabolic singularSPDEs with oscillatory random coefficients. The main challenge lies in the fact that these equations are ill-posed and require a renormalization procedure to construct a pathwise solution theory.

I will focus on the example of the $\varphi^4_2$-equation, a toy model describing the dynamics of thermodynamic fluctuations in two-dimensional ferromagnets near the critical temperature. First, I will present recent results in this direction for periodic coefficients, due to Singh and Hairer, which illustrate how homogenization and renormalization can be combined to obtain an effective, well-posed stochastic PDE. Second, I will discuss the challenges involved in extending these results to the random case and present recent progress on the renormalization of the $\varphi^4_2$-equation in a correlated random environment obtained together with Singh.

Orateur: Nicolas Clozeau