Intervenant : ŁASICA Michał (Institute of Mathematics of the Polish Academy of Sciences)
Titre :
Bounds on singularity of minimizers of Rudin-Osher-Fatemi type functionals

Le : 13/02/2023 de : 09:00 à : 10:00
Coercive convex functionals of linear growth in the gradient of the argument generically attain their minima in the space of functions of bounded variation, i.e. the gradient of the minimizer is a vector measure. One class of such functionals are Rudin-Osher-Fatemi (ROF) type models known for their applications to image processing. ROF-type functionals consist of a regularizing term of linear growth and a lower-order fidelity term measuring the distance between the argument and a given function ("noisy image"). In this talk, I will discuss recent results on control of the Lebesgue-singular part of the gradient of the minimizer in terms of the datum, obtained in collaboration with Z. Grochulska and P. Rybka.
Salle : A707

Intervenant : BRENIER Yann (LMO, Université-Paris-Saclay, CNRS)
Titre :
Une structure variationnelle commune aux équations d'Einstein, Euler et Schrödinger, en termes de transport optimal généralisé avec coût quadratique et densités positives, complexes ou matricielles selon le cas

Le : 13/02/2023 de : 11:00 à : 12:00
Tout est dans le titre.
Salle : A707

Intervenant : Bouchitte - Dolbeault - Ehrlacher - Masnou - Rota Nodari ()
Titre :
Journée de rentrée 2023 du GT CalVa

Le : 16/01/2023 de : 09:00 à : 09:00
Nous vous proposons de nous réunir à l'Université Paris-Cité pour commencer cette année civile en beauté avec une journée d'exposés sur quelques thématiques de Calcul des Variations. Inscription gratuite mais obligatoire sur la page web de la journée : https://indico.math.cnrs.fr/event/8753/
Salle : Amphi Turing, Université Paris-Cité (campus Grands Moulins)

Intervenant : NAZARET Bruno (SAMM, Université Paris 1 Panthéon-Sorbonne)
Titre :
Stabilité dans les inégalités de Gagliardo-Nirenberg-Sobolev

Le : 28/11/2022 de : 11:00 à : 12:00
Après avoir montré que les améliorations d'inégalités reliant fonctionnelles d'entropie et de production d'entropie fournissent naturellement des estimations de stabilité dans les inégalités de Gagliardo-Nirenberg-Sobolev, je montrerais comment une combinaison de méthodes d'entropie et de méthodes spectrales dans l'analyse de flots de diffusion non linéaires permet de les obtenir de manière constructive. Ce travail est issu d'une collaboration avec M. Bonforte, J. Dolbeault et N. Simonov.
Salle : A707

Intervenant : MICHETTI Marco (LMO, Université Paris-Saclay)
Titre :
Maximization of Neumann eigenvalues under diameter constraint

Le : 28/11/2022 de : 09:00 à : 09:00
In this talk we study the maximization problem of the Neumann eigenvalues under diameter constraint in an "optimal" class of domains. We define the profile function f associated to a domain \Omega\subset Rd, assuming that this function is \beta-concave, with 0<\beta\leq 1, we will give sharp upper bounds of the quantity D(\Omega)^2 \mu_k(\Omega) in terms of \beta. These bounds will go to infinity when \beta goes to zero. Giving in this way a geometric characterization of domains for which the diameter is fixed, but the Neumann eigenvalue are large. This will also give a new proof of a result by Kröger, namely sharp upper bounds for D(\Omega)^2 \mu_k(\Omega) when \Omega is convex (that correspond to \beta=(d-1)-1). The proof of this results are based on a maximization problem for relaxed Sturm-Liouville eigenvalues. This talk is based on a joint work with Antoine Henrot.
Salle : A707

Intervenant : BACON Xavier (SAMM, Université Paris 1)
Titre :
An exchange economy problem with transport costs

Le : 26/09/2022 de : 10:00 à : 11:00
In this talk I will present a spatial Pareto maximization problem which takes transport costs into account. The existence of an integrable equilibrium distribution of goods is non trivial and will be presented. Duality techniques will help us to establish a strong duality result which can be interpreted in economics terms. Finally, I will discuss numerical simulations and present an algorithm à la Sinkhorn. This is a joint work with Guillaume Carlier and Bruno Nazaret.
Salle : A707

Intervenant : GALLOUëT Thomas (Mokaplan, INRIA Paris)
Titre :
On a link between Camassa-Holm/$H^{div}$ equation and unbalanced optimal transport

Le : 26/09/2022 de : 09:00 à : 09:00
First we recall the link between Incompressible Euler Equation and Optimal transport (throughout a riemannian submersion) and some of the results that can be deduced using this geometrical link, for instance Brenier generalized geodesics, Polar projection, Lagrangian numerical scheme. We then aim to explain why the couple [Camassa-Holm equation/$H^{div}$/Unbalanced Optimal transport] share the same geometrical structure and discuss which of the previous results can be extended in this case. In particular a definition of a pression for the Camassa-Holm equation naturally arises in this framework.
Salle : A707

Intervenant : BELIVACQUA Giulia (DISMA, Politecnico di Torino)
Titre :
The Kirchhoff-Plateau problem

Le : 11/04/2022 de : 10:00 à : 11:00
The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. Giusteri, Lussardi and Fried in [3] established the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of non-interpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In [1], we use this result to generalize the situation studying a system composed by several rods linked in an arbitrary way and tied by a soap film and we perform some experiments to validate our result.

We also study the Elastic Plateau problem, i.e. the above problem when the boundary is an elastic curve. In [2], we obtain the minimal energy solution of the Plateau problem with elastic boundary as a variational limit of the minima of the Kirchhoff-Plateau problems with a rod boundary when the cross-section of the rod vanishes. The limit boundary is a framed curve that can sustain bending and twisting.

References
[1] G. Bevilacqua, L. Lussardi, A. Marzocchi, Soap film spanning an elastic link, Quart. Appl. Math. 77 (3) (2019), 507–523.
[2] G. Bevilacqua, L. Lussardi, A. Marzocchi, Dimensional reduction of the Kirchhoff-Plateau problem, J. Elasticity 140, 135–148 (2020).
[3] G.G. Giusteri, L. Lussardi, E. Fried, Solution of the Kirchhoff-Plateau problem, J. Nonlinear Sci. 27 (2017), 1043–1063
Salle : A409

Intervenant : OTTO Felix (Max Planck Institut)
Titre :
A variational regularity theory for optimal transportation

Le : 11/04/2022 de : 11:00 à : 12:00
The optimal transportation of one measure into another, leading to the notion of their Wasserstein distance, is a problem in the calculus of variations with a wide range of applications. The regularity theory for the optimal map is subtle and was revolutionized by Caffarelli. This approach relies on the fact that the Euler-Lagrange equation of this variational problem is given by the Monge-Ampère equation. The latter is a prime example of a fully nonlinear (degenerate) elliptic equation, amenable to comparison principle arguments.

We present a purely variational approach to the regularity theory for optimal transportation, introduced with M. Goldman and refined with M. Huesmann. Following De Giorgi's philosophy for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient, through the construction of a variational competitor. This leads to a ``one-step improvement lemma'', and feeds into a Campanato iteration on the $C^{1,\alpha}$-level for the optimal map, capitalizing on affine invariance. This variational approach allows to re-prove the $\epsilon$-regularity result (Figalli-Kim, De Philippis-Figalli) bypassing Caffarelli's theory.

However, the advantage of the variational approach resides in its robustness regarding the regularity of the measures, which can be arbitrary measures, and in terms of the problem formulation, e. g. by its extension to almost minimizers (with M. Prod'homme and T. Ried). The former for instance is crucial in order to tackle the widely popular matching problem (with F. Mattesini and M. Huesmann). The latter is convenient when treating more general than square Euclidean cost functions.
Salle : A409