Séminaire

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**

Séminaire

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**

Séminaire

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

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**

Séminaire

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.

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**