A characterization of the total variation $TV\left(u,\mathrm{\Omega }\right)$ of the Jacobian determinant $detDu$ is obtained for some classes of functions $u:\mathrm{\Omega }\to {\mathbb{R}}^n$ outside the traditional regularity space $W^{1,n}\left(\mathrm{\Omega };{\mathbb{R}}^n\right)$. In particular, explicit formulas are deduced for functions that are locally Lipschitz continuous away from a given one point singularity $x_0\in \mathrm{\Omega }$. Relations between $TV\left(u,\mathrm{\Omega }\right)$ and the distributional determinant $\text{Det}Du$ are established, and an integral representation is obtained for the relaxed energy of certain polyconvex functionals at maps $u\in W^{1,p}\left(\mathrm{\Omega };{\mathbb{R}}^n\right)\cap W^{1,\mathrm{\infty }}\left(\mathrm{\Omega }\left\{x_0\right\};{\mathbb{R}}^n\right)$.

This paper deals with the formulation of the necessary optimality condition for a topology optimization problem of an elastic body in unilateral contact with a rigid foundation. In the contact problem of Tresca, a given friction is governed by an elliptic variational inequality of the second order. The optimization problem consists in finding such topology of the domain occupied by the body that the normal contact stress along the contact boundary of the body is minimized. The topological derivative...

In recent research in the optimization of transportation networks, the problem was formalized as finding the optimal paths to transport a measure y+ onto a measure y- with the same mass. This approach is realistic for simple good distribution networks (water, electric power,. ..) but it is no more realistic when we want to specify who goes where, like in the mailing problem or the optimal urban traffic network problem. In this paper, we present a new framework generalizing the former approathes...

Des liens inattendus ont été récemment mis à jour entre le transport optimal de Monge–Kantorovich et certains problèmes de géométrie riemannienne, en liaison avec la courbure de Ricci. Une des retombées de ces interactions est la naissance d’une théorie “synthétique” des espaces métriques mesurés à courbure de Ricci minorée, venant compléter la théorie classique des espaces métriqes à courbure sectionnelle minorée. Dans ce texte (également fourni aux actes du Séminaire de Théorie Spectrale et Géométrie...

By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a...

We first prove existence and uniqueness of optimal transportation maps for the Monge’s problem associated to a cost function with a strictly convex constraint in the Euclidean plane ℝ2. The cost function coincides with the Euclidean distance if the displacement y − x belongs to a given strictly convex set, and it is infinite otherwise. Secondly, we give a sufficient condition for existence and uniqueness of optimal transportation maps for the original Monge’s problem in ℝ2. Finally, we get existence...

We show that any strictly mean convex translator of dimension n ≥ 3 which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension n ≥ 3 is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular...

For α∈(0, 1) an α-trimming, P∗, of a probability P is a new probability obtained by re-weighting the probability of any Borel set, B, according to a positive weight function, f≤1/(1−α), in the way P∗(B)=∫Bf(x)P(dx). If P, Q are probability measures on euclidean space, we consider the problem of obtaining the best L2-Wasserstein approximation between: (a) a fixed probability and trimmed versions of the other; (b) trimmed versions of both probabilities. These best trimmed approximations naturally...

The aim of this paper is to provide a rigorous variational formulation for the detection of points in 2-d biological images. To this purpose we introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals for which we prove the Γ-convergence to the initial one.

The aim of this paper is to provide a rigorous variational formulation for the detection of points in 2-d biological images. To this purpose we introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals for which we prove the Γ-convergence to the initial one.

In this paper we consider a new kind of Mumford–Shah functional E(u, Ω) for maps u : ℝm → ℝn with m ≥ n. The most important novelty is that the energy features a singular set Su of codimension greater than one, defined through the theory of distributional jacobians. After recalling the basic definitions and some well established results, we prove an approximation property for the energy E(u, Ω) via Γ −convergence, in the same spirit of the work by Ambrosio and Tortorelli [L. Ambrosio and V.M. Tortorelli,...

Nous étudions un théorème de Skorohod pour des mesures vectorielles à valeurs ${ℝ}^d$. En notant $X\left(\mathbb{P}\right)$ la mesure image de $\mathbb{P}$ par la variable aléatoire $X,$ nous donnons des classes de mesures $\mathbb{P}$ et éventuel-lement de variables telles que, si la suite $\left\{X_n\left(\mathbb{P}\right)\right\}$ converge étroitement, il existe une suite $\left\{{\phi }_n\right\},{\phi }_n\left(\mathbb{P}\right)=X_n\left(\mathbb{P}\right)$ qui converge en mesure, éventuel-lement p.s.Le problème de Monge est abordé comme application. Soit $\left|\mathbb{P}\right|$ la mesure variation de $\mathbb{P}$, pour un couple $(\mathbb{P},\mathbb{Q})$ et une fonction coût $c,$ le problème de Monge est l’existence d’une fonction...