We investigate Prékopa-Leindler type inequalities on a Riemannian manifold $M$ equipped with a measure with density $e^{-V}$ where the potential $V$ and the Ricci curvature satisfy ${Hess}_{x}V+{Ric}_x\ge \lambda I$ for all $x\in M$, with some $\lambda \in ℝ$. As in our earlier work [14], the argument uses optimal mass transport on $M$, but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method...

On donne un développement asymptotique du profil iso pé ri mé tri que de ${ℝ}^n$ muni d'une métrique riemannienne périodique, et des conséquences pour le problème de la forme d'équilibre des cristaux.

Fix two points $x,\overline{x}\in S^2$ and two directions (without orientation) $\eta ,\overline{\eta }$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $J\left[\gamma \right]={\int }_0^T\left({}_{\gamma \left(t\right)}(\dot{\gamma }\left(t\right),\dot{\gamma }\left(t\right))+K_{\gamma \left(t\right)}^2{}_{\gamma \left(t\right)}(\dot{\gamma }\left(t\right),\dot{\gamma }\left(t\right))\right)\mathrm{d}t$ along all smooth curves starting from x with direction η and ending in $\overline{x}$ with direction $\overline{\eta }$. Here g is the standard Riemannian metric on S2 and $K_{\gamma }$ is the corresponding geodesic curvature. The interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1). We...

Given a measurable set $A\subset {ℝ}^n$ of positive measure, it is not difficult to show that $|A+A|=\left|2A\right|$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $\left(\right|A+A|-|2A\left|\right)/\left|A\right|$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $\left(\right|A+A|-|2A\left|\right)/\left|A\right|$.

A new approach to irreversible quasistatic fracture growth is given, by means of Young measures. The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases. In the particular situation of constant unloading response, the result contained in [G. Dal Maso and C. Zanini, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 253–279] is recovered. In this case, the convergence of the discrete time approximations is improved....

A new approach to irreversible quasistatic fracture growth is given, by means of Young measures. The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases. In the particular situation of constant unloading response, the result contained in [G. Dal Maso and C. Zanini, Proc. Roy. Soc. Edinburgh Sect. A137 (2007) 253–279] is recovered. In this case, the convergence of the discrete time approximations is improved. ...

This article addresses regularity of optimal transport maps for cost=“squared distance” on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved. Under boundedness and non-vanishing assumptions on the transfered source and target densities we show that optimal maps stay away from the cut-locus (where the cost exhibits singularity), and obtain injectivity and continuity of optimal maps....

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for...

This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is defined via relaxation, and it defines a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted Euclidean...