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...

In questa Nota si risolve il problema di esistenza per un funzionale alla Mumford-Shah in ipotesi più generali rispetto ad altri precedenti lavori sull'argomento. Si dimostra inoltre la locale finitezza delle partizioni ottimali trovate.

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...