Luigi De Pascale

The

We consider the non-nonlinear optimal transportation problem of minimizing the cost functional

\infty

\mathcal{C}_\infty(\lambda) = \operatornamewithlimits{\lambda-ess\,sup}_{(x,y) \in \Omega^2} |y-x|

-Wasserstein Distance: Local Solutions and Existence of Optimal Transport Maps

in the set of probability measures on

The Monge problem for strictly convex norms in Rd

\Omega^2

Asymptotic behavior of non linear eigenvalue problems involving

having prescribed marginals. This corresponds to the question of characterizing the measures that realize the infinite Wasserstein distance. We establish the existence of “local” solutions and characterize this class with the aid of an adequate version of cyclical monotonicity. Moreover, under natural assumptions, we show that local solutions are induced by transport maps.

p-

We prove the existence of an optimal transport map for the Monge problem in a convex bounded subset of R d under the assumptions that the first marginal is absolutely continuous with respect to the Lebesgue measure and that the cost is given by a strictly convex norm. We propose a new approach which does not use disintegration of measures.

Laplacian type operators

We study the asymptotic behaviour of two nonlinear eigenvalue problems which involve

Homogenization of Dirichlet Problems with Convex Bounded Constraints on the Gradient

p

Optimal Shapes and Masses, and Optimal Transportation Problems

-Laplacian-type operators. In the first problem we consider the limit as

Monge's transport problem in the Heisenberg group

p\to\infty