A first analysis of the Monge Problem with vanishing gradient penalization.
The approximation of the Monge problem by a family of functionals with vanishing gradient penalization will be discussed. In particular it will be shown the first order Gamma-limit may select a minimizer different form the monotone one. When the class of minimizers selected by the first order limit is empty an higher order Gamma-limit will be investigated. The density of special classes of transport maps will be investigated as a tool. (Joint work with Jean Louet and Filippo Santambrogio of the Universit\'e de Paris Sud-Orsay).
