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Sharp summability for Monge transport density via interpolation

Luigi De Pascale;  Aldo Pratelli — 2004

ESAIM: Control, Optimisation and Calculus of Variations

Using some results proved in De Pascale and Pratelli [Calc. Var. Partial Differ. Equ. 14 (2002) 249-274] (and De Pascale et al. [Bull. London Math. Soc. 36 (2004) 383-395]) and a suitable interpolation technique, we show that the transport density relative to an L p source is also an L p function for any 1 ≤ p ≤ + ∞ .

The Monge problem for strictly convex norms in ℝ d

Thierry Champion;  Luigi De Pascale — 2010

Journal of the European Mathematical Society

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.

Γ -convergence and absolute minimizers for supremal functionals

Thierry Champion;  Luigi De Pascale;  Francesca Prinari — 2004

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove that the L p approximants naturally associated to a supremal functional Γ -converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer (i.e. local...

-convergence and absolute minimizers for supremal functionals

Thierry Champion;  Luigi De Pascale;  Francesca Prinari — 2010

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove that the approximants naturally associated to a supremal functional -converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer ( local solution)...

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