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Séminaire du LJLL - 06 10 2017 14h00 : G. De Philippis
Guido De Philippis (Ecole Internationale Supérieure d’Etudes Avancées, Trieste)
On the converse of Rademacher’s Theorem and the set of good measures in Lipschitz differentiability spaces
Résumé
Rademacher’s Theorem asserts that every Lipschitz function on R^N is differentiable almost everywhere with respect to the Lebesgue measure. This result has been extended by Pansu to Carnot’s groups and by Cheeger to abstract metric measure spaces which are now called "Lipschitz differentiability spaces". A natural question is then to identify the set of all the "good measures” on metric spaces for which every Lipschitz function is differentiable almost everywhere.
The aim of this talk will be to discuss this issue in increasing generality. In particular we will present a proof of the fact that in R^N Rademacher’s Theorem holds for a measure if and only if this measure is absolutely continuous with respect to the Lebesgue measure. This result is based on a new structural result for measures satisfying a PDE constraint.
We will also show some consequences of this structural result concerning Lipschitz differentiability spaces. We will finally discuss some ongoing work concerning the converse of Pansu’s Theorem.