Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces

Luigi Ambrosio (1) , Nicola Gigli (2) , Giuseppe Savaré (3)

1 SNS - SNS Pisa
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 Dipartimento di matematica F. Casorati

We compare several notion of weak (modulus of) gradient in metric measure spaces. Using tools from optimal transportation theory we prove density in energy of Lipschitz maps independenly of doubling and Poincaré assumptions on the metric measure space.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]; Géométrie métrique [math.MG]

Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré. Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. Revista Matemática Iberoamericana, 2013, 29 (3), pp.969-996. ⟨10.4171/RMI/746⟩. ⟨hal-00769374⟩