Revista Matemática Iberoamericana


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Volume 29, Issue 3, 2013, pp. 969–996
DOI: 10.4171/RMI/746

Published online: 2013-08-04

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

Luigi Ambrosio[1], Nicola Gigli[2] and Giuseppe Savaré[3]

(1) Scuola Normale Superiore, Pisa, Italy
(2) SISSA, Trieste, Italy
(3) Università di Pavia, Italy

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

Keywords: Weak upper gradients, Sobolev functions, optimal transport

Ambrosio Luigi, Gigli Nicola, Savaré Giuseppe: Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. Rev. Mat. Iberoamericana 29 (2013), 969-996. doi: 10.4171/RMI/746