Calculus of Variations and Geometric Measure Theory

E. Bruè - D. Semola

Regularity of Lagrangian flows over $RCD^*(K,N)$ spaces

created by bruè on 10 Feb 2018
modified by semola on 14 Sep 2020


Published Paper

Inserted: 10 feb 2018
Last Updated: 14 sep 2020

Journal: J. Reine Angew. Math.
Volume: 765
Pages: 171–203
Year: 2020


The aim of this note is to provide regularity results for Regular Lagrangian flows of Sobolev vector fields over compact metric measure spaces verifying the Riemannian curvature dimension condition. We first prove, borrowing some ideas already present in the literature, that flows generated by vector fields with bounded symmetric derivative are Lipschitz, providing the natural extension of the standard Cauchy-Lipschitz theorem to this setting. Then we prove a Lusin-type regularity result in the Sobolev case (under the additional assumption that the m.m.s. is Ahlfors regular) therefore extending the already known Euclidean result.