Calculus of Variations and Geometric Measure Theory
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L. Ambrosio - M. Lecumberry - S. Maniglia

Lipschitz regularity and approximate differentiability of the DiPerna-Lions flow

created by ambrosio on 28 Feb 2005
modified on 14 Jun 2005


Submitted Paper

Inserted: 28 feb 2005
Last Updated: 14 jun 2005

Year: 2005


In this paper we show that the flow associated to Sobolev vectorfields, given by the DiPerna-Lions theory, has a kind of Lipschitz regularity property typical of functions in Sobolev spaces: for any $\delta>0$ one can disregard a set of initial points with measure less than $\delta$ to find a Lipschitz flow in the complement in this set. The proof is achieved by introducing suitable averaged difference quotients of the flow and by estimating carefully the right hand side in the transport equations satisfied by them.

Keywords: Approximate differentiability, Flow


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