Absolutely continuous representatives on curves for Sobolev functions

We consider a class of Lipschitz vector fields S :Ω →Rn whose values lie in a suitable cone and we show that the trajectories of the system x = S(x) admit a parametrization that is invertible and Lipschitz with its inverse. As a consequence, every v in W1,1(Ω) admits a representative that is absolutely continuous on almost every trajectory of x = S(x). If S is an arbritrary Lipschitz field the same property does hold locally at every x such that S(x) = 0.  2003 Elsevier Science (USA). All rights reserved.