A continuity method for sweeping processes

We show that the existence and uniqueness of BV continuous sweeping processes can be easily reduced to the Lipschitz continuous case by means of a suitable reparametrization of the associated moving convex set. Moreover we put this approach in the wider framework of rate independent operators acting on curves in metric spaces and we prove an extension theorem for such operators. This abstract theorem is then applied in order to infer continuous dependence of the sweeping process on the data.