Regularization and Graph Approximation of a Discontinuous Evolution

Moreau has shown that a Yosida-type regularization procedure can be used in order to prove the existence of a solution to the so-called sweeping process or evolution problem associated with a moving convex set (see e.g. [Mor 3]). If this set is supposed to be Lipschitz-continuous in the sense of Hausdorff distance, then the solution to the problem it defines is also Lipschitz-continuous with respect to a real variable t. The Yosida or Moreau-Yosida approximants, that is, the absolutely continuous solutions to the regularized differential equations derived from the initial problem, then converge uniformly to the solution to the sweeping process.