Minimizing total variation flow

We prove existence and uniqueness of weak solutions for the minimizing Tota l variation flow with initial data in L-f under Neumann boundary conditions . We prove that the HN-1 measure of the boundaries of level sets of the sol ution decreases with time, as one would expect. We also prove that local ma xima (minima) strictly decrease (increase) their level with time. We shall also consider the Dirichlet problem which presents some particular difficul ties for general initial data in L-1. (C) 2000 Academie des sciences/Editio ns scientifiques et medicales Elsevier SAS.