Minimizing total variation flow

We prove existence and uniqueness of weak solutions for the minimizing Total variation flow with initial data in L(f) under Neumann boundary conditions. We prove that the H(N-1) measure of the boundaries of level sets of the solution decreases with time, as one would expect. We also prove that local maxima (minima) strictly decrease (increase) their level with time. We shall also consider the Dirichlet problem which presents some particular difficulties for general initial data in L(1). (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.