The Dirichlet problem for the total variation flow

We introduce a new concept of solution for the Dirichlet problem for the to tal variational flow named entropy solution. Using Kruzhkov's method of dou bling variables both in space and in time we prove uniqueness and a compari son principle in L-1 for entropy solutions. To prove the existence we use t h nonlinear semigroup theory adn we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions. (C) 200 1 Academic Press.