The Dirichlet problem for the total variation flow

We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L-1 for entropy solutions. To prove the existence we use th nonlinear semigroup theory adn we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions. (C) 2001 Academic Press.