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 Kruzhkovʹs method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L1 for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.