(Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as t →∞. We also prove that local maxima strictly decrease with time; in particular, flat zones immediately decrease their level. We display some numerical experiments illustrating these facts. 1. Introduction. Let Ω be a bounded set in R N with Lipschitz-continuous boundary ∂Ω. We are interested in the problem ∂u Du = div(

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 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.

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Abstract. For a vector field a = ∂J, where J is a convex, coercive and differentiable function on IR N without any growth conditions, we define generalized solution for the nonlinear elliptic equation −diva(∇u) = f in Ω, u = 0 on ∂Ω where Ω is a smooth bounded domain in IR N and f ∈ L 1 (Ω). We prove existence of a generalized solution and show that this solution is in fact an entropy solution. Under assumption of strict monotonicity for a we prove uniqueness of generalized solution. In the proofs of our results we avoid the theory of Sobolev-Orlicz spaces.