The Dirichlet problem for the total variation flow

Andreu, Fuensanta; Ballester, Coloma; Caselles, Vicente; Mazón, José

doi:http://dx.doi.org/10.1006/jfan.2000.3698

The Dirichlet problem for the total variation flow

Citation

Andreu F, Ballester C, Caselles V, Mazón JM. The Dirichlet problem for the total variation flow. J Funct Anal. 2001 Mar 10;180(2):347-403. DOI: 10.1006/jfan.2000.3698

Abstract

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.