On the regularity of the total variation minimizers

We prove regularity results for the unique minimizer of the total variation functional, currently used in image processing analysis since the work by Rudin, Osher and Fatemi. In particular, we show that if the source term f is locally (respectively, globally) Lipschitz, then the solution has the same regularity with local (respectively, global) Lipschitz norm estimated accordingly. The result is proved in any dimension and for any (regular) domain. So far we extend a similar result proved earlier by Caselles, Chambolle and Novaga for dimension N <= 7 and (in case of the global regularity) for convex domains.