Existence of generalized minimizers and of dual solutions for a class of variational problems with linear growth related to image recovery

We continue the analysis of some modifications of the total variation image inpainting method formulated on the space BV (Ω) in the sense that we generalize some of the main results of [13] to the case of vectorvalued images where now we do not impose any structure condition on our density F and the dimension of the domain Ω is arbitrary. Precisely we discuss existence of generalized solutions of the corresponding variational problem and we will also pass to the associated dual variational problem. Among other things, our results are the uniqueness of the absolutely continuous part ∇u of the gradient of BV -solutions u on the entire domain Ω where outside of the damaged region D we even get uniqueness of BV solutions. As remarkable byproducts we further prove new density results for functions of bounded variation and for Sobolev functions.