Existence and almost everywhere regularity of generalized minimizers for a class of variational problems with linear growth related to image inpainting

We continue the analysis of some modifications of the total variation image inpainting method formulated on the space $BV(\Omega)^M$ in the sense that we generalize the main results of [32] to the case that a more general data fitting term is involved. As in [32] we deal with vector-valued images, we do not impose any structure condition on our density $F$ and the dimension of the domain $\Omega$ 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 for which we show unique solvability. Among other things, our results are the uniqueness of the absolutely continuous part $\nabla^a u$ of the gradient of $BV$-solutions $u$ on the entire domain $\Omega$, where outside of the damaged region $D$ we even get uniqueness of $BV$-solutions. Imposing stronger assumptions on our density $F$ and an $L^{\infty}$-condition on our partial observation $f$ we are going to prove a maximum principle for each generalized minimizer and deduce partial $C^{1,\beta}$-regularity of solutions on the entire domain $\Omega$ for all $0<\beta\leq\frac{1}{2}$.