On the solvability in Sobolev spaces and related regularity results for a variant of the TV-image recovery model: the vector-valued case

TL;DR

This paper investigates conditions for solving vector-valued TV-regularization variational problems within Sobolev spaces, extending classical regularity results for image recovery models with linear growth energy densities.

Contribution

It establishes solvability conditions for vector-valued TV models in Sobolev spaces, advancing the understanding of regularity in variational image recovery problems.

Findings

Identifies conditions for classical solvability in Sobolev spaces

Extends TV-regularization analysis to vector-valued functions

Provides new regularity results for linear growth energy densities

Abstract

We study classes of variational problems with energy densities of linear growth acting on vector-valued functions. Our energies are strictly convex variants of the TV-regularization model introduced by Rudin, Osher and Fatemi [15] as a powerful tool in the field of image recovery. In contrast to our previous work we here try to figure out conditions under which we can solve these variational problems in classical spaces, e.g. in the Sobolev class $W^{1,1}$.