Existence of $W^{1,1}$ solutions to a class of variational problems with linear growth

We consider a class of convex integral functionals composed of a term of linear growth in the gradient of the argument, and a fidelity term involving $L^2$ distance from a datum. Such functionals are known to attain their infima in the $BV$ space. Under the assumption of convexity of the domain of integration, we prove that if the datum is in $W^{1,1}$, then the functional has a minimizer in $W^{1,1}$. In fact, the minimizer inherits $W^{1,p}$ regularity from the datum for any $p \in [1, +\infty]$. We infer analogous results for the gradient flow of the underlying functional of linear growth. We admit any convex integrand of linear growth, possibly defined on vector-valued maps.