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

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