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

TL;DR

This paper proves that for convex domains, variational problems with linear growth and $L^2$ fidelity have minimizers in $W^{1,1}$ space, inheriting regularity from the data, with bounds on the singular part of the gradient.

Contribution

It establishes the existence of $W^{1,1}$ solutions for a class of linear growth variational problems on convex domains, extending regularity results.

Findings

Minimizers exist in $W^{1,1}$ when the datum is in $W^{1,1}$.

Minimizers inherit $W^{1,p}$ regularity from the datum.

Quantitative bounds on the singular part of the gradient are obtained.

Abstract

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 that the domain of integration is convex, 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 also obtain a quantitative bound on the singular part of the gradient of the minimizer in the case that the datum is in $BV$. We infer analogous results for the gradient flow of the underlying functional of linear growth. We admit any convex integrand of linear growth.