A Note on Degenerate Variational Problems with Linear Growth

Abstract

Given a class of strictly convex and smooth integrands $f$ with linear growth, we consider the minimization problem ${\int }_{\Omega }f▽u)dx\to$ min and the dual problem with maximizer $\sigma$. Although degenerate problems are studied, the validity of the classical duality relation is proved in the following sense: there exists a generalized minimizer $u\ast \in BV(\Omega ;{\mathbb{R}}^N)$ of the original problem such that $\sigma (x)=▽f({▽}^{a}u\ast )$ holds almost everywhere, where ${▽}^{a}u\ast$ denotes the absolutely continuous part of $▽u\ast$ with respect to the Lebesgue measure. In particular, this relation is also true in regions of degeneracy, i.e. at points $x$ such that $D^{2}f({▽}^{a}u\ast (x))=0$. As an application, we can improve the known regularity results for the dual solution.