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(\nabla u)dx\rightarrow{\normalcolor 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*\in BV(\Omega;\mathbb{R}^{N}) of the original problem such that \sigma(x)=\nabla f(\nabla^{a}u*) holds almost everywhere, where \nabla^{a}u* denotes the absolutely continuous part of \nabla u* 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(\nabla^{a}u*(x))=0. As an appliation, we can improve the known regularity results for the dual solution.