A note on degenerate variational problems with linear growth (Q5949156)

Summary: Given a class of strictly convex and smooth integrands \(f\) with linear growth, we consider the minimization problem \(\int_\Omega f(\nabla u) dx \to \min\) 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 \(u^*\in \text{BV}(W; \mathbb{R}^N)\) 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 application, we can improve the known regularity results for the dual solution.