Given a class of strictly convex and smooth integrands with linear growth, we consider the minimization problem min and the dual problem with maximizer . Although degenerate problems are studied, the validity of the classical duality relation is proved in the following sense: there exists a generalized minimizer of the original problem such that holds almost everywhere, where denotes the absolutely continuous part of with respect to the Lebesgue measure. In particular, this relation is also true in regions of degeneracy, i.e. at points such that . As an application, we can improve the known regularity results for the dual solution.
Cite this article
M. Bildhauer, A Note on Degenerate Variational Problems with Linear Growth. Z. Anal. Anwend. 20 (2001), no. 3, pp. 589–598DOI 10.4171/ZAA/1033