# A Note on Degenerate Variational Problems with Linear Growth

### M. Bildhauer

Universität des Saarlandes, Saarbrücken, Germany

## Abstract

Given a class of strictly convex and smooth integrands $f$ with linear growth, we consider the minimization problem $∫_{Ω}f▽u)dx→$ min and the dual problem with maximizer $σ$. Although degenerate problems are studied, the validity of the classical duality relation is proved in the following sense: there exists a generalized minimizer $u∗∈BV(Ω;R_{N})$ of the original problem such that $σ(x)=▽f(▽_{a}u∗)$ holds almost everywhere, where $▽_{a}u∗$ denotes the absolutely continuous part of $▽u∗$ with respect to the Lebesgue measure. In particular, this relation is also true in regions of degeneracy, i.e. at points $x$ such that $D_{2}f(▽_{a}u∗(x))=0$. 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. 589–598

DOI 10.4171/ZAA/1033