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Zeitschrift für Analysis und ihre Anwendungen

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Volume 20, Issue 3, 2001, pp. 589–598
DOI: 10.4171/ZAA/1033

Published online: 2001-09-30

A Note on Degenerate Variational Problems with Linear Growth

M. Bildhauer

Given a class of strictly convex and smooth integrands $f$ with linear growth, we consider the minimization problem $\int_{\Omega} f \bigtriangledown u) dx \rightarrow$ 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 $\sigma (x) = \bigtriangledown f (\bigtriangledown^a u*)$ holds almost everywhere, where $\bigtriangledown^a u*$ denotes the absolutely continuous part of $\bigtriangledown u*$ with respect to the Lebesgue measure. In particular, this relation is also true in regions of degeneracy, i.e. at points $x$ such that $D^2f(\bigtriangledown ^a u*(x)) = 0$. As an application, we can improve the known regularity results for the dual solution.

Keywords: Degenerate problems, linear growth, duality, regularity

Bildhauer M.: A Note on Degenerate Variational Problems with Linear Growth. Z. Anal. Anwend. 20 (2001), 589-598. doi: 10.4171/ZAA/1033