JournalszaaVol. 20, No. 3pp. 589–598

A Note on Degenerate Variational Problems with Linear Growth

  • M. Bildhauer

    Universität des Saarlandes, Saarbrücken, Germany
A Note on Degenerate Variational Problems with Linear Growth cover

Abstract

Given a class of strictly convex and smooth integrands ff with linear growth, we consider the minimization problem Ωfu)dx\int_{\Omega} f \bigtriangledown u) dx \rightarrow min and the dual problem with maximizer σ\sigma. Although degenerate problems are studied, the validity of the classical duality relation is proved in the following sense: there exists a generalized minimizer uBV(Ω;RN)u* \in BV (\Omega; \mathbb R^N) of the original problem such that σ(x)=f(au)\sigma (x) = \bigtriangledown f (\bigtriangledown^a u*) holds almost everywhere, where au\bigtriangledown^a u* denotes the absolutely continuous part of u\bigtriangledown u* with respect to the Lebesgue measure. In particular, this relation is also true in regions of degeneracy, i.e. at points xx such that D2f(au(x))=0D^2f(\bigtriangledown ^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