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A note on degenerate variational problems with linear growth
URN: urn:nbn:de:bsz:291-scidok-43380
URL: http://scidok.sulb.uni-saarland.de/volltexte/2011/4338/
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Dokument 1.pdf (229 KB)
SWD-Schlagwörter:
Freie Schlagwörter (Englisch):
linear growth , degenerate problems , duality
Institut:
Fachrichtung 6.1 - Mathematik
DDC-Sachgruppe:
Mathematik
Dokumentart:
Preprint (Vorabdruck)
Schriftenreihe:
Preprint / Fachrichtung Mathematik, Universität des Saarlandes
Bandnummer:
30
Sprache:
Englisch
Erstellungsjahr:
2001
Publikationsdatum:
22.11.2011
Kurzfassung auf Englisch:
Given a class of strictly convex and smooth integrands f with linear growth, we consider the minimization problem \int_{\Omega}f(\nabla u)dx\rightarrow{\normalcolor 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)=\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 appliation, we can improve the known regularity results for the dual solution.
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