ESAIM: COCV, Vol. 14, N°2, pp. 211-232
DOI: 10.1051/cocv:2007049

On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications

Lars Diening¹, Josef Málek² and Mark Steinhauer<sup>3

1</sup>  Abteilung für Angewandte Mathematik, Universität Freiburg, Eckerstr. 1, 79104 Freiburg i. Br., Germany; diening@mathematik.uni-freiburg.de
²  Mathematical Institute, Charles University, Sokolovská 83, 18675 Prague 8, Czech Republic; malek@karlin.mff.cuni.cz
³  Mathematical Seminar, University of Bonn, Nussallee 15, 53115 Bonn, Germany; MSteinh@uni-bonn.de

(Received April 14, 2006. Revised August 25, 2006. Published online September 21, 2007.)

Abstract
We study properties of Lipschitz truncations of Sobolev functions with constant and variable exponent. As non-trivial applications we use the Lipschitz truncations to provide a simplified proof of an existence result for incompressible power-law like fluids presented in [Frehse et al., SIAM J. Math. Anal 34 (2003) 1064-1083]. We also establish new existence results to a class of incompressible electro-rheological fluids.

Mathematics Subject Classification. 35J55, 35J65, 35J70, 35Q35, 76D99

Key words: Lipschitz truncation of -functions, existence, weak solution, incompressible fluid, power-law fluid, electro-rheological fluid


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