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Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

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Volume 5, Issue 1


Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients

Verena Bögelein / Frank Duzaar / Jens Habermann / Christoph Scheven
Published Online: 2012-01-18 | DOI: https://doi.org/10.1515/acv.2011.009


We establish partial regularity for solutions to systems modeling electro-rheological fluids in the stationary case. As a model case our result covers the low order regularity of systems of the type

where denotes the symmetric part of the gradient , denotes the pressure, the not necessarily continuous coefficient is a bounded non-negative -function and the variable exponent function fulfills the logarithmic continuity assumption, i.e., we assume that for the modulus of continuity of the exponent function there holds

To be more precise, we prove Hölder continuity of the solution outside of a negligible set. Moreover, we show that and the pressure belong to certain Morrey spaces on the regular set of , i.e., the set where is Hölder continuous. Note that under such weak assumptions partial Hölder continuity for the gradient cannot be expected. Our result is even new if the coefficient is continuous.

Keywords.: Partial Hölder- and Morrey-type regularity; -coefficients; electro-rheological fluids; non-standard growth condition

About the article

Received: 2011-01-30

Accepted: 2011-03-12

Published Online: 2012-01-18

Published in Print: 2012-01-01

Citation Information: Advances in Calculus of Variations, Volume 5, Issue 1, Pages 1–57, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv.2011.009.

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