A lemma on the higher integrability of functions with applications to the regularity theory of two-dimensional generalized Newtonian fluids

Manuscripta Mathematica

Volumn 116, Issue 2, 2005, Pages 135-156

  Bildhauer, Michael ^a   Fuchs, Martin ^a   Zhong, Xiao ^b  

^a SAARLAND UNIVERSITY   (Germany)

^b UNIVERSITY OF JYVÄSKYLÄ   (Finland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 14644430422     PISSN: 00252611     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00229-004-0523-4     Document Type: Article

References (28)

1. Acerbi, E., Mingione, G.: Regularity results for stationary electrorheological fluids. Arch. Rat. Mech. Anal. 164, 213-259 (2002)
2. Adams, R.A.: Sobolev spaces. Academic Press, New York-San Francisco-London, 1975
3. Apushkinskaya, D., Bildhauer, M., Fuchs, M.: Steady states of anisotropic generalized Newtonian fluids. To appear in J. Math. Fluid Mech.
4. Bildhauer, M.: Convex variational problems. Linear, nearly linear and anisotropic growth conditions. Lecture Notes in Mathematics 1818, Springer, Berlin-Heidelberg-New York, 2003
5. Bildhauer, M., Fuchs, M.: Variants of the Stokes Problem: the case of anisotropic potentials. J. Math. Fluid Mech. 5, 364-402 (2003)
6. Bildhauer, M., Fuchs, M.: A regularity result for stationary electrorheological fluids in two dimensions. Math. Meth. Appl. Sciences 27, 1607-1617 (2004)
7. Coscia, A., Mingione, G.: Hölder continuity of the gradient of p(x)-harmonic mappings. C. R. Acad. Sci. Paris Sr. I Math. 328 (4), 363-368 (1999)
8. Esposito, L., Leonetti, F., Mingione, G.: Regularity for minimizers of functionals with p-q growth. Nonlinear Diff. Equ. Appl. 6, 133-148 (1999)
9. Eringen, A.C., Maugin, G.: Electrodynamics of continua. Vol I&II, Springer, New York, 1989
10. Ettwein, F., Elektrorheologische Flüssigkeiten: Existenz starker Lösungen in zweidimensionalen Gebieten. Diplomarbeit, Universität Freiburg, 2002
11. Ettwein, F., Růžička, M.: Existence of strong solutions for electrorheological fluids in two dimensions: steady Dirichlet problem. In: Geometric Analysis and Nonlinear Partial Differential Equations. By S. Hildebrandt and H. Karcher, Springer-Verlag, Berlin-Heidelberg-New York, 2003, pp. 591-602
12. Faraco, D., Koskela, P., Zhong, X.: Mappings of finite distortion: the degree of regularity. Adv. Math. To appear
13. Frehse, J.: Two dimensional variational problems with thin obstacles. Math. Z. 143, 279-288 (1975)
14. Frehse, J., Seregin, G.: Regularity for solutions of variational problems in the deformation theory of plasticity with logarithmic hardening. Proc. St. Petersburg Math. Soc. 5, 184-222 (1998) (in Russian). English translation: Transl. Am. Math. Soc, II, 193, 127-152 (1999)
15. Frehse, J., Seregin, G.: Regularity for solutions of variational problems in the deformation theory of plasticity with logarithmic hardening. Proc. St. Petersburg Math. Soc. 5, 184-222 (1998) (in Russian). English translation: Transl. Am. Math. Soc, II, 193, 127-152 (1999)
16. p-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265-277 (1973)
17. Giaquinta, M.: Multiple integrals in the Calculus of Variations and non-linear elliptic systems. Ann. of Math. Stud. 105, Princeton University Press, Princeton, NJ, 1983
18. Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of the second order. Second Edition, Springer-Verlag, Berlin-Heidelberg, 1983
19. Hutter, K., van de Ven, A.A.F.: Field matter interactions in thermoelastic solids. Lecture Notes in Physics Vol. 88, Springer, Berlin, 1978
20. Iwaniec, T: The Gehring lemma, in P. Duren, J. Heinonen, B. Osgood and B. Palka, Quasiconformal Mappings and Analysis, Springer-Verlag, New-York, 1998
21. Iwaniec, T., Martin, G.: Geometric function theory and nonlinear analysis. Oxford University Press, 2001
22. Kauhanen, J., Koskela, P., Malý, J.: On functions with derivatives in a Lorentz spaces. Manuscripta Math. 100, 87-101 (1999)
23. Lukaszewicz, G.: Micropolar fluids. Modeling and Simulation in Science, Engineering & Technology, Birkhäuser, Boston, 1999
24. Pao, Y.H.: Electromagnetic forces in deformable continua. Mechanics Today (S. Nemat-Nasser, ed.) Vol. 4, Pergamon Press, 1978, pp. 209-306
25. Rajagopal, K.R., Růžička, M.: Mathematical modeling of electrorheological fluids. Cont. Mech. Therm. 13, 59-78 (2001)
26. Růžička, M.: Electrorheological fluids: modeling and mathematical theory. Lecture Notes in Mathematics Vol. 1748, Springer, Berlin-Heidelberg-New York, 2000
27. Stein, E.: Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, NJ, 1970
28. Truesdell, C., Rajagopal, K.R.: An introduction to the mechanics of fluids. Modeling and Simulation in Science, Engineering & Technology, Birkhäuser, Boston, 2000