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On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. (English) Zbl 1143.35037
In this paper the authors recall, survey and strenghten properties of \( W_0^{1,\infty }\)-truncations of \(W_0^{1,p}\)-functions that are useful from the point of view of existence theory concerning nonlinear PDE’s. One illustrates the potential of this tool by establishing the weak stability for the system of \(p\)-Laplace equations with very general right-hand sides. Then one studies the properties of Lipschitz truncations of Sobolev functions with constant and variable exponent. As non-trivial applications the Lipschitz truncations are utilized to provide a simplified proof of an existence result for incompressible power-law like fluids. New existence results are established to a class of incompressible electro-rheological fluids.

35J60 Nonlinear elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35J70 Degenerate elliptic equations
35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76W05 Magnetohydrodynamics and electrohydrodynamics
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[1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal86 (1984) 125-145. Zbl0565.49010 · Zbl 0565.49010 · doi:10.1007/BF00275731
[2] E. Acerbi and N. Fusco, A regularity theorem for minimizers of quasiconvex integrals. Arch. Rational Mech. Anal99 (1987) 261-281. Zbl0627.49007 · Zbl 0627.49007 · doi:10.1007/BF00284509
[3] E. Acerbi and N. Fusco, An approximation lemma for W 1 , p functions, in Material instabilities in continuum mechanics (Edinburgh, 1985-1986), Oxford Sci. Publ., Oxford Univ. Press, New York (1988) 1-5. Zbl0644.46026 · Zbl 0644.46026
[4] L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal19 (1992) 581-597. Zbl0783.35020 · Zbl 0783.35020 · doi:10.1016/0362-546X(92)90023-8
[5] M.E. Bogovskiĭ, Solutions of some problems of vector analysis, associated with the operators div and grad , in Theory of cubature formulas and the application of functional analysis to problems of mathematical physics (Russian)149, Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk (1980) 5-40.
[6] D. Cruz-Uribe, A. Fiorenza and C.J. Neugebauer, The maximal function on variable L p spaces. Ann. Acad. Sci. Fenn. Math28 (2003) 223-238. Zbl1037.42023 · Zbl 1037.42023 · emis:journals/AASF/Vol28/cruz.html · eudml:123518
[7] D. Cruz-Uribe, A. Fiorenza, J.M. Martell and C. Peréz, The boundedness of classical operators on variable L p spaces. Ann. Acad. Sci. Fenn. Math31 (2006) 239-264. · Zbl 1100.42012 · eudml:126704
[8] G. Dal Maso and F. Murat, Almost everywhere convergence of gradients of solutions to nonlinear elliptic systems. Nonlinear Anal31 (1998) 405-412. · Zbl 0890.35039 · doi:10.1016/S0362-546X(96)00317-3
[9] L. Diening, Maximal function on generalized Lebesgue spaces L p ( \cdot ) . Math. Inequal. Appl7 (2004) 245-253. Zbl1071.42014 · Zbl 1071.42014
[10] L. Diening, Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces L p ( \cdot ) and W k , p ( \cdot ) . Math. Nachrichten268 (2004) 31-43. Zbl1065.46024 · Zbl 1065.46024 · doi:10.1002/mana.200310157
[11] L. Diening and P. Hästö, Variable exponent trace spaces. Studia Math (2007) to appear. · Zbl 1134.46016
[12] L. Diening and M. Růžička, Calderón-Zygmund operators on generalized Lebesgue spaces L p ( \cdot ) and problems related to fluid dynamics J. Reine Angew. Math563 (2003) 197-220. Zbl1072.76071 · Zbl 1072.76071 · doi:10.1515/crll.2003.081
[13] G. Dolzmann, N. Hungerbühler and S. Müller, Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right hand side. J. Reine Angew. Math520 (2000) 1-35. Zbl0937.35065 · Zbl 0937.35065 · doi:10.1515/crll.2000.022
[14] F. Duzaar and G. Mingione, The p-harmonic approximation and the regularity of p-harmonic maps. Calc. Var. Partial Diff. Eq20 (2004) 235-256. Zbl1142.35433 · Zbl 1142.35433 · doi:10.1007/s00526-003-0233-x
[15] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, (1992). Zbl0804.28001 · Zbl 0804.28001
[16] X. Fan and D. Zhao, On the spaces L p ( x ) ( \Omega ) and W m , p ( x ) ( \Omega ) . J. Math. Anal. Appl263 (2001) 424-446. · Zbl 1028.46041 · doi:10.1006/jmaa.2000.7617
[17] H. Federer, Geometric Measure Theory Band 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin-Heidelberg-New York (1969). Zbl0176.00801 · Zbl 0176.00801
[18] J. Frehse, J. Málek, and M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal34 (2003) 1064-1083 (electronic). Zbl1050.35080 · Zbl 1050.35080 · doi:10.1137/S0036141002410988
[19] M. Giaquinta, G. Modica and J. Souček, Cartesian currents in the calculus of variations. I, vol. 37 of Ergebnisse der Mathematik. 3. Folge. A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin (1998). Zbl0914.49001 · Zbl 0914.49001
[20] L. Greco, T. Iwaniec and C. Sbordone, Variational integrals of nearly linear growth. Diff. Int. Eq10 (1997) 687-716. Zbl0889.35026 · Zbl 0889.35026
[21] A. Huber, Die Divergenzgleichung in gewichteten Räumen und Flüssigkeiten mit p ( \cdot ) -Struktur. Ph.D. thesis, University of Freiburg, Germany (2005).
[22] O. Kováčik and J. Rákosník, On spaces L p ( x ) and W k , p ( x ) . Czechoslovak Math. J41 (1991) 592-618. · Zbl 0784.46029 · eudml:13956
[23] R. Landes, Quasimonotone versus pseudomonotone. Proc. Roy. Soc. Edinburgh Sect. A126 (1996) 705-717. · Zbl 0863.35033 · doi:10.1017/S0308210500023015
[24] A. Lerner, Some remarks on the Hardy-Littlewood maximal function on variable Lp spaces. Math. Z251 (2005) 509-521. Zbl1092.42009 · Zbl 1092.42009 · doi:10.1007/s00209-005-0818-5
[25] J. Málek and K.R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, in Evolutionary Equations, volume 2 of Handbook of differential equations, C. Dafermos and E. Feireisl Eds., Elsevier B. V. (2005) 371-459. · Zbl 1095.35027
[26] J. Malý and W.P. Ziemer, Fine regularity of solutions of elliptic partial differential equations. American Mathematical Society, Providence, RI (1997). · Zbl 0882.35001
[27] S. Müller, A sharp version of Zhang’s theorem on truncating sequences of gradients. Trans. Amer. Math. Soc351 (1999) 4585-4597. Zbl0942.49013 · Zbl 0942.49013 · doi:10.1090/S0002-9947-99-02520-9
[28] A. Nekvinda, Hardy-Littlewood maximal operator on L p ( x ) ( \Bbb R ) . Math. Inequal. Appl7 (2004) 255-265. · Zbl 1059.42016
[29] P. Pedregal, Parametrized measures and variational principles. Progress in Nonlinear Diff. Eq. Applications, Birkhäuser Verlag, Basel (1997).
[30] L. Pick and M. Růžička, An example of a space L p ( x ) on which the Hardy-Littlewood maximal operator is not bounded. Expo. Math19 (2001) 369-371. Zbl1003.42013 · Zbl 1003.42013 · doi:10.1016/S0723-0869(01)80023-2
[31] K.R. Rajagopal and M. Růžička, On the modeling of electrorheological materials Mech. Res. Commun23 (1996) 401-407. Zbl0890.76007 · Zbl 0890.76007 · doi:10.1016/0093-6413(96)00038-9
[32] K.R. Rajagopal and M. Růžička, Mathematical modeling of electrorheological materials. Cont. Mech. Thermodyn13 (2001) 59-78. Zbl0971.76100 · Zbl 0971.76100 · doi:10.1007/s001610100034
[33] M. Růžička, Electrorheological fluids: modeling and mathematical theory, Lect. Notes Math. 1748. Springer-Verlag, Berlin (2000). · Zbl 0962.76001
[34] K. Zhang, On the Dirichlet problem for a class of quasilinear elliptic systems of partial differential equations in divergence form, in Partial differential equations (Tianjin, 1986), Lect. Notes Math1306 (1988) 262-277. Zbl0672.35026 · Zbl 0672.35026
[35] K. Zhang, Biting theorems for Jacobians and their applications. Ann. Inst. H. Poincaré Anal. Non Linéaire7 (1990) 345-365. · Zbl 0717.49012 · numdam:AIHPC_1990__7_4_345_0 · eudml:78228
[36] K. Zhang, A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa Cl. Sci19 (1992) 313-326. · Zbl 0778.49015 · numdam:ASNSP_1992_4_19_3_313_0 · eudml:84128
[37] K. Zhang, Remarks on perturbated systems with critical growth. Nonlinear Anal18 (1992) 1167-1179. Zbl0786.35061 · Zbl 0786.35061 · doi:10.1016/0362-546X(92)90160-G
[38] W.P. Ziemer. Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics120. Springer-Verlag, Berlin (1989) 308. Zbl0692.46022 · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3
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