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

Steinhauer, Mark; Málek, Josef; Diening, Lars

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 2, pp. 211-232

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.

Classification : 35J55, 35J65, 35J70, 35Q35, 76D99
Keywords: Lipschitz truncation of $W^{1,p}_0/W^{1,p(\cdot )}_0$-functions, existence, weak solution, incompressible fluid, power-law fluid, electro-rheological fluid

[1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal 86 (1984) 125-145. | Zbl | MR

[2] E. Acerbi and N. Fusco, A regularity theorem for minimizers of quasiconvex integrals. Arch. Rational Mech. Anal 99 (1987) 261-281. | Zbl | MR

[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. | Zbl | MR

[4] L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal 19 (1992) 581-597. | Zbl | MR

[5] M.E. Bogovskiĭ, Solutions of some problems of vector analysis, associated with the operators $\mathrm{div}$ and $\mathrm{grad}$, 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. | Zbl | MR

[6] D. Cruz-Uribe, A. Fiorenza and C.J. Neugebauer, The maximal function on variable $L^p$ spaces. Ann. Acad. Sci. Fenn. Math 28 (2003) 223-238. | Zbl | MR

[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. Math 31 (2006) 239-264. | Zbl | MR

[8] G. Dal Maso and F. Murat, Almost everywhere convergence of gradients of solutions to nonlinear elliptic systems. Nonlinear Anal 31 (1998) 405-412. | Zbl | MR

[9] L. Diening, Maximal function on generalized Lebesgue spaces $L^{p(·)}$. Math. Inequal. Appl 7 (2004) 245-253. | Zbl | MR

[10] L. Diening, Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces $L^{p(·)}$ and $W^{k,p(·)}$. Math. Nachrichten 268 (2004) 31-43. | Zbl | MR

[11] L. Diening and P. Hästö, Variable exponent trace spaces. Studia Math (2007) to appear. | Zbl | MR

[12] L. Diening and M. Růžička, Calderón-Zygmund operators on generalized Lebesgue spaces $L^{p(·)}$ and problems related to fluid dynamics J. Reine Angew. Math 563 (2003) 197-220. | Zbl | MR

[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. Math 520 (2000) 1-35. | Zbl | MR

[14] F. Duzaar and G. Mingione, The $p$-harmonic approximation and the regularity of $p$-harmonic maps. Calc. Var. Partial Diff. Eq 20 (2004) 235-256. | Zbl | MR

[15] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, (1992). | Zbl | MR

[16] X. Fan and D. Zhao, On the spaces $L^{p\left(x\right)}\left(\Omega \right)$ and $W^{m,p\left(x\right)}\left(\Omega \right)$. J. Math. Anal. Appl 263 (2001) 424-446. | Zbl | MR

[17] H. Federer, Geometric Measure Theory Band 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin-Heidelberg-New York (1969). | Zbl | MR

[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. Anal 34 (2003) 1064-1083 (electronic). | Zbl | MR

[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). | Zbl | MR

[20] L. Greco, T. Iwaniec and C. Sbordone, Variational integrals of nearly linear growth. Diff. Int. Eq 10 (1997) 687-716. | Zbl | MR

[21] A. Huber, Die Divergenzgleichung in gewichteten Räumen und Flüssigkeiten mit $p(·)$-Struktur. Ph.D. thesis, University of Freiburg, Germany (2005).

[22] O. Kováčik and J. Rákosník, On spaces $L^{p\left(x\right)}$ and $W^{k,p\left(x\right)}$. Czechoslovak Math. J 41 (1991) 592-618. | Zbl

[23] R. Landes, Quasimonotone versus pseudomonotone. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) 705-717. | Zbl | MR

[24] A. Lerner, Some remarks on the Hardy-Littlewood maximal function on variable $L^p$ spaces. Math. Z 251 (2005) 509-521. | Zbl | MR

[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 | MR

[26] J. Malý and W.P. Ziemer, Fine regularity of solutions of elliptic partial differential equations. American Mathematical Society, Providence, RI (1997). | Zbl | MR

[27] S. Müller, A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. Amer. Math. Soc 351 (1999) 4585-4597. | Zbl | MR

[28] A. Nekvinda, Hardy-Littlewood maximal operator on $L^{p\left(x\right)}\left(ℝ\right)$. Math. Inequal. Appl 7 (2004) 255-265. | Zbl | MR

[29] P. Pedregal, Parametrized measures and variational principles. Progress in Nonlinear Diff. Eq. Applications, Birkhäuser Verlag, Basel (1997). | Zbl | MR

[30] L. Pick and M. Růžička, An example of a space $L^{p\left(x\right)}$ on which the Hardy-Littlewood maximal operator is not bounded. Expo. Math 19 (2001) 369-371. | Zbl | MR

[31] K.R. Rajagopal and M. Růžička, On the modeling of electrorheological materials Mech. Res. Commun 23 (1996) 401-407. | Zbl

[32] K.R. Rajagopal and M. Růžička, Mathematical modeling of electrorheological materials. Cont. Mech. Thermodyn 13 (2001) 59-78. | Zbl

[33] M. Růžička, Electrorheological fluids: modeling and mathematical theory, Lect. Notes Math. 1748. Springer-Verlag, Berlin (2000). | Zbl | MR

[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 Math 1306 (1988) 262-277. | Zbl | MR

[35] K. Zhang, Biting theorems for Jacobians and their applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) 345-365. | Zbl | MR | Numdam

[36] K. Zhang, A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa Cl. Sci 19 (1992) 313-326. | Zbl | MR | Numdam

[37] K. Zhang, Remarks on perturbated systems with critical growth. Nonlinear Anal 18 (1992) 1167-1179. | Zbl | MR

[38] W.P. Ziemer. Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics 120. Springer-Verlag, Berlin (1989) 308. | Zbl | MR