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.
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
@article{COCV_2008__14_2_211_0, author = {Steinhauer, Mark and M\'alek, Josef and Diening, Lars}, title = {On {Lipschitz} truncations of {Sobolev} functions (with variable exponent) and their selected applications}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {211--232}, publisher = {EDP-Sciences}, volume = {14}, number = {2}, year = {2008}, doi = {10.1051/cocv:2007049}, mrnumber = {2394508}, zbl = {1143.35037}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2007049/} }
TY - JOUR AU - Steinhauer, Mark AU - Málek, Josef AU - Diening, Lars TI - On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 211 EP - 232 VL - 14 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2007049/ DO - 10.1051/cocv:2007049 LA - en ID - COCV_2008__14_2_211_0 ER -
%0 Journal Article %A Steinhauer, Mark %A Málek, Josef %A Diening, Lars %T On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 211-232 %V 14 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2007049/ %R 10.1051/cocv:2007049 %G en %F COCV_2008__14_2_211_0
Steinhauer, Mark; Málek, Josef; Diening, Lars. On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 2, pp. 211-232. doi : 10.1051/cocv:2007049. http://archive.numdam.org/articles/10.1051/cocv:2007049/
[1] Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal 86 (1984) 125-145. | MR | Zbl
and ,[2] A regularity theorem for minimizers of quasiconvex integrals. Arch. Rational Mech. Anal 99 (1987) 261-281. | MR | Zbl
and ,[3] An approximation lemma for functions, in Material instabilities in continuum mechanics (Edinburgh, 1985-1986), Oxford Sci. Publ., Oxford Univ. Press, New York (1988) 1-5. | MR | Zbl
and ,[4] Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal 19 (1992) 581-597. | MR | Zbl
and ,[5] Solutions of some problems of vector analysis, associated with the operators and , 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. | MR | Zbl
,[6] The maximal function on variable spaces. Ann. Acad. Sci. Fenn. Math 28 (2003) 223-238. | MR | Zbl
, and ,[7] The boundedness of classical operators on variable spaces. Ann. Acad. Sci. Fenn. Math 31 (2006) 239-264. | MR | Zbl
, , and ,[8] Almost everywhere convergence of gradients of solutions to nonlinear elliptic systems. Nonlinear Anal 31 (1998) 405-412. | MR | Zbl
and ,[9] Maximal function on generalized Lebesgue spaces . Math. Inequal. Appl 7 (2004) 245-253. | MR | Zbl
,[10] Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces and . Math. Nachrichten 268 (2004) 31-43. | MR | Zbl
,[11] Variable exponent trace spaces. Studia Math (2007) to appear. | MR | Zbl
and ,[12] Calderón-Zygmund operators on generalized Lebesgue spaces and problems related to fluid dynamics J. Reine Angew. Math 563 (2003) 197-220. | MR | Zbl
and ,[13] Uniqueness and maximal regularity for nonlinear elliptic systems of -Laplace type with measure valued right hand side. J. Reine Angew. Math 520 (2000) 1-35. | MR | Zbl
, and ,[14] The -harmonic approximation and the regularity of -harmonic maps. Calc. Var. Partial Diff. Eq 20 (2004) 235-256. | MR | Zbl
and ,[15] Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, (1992). | MR | Zbl
and ,[16] On the spaces and . J. Math. Anal. Appl 263 (2001) 424-446. | MR | Zbl
and ,[17] Geometric Measure Theory Band 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin-Heidelberg-New York (1969). | MR | Zbl
,[18] 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). | MR | Zbl
, , and ,[19] 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). | MR | Zbl
, and ,[20] Variational integrals of nearly linear growth. Diff. Int. Eq 10 (1997) 687-716. | MR | Zbl
, and ,[21] Die Divergenzgleichung in gewichteten Räumen und Flüssigkeiten mit -Struktur. Ph.D. thesis, University of Freiburg, Germany (2005).
,[22] On spaces and . Czechoslovak Math. J 41 (1991) 592-618. | Zbl
and ,[23] Quasimonotone versus pseudomonotone. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) 705-717. | MR | Zbl
,[24] Some remarks on the Hardy-Littlewood maximal function on variable spaces. Math. Z 251 (2005) 509-521. | MR | Zbl
,[25] 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. | MR | Zbl
and ,[26] Fine regularity of solutions of elliptic partial differential equations. American Mathematical Society, Providence, RI (1997). | MR | Zbl
and ,[27] A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. Amer. Math. Soc 351 (1999) 4585-4597. | MR | Zbl
,[28] Hardy-Littlewood maximal operator on . Math. Inequal. Appl 7 (2004) 255-265. | MR | Zbl
,[29] Parametrized measures and variational principles. Progress in Nonlinear Diff. Eq. Applications, Birkhäuser Verlag, Basel (1997). | MR | Zbl
,[30] An example of a space on which the Hardy-Littlewood maximal operator is not bounded. Expo. Math 19 (2001) 369-371. | MR | Zbl
and ,[31] On the modeling of electrorheological materials Mech. Res. Commun 23 (1996) 401-407. | Zbl
and ,[32] Mathematical modeling of electrorheological materials. Cont. Mech. Thermodyn 13 (2001) 59-78. | Zbl
and ,[33] Electrorheological fluids: modeling and mathematical theory, Lect. Notes Math. 1748. Springer-Verlag, Berlin (2000). | MR | Zbl
,[34] 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. | MR | Zbl
,[35] Biting theorems for Jacobians and their applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) 345-365. | Numdam | MR | Zbl
,[36] A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa Cl. Sci 19 (1992) 313-326. | Numdam | MR | Zbl
,[37] Remarks on perturbated systems with critical growth. Nonlinear Anal 18 (1992) 1167-1179. | MR | Zbl
,[38] Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics 120. Springer-Verlag, Berlin (1989) 308. | MR | Zbl
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