Hostname: page-component-7b7567c8c-kwt9d Total loading time: 0 Render date: 2026-11-15T03:59:51.830Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  20 March 2008

Lars Diening,
Josef Málek and
Mark Steinhauer
Lars Diening
Affiliation:
Abteilung für Angewandte Mathematik, Universität Freiburg, Eckerstr. 1, 79104 Freiburg i. Br., Germany; diening@mathematik.uni-freiburg.de
Josef Málek
Affiliation:
Mathematical Institute, Charles University, Sokolovská 83, 18675 Prague 8, Czech Republic; malek@karlin.mff.cuni.cz
Mark Steinhauer
Affiliation:
Mathematical Seminar, University of Bonn, Nussallee 15, 53115 Bonn, Germany; MSteinh@uni-bonn.de
Get access

Abstract

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. Anal34 (2003) 1064–1083]. We also establish new existence results to a class of incompressible electro-rheological fluids.

Keywords

Lipschitz truncation of $W^{1,p}_0/W^{1,p(\cdot)}_0$ -functionsexistenceweak solutionincompressible fluidpower-law fluidelectro-rheological fluid

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 2 , April 2008, pp. 211 - 232
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Acerbi, E. and Fusco, N., Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal 86 (1984) 125–145. CrossRef
Acerbi, E. and Fusco, N., A regularity theorem for minimizers of quasiconvex integrals. Arch. Rational Mech. Anal 99 (1987) 261–281.
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.
Boccardo, L. and Murat, F., Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal 19 (1992) 581–597. CrossRef
M.E. Bogovskiĭ, Solutions of some problems of vector analysis, associated with the operators ${\rm div}$ and ${\rm grad}$ , 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.
Cruz-Uribe, D., Fiorenza, A. and Neugebauer, C.J., The maximal function on variable $L\sp p$ spaces. Ann. Acad. Sci. Fenn. Math 28 (2003) 223–238.
Cruz-Uribe, D., Fiorenza, A., Martell, J.M. and Peréz, C., The boundedness of classical operators on variable ${L}^p$ spaces. Ann. Acad. Sci. Fenn. Math 31 (2006) 239–264.
Dal Maso, G. and Murat, F., Almost everywhere convergence of gradients of solutions to nonlinear elliptic systems. Nonlinear Anal 31 (1998) 405–412. CrossRef
Diening, L., Maximal function on generalized Lebesgue spaces $L\sp {p(\cdot)}$ . Math. Inequal. Appl 7 (2004) 245–253.
Diening, L., Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces $L^{p(\cdot)}$ and $W^{k,p(\cdot)}$ . Math. Nachrichten 268 (2004) 31–43. CrossRef
L. Diening and P. Hästö, Variable exponent trace spaces. Studia Math (2007) to appear.
L. Diening and M. Růžička, Calderón-Zygmund operators on generalized Lebesgue spaces $L^{p(\cdot)}$ and problems related to fluid dynamics J. Reine Angew. Math 563 (2003) 197–220.
Dolzmann, G., Hungerbühler, N. and Müller, S., 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. CrossRef
Duzaar, F. and Mingione, G., The p-harmonic approximation and the regularity of p-harmonic maps. Calc. Var. Partial Diff. Eq 20 (2004) 235–256. CrossRef
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, (1992).
Fan, X. and Zhao, D., On the spaces ${L}\sp {p(x)}({\Omega})$ and ${W}\sp {m,p(x)}({\Omega})$ . J. Math. Anal. Appl 263 (2001) 424–446. CrossRef
H. Federer, Geometric Measure Theory Band 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin-Heidelberg-New York (1969).
Frehse, J., Málek, J., and Steinhauer, M., 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). CrossRef
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).
Greco, L., Iwaniec, T. and Sbordone, C., Variational integrals of nearly linear growth. Diff. Int. Eq 10 (1997) 687–716.
A. Huber, Die Divergenzgleichung in gewichteten Räumen und Flüssigkeiten mit $p(\cdot)$ -Struktur. Ph.D. thesis, University of Freiburg, Germany (2005).
Kováčik, O. and Rákosník, J., On spaces ${L}\sp {p(x)}$ and ${W}\sp {k,p(x)}$ . Czechoslovak Math. J 41 (1991) 592–618.
Landes, R., Quasimonotone versus pseudomonotone. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) 705–717. CrossRef
Lerner, A., Some remarks on the Hardy-Littlewood maximal function on variable Lp spaces. Math. Z 251 (2005) 509–521. CrossRef
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.
J. Malý and W.P. Ziemer, Fine regularity of solutions of elliptic partial differential equations. American Mathematical Society, Providence, RI (1997).
Müller, S., A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. Amer. Math. Soc 351 (1999) 4585–4597. CrossRef
Nekvinda, A., Hardy-Littlewood maximal operator on $L\sp {p(x)}(\mathbb{R})$ . Math. Inequal. Appl 7 (2004) 255–265.
P. Pedregal, Parametrized measures and variational principles. Progress in Nonlinear Diff. Eq. Applications, Birkhäuser Verlag, Basel (1997).
Pick, L. and Růžička, M., An example of a space ${L}\sp {p(x)}$ on which the Hardy-Littlewood maximal operator is not bounded. Expo. Math 19 (2001) 369–371. CrossRef
K.R. Rajagopal and M. Růžička, On the modeling of electrorheological materials Mech. Res. Commun 23 (1996) 401–407.
Rajagopal, K.R. and Růžička, M., Mathematical modeling of electrorheological materials. Cont. Mech. Thermodyn 13 (2001) 59–78. CrossRef
M. Růžička, Electrorheological fluids: modeling and mathematical theory, Lect. Notes Math. 1748. Springer-Verlag, Berlin (2000).
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.
Zhang, K., Biting theorems for Jacobians and their applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) 345–365. CrossRef
Zhang, K., A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa Cl. Sci 19 (1992) 313–326.
Zhang, K., Remarks on perturbated systems with critical growth. Nonlinear Anal 18 (1992) 1167–1179. CrossRef
W.P. Ziemer. Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics 120. Springer-Verlag, Berlin (1989) 308.