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November  2014, 13(6): 2559-2587. doi: 10.3934/cpaa.2014.13.2559

Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2. 

Department of Mathematics, Shanghai Jiaotong University, Shang hai 200240

3. 

LMAM, School of Mathematical Sciences, Peking University, Bejing 100871

4. 

Department of Mathematics Education, Sangmyung University, Seoul 110--743, South Korea

Received  April 2014 Revised  May 2014 Published  July 2014

In this paper we consider the global gradient estimates for weak solutions of $p(x)$-Laplacian type equation with small BMO coefficients in a $\delta$-Reifenberg flat domain. The modified Vitali covering lemma, good $\lambda$-inequalities, the maximal function technique and the appropriate localization method are the main analytical tools. The global Caldéron--Zygmund theory for such equations is obtained. Moreover, we generalize the regularity estimates in the Lebesgue spaces to the Orlicz spaces.
Citation: Chao Zhang, Lihe Wang, Shulin Zhou, Yun-Ho Kim. Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2559-2587. doi: 10.3934/cpaa.2014.13.2559
References:
[1]

E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids,, \emph{Arch. Ration. Mech. Anal.}, 164 (2002), 213. doi: 10.1007/s00205-002-0208-7.

[2]

E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacean system,, \emph{J. reine angew. Math.}, 584 (2005), 117. doi: 10.1515/crll.2005.2005.584.117.

[3]

R. A. Adams and J. J. F. Fournier, Sobolev spaces,, (2nd edition), (2003).

[4]

K. Astala, T. Iwaniec, P. Koskela and G. Martin, Mappings of BMO-bounded distortion,, \emph{Math. Ann.}, 317 (2000), 703. doi: 10.1007/PL00004420.

[5]

V. Bögelein and M. Parviainen, Self-improving property of nonlinear higher order parabolic systems near the boundary,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 17 (2010), 21. doi: 10.1007/s00030-009-0038-5.

[6]

S. Byun and S. Ryu, Global weighted estimates for the gradient of solutions to nonlinear elliptic equations,, \emph{Ann. Inst. H. Poincaré Anal. Non Linéaire}, 30 (2013), 291. doi: 10.1016/j.anihpc.2012.08.001.

[7]

S. Byun and L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains,, \emph{Comm. Pure Appl. Math.}, 57 (2004), 1283. doi: 10.1002/cpa.20037.

[8]

S. Byun, L. Wang and S. Zhou, Nonlinear elliptic equations with small BMO coefficients in Reifenberg domains,, \emph{J. Funct. Anal.}, 250 (2007), 167. doi: 10.1016/j.jfa.2007.04.021.

[9]

S. Byun, F. Yao and S. Zhou, Gradient estimates in Orlicz space for nonlinear elliptic equations,, \emph{J. Funct. Anal.}, 255 (2008), 1851. doi: 10.1016/j.jfa.2008.09.007.

[10]

L. A. Caffarelli and I. Peral, On $W^{1,p}$ estimates for elliptic equations in divergence form,, \emph{Comm. Pure Appl. Math.}, 51 (1998), 1. doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N.

[11]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration,, \emph{SIAM J. Appl. Math.}, 66 (2006), 1383. doi: 10.1137/050624522.

[12]

E. DiBenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems,, \emph{Amer. J. Math.}, 115 (1993), 1107. doi: 10.2307/2375066.

[13]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents,, Lecture Notes in Mathematics, (2017). doi: 10.1007/978-3-642-18363-8.

[14]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$,, \emph{J. Math. Anal. Appl.}, 263 (2001), 424. doi: 10.1006/jmaa.2000.7617.

[15]

F. Giannetti and A. Passarelli di Napoli, Regularity results for a new class of functionals with non-standard growth conditions,, \emph{J. Differential Equations}, 254 (2013), 1280. doi: 10.1016/j.jde.2012.10.011.

[16]

P. Harjulehto, Variable exponent Sobolev spaces with zero boundary values,, \emph{Math. Bohem.}, 132 (2007), 125.

[17]

T. Iwaniec, Projections onto gradient fields and $L^p$-estimates for degenerated elliptic operators,, \emph{Studia Math.}, 75 (1983), 293.

[18]

T. Iwaniec, $p$-harmonic tensors and quasiregular mappings,, \emph{Ann. Math.}, 136 (1992), 589. doi: 10.2307/2946602.

[19]

T. Iwaniec and A. Verde, On the operator $\mathcalL(f)=f\log|f|$,, \emph{J. Funct. Anal.}, 169 (1999), 391. doi: 10.1006/jfan.1999.3443.

[20]

J. Kinnumen and S. Zhou, A local estimate for nonlinear equations with discontinuous coefficients,, \emph{Comm. Partial Differential Equations}, 24 (1999), 2043. doi: 10.1080/03605309908821494.

[21]

J. Kinnunen and S. Zhou, A boundary estimate for nonlinear equations with discontinuous coefficients,, \emph{Differential Integral Equations}, 14 (2001), 475.

[22]

V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces,, World Scientific, (1991). doi: 10.1142/9789814360302.

[23]

T. Kilpeläinen and P. Koskela, Global integrability of the gradients of solutions to partial differential equations,, \emph{Nonlinear Anal.}, 23 (1994), 899. doi: 10.1016/0362-546X(94)90127-9.

[24]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$,, \emph{Czechoslovak Math. J.}, 41 (1991), 592.

[25]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, \emph{Nonlinear Anal.}, 12 (1988), 1203. doi: 10.1016/0362-546X(88)90053-3.

[26]

T. Mengesha and N. C. Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat domains,, \emph{Arch. Ration. Mech. Anal.}, 203 (2012), 189. doi: 10.1007/s00205-011-0446-7.

[27]

K. Rajagopal and M. Růžička, Mathematical modeling of electro-rheological fluids,, \emph{Contin. Mech. Thermodyn.}, 13 (2001), 59.

[28]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory,, in: Lecture Notes in Mathematics, (1748). doi: 10.1007/BFb0104029.

[29]

J. Musielak, Orlicz Spaces and Modular Spaces,, Springer-Verlag, (1983).

[30]

M. Rao and Z. Ren, Applications of Orlicz Spaces,, Marcel Dekker Inc., (2002). doi: 10.1201/9780203910863.

[31]

E. M. Stein, Harmonic Analysis,, Princeton University Press, (1993).

[32]

L. Wang, F. Yao, S. Zhou and H. Jia, Optimal regularity for the poisson equation,, \emph{Proc. Amer. Math. Soc.}, 137 (2009), 2037. doi: 10.1090/S0002-9939-09-09805-0.

[33]

V. V. Zhikov, On some variational problems,, \emph{Russ. J. Math. Phys.}, 5 (1997), 105.

show all references

References:
[1]

E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids,, \emph{Arch. Ration. Mech. Anal.}, 164 (2002), 213. doi: 10.1007/s00205-002-0208-7.

[2]

E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacean system,, \emph{J. reine angew. Math.}, 584 (2005), 117. doi: 10.1515/crll.2005.2005.584.117.

[3]

R. A. Adams and J. J. F. Fournier, Sobolev spaces,, (2nd edition), (2003).

[4]

K. Astala, T. Iwaniec, P. Koskela and G. Martin, Mappings of BMO-bounded distortion,, \emph{Math. Ann.}, 317 (2000), 703. doi: 10.1007/PL00004420.

[5]

V. Bögelein and M. Parviainen, Self-improving property of nonlinear higher order parabolic systems near the boundary,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 17 (2010), 21. doi: 10.1007/s00030-009-0038-5.

[6]

S. Byun and S. Ryu, Global weighted estimates for the gradient of solutions to nonlinear elliptic equations,, \emph{Ann. Inst. H. Poincaré Anal. Non Linéaire}, 30 (2013), 291. doi: 10.1016/j.anihpc.2012.08.001.

[7]

S. Byun and L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains,, \emph{Comm. Pure Appl. Math.}, 57 (2004), 1283. doi: 10.1002/cpa.20037.

[8]

S. Byun, L. Wang and S. Zhou, Nonlinear elliptic equations with small BMO coefficients in Reifenberg domains,, \emph{J. Funct. Anal.}, 250 (2007), 167. doi: 10.1016/j.jfa.2007.04.021.

[9]

S. Byun, F. Yao and S. Zhou, Gradient estimates in Orlicz space for nonlinear elliptic equations,, \emph{J. Funct. Anal.}, 255 (2008), 1851. doi: 10.1016/j.jfa.2008.09.007.

[10]

L. A. Caffarelli and I. Peral, On $W^{1,p}$ estimates for elliptic equations in divergence form,, \emph{Comm. Pure Appl. Math.}, 51 (1998), 1. doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N.

[11]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration,, \emph{SIAM J. Appl. Math.}, 66 (2006), 1383. doi: 10.1137/050624522.

[12]

E. DiBenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems,, \emph{Amer. J. Math.}, 115 (1993), 1107. doi: 10.2307/2375066.

[13]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents,, Lecture Notes in Mathematics, (2017). doi: 10.1007/978-3-642-18363-8.

[14]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$,, \emph{J. Math. Anal. Appl.}, 263 (2001), 424. doi: 10.1006/jmaa.2000.7617.

[15]

F. Giannetti and A. Passarelli di Napoli, Regularity results for a new class of functionals with non-standard growth conditions,, \emph{J. Differential Equations}, 254 (2013), 1280. doi: 10.1016/j.jde.2012.10.011.

[16]

P. Harjulehto, Variable exponent Sobolev spaces with zero boundary values,, \emph{Math. Bohem.}, 132 (2007), 125.

[17]

T. Iwaniec, Projections onto gradient fields and $L^p$-estimates for degenerated elliptic operators,, \emph{Studia Math.}, 75 (1983), 293.

[18]

T. Iwaniec, $p$-harmonic tensors and quasiregular mappings,, \emph{Ann. Math.}, 136 (1992), 589. doi: 10.2307/2946602.

[19]

T. Iwaniec and A. Verde, On the operator $\mathcalL(f)=f\log|f|$,, \emph{J. Funct. Anal.}, 169 (1999), 391. doi: 10.1006/jfan.1999.3443.

[20]

J. Kinnumen and S. Zhou, A local estimate for nonlinear equations with discontinuous coefficients,, \emph{Comm. Partial Differential Equations}, 24 (1999), 2043. doi: 10.1080/03605309908821494.

[21]

J. Kinnunen and S. Zhou, A boundary estimate for nonlinear equations with discontinuous coefficients,, \emph{Differential Integral Equations}, 14 (2001), 475.

[22]

V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces,, World Scientific, (1991). doi: 10.1142/9789814360302.

[23]

T. Kilpeläinen and P. Koskela, Global integrability of the gradients of solutions to partial differential equations,, \emph{Nonlinear Anal.}, 23 (1994), 899. doi: 10.1016/0362-546X(94)90127-9.

[24]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$,, \emph{Czechoslovak Math. J.}, 41 (1991), 592.

[25]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, \emph{Nonlinear Anal.}, 12 (1988), 1203. doi: 10.1016/0362-546X(88)90053-3.

[26]

T. Mengesha and N. C. Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat domains,, \emph{Arch. Ration. Mech. Anal.}, 203 (2012), 189. doi: 10.1007/s00205-011-0446-7.

[27]

K. Rajagopal and M. Růžička, Mathematical modeling of electro-rheological fluids,, \emph{Contin. Mech. Thermodyn.}, 13 (2001), 59.

[28]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory,, in: Lecture Notes in Mathematics, (1748). doi: 10.1007/BFb0104029.

[29]

J. Musielak, Orlicz Spaces and Modular Spaces,, Springer-Verlag, (1983).

[30]

M. Rao and Z. Ren, Applications of Orlicz Spaces,, Marcel Dekker Inc., (2002). doi: 10.1201/9780203910863.

[31]

E. M. Stein, Harmonic Analysis,, Princeton University Press, (1993).

[32]

L. Wang, F. Yao, S. Zhou and H. Jia, Optimal regularity for the poisson equation,, \emph{Proc. Amer. Math. Soc.}, 137 (2009), 2037. doi: 10.1090/S0002-9939-09-09805-0.

[33]

V. V. Zhikov, On some variational problems,, \emph{Russ. J. Math. Phys.}, 5 (1997), 105.

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