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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:
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Rao, Variable exponent, linear growth functionals in image restoration,, \emph{SIAM J. Appl. Math.}, 66 (2006), 1383. doi: 10.1137/050624522. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] P. Harjulehto, Variable exponent Sobolev spaces with zero boundary values,, \emph{Math. Bohem.}, 132 (2007), 125. Google Scholar [17] T. Iwaniec, Projections onto gradient fields and $L^p$-estimates for degenerated elliptic operators,, \emph{Studia Math.}, 75 (1983), 293. Google Scholar [18] T. Iwaniec, $p$-harmonic tensors and quasiregular mappings,, \emph{Ann. Math.}, 136 (1992), 589. doi: 10.2307/2946602. Google Scholar [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. Google Scholar [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. Google Scholar [21] J. Kinnunen and S. Zhou, A boundary estimate for nonlinear equations with discontinuous coefficients,, \emph{Differential Integral Equations}, 14 (2001), 475. Google Scholar [22] V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces,, World Scientific, (1991). doi: 10.1142/9789814360302. Google Scholar [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. Google Scholar [24] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$,, \emph{Czechoslovak Math. J.}, 41 (1991), 592. Google Scholar [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. Google Scholar [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. Google Scholar [27] K. Rajagopal and M. Růžička, Mathematical modeling of electro-rheological fluids,, \emph{Contin. Mech. Thermodyn.}, 13 (2001), 59. Google Scholar [28] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory,, in: Lecture Notes in Mathematics, (1748). doi: 10.1007/BFb0104029. Google Scholar [29] J. Musielak, Orlicz Spaces and Modular Spaces,, Springer-Verlag, (1983). Google Scholar [30] M. Rao and Z. Ren, Applications of Orlicz Spaces,, Marcel Dekker Inc., (2002). doi: 10.1201/9780203910863. Google Scholar [31] E. M. Stein, Harmonic Analysis,, Princeton University Press, (1993). Google Scholar [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. Google Scholar [33] V. V. Zhikov, On some variational problems,, \emph{Russ. J. Math. Phys.}, 5 (1997), 105. Google Scholar

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. Google Scholar [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. Google Scholar [3] R. A. Adams and J. J. F. Fournier, Sobolev spaces,, (2nd edition), (2003). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] P. Harjulehto, Variable exponent Sobolev spaces with zero boundary values,, \emph{Math. Bohem.}, 132 (2007), 125. Google Scholar [17] T. Iwaniec, Projections onto gradient fields and $L^p$-estimates for degenerated elliptic operators,, \emph{Studia Math.}, 75 (1983), 293. Google Scholar [18] T. Iwaniec, $p$-harmonic tensors and quasiregular mappings,, \emph{Ann. Math.}, 136 (1992), 589. doi: 10.2307/2946602. Google Scholar [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. Google Scholar [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. Google Scholar [21] J. Kinnunen and S. Zhou, A boundary estimate for nonlinear equations with discontinuous coefficients,, \emph{Differential Integral Equations}, 14 (2001), 475. Google Scholar [22] V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces,, World Scientific, (1991). doi: 10.1142/9789814360302. Google Scholar [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. Google Scholar [24] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$,, \emph{Czechoslovak Math. J.}, 41 (1991), 592. Google Scholar [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. Google Scholar [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. Google Scholar [27] K. Rajagopal and M. Růžička, Mathematical modeling of electro-rheological fluids,, \emph{Contin. Mech. Thermodyn.}, 13 (2001), 59. Google Scholar [28] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory,, in: Lecture Notes in Mathematics, (1748). doi: 10.1007/BFb0104029. Google Scholar [29] J. Musielak, Orlicz Spaces and Modular Spaces,, Springer-Verlag, (1983). Google Scholar [30] M. Rao and Z. Ren, Applications of Orlicz Spaces,, Marcel Dekker Inc., (2002). doi: 10.1201/9780203910863. Google Scholar [31] E. M. Stein, Harmonic Analysis,, Princeton University Press, (1993). Google Scholar [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. Google Scholar [33] V. V. Zhikov, On some variational problems,, \emph{Russ. J. Math. Phys.}, 5 (1997), 105. Google Scholar
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