Gradient estimates for the strong p(x)-Laplace equation

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July 2017, 37(7): 4109-4129. doi: 10.3934/dcds.2017175

Gradient estimates for the strong $p(x)$-Laplace equation

Chao Zhang ^1, , Xia Zhang ^2,, and Shulin Zhou ^3,

1.	Department of Mathematics, Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China
2.	Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
3.	LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

* Corresponding author

Received June 2016 Revised February 2017 Published April 2017

We study nonlinear elliptic equations of strong $p(x)$-Laplacian type to obtain an interior Calderón-Zygmund type estimates by finding a correct regularity assumption on the variable exponent $p(x)$. Our proof is based on the maximal function technique and the appropriate localization method.

Keywords: Nonlinear Calderón-Zygmund theory, strong $p(x)$-Laplacian, variable exponents, generalized Lebesgue and Sobolev spaces.

Mathematics Subject Classification: Primary: 35R05, 35J92; Secondary: 35J15, 35J25.

Citation: Chao Zhang, Xia Zhang, Shulin Zhou. Gradient estimates for the strong $p(x)$-Laplace equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175

References:

[1]	E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacean system, J. Reine Angew. Math., 584 (2005), 117-148. doi: 10.1515/crll.2005.2005.584.117.
[2]	T. Adamowicz and P. Hästö, Mappings of finite distortion and PDE with nonstandard growth, Int. Math. Res. Not. IMRN, 10 (2010), 1940-1965. doi: 10.1093/imrn/rnp192.
[3]	T. Adamowicz and P. Hästö, Harnack's inequality and the strong $p(·)$-Laplacian, J. Differential Equations, 250 (2011), 1631-1649. doi: 10.1016/j.jde.2010.10.006.
[4]	K. Astala, T. Iwaniec, P. Koskela and G. Martin, Mappings of BMO-bounded distortion, Math. Ann., 317 (2000), 703-726. doi: 10.1007/PL00004420.
[5]	S. Byun and J. Ok, On $W^{1, q(·)}$-estimates for elliptic equations of $p(x)$-Laplacian type, J. Math. Pures Appl., 106 (2016), 512-545. doi: 10.1016/j.matpur.2016.03.002.
[6]	S. Byun, J. Ok and S. Ryu, Global gradient estimates for elliptic equations of $p(x)$-Laplacian type with BMO nonlinearity, J. Reine Angew. Math., 715 (2016), 1-38. doi: 10.1515/crelle-2014-0004.
[7]	S. Byun and L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math., 57 (2004), 1283-1310. doi: 10.1002/cpa.20037.
[8]	L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-18363-8.
[9]	X. Fan and D. Zhao, On the spaces $L^{p(x)}(Ω)$ and $W^{m, p(x)}(Ω)$, J. Math. Anal. Appl., 263 (2001), 424-446. doi: 10.1006/jmaa.2000.7617.
[10]	T. Iwaniec, $p$-harmonic tensors and quasiregular mappings, Ann. Math., 136 (1992), 589-624. doi: 10.2307/2946602.
[11]	T. Iwaniec and A. Verde, On the operator $\mathcal{L}(f)=f\log\|f\|$, J. Funct. Anal., 169 (1999), 391-420. doi: 10.1006/jfan.1999.3443.
[12]	T. Kilpeläinen and P. Koskela, Global integrability of the gradients of solutions to partial differential equations, Nonlinear Anal., 23 (1994), 899-909. doi: 10.1016/0362-546X(94)90127-9.
[13]	O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618.
[14]	G. M. Lieberman, The natural generalization of the natral conditons of Ladyzenskaja and Ural'tzeva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361. doi: 10.1080/03605309108820761.
[15]	G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.
[16]	E. M. Stein, Harmonic Analysis Princeton University Press, Princeton, NJ, 1993.
[17]	C. Zhang, L. Wang, S. Zhou and Y.-H. Kim, Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains, Commun. Pure Appl. Anal., 13 (2014), 2559-2587. doi: 10.3934/cpaa.2014.13.2559.
[18]	C. Zhang and S. Zhou, Hölder regularity for the gradients of solutions of the strong $p(x)$-Laplacian, J. Math. Anal. Appl., 389 (2012), 1066-1077. doi: 10.1016/j.jmaa.2011.12.047.

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References:

[1]	E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacean system, J. Reine Angew. Math., 584 (2005), 117-148. doi: 10.1515/crll.2005.2005.584.117.
[2]	T. Adamowicz and P. Hästö, Mappings of finite distortion and PDE with nonstandard growth, Int. Math. Res. Not. IMRN, 10 (2010), 1940-1965. doi: 10.1093/imrn/rnp192.
[3]	T. Adamowicz and P. Hästö, Harnack's inequality and the strong $p(·)$-Laplacian, J. Differential Equations, 250 (2011), 1631-1649. doi: 10.1016/j.jde.2010.10.006.
[4]	K. Astala, T. Iwaniec, P. Koskela and G. Martin, Mappings of BMO-bounded distortion, Math. Ann., 317 (2000), 703-726. doi: 10.1007/PL00004420.
[5]	S. Byun and J. Ok, On $W^{1, q(·)}$-estimates for elliptic equations of $p(x)$-Laplacian type, J. Math. Pures Appl., 106 (2016), 512-545. doi: 10.1016/j.matpur.2016.03.002.
[6]	S. Byun, J. Ok and S. Ryu, Global gradient estimates for elliptic equations of $p(x)$-Laplacian type with BMO nonlinearity, J. Reine Angew. Math., 715 (2016), 1-38. doi: 10.1515/crelle-2014-0004.
[7]	S. Byun and L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math., 57 (2004), 1283-1310. doi: 10.1002/cpa.20037.
[8]	L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-18363-8.
[9]	X. Fan and D. Zhao, On the spaces $L^{p(x)}(Ω)$ and $W^{m, p(x)}(Ω)$, J. Math. Anal. Appl., 263 (2001), 424-446. doi: 10.1006/jmaa.2000.7617.
[10]	T. Iwaniec, $p$-harmonic tensors and quasiregular mappings, Ann. Math., 136 (1992), 589-624. doi: 10.2307/2946602.
[11]	T. Iwaniec and A. Verde, On the operator $\mathcal{L}(f)=f\log\|f\|$, J. Funct. Anal., 169 (1999), 391-420. doi: 10.1006/jfan.1999.3443.
[12]	T. Kilpeläinen and P. Koskela, Global integrability of the gradients of solutions to partial differential equations, Nonlinear Anal., 23 (1994), 899-909. doi: 10.1016/0362-546X(94)90127-9.
[13]	O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618.
[14]	G. M. Lieberman, The natural generalization of the natral conditons of Ladyzenskaja and Ural'tzeva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361. doi: 10.1080/03605309108820761.
[15]	G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.
[16]	E. M. Stein, Harmonic Analysis Princeton University Press, Princeton, NJ, 1993.
[17]	C. Zhang, L. Wang, S. Zhou and Y.-H. Kim, Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains, Commun. Pure Appl. Anal., 13 (2014), 2559-2587. doi: 10.3934/cpaa.2014.13.2559.
[18]	C. Zhang and S. Zhou, Hölder regularity for the gradients of solutions of the strong $p(x)$-Laplacian, J. Math. Anal. Appl., 389 (2012), 1066-1077. doi: 10.1016/j.jmaa.2011.12.047.

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