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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, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

T. Iwaniec, $p$-harmonic tensors and quasiregular mappings, Ann. Math., 136 (1992), 589-624.  doi: 10.2307/2946602.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

E. M. Stein, Harmonic Analysis Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

T. Iwaniec, $p$-harmonic tensors and quasiregular mappings, Ann. Math., 136 (1992), 589-624.  doi: 10.2307/2946602.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

E. M. Stein, Harmonic Analysis Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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