American Institute of Mathematical Sciences

Advanced
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.  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

[1]

Sun-Sig Byun, Yumi Cho, Shuang Liang. Calderón-Zygmund estimates for quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020038
[2]

Marius Ionescu, Luke G. Rogers. Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2155-2175. doi: 10.3934/cpaa.2014.13.2155
[3]

Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091
[4]

Melvin Faierman. Fredholm theory for an elliptic differential operator defined on $ \mathbb{R}^n $ and acting on generalized Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1463-1483. doi: 10.3934/cpaa.2020074
[5]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003
[6]

Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure & Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475
[7]

Goro Akagi. Doubly nonlinear parabolic equations involving variable exponents. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 1-16. doi: 10.3934/dcdss.2014.7.1
[8]

Ángel Arroyo, Joonas Heino, Mikko Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-laplacian. Communications on Pure & Applied Analysis, 2017, 16 (3) : 915-944. doi: 10.3934/cpaa.2017044
[9]

Yong Fang, Patrick Foulon, Boris Hasselblatt. Zygmund strong foliations in higher dimension. Journal of Modern Dynamics, 2010, 4 (3) : 549-569. doi: 10.3934/jmd.2010.4.549
[10]

Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 211-233. doi: 10.3934/dcds.2007.19.211
[11]

Miroslav Bulíček, Annegret Glitzky, Matthias Liero. Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 697-713. doi: 10.3934/dcdss.2017035
[12]

Mostafa Bendahmane, Kenneth Hvistendahl Karlsen, Mazen Saad. Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1201-1220. doi: 10.3934/cpaa.2013.12.1201
[13]

Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems & Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026
[14]

Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
[15]

Fabrice Delbary, Kim Knudsen. Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem. Inverse Problems & Imaging, 2014, 8 (4) : 991-1012. doi: 10.3934/ipi.2014.8.991
[16]

CÉSAR E. TORRES LEDESMA. Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$. Communications on Pure & Applied Analysis, 2017, 16 (1) : 99-114. doi: 10.3934/cpaa.2017004
[17]

Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22
[18]

Lujuan Yu. The asymptotic behaviour of the $ p(x) $-Laplacian Steklov eigenvalue problem. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020025
[19]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[20]

C. Fabry, Raul Manásevich. Equations with a $p$-Laplacian and an asymmetric nonlinear term. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 545-557. doi: 10.3934/dcds.2001.7.545

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (67)
  • HTML views (42)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences