Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients

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November 2019, 18(6): 3137-3160. doi: 10.3934/cpaa.2019141

Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients

Feng Zhou ^1,, and Zhenqiu Zhang ^2,

1.	School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong, 250358, China
2.	School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

^* Corresponding author

Received November 2018 Revised November 2018 Published May 2019

Fund Project: The second author is supported by the National Natural Science Foundation of China Grant 11671414.

In this paper we study subquadratic elliptic systems in divergence form with VMO leading coefficients in $ \mathbb{R}^{n} $. We establish pointwise estimates for gradients of local weak solutions to the system by involving the sharp maximal operator. As a consequence, the nonlinear Calderón-Zygmund gradient estimates for $ L^{q} $ and BMO norms are derived.

Keywords: Subquadratic elliptic system, gradient estimate, sharp maximal operator, nonlinear Calderón-Zygmund estimate.

Mathematics Subject Classification: Primary: 35B65; Secondary: 35J60.

Citation: Feng Zhou, Zhenqiu Zhang. Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3137-3160. doi: 10.3934/cpaa.2019141

References:

[1]	L. Beck, Boundary Regularity Results for Local Weak Solutions of Subquadratic Elliptic Systems, Ph.D thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2008.
[2]	D. Breit, A. Cianchi, L. Diening, T. Kuusi and S. Schwarzacher, Pointwise Calderón-Zygmund gradient estimates for the $p$-Laplace system, Journal de Mathématiques Pures et Appliquées, 114 (2018), 146-190. doi: 10.1016/j.matpur.2017.07.011.
[3]	D. Breit, A. Cianchi, L. Diening, T. Kuusi and S. Schwarzacher, The $p$-Laplace system with right-hand side in divergence form: Inner and up to the boundary pointwise estimates, Nonlinear Analysis: Theory, Methods and Applications, 153 (2017), 200-212. doi: 10.1016/j.na.2016.06.011.
[4]	L. Caffarelli and I. Peral, On $W^{1, p}$ estimates for elliptic equations in divergence form, Communications on Pure and Applied Mathematics, 51 (1998), 1-21. doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.0.CO;2-G.
[5]	A. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Mathematica, 88 (1952), 85-139. doi: 10.1007/BF02392130.
[6]	E. DiBenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, American Journal of Mathematics, 115 (1993), 1107-1134. doi: 10.2307/2375066.
[7]	L. Diening and F. Ettwein, Fractional estimates for non-differentiable elliptic systems with general growth, Forum Mathematicum, 20 (2008), 523-556. doi: 10.1515/FORUM.2008.027.
[8]	L. Diening, P. Kaplický and S. Schwarzacher, BMO estimates for the $p$-Laplacian, Nonlinear Analysis, 75 (2012), 637-650. doi: 10.1016/j.na.2011.08.065.
[9]	L. Diening, B. Stroffolini and A. Verde, Everywhere regularity of functionals with $\varphi$-growth, Manuscripta Mathematica, 129 (2009), 449-481. doi: 10.1007/s00229-009-0277-0.
[10]	F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials, American Journal of Mathematics, 133 (2011), 1093-1149. doi: 10.1353/ajm.2011.0023.
[11]	M. Giaquinta and L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, 2nd edition, Scuola Normale Superiore di Pisa, 2012. doi: 10.1007/978-88-7642-443-4.
[12]	E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co. Pte. Ltd., Singapore, 2003. doi: 10.1142/5002.
[13]	T. Iwaniec, Projections onto gradient fields and $L^{p}$-estimates for degenerated elliptic operators, Studia Mathematica, 75 (1983), 293-312. doi: 10.4064/sm-75-3-293-312.
[14]	T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Mathematica, 172 (1994), 137-161. doi: 10.1007/BF02392793.
[15]	T. Kuusi and G. Mingione, Linear potentials in nonlinear potential theory, Archive for Rational Mechanics and Analysis, 207 (2013), 215-246. doi: 10.1007/s00205-012-0562-z.
[16]	T. Kuusi and G. Mingione, A nonlinear Stein theorem, Calculus of Variations and Partial Differential Equations, 51 (2014), 45-86. doi: 10.1007/s00526-013-0666-9.
[17]	G. Mingione, Gradient potential estimates, Journal of the European Mathematical Society, 13 (2011), 459-486. doi: 10.4171/JEMS/258.
[18]	S. Schwarzacher, Hölder-Zygmund estimates for degenerate parabolic systems, Journal of Differential Equations, 256 (2014), 2423-2448. doi: 10.1016/j.jde.2014.01.009.

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References:

[1]	L. Beck, Boundary Regularity Results for Local Weak Solutions of Subquadratic Elliptic Systems, Ph.D thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2008.
[2]	D. Breit, A. Cianchi, L. Diening, T. Kuusi and S. Schwarzacher, Pointwise Calderón-Zygmund gradient estimates for the $p$-Laplace system, Journal de Mathématiques Pures et Appliquées, 114 (2018), 146-190. doi: 10.1016/j.matpur.2017.07.011.
[3]	D. Breit, A. Cianchi, L. Diening, T. Kuusi and S. Schwarzacher, The $p$-Laplace system with right-hand side in divergence form: Inner and up to the boundary pointwise estimates, Nonlinear Analysis: Theory, Methods and Applications, 153 (2017), 200-212. doi: 10.1016/j.na.2016.06.011.
[4]	L. Caffarelli and I. Peral, On $W^{1, p}$ estimates for elliptic equations in divergence form, Communications on Pure and Applied Mathematics, 51 (1998), 1-21. doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.0.CO;2-G.
[5]	A. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Mathematica, 88 (1952), 85-139. doi: 10.1007/BF02392130.
[6]	E. DiBenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, American Journal of Mathematics, 115 (1993), 1107-1134. doi: 10.2307/2375066.
[7]	L. Diening and F. Ettwein, Fractional estimates for non-differentiable elliptic systems with general growth, Forum Mathematicum, 20 (2008), 523-556. doi: 10.1515/FORUM.2008.027.
[8]	L. Diening, P. Kaplický and S. Schwarzacher, BMO estimates for the $p$-Laplacian, Nonlinear Analysis, 75 (2012), 637-650. doi: 10.1016/j.na.2011.08.065.
[9]	L. Diening, B. Stroffolini and A. Verde, Everywhere regularity of functionals with $\varphi$-growth, Manuscripta Mathematica, 129 (2009), 449-481. doi: 10.1007/s00229-009-0277-0.
[10]	F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials, American Journal of Mathematics, 133 (2011), 1093-1149. doi: 10.1353/ajm.2011.0023.
[11]	M. Giaquinta and L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, 2nd edition, Scuola Normale Superiore di Pisa, 2012. doi: 10.1007/978-88-7642-443-4.
[12]	E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co. Pte. Ltd., Singapore, 2003. doi: 10.1142/5002.
[13]	T. Iwaniec, Projections onto gradient fields and $L^{p}$-estimates for degenerated elliptic operators, Studia Mathematica, 75 (1983), 293-312. doi: 10.4064/sm-75-3-293-312.
[14]	T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Mathematica, 172 (1994), 137-161. doi: 10.1007/BF02392793.
[15]	T. Kuusi and G. Mingione, Linear potentials in nonlinear potential theory, Archive for Rational Mechanics and Analysis, 207 (2013), 215-246. doi: 10.1007/s00205-012-0562-z.
[16]	T. Kuusi and G. Mingione, A nonlinear Stein theorem, Calculus of Variations and Partial Differential Equations, 51 (2014), 45-86. doi: 10.1007/s00526-013-0666-9.
[17]	G. Mingione, Gradient potential estimates, Journal of the European Mathematical Society, 13 (2011), 459-486. doi: 10.4171/JEMS/258.
[18]	S. Schwarzacher, Hölder-Zygmund estimates for degenerate parabolic systems, Journal of Differential Equations, 256 (2014), 2423-2448. doi: 10.1016/j.jde.2014.01.009.

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