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Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients
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 |
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.
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. |
show all references
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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