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Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation
Gradient estimates for the strong $p(x)$-Laplace equation
| 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 |
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.
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. |
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. |
| [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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