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$ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China |
In this paper, we establish the $ W^{1,p} $ estimates for solutions of second order elliptic problems with drift terms in bounded Lipschitz domains by using a real variable method. For scalar equations, we prove that the $ W^{1,p} $ estimates hold for $ \frac{3}{2}-\varepsilon<p<3+\varepsilon $ for $ d\geq3 $, and the range for $ p $ is sharp. For elliptic systems, we prove that the $ W^{1,p} $ estimates hold for $ \frac{2d}{d+1}-\varepsilon<p<\frac{2d}{d-1}+\varepsilon $ under the assumption that the Lipschitz constant of the domain is small.
References:
[1] |
P. Auscher and M. Qafsaoui,
Observations on $W^{1, p}$ estimates for divergence elliptic equations with VMO coefficients, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 5 (2002), 487-509.
|
[2] |
L. A. Caffarelli and I. Peral,
On $W^{1, p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51 (1998), 1-21.
doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.0.CO;2-G. |
[3] |
B. Dahlberg and C. Kenig,
Hardy spaces and the Neumann problem in $L^p$ for Laplace's equation in Lipschitz domains, Ann. of Math. (2), 125 (1987), 437-465.
doi: 10.2307/1971407. |
[4] |
G. Di Fazio,
$L^p$ estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. A (7), 10 (1996), 409-420.
|
[5] |
M. Dindoš, The $L^p$ Dirichlet and regularity problems for second order elliptic systems with application to the Lamé system, preprint, arXiv: 2006.13015. |
[6] |
M. Dindoš, S. Hwang and M. Mitrea,
The $L^p$ Dirichlet boundary problem for second order elliptic systems with rough coefficients, Trans. Amer. Math. Soc., 374 (2021), 3659-3701.
doi: 10.1090/tran/8306. |
[7] |
J. Geng,
$W^{1, p}$ estimates for elliptic problems with Neumann boundary conditions in Lipschitz domains, Adv. Math., 229 (2012), 2427-2448.
doi: 10.1016/j.aim.2012.01.004. |
[8] |
J. Geng,
Homogenization of elliptic problems with Neumann boundary conditions in non-smooth domains, Acta Math. Sin. (Engl. Ser.), 34 (2018), 612-628.
doi: 10.1007/s10114-017-7229-5. |
[9] |
J. Geng, Z. Shen and L. Song,
Uniform $W^{1, p}$ estimates for systems of linear elasticity in a periodic medium, J. Funct. Anal., 262 (2012), 1742-1758.
doi: 10.1016/j.jfa.2011.11.023. |
[10] |
M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, 105. Princeton University Press, Princeton, NJ, 1983.
![]() ![]() |
[11] |
D. Jerison and C. Kenig,
The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219.
doi: 10.1006/jfan.1995.1067. |
[12] |
C. Kenig and J. Pipher,
The Dirichlet problem for elliptic equations with drift terms, Publ. Mat., 45 (2001), 199-217.
doi: 10.5565/PUBLMAT_45101_09. |
[13] |
S. Kim and G. Sakellaris,
Green's function for second order elliptic equations with singular lower order coefficients, Comm. Partial Differential Equations, 44 (2019), 228-270.
doi: 10.1080/03605302.2018.1543318. |
[14] |
M. Kontovourkis, On Elliptic Equations with Low-Regularity Divergence-Free Drift Terms and the Steady-State Navier-Stokes Equations in Higher Dimensions, Ph. D thesis, University of Minnesota, 2007. |
[15] |
F. Lin,
On current developments in partial differential equations, Commun. Math. Res., 36 (2020), 1-30.
doi: 10.4208/cmr.2020-0004. |
[16] |
G. Sakellaris, Boundary Value Problems in Lipschitz Domains for Equations with Drifts, Ph. D thesis, The University of Chicago, 2017. |
[17] |
G. Sakellaris,
Boundary value problems in Lipschitz domains for equations with lower order coefficients, Trans. Amer. Math. Soc., 372 (2019), 5947-5989.
doi: 10.1090/tran/7895. |
[18] |
Z. Shen,
Bounds of Riesz transforms on $L^p$ spaces for second order elliptic operators, Ann. Inst. Fourier (Grenoble), 55 (2005), 173-197.
doi: 10.5802/aif.2094. |
[19] |
Z. Shen,
The $L^p$ Dirichlet problem for elliptic systems on Lipschitz domains, Math. Res. Lett., 13 (2006), 143-159.
doi: 10.4310/MRL.2006.v13.n1.a11. |
[20] |
Z. Shen,
The $L^p$ boundary value problems on Lipschitz domains, Adv. Math., 216 (2007), 212-254.
doi: 10.1016/j.aim.2007.05.017. |
[21] |
Z. Shen, Periodic Homogenization of Elliptic Systems, Operator Theory: Advances and Applications, 269. Advances in Partial Differential Equations (Basel). Birkhäuser/Springer, Cham, 2018.
doi: 10.1007/978-3-319-91214-1. |
[22] |
G. Stampacchia,
Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.
doi: 10.5802/aif.204. |
[23] |
Q. Xu,
Uniform regularity estimates in homogenization theory of elliptic system with lower order terms, J. Math. Anal. Appl., 438 (2016), 1066-1107.
doi: 10.1016/j.jmaa.2016.02.011. |
[24] |
Q. Xu,
Uniform regularity estimates in homogenization theory of elliptic systems with lower order terms on the Neumann boundary problem, J. Differential Equations, 261 (2016), 4368-4423.
doi: 10.1016/j.jde.2016.06.027. |
[25] |
Q. Xu, P. Zhao and S. Zhou, The methods of layer potentials for general elliptic homogenization problems in Lipschitz domains, preprint, arXiv: 1801.09220. |
show all references
References:
[1] |
P. Auscher and M. Qafsaoui,
Observations on $W^{1, p}$ estimates for divergence elliptic equations with VMO coefficients, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 5 (2002), 487-509.
|
[2] |
L. A. Caffarelli and I. Peral,
On $W^{1, p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51 (1998), 1-21.
doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.0.CO;2-G. |
[3] |
B. Dahlberg and C. Kenig,
Hardy spaces and the Neumann problem in $L^p$ for Laplace's equation in Lipschitz domains, Ann. of Math. (2), 125 (1987), 437-465.
doi: 10.2307/1971407. |
[4] |
G. Di Fazio,
$L^p$ estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. A (7), 10 (1996), 409-420.
|
[5] |
M. Dindoš, The $L^p$ Dirichlet and regularity problems for second order elliptic systems with application to the Lamé system, preprint, arXiv: 2006.13015. |
[6] |
M. Dindoš, S. Hwang and M. Mitrea,
The $L^p$ Dirichlet boundary problem for second order elliptic systems with rough coefficients, Trans. Amer. Math. Soc., 374 (2021), 3659-3701.
doi: 10.1090/tran/8306. |
[7] |
J. Geng,
$W^{1, p}$ estimates for elliptic problems with Neumann boundary conditions in Lipschitz domains, Adv. Math., 229 (2012), 2427-2448.
doi: 10.1016/j.aim.2012.01.004. |
[8] |
J. Geng,
Homogenization of elliptic problems with Neumann boundary conditions in non-smooth domains, Acta Math. Sin. (Engl. Ser.), 34 (2018), 612-628.
doi: 10.1007/s10114-017-7229-5. |
[9] |
J. Geng, Z. Shen and L. Song,
Uniform $W^{1, p}$ estimates for systems of linear elasticity in a periodic medium, J. Funct. Anal., 262 (2012), 1742-1758.
doi: 10.1016/j.jfa.2011.11.023. |
[10] |
M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, 105. Princeton University Press, Princeton, NJ, 1983.
![]() ![]() |
[11] |
D. Jerison and C. Kenig,
The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219.
doi: 10.1006/jfan.1995.1067. |
[12] |
C. Kenig and J. Pipher,
The Dirichlet problem for elliptic equations with drift terms, Publ. Mat., 45 (2001), 199-217.
doi: 10.5565/PUBLMAT_45101_09. |
[13] |
S. Kim and G. Sakellaris,
Green's function for second order elliptic equations with singular lower order coefficients, Comm. Partial Differential Equations, 44 (2019), 228-270.
doi: 10.1080/03605302.2018.1543318. |
[14] |
M. Kontovourkis, On Elliptic Equations with Low-Regularity Divergence-Free Drift Terms and the Steady-State Navier-Stokes Equations in Higher Dimensions, Ph. D thesis, University of Minnesota, 2007. |
[15] |
F. Lin,
On current developments in partial differential equations, Commun. Math. Res., 36 (2020), 1-30.
doi: 10.4208/cmr.2020-0004. |
[16] |
G. Sakellaris, Boundary Value Problems in Lipschitz Domains for Equations with Drifts, Ph. D thesis, The University of Chicago, 2017. |
[17] |
G. Sakellaris,
Boundary value problems in Lipschitz domains for equations with lower order coefficients, Trans. Amer. Math. Soc., 372 (2019), 5947-5989.
doi: 10.1090/tran/7895. |
[18] |
Z. Shen,
Bounds of Riesz transforms on $L^p$ spaces for second order elliptic operators, Ann. Inst. Fourier (Grenoble), 55 (2005), 173-197.
doi: 10.5802/aif.2094. |
[19] |
Z. Shen,
The $L^p$ Dirichlet problem for elliptic systems on Lipschitz domains, Math. Res. Lett., 13 (2006), 143-159.
doi: 10.4310/MRL.2006.v13.n1.a11. |
[20] |
Z. Shen,
The $L^p$ boundary value problems on Lipschitz domains, Adv. Math., 216 (2007), 212-254.
doi: 10.1016/j.aim.2007.05.017. |
[21] |
Z. Shen, Periodic Homogenization of Elliptic Systems, Operator Theory: Advances and Applications, 269. Advances in Partial Differential Equations (Basel). Birkhäuser/Springer, Cham, 2018.
doi: 10.1007/978-3-319-91214-1. |
[22] |
G. Stampacchia,
Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.
doi: 10.5802/aif.204. |
[23] |
Q. Xu,
Uniform regularity estimates in homogenization theory of elliptic system with lower order terms, J. Math. Anal. Appl., 438 (2016), 1066-1107.
doi: 10.1016/j.jmaa.2016.02.011. |
[24] |
Q. Xu,
Uniform regularity estimates in homogenization theory of elliptic systems with lower order terms on the Neumann boundary problem, J. Differential Equations, 261 (2016), 4368-4423.
doi: 10.1016/j.jde.2016.06.027. |
[25] |
Q. Xu, P. Zhao and S. Zhou, The methods of layer potentials for general elliptic homogenization problems in Lipschitz domains, preprint, arXiv: 1801.09220. |
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