# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021127
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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

Received  April 2021 Revised  July 2021 Early access September 2021

Fund Project: This work was supported in part by the NNSF of China (No. 11971212)

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.

Citation: Bojing Shi. $W^{1, p}$ estimates for elliptic problems with drift terms in Lipschitz domains. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021127
##### 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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] F. Lin, On current developments in partial differential equations, Commun. Math. Res., 36 (2020), 1-30.  doi: 10.4208/cmr.2020-0004.  Google Scholar [16] G. Sakellaris, Boundary Value Problems in Lipschitz Domains for Equations with Drifts, Ph. D thesis, The University of Chicago, 2017.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar

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##### 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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] F. Lin, On current developments in partial differential equations, Commun. Math. Res., 36 (2020), 1-30.  doi: 10.4208/cmr.2020-0004.  Google Scholar [16] G. Sakellaris, Boundary Value Problems in Lipschitz Domains for Equations with Drifts, Ph. D thesis, The University of Chicago, 2017.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar
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