$ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains

Bojing Shi

doi:10.3934/dcds.2021127

Article Contents

2022, Volume 42, Issue 2: 537-553. Doi: 10.3934/dcds.2021127

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$ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains

Bojing Shi

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received: April 2021

Revised: July 2021

Early access: September 2021

Published: February 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J25, 35J57; Secondary: 35B45.

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[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.