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

Bojing Shi

doi:10.3934/dcds.2021127

Previous Article
High and low perturbations of Choquard equations with critical reaction and variable growth
DCDS Home
This Issue
Next Article
Aubry-Mather theory for contact Hamiltonian systems II

doi: 10.3934/dcds.2021127

Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

$ 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

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.

Keywords: $ W^{1, p} $ estimates, elliptic systems, drift terms, real variable method, Lipschitz domains.

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

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

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. 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

[1]	Byungsoo Kang, Hyunseok Kim. W^{1, p}-estimates for elliptic equations with lower order terms. Communications on Pure & Applied Analysis, 2017, 16 (3) : 799-822. doi: 10.3934/cpaa.2017038
[2]	Caiyan Li, Dongsheng Li. $ W^{1,p} $ estimates for elliptic systems on composite material with almost partially BMO coefficients. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3143-3159. doi: 10.3934/cpaa.2021100
[3]	Sallah Eddine Boutiah, Abdelaziz Rhandi, Cristian Tacelli. Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 803-817. doi: 10.3934/dcds.2019033
[4]	Sun-Sig Byun, Lihe Wang. $W^{1,p}$ regularity for the conormal derivative problem with parabolic BMO nonlinearity in reifenberg domains. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 617-637. doi: 10.3934/dcds.2008.20.617
[5]	Antonio Vitolo. $H^{1,p}$-eigenvalues and $L^\infty$-estimates in quasicylindrical domains. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1315-1329. doi: 10.3934/cpaa.2011.10.1315
[6]	Yi Cao, Dong Li, Lihe Wang. The optimal weighted $W^{2, p}$ estimates of elliptic equation with non-compatible conditions. Communications on Pure & Applied Analysis, 2011, 10 (2) : 561-570. doi: 10.3934/cpaa.2011.10.561
[7]	Lele Du. Bounds for subcritical best Sobolev constants in W^{1, p}. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3871-3886. doi: 10.3934/cpaa.2021135
[8]	Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268
[9]	Feng Du, Adriano Cavalcante Bezerra. Estimates for eigenvalues of a system of elliptic equations with drift and of bi-drifting laplacian. Communications on Pure & Applied Analysis, 2017, 6 (2) : 475-491. doi: 10.3934/cpaa.2017024
[10]	Mahamadi Warma. Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1881-1905. doi: 10.3934/cpaa.2013.12.1881
[11]	Alexandre N. Carvalho, Jan W. Cholewa. Strongly damped wave equations in $W^(1,p)_0 (\Omega) \times L^p(\Omega)$. Conference Publications, 2007, 2007 (Special) : 230-239. doi: 10.3934/proc.2007.2007.230
[12]	Yijing Sun. Estimates for extremal values of $-\Delta u= h(x) u^{q}+\lambda W(x) u^{p}$. Communications on Pure & Applied Analysis, 2010, 9 (3) : 751-760. doi: 10.3934/cpaa.2010.9.751
[13]	Junjie Zhang, Shenzhou Zheng, Chunyan Zuo. $ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3305-3318. doi: 10.3934/dcdss.2021080
[14]	Jean Dolbeault, Marta García-Huidobro, Rául Manásevich. Interpolation inequalities in $ \mathrm W^{1,p}( {\mathbb S}^1) $ and carré du champ methods. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 375-394. doi: 10.3934/dcds.2020014
[15]	Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501
[16]	Mirela Kohr, Cornel Pintea, Wolfgang L. Wendland. Stokes-Brinkman transmission problems on Lipschitz and $C^1$ domains in Riemannian manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 493-537. doi: 10.3934/cpaa.2010.9.493
[17]	Mirela Kohr, Cornel Pintea, Wolfgang L. Wendland. Dirichlet - transmission problems for general Brinkman operators on Lipschitz and $C^1$ domains in Riemannian manifolds. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 999-1018. doi: 10.3934/dcdsb.2011.15.999
[18]	Mostafa Bendahmane, Kenneth Hvistendahl Karlsen, Mazen Saad. Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1201-1220. doi: 10.3934/cpaa.2013.12.1201
[19]	Stefano Biagi, Enrico Valdinoci, Eugenio Vecchi. A symmetry result for elliptic systems in punctured domains. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2819-2833. doi: 10.3934/cpaa.2019126
[20]	Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors