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March  2022, 21(3): 959-998. doi: 10.3934/cpaa.2022006

Global regularity estimates for Neumann problems of elliptic operators with coefficients having a BMO anti-symmetric part in NTA domains

1. 

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

2. 

Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

*Corresponding author

Received  August 2021 Revised  November 2021 Published  March 2022 Early access  December 2021

Fund Project: This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11871254, 12071431, 11971058 and 12071197), the National Key Research and Development Program of China (Grant No. 2020YFA0712900) and the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2021-e18)

Let $ n\ge2 $ and $ \Omega\subset\mathbb{R}^n $ be a bounded NTA domain. In this article, the authors study (weighted) global regularity estimates for Neumann boundary value problems of second-order elliptic equations of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in $ \Omega $. Precisely, for any given $ p\in(2,\infty) $, via a weak reverse Hölder inequality with the exponent $ p $, the authors give a sufficient condition for the global $ W^{1,p} $ estimate and the global weighted $ W^{1,q} $ estimate, with $ q\in[2,p] $ and some Muckenhoupt weights, of solutions to Neumann boundary value problems in $ \Omega $. As applications, the authors further obtain global regularity estimates for solutions to Neumann boundary value problems of second-order elliptic equations of divergence form with coefficients consisting of both a small $ \mathrm{BMO} $ symmetric part and a small $ \mathrm{BMO} $ anti-symmetric part, respectively, in bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, $ C^1 $ domains, or (semi-)convex domains, in weighted Lebesgue spaces. The results given in this article improve the known results by weakening the assumption on the coefficient matrix.

Citation: Sibei Yang, Dachun Yang, Wenxian Ma. Global regularity estimates for Neumann problems of elliptic operators with coefficients having a BMO anti-symmetric part in NTA domains. Communications on Pure and Applied Analysis, 2022, 21 (3) : 959-998. doi: 10.3934/cpaa.2022006
References:
[1]

P. Auscher, On Necessary and Sufficient Conditions for Lp-Estimates of Riesz Transforms Associated to Elliptic Operators on ${{\mathbb{R}}^{n}}$ and Related Estimates, Memoirs of the American Mathematical Society, 186 (2007), no. 871, 75 pp. doi: 10.1090/memo/0871.

[2]

P. Auscher and J. M. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights, Adv. Math., 212 (2007), 225-276.  doi: 10.1016/j.aim.2006.10.002.

[3]

P. Auscher and M. Qafsaoui, Observations on W1, p estimates for divergence elliptic equations with VMO coefficients, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 5 (2002), 487-509. 

[4]

A. Banerjee and J. L. Lewis, Gradient bounds for p-harmonic systems with vanishing Neumann (Dirichlet) data in a convex domain, Nonlinear Anal., 100 (2014), 78-85.  doi: 10.1016/j.na.2014.01.009.

[5]

A. Barton and S. Mayboroda, Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces, Memoirs of the American Mathematical Society, 243 (2016), no. 1149,110 pp. doi: 10.1090/memo/1149.

[6]

S. Buckley and P. Koskela, Sobolev - Poincaré implies John, Math. Res. Lett., 2 (1995), 577-593.  doi: 10.4310/MRL.1995.v2.n5.a5.

[7]

T. A. Bui and X. T. Duong, Weighted variable exponent Sobolev estimates for elliptic equations with non-standard growth and measure data, NoDEA Nonlinear Differential Equations Appl., 25 (2018), no. 4, Paper No. 28, 37 pp. doi: 10.1007/s00030-018-0520-z.

[8]

T. A. Bui and X. T. Duong, Weighted Lorentz estimates for parabolic equations with non-standard growth on rough domains, Calc. Var. Partial Differ. Equ., 56 (2017), no. 6, Art. 177, 27 pp. doi: 10.1007/s00526-017-1273-y.

[9]

T. A. Bui and X. T. Duong, Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains, Calc. Var. Partial Differ. Equ., 56 (2017), no. 2, Art. 47, 24 pp. doi: 10.1007/s00526-017-1130-z.

[10]

S. S. Byun and L. Wang, The conormal derivative problem for elliptic equations with BMO coefficients on Reifenberg flat domains, Proc. London Math. Soc., 90 (2005), 245-272.  doi: 10.1112/S0024611504014960.

[11]

S. S. Byun and L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Commun. Pure Appl. Math., 57 (2004), 1283-1310.  doi: 10.1002/cpa.20037.

[12]

L. Caffarelli and I. Peral, On W1, p estimates for elliptic equations in divergence form, Commun. Pure Appl. Math., 51 (1998), 1-21.  doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N.

[13]

D. C. Chang, The dual of Hardy spaces on a bounded domain in ${{\mathbb{R}}^{n}}$, Forum Math., 6 (1994), 65-81.  doi: 10.1515/form.1994.6.65.

[14]

D. C. Chang, G. Dafni and C. Sadosky, A div-curl lemma in BMO on a domain, in Harmonic Analysis, Signal processing, and Complexity, Birkhäuser Boston, Boston, MA, 2005. doi: 10.1007/0-8176-4416-4_5.

[15]

D. C. ChangG. Dafni and H. Yue, A div-curl decomposition for the local Hardy space, Proc. Amer. Math. Soc., 137 (2009), 3369-3377.  doi: 10.1090/S0002-9939-09-09970-5.

[16]

S. Chua, Weighted inequalities on John domains, J. Math. Anal. Appl., 258 (2001), 763-776.  doi: 10.1006/jmaa.2000.7246.

[17]

R. CoifmanP. L. LionsY. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. 

[18]

H. Dong and D. Kim, Elliptic equations in divergence form with partially BMO coefficients, Arch. Ration. Mech. Anal., 196 (2010), 25-70.  doi: 10.1007/s00205-009-0228-7.

[19]

H. Dong and D. Kim, The conormal derivative problem for higher order elliptic systems with irregular coefficients, in: Recent Advances in Harmonic Analysis and Partial Differential Equations, 69-97, Contemp. Math. 581, Amer. Math. Soc., Providence, RI, 2012. doi: 10.1090/conm/581/11534.

[20]

H. Dong and D. Kim, On Lp-estimates for elliptic and parabolic equations with Ap weights, Trans. Amer. Math. Soc., 370 (2018), 5081-5130.  doi: 10.1090/tran/7161.

[21]

H. Dong and Z. Li, The conormal and Robin boundary value problems in nonsmooth domains satisfying a measure condition, J. Funct. Anal., 281 (2021), no. 9, Paper No. 109167, 32 pp. doi: 10.1016/j.jfa.2021.109167.

[22]

H. Dong and T. Phan, Mixed-norm Lp-estimates for non-stationary Stokes systems with singular VMO coefficients and applications, J. Differ. Equ., 276 (2021), 342-367.  doi: 10.1016/j.jde.2020.12.023.

[23]

E. FabesO. Mendez and M. Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains, J. Funct. Anal., 159 (1998), 323-368.  doi: 10.1006/jfan.1998.3316.

[24]

C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math., 129 (1972), 137-193.  doi: 10.1007/BF02392215.

[25]

J. Geng, W1, 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.

[26]

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.

[27] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies 105, Princeton University Press, Princeton, NJ, 1983. 
[28]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.

[29]

L. Grafakos, Classical Fourier Analysis, 3rd edition, Graduate Texts in Mathematics 249, Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.

[30]

D. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. Math., 46 (1982), 80-147.  doi: 10.1016/0001-8708(82)90055-X.

[31]

D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219.  doi: 10.1006/jfan.1995.1067.

[32]

H. JiaD. Li and L. Wang, Global regularity for divergence form elliptic equations on quasiconvex domains, J. Differ. Equ., 249 (2010), 3132-3147.  doi: 10.1016/j.jde.2010.08.015.

[33]

P. W. Jones, Extension theorems for BMO, Indiana Univ. Math. J., 29 (1980), 41-66.  doi: 10.1512/iumj.1980.29.29005.

[34]

P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88.  doi: 10.1007/BF02392869.

[35]

C. E. Kenig and T. Toro, Harmonic measure on locally flat domains, Duke Math. J., 87 (1997), 509-551.  doi: 10.1215/S0012-7094-97-08717-2.

[36]

L. Li and J. Pipher, Boundary behavior of solutions of elliptic operators in divergence form with a BMO anti-symmetric part, Commun. Partial Differ. Equ., 44 (2019), 156-204.  doi: 10.1080/03605302.2018.1542437.

[37]

V. G. Maz'ya and I. E. Verbitsky, Form boundedness of the general second-order differential operator, Commun. Pure Appl. Math., 59 (2006), 1286-1329.  doi: 10.1002/cpa.20122.

[38]

N. G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 189-206. 

[39]

D. MitreaI. MitreaM. Mitrea and L. Yan, Coercive energy estimates for differential forms in semi-convex domains, Commun. Pure Appl. Anal., 9 (2010), 987-1010.  doi: 10.3934/cpaa.2010.9.987.

[40]

D. MitreaM. Mitrea and L. Yan, Boundary value problems for the Laplacian in convex and semiconvex domains, J. Funct. Anal., 258 (2010), 2507-2585.  doi: 10.1016/j.jfa.2010.01.012.

[41]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Translated from the 1967 French original by Gerard Tronel and Alois Kufner, Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-10455-8.

[42]

E. R. Reifenberg, Solution of the Plateau Problem for m-dimensional surfaces of varying topological type, Acta Math., 104 (1960), 1-92.  doi: 10.1007/BF02547186.

[43]

D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc., 207 (1975), 391-405.  doi: 10.2307/1997184.

[44]

G. SereginL. SilvestreV. Šverák and A. Zlatoš, On divergence-free drifts, J. Differ. Equ., 252 (2012), 505-540.  doi: 10.1016/j.jde.2011.08.039.

[45]

Z. Shen, Weighted L2 estimates for elliptic homogenization in Lipschitz domains, arXiv: 2004.03087.

[46]

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.

[47]

Z. Shen, The Lp boundary value problems on Lipschitz domains, Adv. Math., 216 (2007), 212-254.  doi: 10.1016/j.aim.2007.05.017.

[48]

Z. Shen, Bounds of Riesz transforms on Lp spaces for second order elliptic operators, Ann. Inst. Fourier (Grenoble), 55 (2005), 173-197. 

[49] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. 
[50]

T. Toro, Doubling and flatness: geometry of measures, Notices Amer. Math. Soc., 44 (1997), 1087-1094. 

[51]

L. Wang, A geometric approach to the Calderón-Zygmund estimates, Acta Math. Sin. (Engl. Ser.), 19 (2003), 381-396.  doi: 10.1007/s10114-003-0264-4.

[52]

S. YangD. C. ChangD. Yang and W. Yuan, Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains, J. Differ. Equ., 268 (2020), 2510-2550.  doi: 10.1016/j.jde.2019.09.036.

[53]

S. YangD. Yang and W. Yuan, Weighted global regularity estimates for elliptic problems with Robin boundary conditions in Lipschitz domains, J. Differ. Equ., 296 (2021), 512-572.  doi: 10.1016/j.jde.2021.06.010.

[54]

S. Yang, D. Yang and W. Yuan, Global gradient estimates for Dirichlet problems of elliptic operators with a BMO anti-symmetric part, Submitted.

show all references

References:
[1]

P. Auscher, On Necessary and Sufficient Conditions for Lp-Estimates of Riesz Transforms Associated to Elliptic Operators on ${{\mathbb{R}}^{n}}$ and Related Estimates, Memoirs of the American Mathematical Society, 186 (2007), no. 871, 75 pp. doi: 10.1090/memo/0871.

[2]

P. Auscher and J. M. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights, Adv. Math., 212 (2007), 225-276.  doi: 10.1016/j.aim.2006.10.002.

[3]

P. Auscher and M. Qafsaoui, Observations on W1, p estimates for divergence elliptic equations with VMO coefficients, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 5 (2002), 487-509. 

[4]

A. Banerjee and J. L. Lewis, Gradient bounds for p-harmonic systems with vanishing Neumann (Dirichlet) data in a convex domain, Nonlinear Anal., 100 (2014), 78-85.  doi: 10.1016/j.na.2014.01.009.

[5]

A. Barton and S. Mayboroda, Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces, Memoirs of the American Mathematical Society, 243 (2016), no. 1149,110 pp. doi: 10.1090/memo/1149.

[6]

S. Buckley and P. Koskela, Sobolev - Poincaré implies John, Math. Res. Lett., 2 (1995), 577-593.  doi: 10.4310/MRL.1995.v2.n5.a5.

[7]

T. A. Bui and X. T. Duong, Weighted variable exponent Sobolev estimates for elliptic equations with non-standard growth and measure data, NoDEA Nonlinear Differential Equations Appl., 25 (2018), no. 4, Paper No. 28, 37 pp. doi: 10.1007/s00030-018-0520-z.

[8]

T. A. Bui and X. T. Duong, Weighted Lorentz estimates for parabolic equations with non-standard growth on rough domains, Calc. Var. Partial Differ. Equ., 56 (2017), no. 6, Art. 177, 27 pp. doi: 10.1007/s00526-017-1273-y.

[9]

T. A. Bui and X. T. Duong, Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains, Calc. Var. Partial Differ. Equ., 56 (2017), no. 2, Art. 47, 24 pp. doi: 10.1007/s00526-017-1130-z.

[10]

S. S. Byun and L. Wang, The conormal derivative problem for elliptic equations with BMO coefficients on Reifenberg flat domains, Proc. London Math. Soc., 90 (2005), 245-272.  doi: 10.1112/S0024611504014960.

[11]

S. S. Byun and L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Commun. Pure Appl. Math., 57 (2004), 1283-1310.  doi: 10.1002/cpa.20037.

[12]

L. Caffarelli and I. Peral, On W1, p estimates for elliptic equations in divergence form, Commun. Pure Appl. Math., 51 (1998), 1-21.  doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N.

[13]

D. C. Chang, The dual of Hardy spaces on a bounded domain in ${{\mathbb{R}}^{n}}$, Forum Math., 6 (1994), 65-81.  doi: 10.1515/form.1994.6.65.

[14]

D. C. Chang, G. Dafni and C. Sadosky, A div-curl lemma in BMO on a domain, in Harmonic Analysis, Signal processing, and Complexity, Birkhäuser Boston, Boston, MA, 2005. doi: 10.1007/0-8176-4416-4_5.

[15]

D. C. ChangG. Dafni and H. Yue, A div-curl decomposition for the local Hardy space, Proc. Amer. Math. Soc., 137 (2009), 3369-3377.  doi: 10.1090/S0002-9939-09-09970-5.

[16]

S. Chua, Weighted inequalities on John domains, J. Math. Anal. Appl., 258 (2001), 763-776.  doi: 10.1006/jmaa.2000.7246.

[17]

R. CoifmanP. L. LionsY. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. 

[18]

H. Dong and D. Kim, Elliptic equations in divergence form with partially BMO coefficients, Arch. Ration. Mech. Anal., 196 (2010), 25-70.  doi: 10.1007/s00205-009-0228-7.

[19]

H. Dong and D. Kim, The conormal derivative problem for higher order elliptic systems with irregular coefficients, in: Recent Advances in Harmonic Analysis and Partial Differential Equations, 69-97, Contemp. Math. 581, Amer. Math. Soc., Providence, RI, 2012. doi: 10.1090/conm/581/11534.

[20]

H. Dong and D. Kim, On Lp-estimates for elliptic and parabolic equations with Ap weights, Trans. Amer. Math. Soc., 370 (2018), 5081-5130.  doi: 10.1090/tran/7161.

[21]

H. Dong and Z. Li, The conormal and Robin boundary value problems in nonsmooth domains satisfying a measure condition, J. Funct. Anal., 281 (2021), no. 9, Paper No. 109167, 32 pp. doi: 10.1016/j.jfa.2021.109167.

[22]

H. Dong and T. Phan, Mixed-norm Lp-estimates for non-stationary Stokes systems with singular VMO coefficients and applications, J. Differ. Equ., 276 (2021), 342-367.  doi: 10.1016/j.jde.2020.12.023.

[23]

E. FabesO. Mendez and M. Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains, J. Funct. Anal., 159 (1998), 323-368.  doi: 10.1006/jfan.1998.3316.

[24]

C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math., 129 (1972), 137-193.  doi: 10.1007/BF02392215.

[25]

J. Geng, W1, 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.

[26]

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.

[27] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies 105, Princeton University Press, Princeton, NJ, 1983. 
[28]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.

[29]

L. Grafakos, Classical Fourier Analysis, 3rd edition, Graduate Texts in Mathematics 249, Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.

[30]

D. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. Math., 46 (1982), 80-147.  doi: 10.1016/0001-8708(82)90055-X.

[31]

D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219.  doi: 10.1006/jfan.1995.1067.

[32]

H. JiaD. Li and L. Wang, Global regularity for divergence form elliptic equations on quasiconvex domains, J. Differ. Equ., 249 (2010), 3132-3147.  doi: 10.1016/j.jde.2010.08.015.

[33]

P. W. Jones, Extension theorems for BMO, Indiana Univ. Math. J., 29 (1980), 41-66.  doi: 10.1512/iumj.1980.29.29005.

[34]

P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88.  doi: 10.1007/BF02392869.

[35]

C. E. Kenig and T. Toro, Harmonic measure on locally flat domains, Duke Math. J., 87 (1997), 509-551.  doi: 10.1215/S0012-7094-97-08717-2.

[36]

L. Li and J. Pipher, Boundary behavior of solutions of elliptic operators in divergence form with a BMO anti-symmetric part, Commun. Partial Differ. Equ., 44 (2019), 156-204.  doi: 10.1080/03605302.2018.1542437.

[37]

V. G. Maz'ya and I. E. Verbitsky, Form boundedness of the general second-order differential operator, Commun. Pure Appl. Math., 59 (2006), 1286-1329.  doi: 10.1002/cpa.20122.

[38]

N. G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 189-206. 

[39]

D. MitreaI. MitreaM. Mitrea and L. Yan, Coercive energy estimates for differential forms in semi-convex domains, Commun. Pure Appl. Anal., 9 (2010), 987-1010.  doi: 10.3934/cpaa.2010.9.987.

[40]

D. MitreaM. Mitrea and L. Yan, Boundary value problems for the Laplacian in convex and semiconvex domains, J. Funct. Anal., 258 (2010), 2507-2585.  doi: 10.1016/j.jfa.2010.01.012.

[41]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Translated from the 1967 French original by Gerard Tronel and Alois Kufner, Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-10455-8.

[42]

E. R. Reifenberg, Solution of the Plateau Problem for m-dimensional surfaces of varying topological type, Acta Math., 104 (1960), 1-92.  doi: 10.1007/BF02547186.

[43]

D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc., 207 (1975), 391-405.  doi: 10.2307/1997184.

[44]

G. SereginL. SilvestreV. Šverák and A. Zlatoš, On divergence-free drifts, J. Differ. Equ., 252 (2012), 505-540.  doi: 10.1016/j.jde.2011.08.039.

[45]

Z. Shen, Weighted L2 estimates for elliptic homogenization in Lipschitz domains, arXiv: 2004.03087.

[46]

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.

[47]

Z. Shen, The Lp boundary value problems on Lipschitz domains, Adv. Math., 216 (2007), 212-254.  doi: 10.1016/j.aim.2007.05.017.

[48]

Z. Shen, Bounds of Riesz transforms on Lp spaces for second order elliptic operators, Ann. Inst. Fourier (Grenoble), 55 (2005), 173-197. 

[49] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. 
[50]

T. Toro, Doubling and flatness: geometry of measures, Notices Amer. Math. Soc., 44 (1997), 1087-1094. 

[51]

L. Wang, A geometric approach to the Calderón-Zygmund estimates, Acta Math. Sin. (Engl. Ser.), 19 (2003), 381-396.  doi: 10.1007/s10114-003-0264-4.

[52]

S. YangD. C. ChangD. Yang and W. Yuan, Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains, J. Differ. Equ., 268 (2020), 2510-2550.  doi: 10.1016/j.jde.2019.09.036.

[53]

S. YangD. Yang and W. Yuan, Weighted global regularity estimates for elliptic problems with Robin boundary conditions in Lipschitz domains, J. Differ. Equ., 296 (2021), 512-572.  doi: 10.1016/j.jde.2021.06.010.

[54]

S. Yang, D. Yang and W. Yuan, Global gradient estimates for Dirichlet problems of elliptic operators with a BMO anti-symmetric part, Submitted.

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