doi: 10.3934/cpaa.2022006
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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 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 & Applied Analysis, 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

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

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

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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.  Google Scholar

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[45]

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

[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.  Google Scholar

[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.  Google Scholar

[48]

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

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[54]

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

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[45]

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

[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.  Google Scholar

[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.  Google Scholar

[48]

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

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[54]

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

[1]

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