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August  2020, 40(8): 5019-5045. doi: 10.3934/dcds.2020210

$ L^p $ Neumann problems in homogenization of general elliptic operators

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou 710000, China

2. 

School of Mathematical Sciences, Peking University, Beijing 100871, China

* Corresponding author: Qiang Xu

Received  November 2019 Revised  March 2020 Published  May 2020

In this paper, we extend the nontangential maximal function estimates in $ L^p $-norm obtained by C. Kenig, F. Lin and Z. Shen [13] to the nonhomogeneous elliptic operators with rapidly oscillating periodic coefficients. The result relies on a local Lipschitz boundary estimate, which has not been established in [29]. The present argument develops some new techniques to make the Campanato iteration and real methods workable for general elliptic operators. The result is new even for effective operators, as well as general elliptic equations of scalar.

Citation: Li Wang, Qiang Xu, Shulin Zhou. $ L^p $ Neumann problems in homogenization of general elliptic operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 5019-5045. doi: 10.3934/dcds.2020210
References:
[1]

S. Armstrong and J. Mourrat, Lipschitz regularity for elliptic equations with random coefficients, Arch. Ration. Mech. Anal., 219 (2016), 255-348.  doi: 10.1007/s00205-015-0908-4.  Google Scholar

[2]

S. Armstrong and Z. Shen, Lipschitz estimates in almost-periodic homogenization, Comm. Pure Appl. Math., 69 (2016), 1882-1923.  doi: 10.1002/cpa.21616.  Google Scholar

[3]

S. Armstrong and C. Smart, Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. Éc. Norm. Supér., 49 (2016), 423-481.   Google Scholar

[4]

M. Avellaneda and F. Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math., 40 (1987), 803-847.  doi: 10.1002/cpa.3160400607.  Google Scholar

[5]

M. Avellaneda and F. Lin, $L^p$ bounds on singular integrals in homogenization, Comm. Pure Appl. Math., 44 (1991), 897-910.  doi: 10.1002/cpa.3160440805.  Google Scholar

[6]

A. Bensoussan, J.-L. Lions and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, North Holland, 1978.  Google Scholar

[7]

L. 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

[8]

B. Dahlberg and C. Kenig, Hardy spaces and the Neumann problem in $L^p$ for Laplace's equation in Lipschitz domains, Ann. of Math., 125 (1987), 437-465.  doi: 10.2307/1971407.  Google Scholar

[9]

A. GloriaS. Neukamm and F. Otto, Quantification of ergodicity in stochastic homogenization: Optimal bounds via spectral gap on Glauber dynamics, Invent. Math., 199 (2015), 455-515.  doi: 10.1007/s00222-014-0518-z.  Google Scholar

[10]

S. Gu and Q. Xu, Optimal boundary estimates for Stokes systems in homogenization theory, SIAM J. Math. Anal., 49 (2017), 3831-3853.  doi: 10.1137/16M1108571.  Google Scholar

[11]

S. HofmannM. Mitrea and M. Taylor, Symbol calculus for operators of layer potential type on Lipschitz surfaces with VMO normals, and related pseudodifferential operator calculus, Anal. PDE, 8 (2015), 115-181.  doi: 10.2140/apde.2015.8.115.  Google Scholar

[12]

V. Jikov, S. Kozlov and O. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

[13]

C. KenigF. Lin and Z. Shen, Homogenization of elliptic systems with Neumann boundary conditions, J. Amer. Math. Soc., 26 (2013), 901-937.  doi: 10.1090/S0894-0347-2013-00769-9.  Google Scholar

[14]

C. KenigF. Lin and Z. Shen, Convergence rates in $L^2$ for elliptic homogenization problems, Arch. Ration. Mech. Anal., 203 (2012), 1009-1036.  doi: 10.1007/s00205-011-0469-0.  Google Scholar

[15]

C. Kenig and Z. Shen, Layer potential methods for elliptic homogenization problems, Comm. Pure Appl. Math., 64 (2011), 1-44.  doi: 10.1002/cpa.20343.  Google Scholar

[16]

C. Kenig and Z. Shen, Homogenization of elliptic boundary value problems in Lipschitz domains, Math. Ann., 350 (2011), 867-917.  doi: 10.1007/s00208-010-0586-3.  Google Scholar

[17]

A. Kim and Z. Shen, The Neumann problem in $L^p$ on Lipschitz and convex domains, J. Funct. Anal., 255 (2008), 1817-1830.  doi: 10.1016/j.jfa.2008.06.032.  Google Scholar

[18]

F. Lin and Z. Shen, Nodal sets and doubling conditions in elliptic homogenization, Acta Math. Sin., (Engl. Ser.), 35 (2019), 815-831.  doi: 10.1007/s10114-019-8228-5.  Google Scholar

[19]

W. NiuZ. Shen and Y. Xu, Convergence rates and interior estimates in homogenization of higher order elliptic systems, J. Funct. Anal., 274 (2018), 2356-2398.  doi: 10.1016/j.jfa.2018.01.012.  Google Scholar

[20]

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

[21]

Z. Shen, Extrapolation for the $L^p$ Dirichlet Problem in Lipschitz domains, Acta Math. Sin., (Engl. Ser.), 35 (2019), 1074-1084.  doi: 10.1007/s10114-019-8199-6.  Google Scholar

[22]

Z. Shen, Boundary estimates in elliptic homogenization, Anal. PDE, 10 (2017), 653-694.  doi: 10.2140/apde.2017.10.653.  Google Scholar

[23]

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

[24]

Z. Shen, Necessary and sufficient conditions for the solvability of the $L^p$ Dirichlet problem on Lipschitz domains, Math. Ann., 336 (2006), 697-725.  doi: 10.1007/s00208-006-0022-x.  Google Scholar

[25]

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

[26]

T. Suslina, Homogenization in the Sobolev class $H^1(\mathbb{R}^d)$ for second order periodic elliptic operators with the inclusion of first order terms, St. Petersburg Math. J., 22 (2011), 81-162.  doi: 10.1090/S1061-0022-2010-01135-X.  Google Scholar

[27]

Q. Xu, P. Zhao and S. Zhou, The methods of layer potentials for general elliptic homogenization problems in Lipschitz domains, preprint, arXiv: 1801.09220v1. Google Scholar

[28]

Q. Xu, Uniform regularity estimates in homogenization theory of elliptic systems with lower terms, J. Math. Anal. Appl., 438 (2016), 1066-1107.  doi: 10.1016/j.jmaa.2016.02.011.  Google Scholar

[29]

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

[30]

Q. Xu, Convergence rates for general elliptic homogenization problems in Lipschitz domains, SIAM J. Math. Anal., 48 (2016), 3742-3788.  doi: 10.1137/15M1053335.  Google Scholar

[31]

V. Zhikov and S. Pastukhova, On operator estimates for some problems in homogenization theory, Russ. J. Math. Phys., 12 (2005), 515-524.   Google Scholar

[32]

J. Zhuge, Homogenization and boundary layers in domains of finite type, Comm. Partial Differential Equations, 43 (2018), 549-584.  doi: 10.1080/03605302.2018.1446160.  Google Scholar

show all references

References:
[1]

S. Armstrong and J. Mourrat, Lipschitz regularity for elliptic equations with random coefficients, Arch. Ration. Mech. Anal., 219 (2016), 255-348.  doi: 10.1007/s00205-015-0908-4.  Google Scholar

[2]

S. Armstrong and Z. Shen, Lipschitz estimates in almost-periodic homogenization, Comm. Pure Appl. Math., 69 (2016), 1882-1923.  doi: 10.1002/cpa.21616.  Google Scholar

[3]

S. Armstrong and C. Smart, Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. Éc. Norm. Supér., 49 (2016), 423-481.   Google Scholar

[4]

M. Avellaneda and F. Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math., 40 (1987), 803-847.  doi: 10.1002/cpa.3160400607.  Google Scholar

[5]

M. Avellaneda and F. Lin, $L^p$ bounds on singular integrals in homogenization, Comm. Pure Appl. Math., 44 (1991), 897-910.  doi: 10.1002/cpa.3160440805.  Google Scholar

[6]

A. Bensoussan, J.-L. Lions and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, North Holland, 1978.  Google Scholar

[7]

L. 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

[8]

B. Dahlberg and C. Kenig, Hardy spaces and the Neumann problem in $L^p$ for Laplace's equation in Lipschitz domains, Ann. of Math., 125 (1987), 437-465.  doi: 10.2307/1971407.  Google Scholar

[9]

A. GloriaS. Neukamm and F. Otto, Quantification of ergodicity in stochastic homogenization: Optimal bounds via spectral gap on Glauber dynamics, Invent. Math., 199 (2015), 455-515.  doi: 10.1007/s00222-014-0518-z.  Google Scholar

[10]

S. Gu and Q. Xu, Optimal boundary estimates for Stokes systems in homogenization theory, SIAM J. Math. Anal., 49 (2017), 3831-3853.  doi: 10.1137/16M1108571.  Google Scholar

[11]

S. HofmannM. Mitrea and M. Taylor, Symbol calculus for operators of layer potential type on Lipschitz surfaces with VMO normals, and related pseudodifferential operator calculus, Anal. PDE, 8 (2015), 115-181.  doi: 10.2140/apde.2015.8.115.  Google Scholar

[12]

V. Jikov, S. Kozlov and O. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

[13]

C. KenigF. Lin and Z. Shen, Homogenization of elliptic systems with Neumann boundary conditions, J. Amer. Math. Soc., 26 (2013), 901-937.  doi: 10.1090/S0894-0347-2013-00769-9.  Google Scholar

[14]

C. KenigF. Lin and Z. Shen, Convergence rates in $L^2$ for elliptic homogenization problems, Arch. Ration. Mech. Anal., 203 (2012), 1009-1036.  doi: 10.1007/s00205-011-0469-0.  Google Scholar

[15]

C. Kenig and Z. Shen, Layer potential methods for elliptic homogenization problems, Comm. Pure Appl. Math., 64 (2011), 1-44.  doi: 10.1002/cpa.20343.  Google Scholar

[16]

C. Kenig and Z. Shen, Homogenization of elliptic boundary value problems in Lipschitz domains, Math. Ann., 350 (2011), 867-917.  doi: 10.1007/s00208-010-0586-3.  Google Scholar

[17]

A. Kim and Z. Shen, The Neumann problem in $L^p$ on Lipschitz and convex domains, J. Funct. Anal., 255 (2008), 1817-1830.  doi: 10.1016/j.jfa.2008.06.032.  Google Scholar

[18]

F. Lin and Z. Shen, Nodal sets and doubling conditions in elliptic homogenization, Acta Math. Sin., (Engl. Ser.), 35 (2019), 815-831.  doi: 10.1007/s10114-019-8228-5.  Google Scholar

[19]

W. NiuZ. Shen and Y. Xu, Convergence rates and interior estimates in homogenization of higher order elliptic systems, J. Funct. Anal., 274 (2018), 2356-2398.  doi: 10.1016/j.jfa.2018.01.012.  Google Scholar

[20]

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

[21]

Z. Shen, Extrapolation for the $L^p$ Dirichlet Problem in Lipschitz domains, Acta Math. Sin., (Engl. Ser.), 35 (2019), 1074-1084.  doi: 10.1007/s10114-019-8199-6.  Google Scholar

[22]

Z. Shen, Boundary estimates in elliptic homogenization, Anal. PDE, 10 (2017), 653-694.  doi: 10.2140/apde.2017.10.653.  Google Scholar

[23]

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

[24]

Z. Shen, Necessary and sufficient conditions for the solvability of the $L^p$ Dirichlet problem on Lipschitz domains, Math. Ann., 336 (2006), 697-725.  doi: 10.1007/s00208-006-0022-x.  Google Scholar

[25]

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

[26]

T. Suslina, Homogenization in the Sobolev class $H^1(\mathbb{R}^d)$ for second order periodic elliptic operators with the inclusion of first order terms, St. Petersburg Math. J., 22 (2011), 81-162.  doi: 10.1090/S1061-0022-2010-01135-X.  Google Scholar

[27]

Q. Xu, P. Zhao and S. Zhou, The methods of layer potentials for general elliptic homogenization problems in Lipschitz domains, preprint, arXiv: 1801.09220v1. Google Scholar

[28]

Q. Xu, Uniform regularity estimates in homogenization theory of elliptic systems with lower terms, J. Math. Anal. Appl., 438 (2016), 1066-1107.  doi: 10.1016/j.jmaa.2016.02.011.  Google Scholar

[29]

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

[30]

Q. Xu, Convergence rates for general elliptic homogenization problems in Lipschitz domains, SIAM J. Math. Anal., 48 (2016), 3742-3788.  doi: 10.1137/15M1053335.  Google Scholar

[31]

V. Zhikov and S. Pastukhova, On operator estimates for some problems in homogenization theory, Russ. J. Math. Phys., 12 (2005), 515-524.   Google Scholar

[32]

J. Zhuge, Homogenization and boundary layers in domains of finite type, Comm. Partial Differential Equations, 43 (2018), 549-584.  doi: 10.1080/03605302.2018.1446160.  Google Scholar

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