<inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> Neumann problems in homogenization of general elliptic operators

Li Wang; Qiang Xu; Shulin Zhou

doi:10.3934/dcds.2020210

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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

Li Wang ^1, , Qiang Xu ^2,, and Shulin Zhou ^2,

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.

Keywords: Homogenization, Neumann problems, nontangential maximal function estimates.

Mathematics Subject Classification: Primary: 35J57, 35B27; Secondary: 76M50.

Citation: Li Wang, Qiang Xu, Shulin Zhou. $ L^p $ Neumann problems in homogenization of general elliptic operators. Discrete and Continuous Dynamical Systems, 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.
[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.
[3]	S. Armstrong and C. Smart, Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. Éc. Norm. Supér., 49 (2016), 423-481.
[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.
[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.
[6]	A. Bensoussan, J.-L. Lions and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, North Holland, 1978.
[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.
[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.
[9]	A. Gloria, S. 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.
[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.
[11]	S. Hofmann, M. 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.
[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.
[13]	C. Kenig, F. 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.
[14]	C. Kenig, F. 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.
[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.
[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.
[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.
[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.
[19]	W. Niu, Z. 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.
[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.
[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.
[22]	Z. Shen, Boundary estimates in elliptic homogenization, Anal. PDE, 10 (2017), 653-694. doi: 10.2140/apde.2017.10.653.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[31]	V. Zhikov and S. Pastukhova, On operator estimates for some problems in homogenization theory, Russ. J. Math. Phys., 12 (2005), 515-524.
[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.

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.
[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.
[3]	S. Armstrong and C. Smart, Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. Éc. Norm. Supér., 49 (2016), 423-481.
[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.
[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.
[6]	A. Bensoussan, J.-L. Lions and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, North Holland, 1978.
[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.
[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.
[9]	A. Gloria, S. 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.
[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.
[11]	S. Hofmann, M. 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.
[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.
[13]	C. Kenig, F. 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.
[14]	C. Kenig, F. 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.
[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.
[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.
[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.
[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.
[19]	W. Niu, Z. 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.
[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.
[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.
[22]	Z. Shen, Boundary estimates in elliptic homogenization, Anal. PDE, 10 (2017), 653-694. doi: 10.2140/apde.2017.10.653.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[31]	V. Zhikov and S. Pastukhova, On operator estimates for some problems in homogenization theory, Russ. J. Math. Phys., 12 (2005), 515-524.
[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.

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

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