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$ 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 |
In this paper, we extend the nontangential maximal function estimates in $ L^p $-norm obtained by C. Kenig, F. Lin and Z. Shen [
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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