December  2019, 39(12): 6843-6864. doi: 10.3934/dcds.2019234

Homogenization of the boundary value for the Dirichlet problem

1. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea

2. 

Korea Institute for Advanced Study, Seoul 02455, Korea

3. 

Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden

* Corresponding author: Sunghan Kim

Dedicated to Luis Caffarelli for His 70th Birthday

Received  June 2018 Revised  January 2019 Published  June 2019

Fund Project: S. Kim was supported by National Research Foundation of Korea (NRF) grant funded by the Korean government (NRF-2014-Fostering Core Leaders of the Future Basic Science Program). K.-A. Lee was supported by This author was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1701-03. K.-A. Lee also holds a joint appointment with the Research Institute of Mathematics of Seoul National University. H. Shahgholian was supported in part by Swedish Research Council. This project was part of an STINT (Sweden)-NRF (Korea) research cooperation program.

In this paper, we give a mathematically rigorous proof of the averaging behavior of oscillatory surface integrals. Based on ergodic theory, we find a general geometric condition which we call irrational direction dense condition, abbreviated as IDDC, under which the averaging takes place. It should be stressed that IDDC does not imply any control on the curvature of the given surface. As an application, we prove homogenization for elliptic systems with Dirichlet boundary data, in $ C^1 $-domains.

Citation: Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6843-6864. doi: 10.3934/dcds.2019234
References:
[1]

H. Aleksanyan, Slow convergence in periodic homogenization problems for divergence-type elliptic operators, SIAM J. Math. Anal., 48 (2016), 3345-3382.  doi: 10.1137/15M1040165.  Google Scholar

[2]

H. AleksanyanH. Shahgholian and P. Sjölin, Applications of Fourier analysis in homogenization of Dirichlet problem Ⅰ. Pointwise estimates, J. Differential Equations, 254 (2013), 2626-2637.  doi: 10.1016/j.jde.2012.12.017.  Google Scholar

[3]

H. AleksanyanH. Shahgholian and P. Sjölin, Applications of Fourier analysis in homogenization of Dirichlet problem Ⅱ. $L^p$-estimates, Arch. Rational Mech. Anal., 215 (2015), 65-87.  doi: 10.1007/s00205-014-0774-5.  Google Scholar

[4]

S. ArmstrongT. KuusiJ.-C. Mourrat and C. Prange, Quantitative analysis of boundary layers in periodic homogenization, Arch. Rational Mech. Anal., 226 (2017), 695-741.  doi: 10.1007/s00205-017-1142-z.  Google Scholar

[5]

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

[6]

M. Avellaneda and F.-H. Lin, Homogenization of Poisson's kernel and applications to boundary control, J. Math. Pures Appl., 68 (1989), 1-29.   Google Scholar

[7]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, 2$^{nd}$ edition, American Mathematical Society, Providence, Rhode-Island, 1978.  Google Scholar

[8]

S. Choi and I. Kim, Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data, J. Math. Pures Appl., 102 (2014), 419-448.  doi: 10.1016/j.matpur.2013.11.015.  Google Scholar

[9]

S. ChoiI. Kim and K.-A. Lee, Homogenization of Neumann boundary data with fully nonlinear operator, Analysis & PDE, 6 (2013), 951-972.  doi: 10.2140/apde.2013.6.951.  Google Scholar

[10]

H. Dong and S. Kim, Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains, Trans. Amer. Math. Soc., 361 (2009), 3303-3323.  doi: 10.1090/S0002-9947-09-04805-3.  Google Scholar

[11]

W. Feldman, Homogenization of the oscillating Dirichlet boundary condition in general domains, J. Math. Pures Appl., 101 (2014), 599-622.  doi: 10.1016/j.matpur.2013.07.003.  Google Scholar

[12]

D. Gérard-Varet and N. Masmoudi, Homogenization in polygonal domains, J. Eur. Math. Soc., 13 (2011), 1477-1503.  doi: 10.4171/JEMS/286.  Google Scholar

[13]

D. Gérard-Varet and N. Masmoudi, Homogenization and boundary layers, Acta Math., 209 (2012), 1-46.  doi: 10.1007/s11511-012-0083-5.  Google Scholar

[14]

S. Hofmann and S. Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math., 124 (2007), 139-172.  doi: 10.1007/s00229-007-0107-1.  Google Scholar

[15]

C. KenigF.-H. Lin and Z. Shen, Periodic homogenization of Green and Neumann functions, Comm. Pure Appl. Math., 67 (2014), 1219-1262.  doi: 10.1002/cpa.21482.  Google Scholar

[16]

C. Prange, Asymptotic analysis of boundary layer correctors in periodic homogenization, SIAM J. Math. Anal., 45 (2013), 345-387.  doi: 10.1137/120876502.  Google Scholar

[17]

E. Stein, Harmonic Analysis : Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Vol. 43, Princeton University Press, New Jersey, 1993.  Google Scholar

[18]

Z. Shen and J. Zhuge, Boundary layers in periodic homogenization of Neumann problems, Comm. Pure Appl. Math., 71 (2018), 2163-2219.  doi: 10.1002/cpa.21740.  Google Scholar

[19]

J. Zhuge, Homogenization and boundary layers in domains of finite type, Comm. Partial Differential Equations, (2018), 1–36. doi: 10.1080/03605302.2018.1446160.  Google Scholar

show all references

References:
[1]

H. Aleksanyan, Slow convergence in periodic homogenization problems for divergence-type elliptic operators, SIAM J. Math. Anal., 48 (2016), 3345-3382.  doi: 10.1137/15M1040165.  Google Scholar

[2]

H. AleksanyanH. Shahgholian and P. Sjölin, Applications of Fourier analysis in homogenization of Dirichlet problem Ⅰ. Pointwise estimates, J. Differential Equations, 254 (2013), 2626-2637.  doi: 10.1016/j.jde.2012.12.017.  Google Scholar

[3]

H. AleksanyanH. Shahgholian and P. Sjölin, Applications of Fourier analysis in homogenization of Dirichlet problem Ⅱ. $L^p$-estimates, Arch. Rational Mech. Anal., 215 (2015), 65-87.  doi: 10.1007/s00205-014-0774-5.  Google Scholar

[4]

S. ArmstrongT. KuusiJ.-C. Mourrat and C. Prange, Quantitative analysis of boundary layers in periodic homogenization, Arch. Rational Mech. Anal., 226 (2017), 695-741.  doi: 10.1007/s00205-017-1142-z.  Google Scholar

[5]

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

[6]

M. Avellaneda and F.-H. Lin, Homogenization of Poisson's kernel and applications to boundary control, J. Math. Pures Appl., 68 (1989), 1-29.   Google Scholar

[7]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, 2$^{nd}$ edition, American Mathematical Society, Providence, Rhode-Island, 1978.  Google Scholar

[8]

S. Choi and I. Kim, Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data, J. Math. Pures Appl., 102 (2014), 419-448.  doi: 10.1016/j.matpur.2013.11.015.  Google Scholar

[9]

S. ChoiI. Kim and K.-A. Lee, Homogenization of Neumann boundary data with fully nonlinear operator, Analysis & PDE, 6 (2013), 951-972.  doi: 10.2140/apde.2013.6.951.  Google Scholar

[10]

H. Dong and S. Kim, Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains, Trans. Amer. Math. Soc., 361 (2009), 3303-3323.  doi: 10.1090/S0002-9947-09-04805-3.  Google Scholar

[11]

W. Feldman, Homogenization of the oscillating Dirichlet boundary condition in general domains, J. Math. Pures Appl., 101 (2014), 599-622.  doi: 10.1016/j.matpur.2013.07.003.  Google Scholar

[12]

D. Gérard-Varet and N. Masmoudi, Homogenization in polygonal domains, J. Eur. Math. Soc., 13 (2011), 1477-1503.  doi: 10.4171/JEMS/286.  Google Scholar

[13]

D. Gérard-Varet and N. Masmoudi, Homogenization and boundary layers, Acta Math., 209 (2012), 1-46.  doi: 10.1007/s11511-012-0083-5.  Google Scholar

[14]

S. Hofmann and S. Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math., 124 (2007), 139-172.  doi: 10.1007/s00229-007-0107-1.  Google Scholar

[15]

C. KenigF.-H. Lin and Z. Shen, Periodic homogenization of Green and Neumann functions, Comm. Pure Appl. Math., 67 (2014), 1219-1262.  doi: 10.1002/cpa.21482.  Google Scholar

[16]

C. Prange, Asymptotic analysis of boundary layer correctors in periodic homogenization, SIAM J. Math. Anal., 45 (2013), 345-387.  doi: 10.1137/120876502.  Google Scholar

[17]

E. Stein, Harmonic Analysis : Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Vol. 43, Princeton University Press, New Jersey, 1993.  Google Scholar

[18]

Z. Shen and J. Zhuge, Boundary layers in periodic homogenization of Neumann problems, Comm. Pure Appl. Math., 71 (2018), 2163-2219.  doi: 10.1002/cpa.21740.  Google Scholar

[19]

J. Zhuge, Homogenization and boundary layers in domains of finite type, Comm. Partial Differential Equations, (2018), 1–36. doi: 10.1080/03605302.2018.1446160.  Google Scholar

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