Homogenization of the boundary value for the Dirichlet problem

Sunghan Kim; Ki-Ahm Lee; Henrik Shahgholian

doi:10.3934/dcds.2019234

Article Contents

2019, Volume 39, Issue 12: 6843-6864. Doi: 10.3934/dcds.2019234

This issue Previous Article Free boundary problems associated with cancer treatment by combination therapy Next Article Regularity results for the equation

$ u_{11}u_{22} = 1 $

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
^* Corresponding author: Sunghan Kim

Dedicated to Luis Caffarelli for His 70th Birthday

Received: June 2018

Revised: January 2019

Published: September 2019

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35B27, 35J57; Secondary: 42B20, 28D99.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[2]	H. Aleksanyan, H. 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.
[3]	H. Aleksanyan, H. 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.
[4]	S. Armstrong, T. Kuusi, J.-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.
[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.
[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.
[7]	A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, 2$^{nd}$ edition, American Mathematical Society, Providence, Rhode-Island, 1978.
[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.
[9]	S. Choi, I. 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.
[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.
[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.
[12]	D. Gérard-Varet and N. Masmoudi, Homogenization in polygonal domains, J. Eur. Math. Soc., 13 (2011), 1477-1503. doi: 10.4171/JEMS/286.
[13]	D. Gérard-Varet and N. Masmoudi, Homogenization and boundary layers, Acta Math., 209 (2012), 1-46. doi: 10.1007/s11511-012-0083-5.
[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.
[15]	C. Kenig, F.-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.
[16]	C. Prange, Asymptotic analysis of boundary layer correctors in periodic homogenization, SIAM J. Math. Anal., 45 (2013), 345-387. doi: 10.1137/120876502.
[17]	E. Stein, Harmonic Analysis : Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Vol. 43, Princeton University Press, New Jersey, 1993.
[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.
[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.