Uniform Estimates in Homogenization: Compactness Methods and Applications
Journées équations aux dérivées partielles (2014), article no. 7, 25 p.

The purpose of this note is to explain how to use compactness to get uniform estimates in the homogenization of elliptic systems with or without oscillating boundary. Along with new results in this direction, we highlight some important applications to pointwise estimates of Green and Poisson kernels, to the homogenization of boundary layer systems and to the boundary control of composite materials.

DOI : https://doi.org/10.5802/jedp.110
Keywords: Homogenization, compactness methods, boundary layers, potential theory, Green kernel, Poisson kernel, control of distributed systems
@article{JEDP_2014____A7_0,
author = {Prange, Christophe},
title = {Uniform Estimates in Homogenization: Compactness Methods and Applications},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2014},
doi = {10.5802/jedp.110},
language = {en},
url = {http://www.numdam.org/item/JEDP_2014____A7_0}
}

Prange, Christophe. Uniform Estimates in Homogenization: Compactness Methods and Applications. Journées équations aux dérivées partielles (2014), article  no. 7, 25 p. doi : 10.5802/jedp.110. http://www.numdam.org/item/JEDP_2014____A7_0/

[1] Almgren, F. J. Jr. Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math. (2), Tome 87 (1968), pp. 321-391 | MR 225243 | Zbl 0162.24703

[2] Armstrong, S. N.; Shen, Z. Lipschitz estimates in almost-periodic homogenization, ArXiv e-prints (2014)

[3] Armstrong, S. N.; Smart, C. K. Quantitative stochastic homogenization of convex integral functionals, ArXiv e-prints (2014)

[4] Avellaneda, M.; Lin, F.-H. Compactness methods in the theory of homogenization, Comm. Pure Appl. Math, Tome 40 (1987) no. 6, pp. 803-847 | MR 910954 | Zbl 0632.35018

[5] Avellaneda, M.; Lin, F.-H. Counterexamples related to high-frequency oscillation of Poisson’s kernel, Appl. Math. Optim., Tome 15 (1987) no. 2, pp. 109-119 | MR 868902 | Zbl 0662.35028

[6] Avellaneda, M.; Lin, F.-H. Homogenization of elliptic problems with ${L}^{p}$ boundary data, Appl. Math. Optim., Tome 15 (1987) no. 2, pp. 93-107 | MR 868901 | Zbl 0644.35034

[7] Avellaneda, M.; Lin, F.-H. Compactness methods in the theory of homogenization. II. Equations in nondivergence form, Comm. Pure Appl. Math., Tome 42 (1989) no. 2, pp. 139-172 | MR 978702 | Zbl 0645.35019

[8] Avellaneda, M.; Lin, F.-H. Homogenization of Poisson’s kernel and applications to boundary control, J. Math. Pures Appl. (9), Tome 68 (1989) no. 1, pp. 1-29 | MR 985952 | Zbl 0617.35014

[9] Avellaneda, M.; Lin, F.-H. ${L}^{p}$ bounds on singular integrals in homogenization, Comm. Pure Appl. Math., Tome 44 (1991) no. 8-9, pp. 897-910 | MR 1127038 | Zbl 0761.42008

[10] Bensoussan, A.; Lions, J. L.; Papanicolaou, G. Asymptotic analysis for periodic structures, North-Holland Publishing Co., Amsterdam, Studies in Mathematics and its Applications, Tome 5 (1978) | MR 503330 | Zbl 1229.35001

[11] Bombieri, E. Regularity theory for almost minimal currents, Arch. Rational Mech. Anal., Tome 78 (1982) no. 2, pp. 99-130 | MR 648941 | Zbl 0485.49024

[12] Caffarelli, L. A. Compactness methods in free boundary problems, Comm. Partial Differential Equations, Tome 5 (1980) no. 4, pp. 427-448 | MR 567780 | Zbl 0437.35070

[13] Choi, S.; Kim, I. C. Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data, J. Math. Pures Appl. (9), Tome 102 (2014) no. 2, pp. 419-448 | MR 3227328

[14] Cioranescu, D.; Donato, P. An Introduction to Homogenization, Oxford University Press, Oxford Lecture Series in Mathematics and its Applications (1999) | MR 1765047 | Zbl 0939.35001

[15] De Giorgi, E. Frontiere orientate di misura minima, Editrice Tecnico Scientifica, Pisa, Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61 (1961), pp. 57 | MR 179651

[16] Evans, L. C. Quasiconvexity and partial regularity in the calculus of variations, Arch. Rational Mech. Anal., Tome 95 (1986) no. 3, pp. 227-252 | MR 853966 | Zbl 0627.49006

[17] Feldman, W. M. Homogenization of the oscillating Dirichlet boundary condition in general domains, J. Math. Pures Appl. (9), Tome 101 (2014) no. 5, pp. 599-622 | MR 3192425 | Zbl 1293.35109

[18] Feldman, W. M.; Kim, I.; Souganidis, P. E. Quantitative Homogenization of Elliptic PDE with Random Oscillatory Boundary Data, ArXiv e-prints (2014)

[19] Geng, J.; Shen, Z. Uniform Regularity Estimates in Parabolic Homogenization, ArXiv e-prints (2013)

[20] Geng, J.; Shen, Z.; Song, L. Uniform ${W}^{1,p}$ estimates for systems of linear elasticity in a periodic medium, J. Funct. Anal., Tome 262 (2012) no. 4, pp. 1742-1758 | MR 2873858 | Zbl 1236.35035

[21] Gérard-Varet, D. The Navier wall law at a boundary with random roughness, Comm. Math. Phys., Tome 286 (2009) no. 1, pp. 81-110 | MR 2470924 | Zbl 1176.35127

[22] Gérard-Varet, D.; Masmoudi, N. Relevance of the slip condition for fluid flows near an irregular boundary, Comm. Math. Phys., Tome 295 (2010) no. 1, pp. 99-137 | MR 2585993 | Zbl 1193.35130

[23] Gérard-Varet, D.; Masmoudi, N. Homogenization in polygonal domains, J. Eur. Math. Soc., Tome 13 (2011), pp. 1477-1503 | MR 2825170 | Zbl 1228.35100

[24] Gérard-Varet, D.; Masmoudi, N. Homogenization and boundary layers, Acta Math., Tome 209 (2012) no. 1, pp. 133-178 | MR 2979511 | Zbl 1259.35024

[25] Giaquinta, M. Multiple integrals in the calculus of variations and nonlinear elliptic systems, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 105 (1983) | MR 717034 | Zbl 0516.49003

[26] Gloria, A.; Neukamm, S.; Otto, F. A regularity theory for random elliptic operators, ArXiv e-prints (2014) | MR 3177848

[27] Kenig, C. Weighted ${H}^{p}$ spaces on Lipschitz domains, Amer. J. Math., Tome 102 (1980) no. 1, pp. 129-163 | MR 556889 | Zbl 0434.42024

[28] Kenig, C.; Lin, F.-H.; Shen, Z. Homogenization of elliptic systems with Neumann boundary conditions, J. Amer. Math. Soc., Tome 26 (2013) no. 4, pp. 901-937 | MR 3073881 | Zbl 1277.35166

[29] Kenig, C.; Lin, F.-H; Shen, Z. Homogenization of Green and Neumann Functions (2014) (to appear in Communications in Pure and Applied Mathematics)

[30] Kenig, C.; Prange, C. Uniform Lipschitz Estimates in Bumpy Half-Spaces, ArXiv e-prints (2014)

[31] Kenig, C.; Shen, Z. Homogenization of elliptic boundary value problems in Lipschitz domains, Math. Ann., Tome 350 (2011) no. 4, pp. 867-917 | MR 2818717 | Zbl 1223.35139

[32] Kenig, C.; Shen, Zhongwei Layer potential methods for elliptic homogenization problems, Comm. Pure Appl. Math., Tome 64 (2011) no. 1, pp. 1-44 | MR 2743875 | Zbl 1213.35063

[33] Lions, J. L. Asymptotic problems in distributed systems (1985) (IMA Preprint Series)

[34] Moskow, S.; Vogelius, M. First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof, Proc. Roy. Soc. Edinburgh Sect. A, Tome 127 (1997) no. 6, pp. 1263-1299 | MR 1489436 | Zbl 0888.35011

[35] Murat, F.; Tartar, L. $H$-convergence, Topics in the mathematical modelling of composite materials, Birkhäuser Boston, Boston, MA (Progr. Nonlinear Differential Equations Appl.) Tome 31 (1997), pp. 21-43 | MR 1493039 | Zbl 0920.35019

[36] Prange, C. Asymptotic analysis of boundary layer correctors in periodic homogenization, SIAM J. Math. Anal., Tome 45 (2013) no. 1, pp. 345-387 | MR 3032981 | Zbl 1270.35067

[37] Shen, Z. ${W}^{1,p}$ estimates for elliptic homogenization problems in nonsmooth domains, Indiana Univ. Math. J., Tome 57 (2008) no. 5, pp. 2283-2298 | MR 2463969 | Zbl 1166.35013

[38] Shen, Z. Convergence Rates and Hölder Estimates in Almost-Periodic Homogenization of Elliptic Systems, ArXiv e-prints (2014)