Uniform Estimates in Homogenization: Compactness Methods and Applications
- [1] Department of Mathematics The University of Chicago Eckhart Hall 325 5734 S. University Avenue Chicago, Illinois 60637, USA
Journées Équations aux dérivées partielles (2014)
- page 1-25
- ISSN: 0752-0360
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topPrange, Christophe. "Uniform Estimates in Homogenization: Compactness Methods and Applications." Journées Équations aux dérivées partielles (2014): 1-25. <http://eudml.org/doc/275629>.
@article{Prange2014,
abstract = {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.},
affiliation = {Department of Mathematics The University of Chicago Eckhart Hall 325 5734 S. University Avenue Chicago, Illinois 60637, USA},
author = {Prange, Christophe},
journal = {Journées Équations aux dérivées partielles},
keywords = {Homogenization; compactness methods; boundary layers; potential theory; Green kernel; Poisson kernel; control of distributed systems},
language = {eng},
pages = {1-25},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Uniform Estimates in Homogenization: Compactness Methods and Applications},
url = {http://eudml.org/doc/275629},
year = {2014},
}
TY - JOUR
AU - Prange, Christophe
TI - Uniform Estimates in Homogenization: Compactness Methods and Applications
JO - Journées Équations aux dérivées partielles
PY - 2014
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 25
AB - 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.
LA - eng
KW - Homogenization; compactness methods; boundary layers; potential theory; Green kernel; Poisson kernel; control of distributed systems
UR - http://eudml.org/doc/275629
ER -
References
top- F. J. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math. (2) 87 (1968), 321-391 Zbl0162.24703MR225243
- S. N. Armstrong, Z. Shen, Lipschitz estimates in almost-periodic homogenization, ArXiv e-prints (2014)
- S. N. Armstrong, C. K. Smart, Quantitative stochastic homogenization of convex integral functionals, ArXiv e-prints (2014)
- M. Avellaneda, F.-H. Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math 40 (1987), 803-847 Zbl0632.35018MR910954
- M. Avellaneda, F.-H. Lin, Counterexamples related to high-frequency oscillation of Poisson’s kernel, Appl. Math. Optim. 15 (1987), 109-119 Zbl0662.35028MR868902
- M. Avellaneda, F.-H. Lin, Homogenization of elliptic problems with boundary data, Appl. Math. Optim. 15 (1987), 93-107 Zbl0644.35034MR868901
- M. Avellaneda, F.-H. Lin, Compactness methods in the theory of homogenization. II. Equations in nondivergence form, Comm. Pure Appl. Math. 42 (1989), 139-172 Zbl0645.35019MR978702
- M. Avellaneda, F.-H. Lin, Homogenization of Poisson’s kernel and applications to boundary control, J. Math. Pures Appl. (9) 68 (1989), 1-29 Zbl0617.35014MR985952
- M. Avellaneda, F.-H. Lin, bounds on singular integrals in homogenization, Comm. Pure Appl. Math. 44 (1991), 897-910 Zbl0761.42008MR1127038
- A. Bensoussan, J. L. Lions, G. Papanicolaou, Asymptotic analysis for periodic structures, 5 (1978), North-Holland Publishing Co., Amsterdam Zbl1229.35001MR503330
- E. Bombieri, Regularity theory for almost minimal currents, Arch. Rational Mech. Anal. 78 (1982), 99-130 Zbl0485.49024MR648941
- L. A. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Differential Equations 5 (1980), 427-448 Zbl0437.35070MR567780
- S. Choi, I. C. Kim, Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data, J. Math. Pures Appl. (9) 102 (2014), 419-448 Zbl1329.35046MR3227328
- D. Cioranescu, P. Donato, An Introduction to Homogenization, (1999), Oxford University Press Zbl0939.35001MR1765047
- E. De Giorgi, Frontiere orientate di misura minima, (1961), Editrice Tecnico Scientifica, Pisa MR179651
- L. C. Evans, Quasiconvexity and partial regularity in the calculus of variations, Arch. Rational Mech. Anal. 95 (1986), 227-252 Zbl0627.49006MR853966
- W. M. Feldman, Homogenization of the oscillating Dirichlet boundary condition in general domains, J. Math. Pures Appl. (9) 101 (2014), 599-622 Zbl1293.35109MR3192425
- W. M. Feldman, I. Kim, P. E. Souganidis, Quantitative Homogenization of Elliptic PDE with Random Oscillatory Boundary Data, ArXiv e-prints (2014)
- J. Geng, Z. Shen, Uniform Regularity Estimates in Parabolic Homogenization, ArXiv e-prints (2013) Zbl1325.35081
- J. Geng, Z. Shen, L. Song, Uniform estimates for systems of linear elasticity in a periodic medium, J. Funct. Anal. 262 (2012), 1742-1758 Zbl1236.35035MR2873858
- D. Gérard-Varet, The Navier wall law at a boundary with random roughness, Comm. Math. Phys. 286 (2009), 81-110 Zbl1176.35127MR2470924
- D. Gérard-Varet, N. Masmoudi, Relevance of the slip condition for fluid flows near an irregular boundary, Comm. Math. Phys. 295 (2010), 99-137 Zbl1193.35130MR2585993
- D. Gérard-Varet, N. Masmoudi, Homogenization in polygonal domains, J. Eur. Math. Soc. 13 (2011), 1477-1503 Zbl1228.35100MR2825170
- D. Gérard-Varet, N. Masmoudi, Homogenization and boundary layers, Acta Math. 209 (2012), 133-178 Zbl1259.35024MR2979511
- M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, 105 (1983), Princeton University Press, Princeton, NJ Zbl0516.49003MR717034
- A. Gloria, S. Neukamm, F. Otto, A regularity theory for random elliptic operators, ArXiv e-prints (2014) Zbl1307.35029MR3177848
- C. Kenig, Weighted spaces on Lipschitz domains, Amer. J. Math. 102 (1980), 129-163 Zbl0434.42024MR556889
- C. Kenig, F.-H. Lin, Z. Shen, Homogenization of elliptic systems with Neumann boundary conditions, J. Amer. Math. Soc. 26 (2013), 901-937 Zbl1277.35166MR3073881
- C. Kenig, F.-H Lin, Z. Shen, Homogenization of Green and Neumann Functions, (2014) Zbl1300.35030
- C. Kenig, C. Prange, Uniform Lipschitz Estimates in Bumpy Half-Spaces, ArXiv e-prints (2014) Zbl1317.35043
- C. Kenig, Z. Shen, Homogenization of elliptic boundary value problems in Lipschitz domains, Math. Ann. 350 (2011), 867-917 Zbl1223.35139MR2818717
- C. Kenig, Zhongwei Shen, Layer potential methods for elliptic homogenization problems, Comm. Pure Appl. Math. 64 (2011), 1-44 Zbl1213.35063MR2743875
- J. L. Lions, Asymptotic problems in distributed systems, (1985)
- S. Moskow, M. Vogelius, First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 1263-1299 Zbl0888.35011MR1489436
- F. Murat, L. Tartar, -convergence, Topics in the mathematical modelling of composite materials 31 (1997), 21-43, Birkhäuser Boston, Boston, MA Zbl0920.35019MR1493039
- C. Prange, Asymptotic analysis of boundary layer correctors in periodic homogenization, SIAM J. Math. Anal. 45 (2013), 345-387 Zbl1270.35067MR3032981
- Z. Shen, estimates for elliptic homogenization problems in nonsmooth domains, Indiana Univ. Math. J. 57 (2008), 2283-2298 Zbl1166.35013MR2463969
- Z. Shen, Convergence Rates and Hölder Estimates in Almost-Periodic Homogenization of Elliptic Systems, ArXiv e-prints (2014)
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