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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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)