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Homogenization of elliptic systems with Neumann boundary conditions


Authors: Carlos E. Kenig, Fanghua Lin and Zhongwei Shen
Journal: J. Amer. Math. Soc. 26 (2013), 901-937
MSC (2010): Primary 35J57
DOI: https://doi.org/10.1090/S0894-0347-2013-00769-9
Published electronically: March 27, 2013
MathSciNet review: 3073881
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Abstract | References | Similar Articles | Additional Information

Abstract: For a family of second-order elliptic systems with rapidly oscillating periodic coefficients in a $ C^{1,\alpha }$ domain, we establish uniform $ W^{1,p}$ estimates, Lipschitz estimates, and nontangential maximal function estimates on solutions with Neumann boundary conditions.


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  • [1] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727. MR 0125307 (23 #A2610)
  • [2] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35-92. MR 0162050 (28 #5252)
  • [3] Marco Avellaneda and Fang-Hua Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math. 40 (1987), no. 6, 803-847. MR 910954 (88i:35019), https://doi.org/10.1002/cpa.3160400607
  • [4] Marco Avellaneda and Fang-Hua Lin, Homogenization of elliptic problems with $ L^p$ boundary data, Appl. Math. Optim. 15 (1987), no. 2, 93-107. MR 868901 (88m:35014), https://doi.org/10.1007/BF01442648
  • [5] Marco Avellaneda and Fang-Hua Lin, Compactness methods in the theory of homogenization. II. Equations in nondivergence form, Comm. Pure Appl. Math. 42 (1989), no. 2, 139-172. MR 978702 (90c:35035), https://doi.org/10.1002/cpa.3160420203
  • [6] Marco Avellaneda and Fang-Hua Lin, Homogenization of Poisson's kernel and applications to boundary control, J. Math. Pures Appl. (9) 68 (1989), no. 1, 1-29. MR 985952 (90g:35016)
  • [7] M. Avellaneda and Fang-Hua Lin, $ L^p$ bounds on singular integrals in homogenization, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 897-910. MR 1127038 (92j:42015), https://doi.org/10.1002/cpa.3160440805
  • [8] Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam, 1978. MR 503330 (82h:35001)
  • [9] L. A. Caffarelli and I. Peral, On $ W^{1,p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), no. 1, 1-21. MR 1486629 (99c:35053), https://doi.org/10.1002/(SICI)1097-0312(199801)51:1$ <$1::AID-CPA1$ >$3.3.CO;2-N
  • [10] G. A. Chechkin, A. L. Piatnitski, and A. S. Shamaev, Homogenization, Translations of Mathematical Monographs, vol. 234, American Mathematical Society, Providence, RI, 2007. Methods and applications; Translated from the 2007 Russian original by Tamara Rozhkovskaya. MR 2337848 (2008j:35013)
  • [11] B. Dahlberg, personal communication (1990).
  • [12] Björn E. J. Dahlberg and Carlos E. Kenig, Hardy spaces and the Neumann problem in $ L^p$ for Laplace's equation in Lipschitz domains, Ann. of Math. (2) 125 (1987), no. 3, 437-465. MR 890159 (88d:35044), https://doi.org/10.2307/1971407
  • [13] A. F. M. ter Elst, Derek W. Robinson, and Adam Sikora, On second-order periodic elliptic operators in divergence form, Math. Z. 238 (2001), no. 3, 569-637. MR 1869699 (2003b:35087), https://doi.org/10.1007/s002090100268
  • [14] C. Fefferman and E. M. Stein, $ H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137-193. MR 0447953 (56 #6263)
  • [15] Jun Geng, $ W^{1,p}$ estimates for elliptic problems with Neumann boundary conditions in Lipschitz domains, Adv. Math. 229 (2012), no. 4, 2427-2448. MR 2880228, https://doi.org/10.1016/j.aim.2012.01.004
  • [16] Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034 (86b:49003)
  • [17] Steve Hofmann and Seick Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math. 124 (2007), no. 2, 139-172. MR 2341783 (2008k:35110), https://doi.org/10.1007/s00229-007-0107-1
  • [18] V. V. Jikov, S. M. Kozlov, and O. A. Oleĭnik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosifyan]. MR 1329546 (96h:35003b)
  • [19] Carlos E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 83, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1994. MR 1282720 (96a:35040)
  • [20] C. Kenig, F. Lin, and Z. Shen, Periodic homogenization of Green and Neumann functions, Comm. Pure Appl. Math. (to appear).
  • [21] Carlos E. Kenig and Jill Pipher, The Neumann problem for elliptic equations with nonsmooth coefficients, Invent. Math. 113 (1993), no. 3, 447-509. MR 1231834 (95b:35046), https://doi.org/10.1007/BF01244315
  • [22] Carlos E. Kenig and Zhongwei Shen, Homogenization of elliptic boundary value problems in Lipschitz domains, Math. Ann. 350 (2011), no. 4, 867-917. MR 2818717 (2012m:35039), https://doi.org/10.1007/s00208-010-0586-3
  • [23] Carlos E. Kenig and Zhongwei Shen, Layer potential methods for elliptic homogenization problems, Comm. Pure Appl. Math. 64 (2011), no. 1, 1-44. MR 2743875 (2011i:35009), https://doi.org/10.1002/cpa.20343
  • [24] Joel Kilty and Zhongwei Shen, The $ L^p$ regularity problem on Lipschitz domains, Trans. Amer. Math. Soc. 363 (2011), no. 3, 1241-1264. MR 2737264 (2012a:35072), https://doi.org/10.1090/S0002-9947-2010-05076-7
  • [25] Aekyoung Shin Kim and Zhongwei Shen, The Neumann problem in $ L^p$ on Lipschitz and convex domains, J. Funct. Anal. 255 (2008), no. 7, 1817-1830. MR 2442084 (2009m:35065), https://doi.org/10.1016/j.jfa.2008.06.032
  • [26] J.-L. Lions, Asymptotic problems in distributed systems, Metastability and incompletely posed problems (Minneapolis, Minn., 1985), IMA Vol. Math. Appl., vol. 3, Springer, New York, 1987, pp. 241-258. MR 870019 (87m:93076), https://doi.org/10.1007/978-1-4613-8704-6_14
  • [27] J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev. 30 (1988), no. 1, 1-68. MR 931277 (89e:93019), https://doi.org/10.1137/1030001
  • [28] O. A. Oleĭnik, A. S. Shamaev, and G. A. Yosifian, Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications, vol. 26, North-Holland Publishing Co., Amsterdam, 1992. MR 1195131 (93k:35025)
  • [29] Zhongwei Shen, Bounds of Riesz transforms on $ L^p$ spaces for second order elliptic operators, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 173-197 (English, with English and French summaries). MR 2141694 (2006a:35045)
  • [30] Zhongwei Shen, Necessary and sufficient conditions for the solvability of the $ L^p$ Dirichlet problem on Lipschitz domains, Math. Ann. 336 (2006), no. 3, 697-725. MR 2249765 (2008e:35059), https://doi.org/10.1007/s00208-006-0022-x
  • [31] Zhongwei Shen, The $ L^p$ boundary value problems on Lipschitz domains, Adv. Math. 216 (2007), no. 1, 212-254. MR 2353255 (2009a:35064), https://doi.org/10.1016/j.aim.2007.05.017
  • [32] Michael E. Taylor, Tools for PDE, Mathematical Surveys and Monographs, vol. 81, American Mathematical Society, Providence, RI, 2000. Pseudodifferential operators, paradifferential operators, and layer potentials. MR 1766415 (2001g:35004)
  • [33] Gregory Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572-611. MR 769382 (86e:35038), https://doi.org/10.1016/0022-1236(84)90066-1

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Additional Information

Carlos E. Kenig
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: cek@math.uchicago.edu

Fanghua Lin
Affiliation: Courant Institute of Mathematical Sciences, New York University, New York, New York 10012
Email: linf@cims.nyu.edu

Zhongwei Shen
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: zshen2@email.uky.edu

DOI: https://doi.org/10.1090/S0894-0347-2013-00769-9
Received by editor(s): October 28, 2010
Received by editor(s) in revised form: February 26, 2013
Published electronically: March 27, 2013
Additional Notes: The first author was supported in part by NSF grant DMS-0968472
The second author was supported in part by NSF grant DMS-0700517
The third author was supported in part by NSF grant DMS-0855294
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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