Continuity Properties for Boundary Data Homogenization Problems

Abstract

In this note we study periodic homogenization of Dirichlet problem for divergence type elliptic systems when both the coefficients and the boundary data are oscillating. One of the key difficulties here is the determination of the fixed boundary data corresponding to the limiting (homogenized) problem. This issue has been addressed in recent papers by Gérard-Varet and Masmoudi (Acta Math. 209:133–178, 2012), and by Prange (SIAM J. Math. Anal. 45(1):345–387, 2012), however, not much is known about the regularity of this fixed data. The main objective of this note is to initiate a study of this problem, and to prove several regularity results in this connection.

References

1. Aleksanyan, H.: Slow convergence in periodic homogenization problems for divergence-type elliptic operators. SIAM J. Math. Anal. 48(5), 3345–3382 (2016)

   MathSciNet  Article  MATH  Google Scholar

2. Aleksanyan, H., Shahgholian, H., Sjölin, P.: Applications of Fourier analysis in homogenization of Dirichlet problem. \(L^p\) estimates. Arch. Ration. Mech. Anal. (ARMA) 215(1), 65–87 (2015)

   MathSciNet  Article  MATH  Google Scholar

3. Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. AMS, Providence (2011)

   Book  MATH  Google Scholar

4. Bott, R., Milnor, J.: On the parallelizabilty of spheres. Bull. AMS 64, 87–89 (1958)

   Article  MATH  Google Scholar

5. Cioranescu, D., Donato, P.: An introduction to homogenization. Oxford Lecture Series in Mathematics and its Applications, vol. 17. The Clarendon Press, Oxford University Press, New York (1999)

6. Dong, H., Kim, S.: Green's matrices for second order elliptic systems with measurable coefficients in two dimensional domains. Trans. Am. Math. Soc. 361, 3303–3323 (2009)

   MathSciNet  Article  MATH  Google Scholar

7. Feldman, W., Kim, I.: Continuity and discontinuity of the boundary layer tail. arXiv:1502.00966 (2015)

8. Gérard-Varet, D., Masmoudi, N.: Homogenization and boundary layers. Acta Math. 209, 133–178 (2012)

   MathSciNet  Article  MATH  Google Scholar

9. Gérard-Varet, D., Masmoudi, N.: Homogenization in polygonal domains. J. Eur. Math. Soc. (JEMS) 13(5), 1477–1503 (2011)

   MathSciNet  Article  MATH  Google Scholar

10. Giaquinta, M., Martinazzi, L.: An introduction to the regularity theory for elliptic systems. Harmonic maps and minimal graphs. Scuola Normale Superiore Pisa (Lecture Notes) (2012)

11. Hong, Y.P., Pan, C.-T.: A lower bound for the smallest singular value. Linear Algebra Appl. 172, 27–32 (1992)

    MathSciNet  Article  MATH  Google Scholar

12. Hofmann, S., Kim, S.: The Green function estimates for strongly elliptic systems of second order. Manuscr. Math. 124, 139–172 (2007)

    MathSciNet  Article  MATH  Google Scholar

13. Kenig, C.E., Lin, F., Shen, Z.: Periodic homogenization of Green and Neumann functions. Commun. Pure Appl. Math. 67(8), 1219–1262 (2014)

    MathSciNet  Article  MATH  Google Scholar

14. Kervaire, M., Milnor, J.: Groups of homotopy spheres: I. Ann. Math. 77(3), 504–537 (1963)

    MathSciNet  Article  MATH  Google Scholar

15. Lions, J.L.: Some methods in mathematical analysis of systems and their control. Science Press, Gordon and Breach, Beijing, New York (1981)

    MATH  Google Scholar

16. Neuss-Radu, M.: The boundary behavior of a composite material. Math. Model. Numer. Anal. 35(3), 407–435 (2001)

    MathSciNet  Article  MATH  Google Scholar

17. Prange, C.: Asymptotic analysis of boundary layer correctors in periodic homogenization. SIAM J. Math. Anal. 45(1), 345–387 (2012)

    MathSciNet  Article  MATH  Google Scholar

18. Tartar, L.: The general theory of homogenization: a personalized introduction. Lecture Notes of the Unione Matematica Italiana, vol. 7. Springer, Berlin, Heidelberg (2009)

19. Stein, E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar

20. Šubin, M.: Differential and psedudifferential operators in spaces of almost periodic functions. Math. Sb. (N.S.) 95(137), 560–587 (1974)

    MATH  Google Scholar

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Author notes

1. Hayk Aleksanyan

   Present address: Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden

Authors and Affiliations

1. School of Mathematics, The University of Edinburgh, JCMB The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK

   Hayk Aleksanyan

Corresponding author

Correspondence to Hayk Aleksanyan.

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Aleksanyan, H. Regularity of boundary data in periodic homogenization of elliptic systems in layered media. manuscripta math. 154, 225–256 (2017). https://doi.org/10.1007/s00229-016-0905-4

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• Received: 15 September 2015

• Accepted: 01 December 2016

• Published: 21 December 2016

• Issue Date: September 2017

• DOI : https://doi.org/10.1007/s00229-016-0905-4

Mathematics Subject Classification

• Primary 35B27
• Secondary 35B40
• 35J08
• 35J57
• 42B05

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