A Singularly Perturbed Dirichlet Problem for the Poisson Equation in a Periodically Perforated Domain. A Functional Analytic Approach

Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 229)

Abstract

Let Ω be a sufficiently regular bounded open connected subset of \( \mathbb{R}^{n} \) such that 0 ϵ Ω and that \( \mathbb{R}^{n}\setminus \rm {cl}\Omega \) is connected. Then we take \( (q_{11},...,q_{nn})\in]0,+\infty{[^{n}}\; \rm {and} \;p \in Q \equiv \prod\nolimits^{n}_{j=1}]0,q_{jj}[.\) If є is a small positive number, then we define the periodically perforated domain \( \mathbb{S}[\Omega_{p,\epsilon}]^{-} \equiv \mathbb{R}^{n} \setminus \cup_{z\in\mathbb{Z}^{n}}\rm {cl}(p+\epsilon\Omega\;+\;\sum\nolimits^{n}_{j=1}(q_{jj}z_{j})e_{j}) \), where \(\left\{e_{1},...,e_{n}\right\}\) is the canonical basis of \( \mathbb{R}^{n}\). For є small and positive, we introduce a particular Dirichlet problem for the Poisson equation in the set \( \mathbb{S}[\Omega_{p,\epsilon}]^{-}\). . Namely, we consider a Dirichlet condition on the boundary of the set \( p \; + \; \epsilon\Omega\) , together with a periodicity condition. Then we show real analytic continuation properties of the solution as a function of є , of the Dirichlet datum on \( p \; + \; \epsilon\partial\Omega\) , and of the Poisson datum, around a degenerate triple with є = 0.

Keywords

Dirichlet problem singularly perturbed domain Poisson equation periodically perforated domain real analytic continuation in Banach space. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Ammari and H. Kang, Polarization and moment tensors, volume 162 of Applied Mathematical Sciences. Springer, New York, 2007.Google Scholar
  2. [2]
    L.P. Castro and E. Pesetskaya, A transmission problem with imperfect contact for an unbounded multiply connected domain. Math. Methods Appl. Sci. 33 (2010), 517–526.MathSciNetMATHGoogle Scholar
  3. [3]
    L.P. Castro, E. Pesetskaya, and S.V. Rogosin, Effective conductivity of a composite material with non-ideal contact conditions. Complex Var. Elliptic Equ. 54 (2009), 1085–1100.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    D. Cioranescu and F. Murat, Un terme étrange venud’ ailleurs. In Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. II (Paris, 1979/1980), volume 60 of Res. Notes in Math., pages 98–138, 389–390. Pitman, Boston, Mass., 1982.Google Scholar
  5. [5]
    D. Cioranescu and F. Murat, Un terme étrange venud’ ailleurs. II. In Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. III (Paris, 1980/1981), volume 70 of Res. Notes in Math., pages 154–178, 425–426. Pitman, Boston, Mass., 1982.Google Scholar
  6. [6]
    M. Dalla Riva, Stokes flow in a singularly perturbed exterior domain. Complex Var. Elliptic Equ., (to appear).Google Scholar
  7. [7]
    M. Dalla Riva and M. Lanza de Cristoforis, Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem: a functional analytic approach. Complex Var. Elliptic Equ. 55 (2010), 771–794.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    M. Dalla Riva and M. Lanza de Cristoforis, A singularly perturbed nonlinear traction boundary value problem for linearized elastostatics. A functional analytic approach. Analysis (Munich) 30 (2010), 67–92.MathSciNetMATHGoogle Scholar
  9. [9]
    M. Dalla Riva, M. Lanza de Cristoforis, and P. Musolino, Analytic dependence of volume potentials corresponding to parametric kernels in Roumieu classes. Typewritten manuscript, 2012.Google Scholar
  10. [10]
    P. Drygas and V. Mityushev, Effective conductivity of unidirectional cylinders with interfacial resistance, Quart. J. Mech. Appl. Math. 62 (2009), 235–262.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1983.Google Scholar
  12. [12]
    Eh.I. Grigolyuk and L.A. Fil’shtinskij, Periodic piecewise homogeneous elastic structures, (in Russian), Nauka, Moscow, 1992.Google Scholar
  13. [13]
    V. Kozlov, V. Maz’ya, and A. Movchan, Asymptotic analysis of fields in multistructures. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1999.Google Scholar
  14. [14]
    M. Lanza de Cristoforis, Properties and pathologies of the composition and inversion operators in Schauder spaces. Rend. Accad. Naz. Sci. XL Mem. Mat. 15 (1991), 93–109.MathSciNetMATHGoogle Scholar
  15. [15]
    M. Lanza de Cristoforis, Differentiability properties of a composition operator. Rend. Circ. Mat. Palermo (2) Suppl. 56 (1998), 157–165.Google Scholar
  16. [16]
    M. Lanza de Cristoforis, Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces. Comput. Methods Funct. Theory 2 (2002), 1–27.MathSciNetMATHGoogle Scholar
  17. [17]
    M. Lanza de Cristoforis, Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole, and relative capacity. In Complex analysis and dynamical systems, Proc. Conf. Karmiel, June 1922, 2001, volume 364 of Contemp. Math., pages 155–167. Amer. Math. Soc., Providence, RI, 2004.Google Scholar
  18. [18]
    M. Lanza de Cristoforis, A domain perturbation problem for the Poisson equation. Complex Var. Theory Appl. 50 (2005), 851–867.MathSciNetMATHGoogle Scholar
  19. [19]
    M. Lanza de Cristoforis, Perturbation problems in potential theory, a functional analytic approach. J. Appl. Funct. Anal. 2 (2007), 197–222.MathSciNetMATHGoogle Scholar
  20. [20]
    M. Lanza de Cristoforis, Asymptotic behavior of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole. A functional analytic approach. Analysis (Munich) 28 (2008), 63–93.MathSciNetMATHGoogle Scholar
  21. [21]
    M. Lanza de Cristoforis, A singular domain perturbation problem for the Poisson equation. In More progresses in analysis, Proceedings of the 5th international ISAAC congress, Catania, Italy, July 2530, 2005, pages 955–965. World Scientific, Hackensack, NJ, 2009.Google Scholar
  22. [22]
    M. Lanza de Cristoforis and P. Musolino, A perturbation result for periodic layer potentials of general second-order differential operators with constant coefficients. Far East J. Math. Sci. (FJMS) 52 (2011), 75–120.MathSciNetMATHGoogle Scholar
  23. [23]
    M. Lanza de Cristoforis and P. Musolino, A real analyticity result for a nonlinear integral operator. J. Integral Equations Appl., (to appear).Google Scholar
  24. [24]
    M. Lanza de Cristoforis and P. Musolino, A singularly perturbed nonlinear Robin problem in a periodically perforated domain: a functional analytic approach. Complex Var. Elliptic Equ., (to appear).Google Scholar
  25. [25]
    M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density. J. Integral Equations Appl. 16 (2004), 137–174.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    V. Maz’ya, S. Nazarov, and B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vols. I, II, volumes 111, 112 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 2000.Google Scholar
  27. [27]
    C. Miranda, Sulle proprietà di regolarità di certe trasformazioni integrali. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I 7 (1965), 303–336.MathSciNetMATHGoogle Scholar
  28. [28]
    V. Mityushev and P.M. Adler, Longitudinal permeability of spatially periodic rectangular arrays of circular cylinders. I. A single cylinder in the unit cell. ZAMM Z. Angew. Math. Mech. 82 (2002), 335–345.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    P. Musolino, A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach. Math. Methods Appl. Sci. 35 (2012), 334–349.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    S. Ozawa, Electrostatic capacity and eigenvalues of the Laplacian. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), 53–62.MathSciNetMATHGoogle Scholar
  31. [31]
    L. Preciso, Perturbation analysis of the conformal sewing problem and related problems. PhD Dissertation, University of Padova, 1998.Google Scholar
  32. [32]
    L. Preciso, Regularity of the composition and of the inversion operator and perturbation analysis of the conformal sewing problem in Roumieu type spaces. Tr. Inst. Mat. Minsk 5 (2000), 99–104.MATHGoogle Scholar
  33. [33]
    G. Prodi and A. Ambrosetti, Analisi non lineare. I quaderno. Editrice Tecnico Scientifica, Pisa, 1973.Google Scholar
  34. [34]
    S. Rogosin, M. Dubatovskaya, and E. Pesetskaya, Eisenstein sums and functions and their application at the study of heat conduction in composites. Siauliai Math. Semin. 4 (2009), 167–187.MathSciNetGoogle Scholar
  35. [35]
    G.M. Troianiello, Elliptic differential equations and obstacle problems. TheUniversity Series in Mathematics. Plenum Press, New York, 1987.Google Scholar
  36. [36]
    M.J. Ward and J.B. Keller, Strong localized perturbations of eigenvalue problems. SIAM J. Appl. Math. 53 (1993), 770–798.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Dipartimento di Matematica Pura ed ApplicataUniversità degli Studi di PadovaPadovaItaly

Personalised recommendations