Annali di Matematica Pura ed Applicata (1923 -)

, Volume 197, Issue 4, pp 1089–1116 | Cite as

Analytic dependence of a periodic analog of a fundamental solution upon the periodicity parameters

  • M. Lanza de Cristoforis
  • P. Musolino


We prove an analyticity result in Sobolev–Bessel potential spaces for the periodic analog of the fundamental solution of a general elliptic partial differential operator upon the parameters which determine the periodicity cell. Then we show concrete applications to the Helmholtz and the Laplace operators. In particular, we show that the periodic analogs of the fundamental solution of the Helmholtz and of the Laplace operator are jointly analytic in the spatial variable and in the parameters which determine the size of the periodicity cell. The analysis of the present paper is motivated by the application of the potential theoretic method to periodic anisotropic boundary value problems in which the “degree of anisotropy” is a parameter of the problem.


Periodic fundamental solution Elliptic differential equation Real analytic dependence Helmholtz equation Laplace equation 

Mathematics Subject Classification

47H30 42B99 31B10 45A05 35J25 


  1. 1.
    Ammari, H., Kang, H.: Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory. Springer, New York (2007)zbMATHGoogle Scholar
  2. 2.
    Ammari, H., Kang, H., Lim, M.: Effective parameters of elastic composites. Indiana Univ. Math. J. 55, 903–922 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ammari, H., Kang, H., Touibi, K.: Boundary layer techniques for deriving the effective properties of composite materials. Asymptot. Anal. 41, 119–140 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Arens, T., Sandfort, K., Schmitt, S., Lechleiter, A.: Analysing Ewald’s method for the evaluation of Green’s functions for periodic media. IMA J. Appl. Math. 78, 405–431 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berman, C.L., Greengard, L.: A renormalization method for the evaluation of lattice sums. J. Math. Phys. 35(2), 6036–6048 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Böhme, R., Tomi, L.: Zur Struktur der Lösungsmenge des Plateauproblems. Math. Z. 133, 1–29 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dalla Riva, M.: A family of fundamental solutions of elliptic partial differential operators with real constant coefficients. Integral Equ. Oper. Theory 76, 1–23 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dalla Riva, M., Lanza de Cristoforis, M., Musolino, P.: Analytic dependence of volume potentials corresponding to parametric families of fundamental solutions. Integral Equ. Oper. Theory 82, 371–393 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dalla Riva, M., Morais, J., Musolino, P.: A family of fundamental solutions of elliptic partial differential operators with quaternion constant coefficients. Math. Methods Appl. Sci. 36, 1569–1582 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dalla Riva, M., Musolino, P.: A singularly perturbed nonlinear traction problem in a periodically perforated domain: a functional analytic approach. Math. Methods Appl. Sci. 37, 106–122 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)CrossRefzbMATHGoogle Scholar
  12. 12.
    Folland, G.B.: Introduction to Partial Differential Equations. Princeton University Press, Princeton (1995)zbMATHGoogle Scholar
  13. 13.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  14. 14.
    Henry, D.: Topics in Nonlinear Analysis. Trabalho de Matemática, vol. 192, Brasilia (1982)Google Scholar
  15. 15.
    Lanza de Cristoforis, M.: Properties and pathologies of the composition and inversion operators in Schauder spaces. Acc. Naz. delle Sci. detta dei XL 15, 93–109 (1991)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Lanza de Cristoforis, M., Musolino, P.: A perturbation result for periodic layer potentials of general second order differential operators with constant coefficients. Far East J. Math. Sci. 52, 75–120 (2011)Google Scholar
  17. 17.
    Lanza de Cristoforis, M., Musolino, P.: A real analyticity result for a nonlinear integral operator. J. Integral Equ. Appl. 25, 21–46 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lanza de Cristoforis, M., Musolino, P.: A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach. Comm. Pure Appl. Anal. 13, 2509–2542 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lanza de Cristoforis, M., Rossi, L.: Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density. J. Integral Equ. Appl. 16, 137–174 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lin, C.S., Wang, C.L.: Elliptic functions, Green functions and the mean field equations on tori. Ann. Math. 172, 911–954 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications, vol. I. Springer, New York (1972)CrossRefzbMATHGoogle Scholar
  22. 22.
    Mamode, M.: Fundamental solution of the Laplacian on flat tori and boundary value problems for the planar Poisson equation in rectangles. Bound. Value Probl. 2014, 221 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  24. 24.
    Mityushev, V., Adler, P.M.: Longitudinal permeability of spatially periodic rectangular arrays of circular cylinders. I. A single cylinder in the unit cell. ZAMM Z. Angew. Math. Mech. 82, 335–345 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Schmeisser, H.J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. Akademische Verlagsgesellschaft Geest & Portig K.-G, Leipzig (1987)zbMATHGoogle Scholar
  26. 26.
    Tornberg, A.K., Greengard, L.: A fast multipole method for the three-dimensional Stokes equations. J. Comput. Phys. 227, 1613–1619 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Triebel, H.: Theory of Function Spaces. Birkhäuser Verlag, Basel (1983)CrossRefzbMATHGoogle Scholar
  28. 28.
    Valent, T.: Boundary Value Problems of Finite Elasticity. Local Theorems on Existence, Uniqueness and Analytic Dependence on Data. Springer, New York (1988)CrossRefzbMATHGoogle Scholar

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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di PadovaPaduaItaly
  2. 2.Department of MathematicsAberystwyth UniversityCeredigionWales, UK

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