Advertisement

A Family of Fundamental Solutions of Elliptic Partial Differential Operators with Real Constant Coefficients

  • Matteo Dalla RivaEmail author
Article

Abstract

We present a construction of a family of fundamental solutions for elliptic partial differential operators with real constant coefficients. The elements of such a family are expressed by means of jointly real analytic functions of the coefficients of the operators and of the spatial variable. The aim is to write detailed expressions for such functions. Such expressions are then exploited to prove regularity properties in the frame of Schauder spaces and jump properties of the corresponding single layer potentials.

Mathematics Subject Classification (2010)

35E05 35B20 

Keywords

Fundamental solutions elliptic partial differential operators with real constant coefficients layer potentials 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boman J.: Differentiability of a function and of its composition with functions of one variable. Math. Scand. 20, 249–268 (1967)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cartan, H.: Differential calculus. Hermann, Paris; Houghton Mifflin Co., Boston, Mass. (1971)Google Scholar
  3. 3.
    Cialdea A.: A general theory of hypersurface potentials. Ann. Mat. Pura Appl. 168(4), 37–61 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cialdea A.: Completeness theorems for elliptic equations of higher order with constants coefficients. Georgian Math. J. 14, 81–97 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Costabel, M., Le Louër, F.: Shape derivatives of boundary integral operators in electromagnetic scattering. Part I: shape differentiability of pseudo-homogeneous boundary integral operators. Integr. Equ. Oper. Theory 72, 509–535 (2012)Google Scholar
  6. 6.
    Costabel, M., Le Louër, F.: Shape derivatives of boundary integral operators in electromagnetic scattering. Part II: application to scattering by a homogeneous dielectric obstacle. Integr. Equ. Oper. Theory 73, 17–48 (2012)Google Scholar
  7. 7.
    Dalla Riva, M.: Potential theoretic methods for the analysis of singularly perturbed problems in linearized elasticity. Phd dissertation, University of Padova (Italy), supervisor M. Lanza de Cristoforis (2007)Google Scholar
  8. 8.
    Dalla Riva, M.: The layer potentials of some partial differential operators: real analytic dependence upon perturbations. In: Further Progress in Analysis, pp. 208–217. World Scientific Publishing, Hackensack (2009)Google Scholar
  9. 9.
    Dalla Riva M., Lanza de Cristoforis M.: A perturbation result for the layer potentials of general second order differential operators with constant coefficients. J. Appl. Funct. Anal. 5, 10–30 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dalla Riva, M., Morais, J., Musolino, P.: A family of fundamental solutions of elliptic partial differential operators with quaternion constant coefficients. Math. Meth. Appl. Sci, to appear. doi: 10.1002/mma.2706
  11. 11.
    Deimling K.: Nonlinear Functional Analysis. Springer, Berlin (1985)zbMATHCrossRefGoogle 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. In: Classics in Mathematics. Springer, Berlin (2001)Google Scholar
  14. 14.
    John F.: Plane Waves and Spherical Means Applied To Partial Differential Equations. Interscience Publishers, New York-London (1955)zbMATHGoogle Scholar
  15. 15.
    Lanza de Cristoforis M.: A domain perturbation problem for the Poisson equation. Complex Var. Theory Appl. 50, 851–867 (2005)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Lanza de Cristoforis M.: 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, 63–93 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    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)MathSciNetzbMATHGoogle Scholar
  18. 18.
    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. Integr. Equ. Appl. 16, 137–174 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Lanza de Cristoforis, M., Rossi, L.: Real analytic dependence of simple and double layer potentials for the Helmholtz equation upon perturbation of the support and of the density. In: Analytic Methods of Analysis and Differential Equations: AMADE 2006, pp. 193–220. Cambridge Sci. Publ., Cambridge (2008)Google Scholar
  20. 20.
    Mantlik F.: Partial differential operators depending analytically on a parameter. Ann. Inst. Fourier (Grenoble) 41, 577–599 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Mantlik F.: Fundamental solutions for hypoelliptic differential operators depending analytically on a parameter. Trans. Am. Math. Soc 334, 245–257 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Mitrea, I.: Mapping properties of layer potentials associated with higher-order elliptic operators in Lipschitz domains. In: Topics in Operator Theory. Systems and Mathematical Physics, vol. 2, pp. 363–407. Oper. Theory Adv. Appl. vol. 203. Birkhäuser, Basel (2010)Google Scholar
  23. 23.
    Miranda C.: Sulle proprietà di regolarità di certe trasformazioni integrali. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I 7(8), 303–336 (1965)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Potthast R.: Fréchet differentiability of boundary integral operators in inverse acoustic scattering. Inverse Probl. 10, 431–447 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Potthast R.: Domain derivatives in electromagnetic scattering. Math. Methods Appl. Sci. 19, 1157–1175 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Potthast R.: Fréchet differentiability of the solution to the acoustic Neumann scattering problem with respect to the domain. J. Inverse Ill-Posed Probl. 4, 67–84 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Trèves F.: Fundamental solutions of linear partial differential equations with constant coefficients depending on parameters. Am. J. Math. 84, 561–577 (1962)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Centro de Investigação e Desenvolvimento em Matemática e Aplicaçoẽs (CIDMA)Universidade de AveiroAveiroPortugal

Personalised recommendations