Acta Applicandae Mathematicae

, Volume 102, Issue 2–3, pp 281–318 | Cite as

Fundamental Solutions in the Colombeau Framework: Applications to Solvability and Regularity Theory

  • Claudia Garetto


In this article we introduce the notion of fundamental solution in the Colombeau context as an element of the dual \(\mathcal {L}(\ensuremath {\mathcal {G}_{\mathrm{c}}}(\mathbb {R}^{n}),\widetilde {\mathbb {C}})\) . After having proved the existence of a fundamental solution for a large class of partial differential operators with constant Colombeau coefficients, we investigate the relationships between fundamental solutions in \(\mathcal {L}(\ensuremath {\mathcal {G}_{\mathrm{c}}}(\mathbb {R}^{n}),\widetilde {\mathbb {C}})\) , Colombeau solvability and \(\ensuremath {\mathcal {G}}\) - and \(\ensuremath {\ensuremath {\mathcal {G}}^{\infty}}\) -hypoellipticity respectively.


Algebras of generalized functions Fundamental solutions Regularity theory 

Mathematics Subject Classification (2000)

46F30 35E05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chazarain, J., Pirou, A.: Introduction to the theory of linear partial differential equations. In: Studies in Mathematics and Its Applications. North Holland, Amsterdam (1982) Google Scholar
  2. 2.
    Friedlander, G., Joshi, M.: Introduction to the Theory of Distributions, 2nd edn. Cambridge University Press, New York (1998) Google Scholar
  3. 3.
    Garetto, C.: Pseudo-differential operators in algebras of generalized functions and global hypoellipticity. Acta Appl. Math. 80(2), 123–174 (2004) CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Garetto, C.: Pseudodifferential operators with generalized symbols and regularity theory. Ph.D. thesis, University of Torino (2004) Google Scholar
  5. 5.
    Garetto, C.: Topological structures in Colombeau algebras: topological \(\widetilde {\mathbb {C}}\) -modules and duality theory. Acta. Appl. Math. 88(1), 81–123 (2005) CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Garetto, C.: Topological structures in Colombeau algebras: investigation of the duals of \({\ensuremath {\mathcal {G}_{\mathrm{c}}}(\Omega )}\) , \({\ensuremath {\mathcal {G}}(\Omega )}\) and \({\ensuremath {\mathcal {G}}_{{\,}\atop {\hskip -4pt\scriptstyle {S}}}\!(\mathbb {R}^{n})}\) . Mon.hefte Math. 146(3), 203–226 (2005) CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Garetto, C.: Microlocal analysis in the dual of a Colombeau algebra: generalized wave front sets and noncharacteristic regularity. N.Y. J. Math. 12, 275–318 (2006) MathSciNetMATHGoogle Scholar
  8. 8.
    Garetto, C.: Closed graph and open mapping theorems for topological \(\widetilde {\mathbb {C}}\) -modules and applications. Math. Nac. (2008, to appear). arXiv:math.FA/0608087(v2) Google Scholar
  9. 9.
    Garetto, C., Hörmann, G.: Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities. Proc. Edinb. Math. Soc. 48(3), 603–629 (2005) CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Garetto, C., Hörmann, G.: Duality theory and pseudodifferential techniques for Colombeau algebras: generalized kernels and microlocal analysis. Bull. Cl. Sci. Math. Nat. Sci. Math. 31, 115–136 (2006) CrossRefGoogle Scholar
  11. 11.
    Garetto, C., Gramchev, T., Oberguggenberger, M.: Pseudodifferential operators with generalized symbols and regularity theory. Electr. J. Differ. Equ. 2005(116), 1–43 (2003) Google Scholar
  12. 12.
    Garetto, C., Hörmann, G., Oberguggenberger, M.: Generalized oscillatory integrals and Fourier integral operators. Proc. Edinb. Math. Soc. (2008, to appear). arXiv:math.AP/0607706 Google Scholar
  13. 13.
    Hirsch, M.W., Smale, S.: Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, New York (1974) MATHGoogle Scholar
  14. 14.
    Hörmander, L.: Linear Partial Differential Operators. Springer, Berlin (1963) MATHGoogle Scholar
  15. 15.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. II. Springer, New York (1983) Google Scholar
  16. 16.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. 1, 2nd edn. Springer, New York (1990) Google Scholar
  17. 17.
    Hörmann, G., Oberguggenberger, M.: Elliptic regularity and solvability for partial differential equations with Colombeau coefficients. Electr. J. Differ. Equ. 2004(14), 1–30 (2004) Google Scholar
  18. 18.
    Hörmann, G., Oberguggenberger, M., Pilipovic, S.: Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients. Trans. Am. Math. Soc. 358(8), 3363–3383 (2006) CrossRefMATHGoogle Scholar
  19. 19.
    Nedeljkov, M., Pilipović, S.: Paley-Wiener type theorems for Colombeau’s generalized functions. J. Math. Anal. Appl. 195, 108–122 (1995) CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Nedeljkov, M., Pilipović, S.: Hypoelliptic differential operators with generalized constant coefficients. Proc. Edinb. Math. Soc. 41, 47–60 (1998) MATHGoogle Scholar
  21. 21.
    Nedeljkov, M., Pilipović, S., Scarpalézos, D.: The Linear Theory of Colombeau Generalized Functions. Pitman Research Notes in Mathematics, vol. 385. Longman, Harlow (1998) MATHGoogle Scholar
  22. 22.
    Oberguggenberger, M.: Multiplication of Distributions and Applications to Partial Differential Equations. Pitman Research Notes in Mathematics, vol. 259. Longman, Harlow (1992) MATHGoogle Scholar
  23. 23.
    Scarpalézos, D.: Topologies dans les espaces de nouvelles fonctions généralisées de Colombeau. \({\widetilde{ \mathbb {C}}}\) -modules topologiques. Université Paris 7 (1992) Google Scholar
  24. 24.
    Scarpalézos, D.: Some remarks on functoriality of Colombeau’s construction; topological and microlocal aspects and applications. Integral Transforms Spec. Funct. 6(1–4), 295–307 (1998) CrossRefMATHGoogle Scholar
  25. 25.
    Scarpalézos, D.: Colombeau’s generalized functions: topological structures; microlocal properties. A simplified point of view. I. Bull. Cl. Sci. Math. Nat. Sci. Math. 25, 89–114 (2000) Google Scholar
  26. 26.
    Soraggi, R.L.: Fourier analysis on Colombeau’s algebra of generalized functions. J. Anal. Math. 69, 201–227 (1996) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Institut für Grundlagen der Bauingenieurwissenschaften, Arbeitsbereich für Technische MathematikLeopold-Franzens-Universität InnsbruckInnsbruckAustria

Personalised recommendations