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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
Article

Abstract

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.

Keywords

Algebras of generalized functions Fundamental solutions Regularity theory 

Mathematics Subject Classification (2000)

46F30 35E05 

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References

  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) CrossRefMathSciNetzbMATHGoogle 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) CrossRefMathSciNetzbMATHGoogle 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) CrossRefMathSciNetzbMATHGoogle 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) MathSciNetzbMATHGoogle 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) CrossRefMathSciNetzbMATHGoogle 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) zbMATHGoogle Scholar
  14. 14.
    Hörmander, L.: Linear Partial Differential Operators. Springer, Berlin (1963) zbMATHGoogle 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) CrossRefzbMATHGoogle Scholar
  19. 19.
    Nedeljkov, M., Pilipović, S.: Paley-Wiener type theorems for Colombeau’s generalized functions. J. Math. Anal. Appl. 195, 108–122 (1995) CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Nedeljkov, M., Pilipović, S.: Hypoelliptic differential operators with generalized constant coefficients. Proc. Edinb. Math. Soc. 41, 47–60 (1998) zbMATHGoogle 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) zbMATHGoogle Scholar
  22. 22.
    Oberguggenberger, M.: Multiplication of Distributions and Applications to Partial Differential Equations. Pitman Research Notes in Mathematics, vol. 259. Longman, Harlow (1992) zbMATHGoogle 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) CrossRefzbMATHGoogle 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) MathSciNetzbMATHCrossRefGoogle 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

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