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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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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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