Electron. J. Diff. Eqns., Vol. 2004(2004), No. 14, pp. 1-30.

Elliptic regularity and solvability for partial differential equations with Colombeau coefficients

Günther Hörmann & Michael Oberguggenberger

This paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new notions of ellipticity and hypoellipticity, study their interrelation, and give a number of new examples and counterexamples. Using the concept of $\mathcal{G}^\infty$-regularity of generalized functions, we derive a general global regularity result in the case of operators with constant generalized coefficients, a more specialized result for second order operators, and a microlocal regularity result for certain first order operators with variable generalized coefficients. We also prove a global solvability result for operators with constant generalized coefficients and compactly supported Colombeau generalized functions as right hand sides.

Submitted July 8, 2003. Published February 3, 2004
Math Subject Classifications: 46F30, 35D05, 35D10
Key Words: Algebras of generalized functions, regularity, solvability

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Günther Hörmann
Institut für Mathematik, Universität Wien
Wien, Austria
email: Guenther.Hoermann@uibk.ac.at
Michael Oberguggenberger
Institut fur Technische Mathematik
Geometrie und Bauinformatik
Universität Innsbruck
Innsbruck, Austria
email: Michael.Oberguggenberger@uibk.ac.at

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