Elliptic regularity and solvability for partial differential equations with Colombeau coefficients
By G. Hörmann, Michael Oberguggenberger. 2003
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The 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 $\G^\infty$-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.
Subfields: Applied Mathematics, Mathematical Physics, Philosophy
View Paper: https://www.semanticscholar.org/paper/9753b09ce1634da818cc5750e1fed99871b2a93a
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In Applied Mathematics
In Mathematical Physics
In Philosophy
Citations in the corpus
- Pseudodifferential operators with generalized symbols and regularity theory (2005, methodology): “As an application, we formulate a sufficient condition of hypoellipticity in this setting, which leads to regularity results for generalized pseudodifferential equations.”
- Topological Structures in Colombeau Algebras: Topological $\widetilde{\mathbb{C}}$ -modules and Duality Theory (2005, background): “These investigations have produced a theory of point values in the Colombeau algebra G(Ω) and results of invertibility and positivity in the ring of constant generalized functions C̃ [19, 38, 39] but also microlocal analysis in Colombeau algebras and regularity theory for generalized solutions to pa”