scholarly journals Colombeau Generalized Functions and Solvability of Differential Operators

2006 ◽  
pp. 467-477
Author(s):  
Khaled Benmeriem ◽  
Chikh Bouzar
Keyword(s):  
Differential Operators ◽  
Generalized Functions ◽  
Colombeau Generalized Functions

scholarly journals Hypoelliptic differential operators with generalized constant coefficients

1998 ◽  
Vol 41 (1) ◽  
pp. 47-60 ◽  
Author(s):  
M. Nedeljkov ◽  
S. Pilipović
Keyword(s):  
Small Parameter ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Necessary And Sufficient Conditions ◽  
Order Estimate ◽  
Colombeau Generalized Functions ◽  
Constant Coefficients ◽  
Necessary And Sufficient

The space of Colombeau generalized functions is used as a frame for the study of hypoellipticity of a family of differential operators whose coefficients depend on a small parameter ε.There are given necessary and sufficient conditions for the hypoellipticity of a family of differential operators with constant coefficients which depend on ε and behave like powers of ε as ε→0. The solutions of such family of equations should also satisfy the power order estimate with respect to ε.

scholarly journals Algebras of generalized functions with smooth parameter dependence

2012 ◽  
Vol 55 (1) ◽  
pp. 105-124 ◽  
Author(s):  
Annegret Burtscher ◽  
Michael Kunzinger
Keyword(s):  
Systematic Study ◽  
Generalized Functions ◽  
Parameter Dependence ◽  
Order Structure ◽  
Colombeau Generalized Functions ◽  
Algebras Of Generalized Functions ◽  
Algebraic Properties

AbstractWe show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring $\tilde{\mathbb{K}}_\mathrm{sm}$ of generalized numbers in this unified setting. In particular, we investigate the ring and order structure of $\tilde{\mathbb{K}}_\mathrm{sm}$ and establish some properties of its ideals.

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