Pseudo-Differential Operators in Algebras of Generalized Functions and Global Hypoellipticity

By Claudia Garetto. 2005

53 citations recorded by the source corpus.

The aim of this work is to develop a global calculus for pseudo-differential operators acting on suitable algebras of generalized functions. In particular, a condition of global hypoellipticity of the symbols gives a result of regularity for the corresponding pseudo-differential equations. This calculus and this frame are proposed as tools for the study in Colombeau algebras of partial differential equations globally defined on Rn.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b82715edb885f317ca0da2371564fa3502673aca

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In Mathematical Physics

1. Claudia Garetto
2. G. Hörmann
3. Todor Gramchev
4. Michael Oberguggenberger
5. Stevan Pilipovic

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.”
• Elliptic regularity and solvability for partial differential equations with Colombeau coefficients (2003, background): “In terms of the microlocal point of view - what should be a non-characteristic direction? What sort of regularity results are to be expected? While regularity theory of Colombeau coefficients with classical, constant coefficients was settled in [16], there is current interest in pseudodifferential o”
• Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients (2003, background): “Recent related research in Colombeau regularity theory has shown progress in a diversity of directions, including such topics as pseudodifferential operators with generalized symbols and case studies in microlocal analysis of nonlinear singularity propagation [ 4 , 10, 11, 15].”
• Topological Structures in Colombeau Algebras: Topological $\widetilde{\mathbb{C}}$ -modules and Duality Theory (2005, methodology): “Well-known Colombeau algebras as C̃, G(Ω) [19] and G S (R) [13, 15] are recognized to be special cases.”
• MICROLOCAL ANALYSIS OF GENERALIZED FUNCTIONS: PSEUDODIFFERENTIAL TECHNIQUES AND PROPAGATION OF SINGULARITIES (2004, methodology): “This is a crucial difference from more general definitions of ellipticity given in [6,7,16], whereas a similar condition has already been used in [15, § 6] in a special case.”
• Topological Structures in Colombeau Algebras: Investigation of the Duals of ${\cal G}_{{\rm c}}(\Omega)$, ${\cal G}(\Omega)$ and ${\cal G}_{{\cal S}}({\Bbb R}^n)$ (2005, background): “S -regularity which improve the ones obtained in [ 4 ].”