Microlocal Asymptotic Analysis in Algebras of Generalized Functions

Antoine Delcroix (1, 2) , Jean-André Marti (3) , Michael Oberguggenberger (4)

1 CRREF - Centre de recherches et de ressources en éducation et formation
2 AOC - Analyse Optimisation Controle
3 GTSI - Groupe de Technologie des Surfaces et Interfaces [URp2_2]
4 ISBE - Institute of Basic Sciences in Engineering

We introduce a new type of local and microlocal asymptotic analysis in algebras of generalized functions, based on the presheaf properties of those algebras and on the properties of their elements with respect to a regularizing parameter. Contrary to the more classical frequential analysis based on the Fourier transform, we can describe a singular asymptotic spectrum which has good properties with respect to nonlinear operations. In this spirit we give several examples of propagation of singularities through nonlinear operators.

Mots clés: microlocal analysis; generalized functions; nonlinear operators; presheaf; propagation of singularities; singular spectrum

Domaines: Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]

Antoine Delcroix, Jean-André Marti, Michael Oberguggenberger. Microlocal Asymptotic Analysis in Algebras of Generalized Functions. Asymptotic Analysis, 2008, 59 (1-2), pp.83-107. ⟨10.3233/ASY-2008-0885⟩. ⟨hal-00140656⟩