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Resurgent analysis and differential equations with singularities. (English) Zbl 0847.35025
Lumer, G√ľnter (ed.) et al., Partial differential equations. Models in physics and biology. Contributions to the conference, held in Han-sur-Lesse, Belgium, in December 1993. Berlin: Akademie Verlag. Math. Res. 82, 401-418 (1994).
The conormal asymptotic used previously by the first author in the study of qualitative behavior of the solutions of PDE near its singularities is treated and generalized here in terms of some space of functions called resurgent functions of power type. It seems that this setting is a synthesis of ideas comming from Ecalle’s resurgent variable theory and the first author’s conormal asymptotic. In fact, here the problem is how to construct asymptotics for the solutions of PDE, and the motivation is that even for simple equations like \(\sqrt{- \Delta u}+ u= f\) on a manifold with conical singularities we need more general expansions. A notion of a differential operator \(\widehat P\) with power singularities on a smooth manifold \(M\) is introduced and equations with resurgent right-hand part \(\widehat P u= f\) are studied. The resurgent structure of the solutions and a technique of its construction is developed. As an example elliptic equations on a cone and on a manifold with an edge are given.
For the entire collection see [Zbl 0809.00019].

35C20 Asymptotic expansions of solutions to PDEs