An Improved Setting for Generalized Functions: Fine Ultrafunctions

Ultrafunctions are a particular class of functions defined on a Non Archimedean field E superset of R. They have been introduced and studied in some previous works (Benci, in Adv Nonlinear Stud 13:461-486, 2013; Benci and Luperi Baglini, in Electron J Differ Equ Conf 21:11-21, 2014; Benci et al., in Adv Nonlinear Anal 10. https: //doi . org/10.1515/anona-2017-0225. 2; Benci et al., in Adv. Nonlinear Anal 9, 2018). In this paper we develop the notion of fine ultrafunctions which improves the older definitions in many crucial points. Some applications are given to show how ultrafunctions can be applied in studing Partial Differential Equations. In particular, it is possible to prove the existence of ultrafunction solutions to ill posed evolution poblems.

Automated summary

This paper introduces ultrafunctions and their definition and study on a Non Archimedean field, presents an improved concept of fine ultrafunctions, and provides applications of ultrafunctions in studying Partial Differential Equations, particularly in solving ill-posed evolution problems.