An Improved Setting for Generalized Functions: Fine Ultrafunctions

Ultrafunctions are a particular class of functions defined on a Non Archimedean field E ⊃ 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.