An Improved Setting for Generalized Functions: Fine Ultrafunctions

作     者：Benci, Vieri 

作者机构：Univ Pisa Dipartimento Matemat Via F Buonarroti 1-C Pisa Italy 

出 版 物：《MILAN JOURNAL OF MATHEMATICS》 (米兰数学杂志)

年 卷 期：2022年第90卷第2期

页      面：575-646页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：Universita di Pisa within the CRUI-CARE Agreement 

主　　题：Partial differential equations Generalized functions Distributions Non archimedean mathematics Non standard analysis Ill posed problems 

摘      要：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.