The Caccioppoli ultrafunctions

Advances in Nonlinear Analysis

article

Vieri Benci¹  Luigi Carlo Berselli¹  Carlo Romano Grisanti¹ 

[1] Dipartimento di Matematica, Università degli Studi di Pisa

关键词: Ultrafunctions;    non-Archimedean mathematics;    nonstandard analysis;    delta function;    distributions;   

DOI  :  10.1515/anona-2017-0225

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

Ultrafunctions are a particular class of functions defined on a hyperreal field ℝ ∗ ⊃ ℝ {\mathbb{R}^{\ast}\supset\mathbb{R}} . They have been introduced and studied in some previous works [2, 6, 7]. In this paper we introduce a particular space of ultrafunctions which has special properties, especially in term of localization of functions together with their derivatives. An appropriate notion of integral is then introduced which allows to extend in a consistent way the integration by parts formula, the Gauss theorem and the notion of perimeter. This new space we introduce, seems suitable for applications to Partial Differential Equations and Calculus of Variations. This fact will be illustrated by a simple, but meaningful example.

【 授权许可】

CC BY