The Caccioppoli Ultrafunctions

TL;DR

This paper introduces a new space of ultrafunctions with enhanced localization and integral properties, potentially useful for PDEs and calculus of variations, demonstrated through a simple example.

Contribution

It defines a specialized ultrafunction space with improved localization and integral properties, extending classical theorems in a hyperreal setting.

Findings

Extended integration by parts and Gauss theorem for ultrafunctions

New ultrafunction space with localized functions and derivatives

Potential applications to PDEs and calculus of variations

Abstract

Ultrafunctions are a particular class of functions defined on a hyperreal field $\mathbb{R}^{\ast}\supset\mathbb{R}$. They have been introduced and studied in some previous works. 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 PDE's and Calculus of Variations. This fact will be illustrated by a simple, but meaningful example.