Abstract
AbstractUltrafunctions are a particular class of functions defined on a hyperreal fieldℝ∗⊃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.
Highlights
The Caccioppoli ultrafunctions can be considered as a kind generalized functions
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 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
Summary
The Caccioppoli ultrafunctions can be considered as a kind generalized functions. In many circumstances, the notion of real function is not sufficient to the needs of a theory and it is necessary to extend it. The main novelty of this paper is that we introduce the space of Caccioppoli ultrafunctions They satisfy special properties which are very powerful in applications to Partial Differential Equations and Calculus of Variations. The most relevant point, which is not present in the previous approaches to ultrafunctions, is that we are able the extend the notion of partial derivative so that it is a local operator and it satisfies the usual formula valid when integrating by parts, at the price of a suitable extension of the integral as well. We can prove the integration by parts rule (1.1) and the Gauss’ divergence theorem (with the appropriate extension ∫◻ of the usual integral), which are the main tools used in the applications These results are a development of the theory previously introduced in [9] and [11]. In the last section we present some very simple examples to show that the ultrafunctions can be used to perform a precise mathematical analysis of problems which are not tractable via the distributions
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