A generalization of Gauss’ divergence theorem

Abstract

This paper is devoted to the proof Gauss' divergence theorem in the framework of "ultrafunctions". They are a new kind of generalized functions, which have been introduced recently in [2] and developed in [4], [5] and [6]. Their peculiarity is that they are based on a non-Archimedean field, namely on a field which contains infinite and infinitesimal numbers. Ultrafunctions have been introduced to provide generalized solutions to equations which do not have any solutions, not even among the distributions.

The Caccioppoli ultrafunctions

“…Nevertheless, we can prove the integration by parts rule (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 [8] and [10].…”

Generalized solutions in PDEs and the Burgers' equation

“…In this section we recall the basic definitions and facts regarding non-Archimedean fields, following an approach that has been introduced in [13] (see also [4,6,7,8,9,10,11,12]). In the following, K will denote an ordered field.…”

“…Ultrafunctions are a family of generalized functions defined on the field of hyperreals, which are a well known extension of the reals. They have been introduced in [4], and also studied in [7,8,10,11,12,13]. The non-Archimedean setting in which we will work (which is a reformulation, in a topological language, of the ultrapower approach to NSA of Keisler) is introduced in Section 2.…”

“…A classical way to do this is to introduce the notion of weak solution; another approach is to use generalized functions. Ultrafunctions are a particular class of generalized functions that has been previously introduced and used to define generalized solutions of stationary problems in [4,7,9,11,12]. In this paper we generalize this notion in order to study also evolution problems.…”

Generalized solutions of variational problems and applications

“…In this paper we deal with ultrafunctions, which are a kind of generalized functions that have been introduced recently in [1] and developed in [2,[4][5][6][7][8][9][10][11].…”