A Topological Approach to Non-Archimedean Mathematics
Non-Archimedean mathematics (in particular, nonstandard analysis) allows to construct some useful models to study certain phenomena arising in PDE’s; for example, it allows to construct generalized solutions of differential equations and variational probl
- PDF / 292,819 Bytes
- 24 Pages / 439.37 x 666.142 pts Page_size
- 67 Downloads / 165 Views
Abstract Non-Archimedean mathematics (in particular, nonstandard analysis) allows to construct some useful models to study certain phenomena arising in PDE’s; for example, it allows to construct generalized solutions of differential equations and variational problems that have no classical solution. In this paper we introduce certain notions of Non-Archimedean mathematics (and of nonstandard analysis) by means of an elementary topological approach; in particular, we construct Non-Archimedean extensions of the reals as appropriate topological completions of R. Our approach is based on the notion of Λ-limit for real functions, and it is called Λ-theory. It can be seen as a topological generalization of the α-theory presented in [6], and as an alternative topological presentation of the ultrapower construction of nonstandard extensions (in the sense of [21]). To motivate the use of Λ-theory for applications we show how to use it to solve a minimization problem of calculus of variations (that does not have classical solutions) by means of a particular family of generalized functions, called ultrafunctions. Keywords Non-Archimedean mathematics · Nonstandard analysis functions · Generalized functions · Ultrafunctions
·
Limits of
MSC 2010: 26E30 · 26E35 · 54A20
V. Benci (B) Dipartimento di Matematica, Università degli Studi di Pisa, Via F. Buonarroti 1/c, 56127 Pisa, Italy e-mail: [email protected] L. Luperi Baglini Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria e-mail: [email protected] © Springer International Publishing Switzerland 2016 F. Gazzola et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer Proceedings in Mathematics & Statistics 176, DOI 10.1007/978-3-319-41538-3_2
17
18
V. Benci and L. Luperi Baglini
1 Introduction In a previous series of papers [5, 9–14] we have introduced and studied a new family of generalized functions called ultrafunctions and its applications to certain problems in mathematical analysis, including some applications to PDE’s in [14]. The development of a rigorous study of (a large class of) PDE’s in ultrafunction theory is the object of [15], where we exemplify our approach by studying in detail Burgers’ equation. Henceforth, it is our feeling that many problems in PDE’s theory could be fruitfully studied by means of the theory of ultrafunctions. However, one might have the impression that a drawback of our approach is the use of the machinery of NSA, which is not a “common working tool” for most analysts. Even if NSA has already been applied to many different fields of mathematics (such as functional analysis, probability theory, combinatorial number theory, mathematical physics and so on) to obtain important results, the original formalism of Robinson, based on model theory (see e.g. [25]), appears too technical to many researchers, and not directly usable by most mathematicians. Since Robinson’s work first appeared, a simpler semantic approach (due to Robinson himself and Elias Zakon)
Data Loading...
