Searcharxiv⌕ Search

arXiv subjects

Vieri Benci

Publications and source records attributed to Vieri Benci.

At least 19 recordsLinked to original sources
arXiv2026

Generalized functions in a Non-Archimedean field

Ultrafunctions are generalized functions defined on a non-Archimedean field extending R. They provide a framework for treating problems that fall outside the classical theory of distributions. In this paper we introduce an improved notion of ultrafunctions which refines the previous constructions. This new approach allows a more flexible functional framework than the one used in other papers and yields a more flexible functional setting. In particular, this approach allows us to prove the existence of ultrafunction solutions for several classes of ill-posed evolution problems arising in partial differential equations.

math.AP↗
arXiv2025

Numbers and numerosities

We develop new aspects of the the of numerosity theory; more exactly, we emphasize its relation with the ordinal numbers, cardinal numbers, hyperreal numbers and surreal numbers. In particular, we combine the notion of numerosity with the idea of continuum and we get a definition of Euclidean line which includes all the sets of infinite numbers mentioned above.

math.AP↗
arXiv2023

Lusternik-Schnirelman and Morse theory for the Van der Waals-Cahn-Hilliard equation with volume constraint

We give a multiplicity result for solutions of the Van der Waals-Cahn-Hilliard two-phase transition equation with volume constraints on a closed Riemannian manifold. Our proof employs some results from the classical Lusternik--Schnirelman and Morse theory, together with a technique, the so-called \emph{photography method}, which allows us to obtain lower bounds on the number of solutions in terms of topological invariants of the underlying manifold. The setup for the photography method employs recent results from Riemannian isoperimetry for small volumes.

math.AP↗
arXiv2022

An improved setting for generalized functions: fine ultrafunctions

Ultrafunctions are a particular class of functions defined on a Non Archimedean field E. They have been introduced and studied in some previous works. In this paper we develop the notion of fine ultrafunctions which improves the older definitions in many crucial points. Some applications are given to show how ultrafunctions can be applied in studing Partial Differential Equations. In particular, it is possible to prove the existence of ultrafunction solutions to ill posed evolution poblems.

math.AP↗
arXiv2021

The Euclidean Universe

In this paper, we introduce a mathematical structure called Euclidean Universe. This structure provides a basic framework for Non-Archimedean Mathematics and in particular for Nonstandard Analysis.

math.GM↗
arXiv2020

A classical model for the Maxwell equations coupled with matter

We present a simple model of interaction of the Maxwell equations with a matter field defined by the Klein-Gordon equation. A simple linear interaction and a nonlinear perurbation produce solutions of the equations containing hylomorphic solitons, namely stable, solitary waves whose existence is related to the ratio energy/charge. These solitons, at low energy, behave as poinwise charged particles in an electromagnetic field.

math-ph↗
arXiv2020

The Euclidean numbers

We introduce axiomatically a Nonarchimedean field E, called the field of the Euclidean numbers, where a transfinite sum is defined that is indicized by ordinal numbers less than the first inaccessible Ω. Thanks to this sum, E becomes a saturated hyperreal field isomorphic to the so called Kiesler field of cardinality Ω, and suitable topologies can be put on E and on Ω \cup {Ω} so as to obtain the transfinite sums as limits of a suitable class of their finite subsums. Moreover there is a natural isomorphic embedding into E of the semiring Ω equipped by the natural sum and product. Finally a notion of numerosity satisfying all Euclidean common notions is given, whose values are nonnegative nonstandard integers of E. Then E can be charachterized as the hyperreal field generated by the real numbers and together with the semiring of numerosities (and this explains the name Euclidean numbers).

math.LO↗
arXiv2019

Van der Waals--Allen--Cahn--Hilliard equation with a volume constraint

We give multiplicity results for the solutions of a nonlinear elliptic equation, with an asymmetric double well potential of Van der Waals-Allen--Cahn--Hilliard type, satisfying a linear volume constraint, on a bounded Lipschitz domain $Ω\subset\mathds R^N$. The number of solutions is estimated in terms of topological and homological invariants of the underlying domain $Ω$.

math.AP↗
arXiv2019

Infinitesimal and Infinite Numbers as an Approach to Quantum Mechanics

Non-Archimedean mathematics is an approach based on fields which contain infinitesimal and infinite elements. Within this approach, we construct a space of a particular class of generalized functions, ultrafunctions. The space of ultrafunctions can be used as a richer framework for a description of a physical system in quantum mechanics. In this paper, we provide a discussion of the space of ultrafunctions and its advantages in the applications of quantum mechanics, particularly for the Schrödinger equation for a Hamiltonian with the delta function potential.

math-ph↗
arXiv2018

Non-Archimedean Mathematics and the formalism of Quantum Mechanics

This paper is divided in four parts. In the introduction, we discuss the program and the motivations of this paper. In section 2, we introduce the non-Archimedean field of Euclidean numbers E and we present a summary of the theory of Λ-limits which can be considered as a different approach to nonstandard methods. In the third part (section 3), we define axiomatically the space of ultrafunctions which are a kind of generalized function based on the field of Euclidean numbers E. Finally, we describe an application of the previus theory to the formalism of classical Quantum Mecanics.

math-ph↗
arXiv2018

Generalized solutions of variational problems and applications

Ultrafunctions are a particular class of generalized functions defined on a hyperreal field $\mathbb{R}^{*}\supset\mathbb{R}$ that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study the relationships between these generalized solutions and classical minimizing sequences. Finally, we study some examples to highlight the potential of this approach.

math.FA↗
arXiv2017

The Caccioppoli Ultrafunctions

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.

math.AP↗
arXiv2016

Existence of torsional solitons in a beam model of suspension bridge

This paper studies the existence of solitons, namely stable solitary waves, in a suspension bridge. The bridge is modeled as a degenerate plate, that is, a central beam with cross sections, and displays two degrees of freedom: the vertical displacement of the beam and the torsional angles of the cross sections. Under fairy general assumptions, we prove the existence of solitons. Under the additional assumption of large tension in the sustaining cables, we prove that these solitons have a nontrivial torsional component. This appears relevant for the security since several suspension bridges collapsed due to torsional oscillations.

math.AP↗
arXiv2016

Generalized solutions in PDE's and the Burgers' equation

In many situations, the notion of function is not sufficient and it needs to be extended. 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. In particular, we introduce the notion of Generalized Ultrafunction Solution (GUS) for a large family of PDE's, and we confront it with classical strong and weak solutions. Moreover, we prove an existence and uniqueness result of GUS's for a large family of PDE's, including the nonlinear Schroedinger equation and the nonlinear wave equation. Finally, we study in detail GUS's of Burgers' equation, proving that (in a precise sense) the GUS's of this equation provide a description of the phenomenon at microscopic level.

math.AP↗
arXiv2015

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 problems that have no classical solution. In this paper we introduce certain notions of non-Archimedean mathematics (in particular, 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 $\mathbb{R}$. 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 \cite{BDN2003}, and as an alternative topological presentation of the ultrapower construction of nonstandard extensions (in the sense of \cite{keisler}). 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.

math.LO↗
arXiv2015

Tempered ultrafunctions

Ultrafunctions are a particular class of functions defined on some non- Archimedean field. They provide generalized solutions to functional equa- tions which do not have any solutions among the real functions or the distributions. In this paper we introduce a new class of ultrafunctions, called tempered ultrafunctions, which are somewhat related to the tem- pered distributions and present some interesting peculiarities.

math.FA↗
arXiv2015

Towards a Morse theory on Banach spaces via ultrafunctions

Morse Theory on Banach spaces would be a useful tool in nonlinear analysis but its development is hindered by many technical problems. In this paper we present an approach based on a new notion of generalized functions called \textquotedblleft ultrafunctions\textquotedblright\ which solves some of the technical questions involved.

math.FA↗