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An Improved Setting for Generalized Functions: Fine Ultrafunctions

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An Improved Setting for Generalized Functions: Fine Ultrafunctions An improved setting for generalized functions: fine ultrafunctions Vieri Benci ∗ May 5, 2021 Abstract Ultrafunctions are a particular class of functions defined on a Non Archimedean field E ⊃ R. They have been introduced and studied in some previous works ([6],[8],[14],..,[22]). 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. Keywords. Partial Differential Equations, generalized functions, distri- butions, Non Archimedean Mathematics, Non Standard Analysis, ill posed problems. Mathematics Subject Classification (2020): 35A01, 03H05, 46T30 Contents 1 Introduction 2 1.1 FewremarksonNon-ArchimedeanMathematics . 4 1.2 Fewremarksongeneralizedfunction . 5 1.3 Notations ............................... 5 arXiv:2105.01490v1 [math.AP] 4 May 2021 2 Preliminary notions 7 2.1 NonArchimedeanFields....................... 8 2.2 Λ-theory ............................... 9 2.3 Extensionofsetsandfunctions . 12 2.4 Hyperfinitesets............................ 13 2.5 Gridfunctions ............................ 14 ∗Dipartimento di Matematica, Universit`adegli Studi di Pisa, Via F. Buonarroti 1/c, Pisa, ITALY 1 3 Ultrafunctions 15 3.1 Definitionofultrafunctions . 15 3.2 Definitionoffineultrafunctions . 18 3.3 The pointwise scalar product of ultrafunctions . 21 3.4 Regularandsmoothultrafunctions . 22 3.5 Time-dependentultrafunctions . 24 4 Basicpropertiesofultrafunctions 27 4.1 The pointwise integral of 1 ultrafunctions . 27 4.2 Ultrafunctionsandmeasures.L . 29 4.3 The vicinity ofaset ......................... 31 4.4 TheGauss’divergencetheorem . 33 4.5 Ultrafunctions and distributions . 35 5 Some applications 38 5.1 Secondorderequationsindivergenceform . 39 5.2 TheNeumannboundaryconditions. 44 5.3 Regularweaksolutions.. .. .. .. .. .. .. .. .. .. 45 5.4 Calculus of variations . 46 5.5 Evolutionproblems.......................... 48 5.6 Someexamplesofevolutionproblems. 50 5.7 Linearproblems............................ 55 6 A model for ultrafunctions 56 6.1 Hyperfinitestepfunctions . 57 6.2 σ-bases ................................ 60 1 6.3 The spaces WΛ and WΛ ....................... 62 6.4 The partitions η and η ..................... 63 W PN P◦ 6.5 The spaces VΛ(E ) and V (Γ) ................... 66 6.6 Thepointwiseintegral ........................ 68 6.7 Definition of the generalized derivative . 69 7 Conclusive remarks 71 1 Introduction 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 ultrafunctions are a kind of generalized functions based on a field E containing the field of real numbers R. The field E (called field of Euclidean numbers) is a peculiar hyperreal field which satisfies some properties useful for the purposes of this paper. The ultrafunctions provide generalized solutions to certain equations which do not have any solution, not even among the distributions. We list some of the main properties of the ultrafunctions: 2 • the ultrafunctions are defined on a set Γ, RN Γ EN , ⊂ ⊂ and take values in E; actually they form an algebra V ◦ over the field E; • to every function f : RN R corresponds a unique ultrafunction → f ◦ :Γ E → that extends f to Γ and satisfies suitable properties described below; • there exists a linear functional : V ◦ E → I called pointwise integral such that f C0 RN , ∀ ∈ c f ◦(x)dx = f(x)dx I Z • there are N operators D : V ◦ V ◦, i =1, ..., N i → called generalized partial derivatives such that f C1 RN , ∀ ∈ c ∂f ◦ = D f ◦ ∂x i i • to every distribution T ′ corresponds an ultrafunction T ◦ such that ϕ ∈ D ∀ ∈ D T ◦(x)ϕ◦dx = T, ϕ h i I and ∂T ◦ = D T ◦ ∂x i i • if u is the solution of a PDE, then u◦ is the solution of the same PDE ”translated” in the framework of ultrafunctions. • Γ is a hyperfinite set (see section 2.4) so that we have enough compactness to prove the existence of a solution for a very large class of equations which incluses many ill posed problems. The ultrafunctions have been recently introduced in [6] and developed in [8],[14],..,[22]. In these papers different models of ultrafuctions have been ana- lyzed and several applications have been provided. In this paper we introduce 3 an improved model: the space of fine ultrafunctions. The fine ultrafunctions form an algebra in which the pointwise integral and the generalized derivative satisfy most of the familiar properties that are consistent with the algebraic structure of V ◦. In particular, these properties allow to solve many evolution problem in the space C1(E, V ◦) (see sections 3.5 and 5.5). This paper is organized as follow: in the rest of this introduction, we frame the theory of ultrafunction and expose our point of view on Non-Archimedean Mathematics and on the notion of generalized functions. In section 2, we present the preliminary material necessary to the rest of the paper. In particular we present an approach to Non Standard Analysis (NSA) suitable for the theory of ultrafunctions. This approach is based on the notion of Λ-limit (see also [6] and [12]) which leads to the field of Euclidean numbers (see also [23]). This part has been written in such a way to be understood also by a reader who is not familiar with NSA. In section 3 we recall the notion of ultrafunction, we define the spaces of fine ultrafunctions and of time dependent ultrafunctions. The main properties of the fine ultrafunctions are examined in section 4. Section 5 is devoted to some applications that exemplify the use of ultra- functions in PDE’s. Section 6 is devoted to the proof that the definition of ultrafunctions is consistent. In fact, even if this definition is based on notions which appear quite natural, the consistency of these notions is a delicate issue. We prove this consistency by the construction of a very involved model; we do not know if a simpler model exists. This section is very technical and we assume the reader to be used with the techniques of NSA. 1.1 Few remarks on Non-Archimedean Mathematics The scientific community has always accepted new mathematical entities, espe- cially if these are useful in the modeling of natural phenomena and in solving the problems posed by the technique. Some of these entities are the infinitesi- mals that have been a carrier of the modern science since the discovery of the infinitesimal calculus at the end of XVII century. But despite the successes achieved with their employment, they have been opposed and even fought by a considerable part of the scientific community. The Jesuits in Italy and part of the Royal Society in England fought the spread of these ”subversive notions”. Sometimes it is said that people opposed infinitesimals because of their lack of rigor, but this argument convinces me little (see e.g. [3, 24, 7]). At the end of the 19th century they were placed on a more rigorous basis thanks to the works of Du Bois-Reymond [27], Veronese [41], Levi-Civita [32] and others, neverthe- less they were fought (and defeated) by the likes of Russell (see e.g. [34]) and Peano [35]. Also the reception of the Non Standard Analysis created in the ’60s by Robinson has not been as good as it deserved, even though a minority of 4 mathematicians of the highest level has elaborated interesting theories based on it (see e.g. [1], [33], [39]). Personally, I am convinced that the Non-Archimedean Mathematics is branch of mathematics very rich and allows to construct models of the real world in a more efficient way. Actually, this is the main motivation of this paper. 1.2 Few remarks on generalized function The intensive use of the Laplace transform in engineering led to the heuristic use of symbolic methods, called operational calculus. An influential book on operational calculus was Oliver Heaviside’s Electromagnetic Theory of 1899 [29]. This is the first time that generalized functions appeared. When the Lebesgue integral was introduced, the notion of generalized function became central to mathematics since the notion of function was replaced by something defined almost everywhere and not pointwise. During the late 1920s and 1930s further steps were taken, very important to future work. The Dirac delta function was boldly defined by Paul Dirac as a central aspect of his scientific formalism. Jean Leray and Sergei Sobolev, working in partial differential equations, defined the first adequate theory of generalized functions and generalized derivative in order to work with weak solutions of partial differential equations. Sobolev’s work was further developed in an extended form by Laurent Schwartz. Today, among people working in partial differential equations, the theory of distributions of L. Schwartz is the most commonly used, but also other notions of generalized functions have been introduced by J.F. Colombeau [26] and M. Sato [36]. After the discovery of Non Standard Analysis, many models of generalized functions based on hyperreal fields appeared. The existence of infinite and infinitesimal numbers allows to relate the delta of Dirac δ to a function which takes an infinite value in a neighborhhod of 0 and vanishes in the other points. 2 So, in this context, expression such as √δa or δa make absolutely sense. The literature in this context is quite large and, without the hope to be exaustve, we refer to the following papers and their references: Albeverio , Fenstad, Hoegh-Krohn [1], Nelson [33],
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