arXiv:2105.01490v1 [math.AP] 4 May 2021 rlmnr notions Preliminary 2 Introduction 1 Contents ITALY ∗ nipoe etn o eeaie functions: generalized for setting improved An . rdfntos...... 14 . . 13 9 ...... 12 ...... 8 ...... functions . . . . Grid . . . . . sets . 2.5 functions Hyperfinite . and . . sets . of . 2.4 Extension . . . 2.3 Λ-theory . Fields Archimedean 2.2 Non 2.1 . oain 5 ...... 5 ...... 4 ...... function Notations . generalized on remarks 1.3 Mathematics Few Non-Archimedean on remarks 1.2 Few 1.1 iatmnod aeaia nvri` el td iPis di Studi Universit`a degli Matematica, di Dipartimento uin,NnAcieenMteais o tnadAnalys problems. Standard Non Mathematics, Archimedean Non butions, Keywords poblems. evolution a eapidi tdn ata ieeta qain.In of existence Equations. the ultr Differential prove how to Partial show possible studing to is in it given applied are be applications can defini Some paper older the this points. improves which crucial In ultrafunctions fine of ([6],[8],[14],..,[22]). notion works previous some rhmda field Archimedean ahmtc ujc lsicto 22) 50,0H5 4 03H05, 35A01, (2020): Classification Subject Mathematics lrfntosaeapriua ls ffntosdfie o defined functions of class particular a are Ultrafunctions ata ieeta qain,gnrlzdfntos d functions, generalized Equations, Differential Partial . E n ultrafunctions fine ⊃ R hyhv enitoue n tde in studied and introduced been have They . a ,2021 5, May ir Benci Vieri Abstract 1 lrfnto solutions ultrafunction ∗ ,VaF unroi1c Pisa, 1/c, Buonarroti F. Via a, edvlpthe develop we in nmany in tions oilposed ill to s l posed ill is, particular, afunctions 6T30 Non a n istri- 7 2 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], Arkeryd, Cutland, Henson [2], Bottazzi [25], Todorov [40].

1.3 Notations For the sake of the reader, we list the main notation used in this paper. If X is any set and Ω is a measurable subset of RN , then

• ℘ (X) denotes the power set of X and ℘fin (X) denotes the family of finite subsets of X; • X will denote the cardinality of X; | | • B (x )= x RN x x < r ; r 0 ∈ | | − 0| 

5 • N (Ω) = x RN dist(x, Ω) <ε ; ε ∈ | • int(Ω) denotes the interior part of Ω and Ω denotes the closure of Ω; • F (X, Y ) denotes the set of all functions from X to Y and F (Ω) = F (Ω, R);

• if W is any function space, then Wc will denote de function space of functions in W having compact support; • if W F RN is any function space, then W (Ω) will denote the space of their⊂ restriction to Ω.
• if W is a generic function space, its topological dual will be denoted by W ′ and the pairing by , ; h· ·iW • 0 C0 Ω denotes the set of continuous functions defined on Ω which vanish on ∂Ω;
• Ck (Ω) denotes the set of functions defined on Ω which have continuous derivatives up to the order k; • p p p (Ω) ( loc (Ω)) denotes the set of the functions u such that u is inte- grableL (locallyL integrable) functions in Ω; | | • 1 1 L (Ω) (Lloc (Ω)) denotes the usual equivalence classes of integrable (lo- cally integrable) functions in Ω; these classes will be denoted by:

1 [f] 1 = g (Ω) g(x)= f(x) a.e. ; L ∈ Lloc |  • Hk,p (Ω) denotes the usual Sobolev space of functions defined on Ω; • D (Ω) denotes the set of the infinitely differentiable functions with compact support defined on Ω; D′ (Ω) denotes the topological dual of D (Ω), namely the set of distributions on Ω; • if Ω = RN , when no ambiguity is possible, we will write D, D′, L1, 1, ... instead of D RN , D′ RN , L1 RN , 1 RN , ... L L • if X is any topological
vector
space
, X will denote
the duality between X and its topological dual X′; h· ·i • E will denote the field of Euclidean number which will be defined in section 2.2; • for any ξ EN ,ρ E, we set B (ξ)= x EN : x ξ <ρ ; ∈ ∈ ρ ∈ | − | • supp(f) denotes the usual notion of support of a function or a distribution in RN ; • supp∗(f) denotes the usual notion of support of an internal function or a distribution in EN ;

6 • supp(f)= x Γ: f(x) =0 denotes the support of a grid funcion { ∈ 6 } • mon(x)= y EN : x y where x y means that x y is infinitesimal; { ∈ ∼ } ∼ − • gal(x)= y EN : x y is finite ; { ∈ − } • we denote by χX the indicator (or characteristic) function of X, namely

1 if x X χ (x)= ∈ X 0 if x/ X ; ( ∈ If X = a and a is an atom, then, in order to simplify the notation, we { } will write χa(x) instead of χ{a}(x). • = (∂ , ..., ∂ ) denotes the usual gradient of standard functions; ∇ 1 N • D = (D1, ..., DN ) will denote the extension of the gradient in the sense of the ultrafunctions;

N • φ will denote the usual divergence of standard vector fields φ C1 ; ∇ · ∈ • D φ will denote the extension of the divergence in the sense of ultrafun  c- tions;· • ∆ denotes the usual Laplace operator of standard functions; • D2 will denote the extension of the Laplace operator in the sense of ultra- functions.

∞ • ρ(x) will denote a C -bell function having support in B1(0) and such that ρ(x)dx =1, for example Z 1 1− 2 e 1−|x| ρ(x)= 1 χB1(0)(x) (1) 1− 2 · e 1−|y| dy Z B1(0) 2 Preliminary notions

As we have already remarked in the introduction, in this section, we present the material necessary to the rest of the paper. In particular we present an approach to NSA based on Λ-theory. This part has been written in such a way to be understood also by a reader who is not familiar with NSA. Λ-theory can be considered a different approach to Nonstandard Analysis. It can be introduced via the notion of Λ-limit, and it can be easily used for the purposes of this paper.

7 2.1 Non Archimedean Fields Here, we recall the basic definitions and some facts regarding Non-Archimedean fields. In the following, K will denote an ordered field. We recall that such a field contains (a copy of) the rational numbers. Its elements will be called numbers. Definition 2.1. Let K be an ordered field. Let ξ K. We say that: ∈ • ξ is infinitesimal if, for all positive n N, ξ < 1 ; ∈ | | n • ξ is finite if there exists n N such that ξ n (equivalently, if ξ is not finite). ∈ | | Definition 2.2. An ordered field K is called Non-Archimedean if it contains an infinite number. It’s easily seen that all infinitesimal are finite, that the inverse of an infinite number is a nonzero infinitesimal number. Infinitesimal numbers can be used to formalize a new notion of ”closeness”: Definition 2.3. We say that two numbers ξ, ζ K are infinitely close if ξ ζ is infinitesimal. In this case we write ξ ζ. ∈ − ∼ Clearly, the relation ” ” of infinite closeness is an equivalence relation. ∼ Theorem 2.4. If K R,(K = R) is an ordered field, then it is Non-Archimedean and every finite number⊃ ξ K6 is infinitely close to a unique real number r ξ, called the the standard part∈ of ξ. ∼ Given a finite number ξ, we denote it standard part by st(ξ). Definition 2.5. Let K be a superreal field, and ξ K a number. The monad of ξ is the set of all numbers that are infinitely close∈ to it:

mon(ξ)= ζ K : ξ ζ , { ∈ ∼ } and the galaxy of ξ is the set of all numbers that are finitely close to it:

gal(ξ)= ζ K : ξ ζ is finite . { ∈ − } By definition, it follows that the set of infinitesimal numbers is mon(0) and that the set of finite numbers is gal(0). Moreover, the standard part can be regarded as a function: st : gal(0) R. (2) → Moreover, with some abuse of notation, we will set

+ if ξ is a positive infinite number; st (ξ)= ∞ if ξ is a negative infinite number.  −∞

8 2.2 Λ-theory In order to construct a space of ultrafunctions it is useful to take the set Λ sufficiently large; for example a superstructure over R defined as follows:

Λ= V∞(R)= Vn(R), N n[∈ where the sets Vn(R) are defined by induction:

V0(R)= R and, for every n N, ∈ V (R)= V (R) ℘ (V (R)) . (3) n+1 n ∪ n Identifying the couples with the Kuratowski pairs and the functions and the relations with their graphs, it follows that V∞(R) contains almost every math- ematical entities used in PDE’s. A function ϕ : L E will be called net (with values in E). → Let L = ℘fin (Λ) be the family of finite subsets of Λ. L equipped with the partial order structure ” ” is a directed set; hence if ϕ is a net with value in a topological space the notion⊂ of limit is well defined: for example if ϕ : L R, we set → L = lim ϕ(λ) (4) λ→Λ if and only if, ε R+, λ L, such that λ λ , ∀ ∈ ∃ 0 ∈ ∀ ⊃ 0 ϕ(λ) L ε | − | ≤ Notice that in the notation (4), Λ can be regarded as the ”point at infinity” of L. A typical example of a limit of a net defined on L is provided by the definition of the Cauchy integral:

b f(x)dx = lim f(x)(x+ x); x+ = min y R λ y > x . a λ→Λ − { ∈ ∩ | } Z x∈[Xa,b]∩λ Now we will introduce axiomatically a new notion of limit: Axiom 2.6. There is a field E R, called field of Euclidean numbers, such that every net ⊃ ϕ : L V (R), n N, → n ∈ has a unique Λ-limit lim ϕ(λ) Vn(E) λ↑Λ ∈ which satisfies the following properties:

9 1. if eventually ϕ(λ)= ψ(λ),1 then

lim ϕ(λ) = lim ψ(λ); λ↑Λ λ↑Λ

2. if ϕ1(λ), ..., ϕn(λ) are nets, then

lim ϕ1(λ), ..., ϕn(λ) = lim ϕ1(λ), ..., lim ϕn(λ) ; λ↑Λ { } λ↑Λ λ↑Λ  

3. if Eλ is a net of sets, then

lim Eλ = lim ϕ(λ) λ L, ϕ(λ) Eλ ; λ↑Λ λ↑Λ | ∀ ∈ ∈   4. we have that

E := lim xλ λ L, xλ R (5) λ↑Λ | ∀ ∈ ∈   and if x ,y R, then, λ λ ∈

lim (xλ + yλ) = lim xλ + lim yλ, λ↑Λ λ↑Λ λ↑Λ

lim (xλ yλ) = lim xλ lim yλ; λ↑Λ · λ↑Λ · λ↑Λ

xλ yλ lim xλ lim yλ. (6) ≥ ⇒ λ↑Λ ≥ λ↑Λ

Notice that in order to distinguish the limit (2.7) (which we will call Cauchy limit) from the Λ-limit, we have used the symbols ”λ Λ” and ”λ Λ” respectively. → ↑ In the rest of this section, we will make some remarks for the readers who are not familiar with NSA. The points 1,2,4 are not surprising since we expect to be satisfied by any notion of limit provided the target space be equipped with a reasonable topology. The point 3 can be considered as a definition. In axiom 2.6, the new (and, for someone, surprising) fact is that every net has a Λ-limit. Nevertheless Axiom 2.6 is not contradictory and a model for it can be constructed in ZFC (see e.g. [6] or [23]). Probably the first question which a newcomer to the world of NSA would ask is the following: what is the limit of a sequence such that ϕ(λ) := ( 1)|λ| since it takes the values +1 if λ is even or 1 if λ is odd. Let us see− what Axiom 2.6 tells us. By 2.6.2 and| | 2.6.1, − | |

lim 1, 1 = lim 1, lim ( 1) = 1, 1 λ↑Λ { − } λ↑Λ λ↑Λ − { − }   1 We say that a relation ϕ(λ)Rψ(λ) eventually holds if ∃λ0 ∈ L such that ∀λ ⊃ λ0, ϕ(λ)Rψ(λ).

10 and by 2.6.3, we have that lim( 1)|λ| lim 1, 1 ; λ↑Λ − ∈ λ↑Λ { − } and hence lim( 1)|λ| 1, 1 λ↑Λ − ∈{ − } |λ| |λ| Then either limλ↑Λ( 1) = 1 or limλ↑Λ( 1) = 1. Which alternative occurs cannot be deduced by− axiom 2.6; each alternative− − can be added as an indepen- dent axiom. In the models constructed in [23] this limit is +1; however this and similar questions are not relevant for this paper and we refer to the mentioned references for a deeper discussion of this point (in particular see [12] and [23]). Axiom 2.6 is sufficient for our applications. The second question which a newcomer would ask is about the limit of a divergent sequence such that ϕ(λ) := λ N . Let us put | ∩ | α := lim λ N (7) λ↑Λ | ∩ | What can we say about α? By (6), α / R. In order to give a feeling of the ”meaning” of α, we will put it in relation∈ with other infinite numbers. If E Λ R, we put ∈ \ num (E) = lim E λ . (8) λ↑Λ | ∩ | If E is a finite set, the sequence is eventually equal to the number of elements of E; then, by axiom 2.6.1, num (E)= E N. | | ∈ If E is an infinite set, num (E) / N. Hence, the limits as (8) give mathematical entities that extends the notion∈ of ”number of elements of a set” to infinite sets and it is legitimate to call them ”infinite numbers”. The infinite number num (E) is called numerosity of E. The theory of numerosities can be considered as an extension of the Cantorian theory of cardinal and ordinal numbers. The reader interested to the details and the developments of this theory is referred to [4, 11, 13, 23, 9]. If a real net xλ admits the Cauchy limit, the relation between the two limits is given by the following identity:

lim xλ = st lim xλ (9) λ→Λ λ↑Λ   An other important relation between the two limits is the following: Proposition 2.7. If lim xλ = ξ E λ↑Λ ∈ and ξ is bounded, then there exist a sequence λ L such that n ∈ lim xλ = st(ξ). n→∞

11 Proof: Set x = st(ξ) and for every n N, take λ such that x 0 ∈ n λn ∈ B1/n(x). 

Remark 2.8. As we have already remarked, the field of Euclidean numbers is a hyperreal field in the sense of Non Standard Analysis. We do not use the name ”hyperreal numers” to emphasize the fact that E has been defined by the notion of Λ-limit and hence it satisfies some properties which are not shared by other hyperreal fields. These properties are relevant in the definitions of ultrafunctions. The explanation of the choice of the name ”Euclidean numbers” can be found in [9].

2.3 Extension of sets and functions In this section we recall some basic notions of Non Standard Analysis presented in the framework of Λ-theory. Given a set A Λ, we define ∈ A∗ = lim ϕ(λ) λ, ϕ(λ) A ; (10) λ↑Λ | ∀ ∈   Following Keisler [30], A∗ will be called the natural extension of A. By (5), we have that R∗ = E. If we identify a relation R or a function f with its graph, then, by (10) R∗ and f ∗ are well defined. In particular any function

f : A B, A,B Λ, → ∈ can be extended to A∗ and we have that

∗ f lim xλ = lim f (xλ) ; (11) λ↑Λ λ↑Λ   the function f ∗ : A∗ B∗, → will be called natural extension of f. More in general, if

u : A B λ → is a net of functions, we have that for any x = lim , x B, λ↑Λ λ ∈

u(x) = lim uλ (xλ) (12) λ↑Λ

Example: Let f : RN R be a differentiable function, then → ∗ ∗ ∗ ∂i f = (∂if) where the operator ∂ 1 N 0 ∂i = : C R C (R) ∂xi →
12 is regarded as a function between functional spaces and hence

∗ ∂∗ : C1 RN C0 (R)∗ . i → Following the current literature in
NSA, we give the following definition:

Definition 2.9. A set EΛ obtained as Λ-limit of a net of sets Eλ Λ is called internal. ∈ In particular, if E Λ, if you compare Axiom 2.6.3 with (10), then you see that ∈ ∗ E = lim Eλ λ↑Λ in the case in which Eλ is the net identically equal to E. Let us see an example of external set i.e. of a set which is not internal. By axiom 2.6.3, the set E∗ contains a unique ”copy” x∗ of every element x E. Now set ∈ Eσ := x∗ E∗ x E . (13) { ∈ | ∈ } We have that Eσ E∗ and the equality holds if and only if E is finite. It is easy to see that if ⊆E is infinite E and Eσ are external.

Example: Let E = C0 (R) ; then C0 (R)σ C0 (R)∗ . For example, if we take ⊂ ∗ sin (x) = lim sin (xλ) , x = lim xλ λ↑Λ λ↑Λ and ∗ sin (αx) = lim sin( λ N xλ), λ↑Λ | ∩ | · we have that sin∗ (x) C0 (R)σ C0 (R)∗ and sin∗ (αx) C0 (R)∗ C0 (R)σ . ∈ ⊂ ∈ \ 2.4 Hyperfinite sets An other fundamental notion in NSA is the following:

Definition 2.10. We say that a set F Λ is hyperfinite if there is a net F of finite sets such that ∈ { λ}λ∈Λ

F = lim Fλ = lim xλ xλ Fλ λ↑Λ λ↑Λ | ∈   The hyperfinite sets share many properties of finite sets. For example, a hyperfinite set F E has a maximum xM and a minimum xm respectively given by ⊂ xM = lim max Fλ; xm = lim min Fλ λ↑Λ λ↑Λ

13 Moreover, it is possible to ”add” the elements of an hyperfinite set of num- bers. If F is an hyperfinite set of numbers, the hyperfinite sum of the elements of F is defined as follows:

x = lim x. λ↑Λ xX∈F xX∈Fλ One of the advantage to use the field of Euclidean numbers rather than a generic hyperreal field lies in the possibility to associate a unique hyperfinite set E⊛ to any set E V (R) according to the following definition: ∈ ∞ Definition 2.11. Given a set E Λ, the set ⊂ E⊛ := lim (E λ) λ↑Λ ∩ is called hyperfinite extension of E.

If F = limλ↑Λ Fλ is a hyperfinite set, its hypercardinality is defined by

F := lim Fλ | | λ↑Λ | | To any set E we can associate the sets Eσ, E⊛ and E∗ which are ordered as follows: Eσ E⊛ E∗; ⊆ ⊆ the inclusions are strict if and only if the set E is infinite. Notice that the hypercardinality of E⊛, defined by

E⊛ = lim E λ , λ↑Λ | ∩ | is the numerosity of E as it has been defined by (8).

2.5 Grid functions Definition 2.12. A hyperfinite set Γ such that RN Γ EN is called hyper- finite grid. ⊂ ⊂ ⊛ For example the set RN is an hyperfinite grid. Definition 2.13. A space of
grid functions is a family G(Γ) of internal func- tions u :Γ R → If w F(RN )∗, the restriction of w to Γ is a grid function which we will denote by∈ w◦ namely, if w = lim w and x = lim x Γ, we have that λ↑Λ λ λ↑Λ λ ∈ ◦ w (x) = lim wλ(xλ). (14) λ↑Λ For every a Γ, ∈ χ (x) G(Γ) a ∈

14 is a grid function, and hence every grid function can be represented by the following sum:

u(x)= u(a)χa(x) (15) aX∈Γ E namely χa a∈Γ is a basis for G(Γ) considered as a vector space over . Given f F(R{N ),}we will write f ◦ instead of (f ∗)◦, namely ∈ ◦ ∗ f (x) := lim f(xλ)= f (a)χa(x). (16) λ↑Λ aX∈Γ so, G(Γ) contains a unique copy f ◦ of every function f F(RN ). If a function, such as 1/ x is not defined in some point and x Γ, we∈ put (1/ x )◦ equal to 0 for x = 0;| in| general, if Ω is a subset of RN and ∈f is defined in Ω,| | we set

◦ ∗ f (x)= f (a)χa(x) (17) ◦ aX∈Ω where for every set E RN , we define ⊂ E◦ = E∗ Γ. (18) ∩ 3 Ultrafunctions

If we have a differential equation, it is relatively easy to find an approximated solution in a suitable space of grid functions. If this equation has a ”classic” solution, this solution, in some sense, approximates the classic solution. Then, if we take a grid, the ”grid solutions” is almost equal to the classic solution. How- ever the ”grid solutions” cannot be considered as generalizations of the classic solutions since they do not coincide with them. The theory of ultrafunctions is based on the idea of a space of functions (defined on a hyperfinite grid) in which the generalized derivative and the generalized integral coincide with the usual 1 0 ones for every function f in C and in Cc respectively. This fact implies that a ”ultrafunction solution” coincides with the classical one if the latter exists and hence it is legitimate to be considered a generalized solution.

3.1 Definition of ultrafunctions Let V = V RN be a function space such that
D RN V 1 RN ⊂ ⊂ Lloc and let Vλ L be a net of finite
dimensional subspaces
of V such that { }λ∈

Vλ = V. L λ[∈

15 Now we set

N VΛ = VΛ E = lim Vλ = lim uλ uλ Vλ ; λ↑Λ λ↑Λ | ∈  
V is an internal vector space of hyperfinite dimension. Clearly V V ∗ since Λ λ ⊂

∗ V = lim uλ uλ V . λ↑Λ | ∈  

The space VΛ allows to equip a space grid functions G(Γ) of a richer structure:

◦ Definition 3.1. A space of ultrafunctions V (Γ) modelled on VΛ is a family of grid functions G(Γ) such that the restriction map

◦ : V V ◦(Γ) (19) Λ → is an internal isomorphism between hyperfinite dimensional vector spaces. So, u V ◦(Γ) if and only if ∈ ◦ u = lim uλ with uλ Vλ. λ↑Λ ∈   By (14), the above equation can be reformulated as follows: u V ◦(Γ) if and only if, x = lim x Γ, there exists a net u V RN such∈ that ∀ λ↑Λ λ ∈ λ ∈ λ
u(x) = lim uλ(xλ). λ↑Λ

◦ In the following of this paper, if u V (Γ), such a net will be denoted by uλ. ∈ N We denote by σa(x) the only function in VΛ E such that

◦
σa = χa (20) EN Clearly σa(x) a∈Γ is a basis of VΛ which will be called σ-basis. The {σ ’s allow} to write the inverse of the map (19) a
( ) : V ◦(Γ) V EN (21) · Λ → Λ as follows: if u V ◦(Γ),
∈

uΛ(x) := u(a)σa(x). aX∈Γ N If f F E , in order to simplify the notation, we will write fΛ instead of (f ◦) ∈, namely we have that Λ
∗ fΛ(x)= f (a)σa(x). aX∈Γ

16 Notice that f = f ∗ f V RN (22) Λ ⇔ ∈ ∗ More in general, if w F EN , with some abuse
of notation, we will write ∈
wΛ(x)= w(a)σa(x), (23) aX∈Γ In this case, the map ∗ ( ) : F RN V EN (24) · Λ → Λ is just a projection.

∗ If u V ◦ (Γ) and u 1 RN , the integral can be defined as follows: ∈ Λ ∈ L ∗
u(x)dx := uΛ(x)dx = lim uλ(x)dx; (25) λ↑Λ I Z Z moreover, we define the integral of u over a set Ω◦ := Ω∗ Γ as follows: ∩ ∗ u(x)dx := uΛ(x)dx = lim uλdx (26) Ω∗ λ↑Λ Ω IΩ◦ Z Z provided that Ω be measurable. We will refer to

: V ◦ (Γ) E → I as to the pointwise integral. The reason of this name is due to the fact that (15) and (25) imply that

u(x)dx = u(a)d(a) (27) I aX∈Γ where ∗ d(a) := χa(x)dx = σa(x)dx. (28) I Z We may think of d(a) as of the ”measure” of a the point a Γ. The pointwise integral extends the usual Lebesgue integral from V to V∈◦, more exactly, if f V 1, then ∈ ∩ L f ◦(x)dx = f(x)dx (29) I Z However the equality above is not true for every Lebesgue integrable function. In fact, if a RN , ∈ χa(x)dx =0 Z but, by (27), we have that ◦ χa(x)dx > 0 I

17 at least for some a RN . This fact is quite natural, in fact when we work in a non-Archimedean∈ world infinitesimals matter and cannot be forgotten as the Riemann and the Lebesgue integrals do. Also the above inequality shows that it is necessary to use a different symbol to distinguish the pointwise integral from the Lebesgue integral (here we have used ). ◦ 1 I Given u V , if uλ C Vλ, it is natural to define the partial derivative in a point x∈= lim x∈ Γ,∩ as follows λ↑Λ λ ∈ ∗ ◦ Diu(x) := [∂i uΛ(x)] = lim ∂iuλ(xλ). (30) λ↑Λ

So the derivative of an ultrafunction is the restriction to Γ of the natural exten- sion of the usual derivative. 1 In particular, if we choose V = Cc , we have that the integral and the deriva- tive are defined for every ultrafunction in V ◦.

3.2 Definition of fine ultrafunctions There are many ultrafunctions spaces which depend on the choice of the space V, the net Vλ and the grid Γ. However there are some basic properties which should be{ satisfied} by ”good” ultrafunctions which make the theory rich and flexible. Such ultrafunctions will be called fine. Roughly speaking a space of ultrafunctions is fine if many of the properties of standard functions are satisfied. Definition 3.2. A space of ultrafunctions V ◦ is called fine if

0 1. V = C0 and Vλ λ∈Λ is a net of functions which satisfies the following property: if we{ set}

U 0 = span u, w, ∂ w, ..., ∂ w u C0 λ; w C1 λ (31) λ 1 N | ∈ c ∩ ∈ c ∩ then  u, v U 0 uv V , ∈ λ ⇒ ∈ λ 2. there is a linear internal functional

: V ◦ E → I called pointwise integral which satisfies the following properties:

(a) if u = lim u , u V , then λ↑Λ λ λ ∈ λ

u(x)dx = lim uλ(x)dx. (32) λ↑Λ I Z (b) for every a Γ, ∈ χa(x)dx > 0; (33) I

18 3. there are N internal operators

D : V ◦ V ◦, i =1, ..., N i → called generalized partial derivatives such that the following properties are fulfilled:

(a) if u = lim u , u C1 λ, then, λ↑Λ λ λ ∈ c ∩ ◦ ∗ ◦ Diu = (∂i uΛ) = lim ∂iuλ ; λ↑Λ   (b) for every u, v V ◦, ∈

D u(x)v(x)dx = u(x)D v(x)dx; (34) i − i I I (c) for every u V ◦, ∈ D u =0 u =0; (35) i ⇔ (d) for every a Γ, ∈ supp [D χ ] mon(a). (36) i a ⊂

Now some comments on Def. 3.2. Assumptions (3a) specifies the choice of the space V and states that Vλ is 0 1 sufficiently large with respect to Cc λ and Cc λ. This is a technical assumption which simplifies the definition of regular∩ ultrafunctions∩ (see section 3.4). Assumption (2a) is nothing else but the definition of the pointwise integral. By (2b), we have that f V, ∀ ∈

f ◦(x)dx = f(x)dx. I Z Assumption (2b) is quite natural, but it is not satisfied by every space of ul- trafunctions; it is very important and, among other things, it implies that the bilinear form (u, v) uv dx defines a scalar product (see section 3.3). 7→ By Property (3a), forI every function f C1 and every x RN , ∈ c ∈ ◦ Dif (x)= ∂if(x); (37) hence the generalized derivative extends the usual derivative and by (16) we have that, x Γ, ∀ ∈ ◦ ∗ ∗ Dif (x)= ∂i f (a)χa(x). aX∈Γ

19 Property (3b) states a formula which is of primary importance in the theory of weak derivatives, distributions, calculus of variations etc. Usually this formula is deduced by the Leibniz rule

D(fg)= Dfg + fDg

However, it is inconsistent to assume that Leibnitz rule be satisfied by every pairs of ultrafunction (see (44) and the discussion at the beginning of Section 3.4). Nevertheless the identity (34) holds for the fine ultrafunctions. In particular, by (34) and (3a), we have that u V ◦, ϕ C1, ∀ ∈ ∀ ∈ c

D uϕ◦dx = uD ϕ◦dx = u (∂ ϕ) ◦dx; i − i − · i I I I this equality relates the generalized derivative to the notion of weak derivative. Property (3c) is expected, since uΛ has compact support. This property holds for all the ultrafunctions in V ◦ and since V ◦ is a vector space of hyperfinite dimension, the fact that ker (Di)= 0 implies that Di is an invertible operator. Also property (36) is a natural{ request.} By virtue of this property the equality (37) holds for every f C1 and not only for functions in C1. ∈ c While the construction of a generic space of ultrafunction is a relatively easy task, the construction of a space of fine ultrafunctions is much more delicate. We have the following theorem which is one of the main results of this paper: Theorem 3.3. The requests of Def. 3.2 are consistent. In particular, there exists a space of fine ultrafuctions in which the generalized derivative D can be approximated as follows: u, v V ◦ ∀ ∈ ∗ uΛ(x + ηei) uΛ(x ηei) D uv dx − − v (x)dx ε u 2 v 2 , i − 2η Λ ≤ k ΛkL ·k ΛkL I Z  

where η and ε are suitable infinitesimal independent of u and v and e1, ..., eN is the canonical basis in RN . { } The proof of this theorem is rather involved and it will be given in section 6 (see Th. 6.21). The rest of this section and section 4 will be devoted in showing that a fine space ultrafunctions provides a quite rich structure and many interesting and natural properties can be proved in a relative simple way. Remark 3.4. A space of ultrafunctions cannot be uniquely defined, since its existence depends on E and hence on Zorn’s Lemma. However, if we exclude the choice of the space V , the properties required in Def. 3.2 are quite natural; hence, if a physical phenomenon is modelled by ultrafunctions, the properties which can be deduced can be considered reliable. From now on, we will treat only with fine ultrafunctions and the word ”fine” will be usually omitted.

20 3.3 The pointwise scalar product of ultrafunctions By (33), we have that d(a) > 0 and hence, the pointwise integral allows to defines the following scalar product which we will call pointwise scalar product:

u(x)v(x)dx = u(x)v(x)d(x). (38) I xX∈Γ If f,g V, we have that ∈ ∗ f ◦g◦dx = f ∗g∗dx = fg dx; (39) I Z Z ◦ however we must be careful since, for some u, v V such that uΛvΛ / VΛ, we might have that ∈ ∈ ∗ uv dx = u v dx. (40) 6 Λ Λ I Z even if these to quantity are not too different. In any case, setting

0 0 U := lim Uλ, (41) λ↑Λ

0 where Uλ is given by (31), ∗ 0 u, v U uv dx = lim uλvλdx = uΛvΛdx. ∈ ⇒ λ↑Λ I Z Z The pointwise product provides the pointwise (Euclidean) norm of an ultrafunction:

1 2 1 2 u = u(a) 2d(a) = u(x) 2dx . k k | | ! | | aX∈Γ I  Also, we can define other norms which might be useful in the applications: for p [1, ) , we set ∈ ∞ 1 1 p p p p u p = u(a) d(a) = u(x) dx k k | | ! | | aX∈Γ I  and, for p = , obviously, we set ∞ u = max u(a) . k k∞ a∈Γ | | Notice that all these norms are equivalent in the sense that given any two norms u and u , there exists two numbers m and M such that u V ◦, k kp k kq ∀ ∈ u m k kp M (42) ≤ u ≤ k kq

21 Of course, if p = q, m is an infinitesimal number and M is an infinite number. The scalar6 product (38) also allows to define the delta (or the Dirac) ultrafunction: for every a Γ, ∈ χ (x) δ (x)= a . (43) a d(a)

As it is natural to expect, for every u V ◦, we have that ∈ χ (x) δ (x)u(x)dx = u(x)δ (x)d(x)= u(x) a d(x)= u(a) a a d(a) I xX∈Γ xX∈Γ The delta ultrafunctions are orthogonal with each other with respect to the pointwise scalar product; hence, if normalized, they provide an orthonormal basis, called delta-basis, given by

χa δa = a∈Γ ( d(a)) np o a∈Γ So, the identity (15) can be rewritten as follows:p

u(x)= u(ξ) δa(ξ)dξ δa(x). a∈Γ   X I p p 3.4 Regular and smooth ultrafunctions As we already remarked the Leibniz rule does not hold for ultrafunctions and it is not possible to define a generalized derivative which has this property. It is easy to check this fact for idempotent functions, in fact by the Leibniz rule, we should have 2 DχE =2χEDχE 2 and since χE = χE, we deduce that

DχE =2χEDχE and hence, for every x E, ∈ DχE = 0 (44) and this fact contradicts any reasonable generalization of the notion of deriva- tive. Actually the Schwartz impossibility theorem states that the Leibniz rule cannot be satisfied by any algebra which contains, not only the idempotent functions, but also the continuous functions (see [38],[17]). So it is interesting to investigate the subspaces of ultrafunctions for which the Leibniz rule holds and more in general to determine spaces in which many of the usual properties of smooth functions be satisfied. This is important for the applications when we want to study the qualitative properties (and in particular the regularity) of the solutions of an equation.

22 Definition 3.5. For m> 1, we define by induction

U m := u U m−1 i, D u U m−1 ∈ | ∀ i ∈ (we recall that U 0 has been defined by (41) and (31)). If u U m, we will say that u is m-regular. ∈ Let us see the main properties of the spaces U m. Theorem 3.6. The spaces of m-regular ultrafunctions satisfy the following properties:

1. if u U m, m 1, then ∈ ≥ ∗ ◦ Diu = (∂i uΛ) ; (45) 2. if f Cm, then, m 1, f ◦ U m; ∈ c ∀ ≥ ∈ 3. if u, v U 0, ∈ (uv)Λ = uΛvΛ; 4. if u,v,uv U 1 then the Leibniz rule holds: ∈

Di (uv)= Diuv + uDiv.

m 1 1 Proof: 1-If u U U , then uλ Cc λ and the conclusion follows from Def. 3.2.3a. ∈ ⊂ ∈ ∩ 1 2 - We prove (45) by induction. For m = 1, f Cc , then by Def. 3.2.3a, ◦ ∈ m−1 Dif = (∂if) and (45) is satisfied. For m 2, since ∂if Cc , by (45), ◦ m−1 ◦ ◦ ≥ ◦ m∈−1 ◦ m (∂if) U and by (1) (∂if) = Dif ; hence Dif U . So f U . 3 - If∈ u, v U 0, then u, v U 0, and so, by Def. 3.2.1,∈ u v V . Hence,∈ ∈ ∈ λ λ λ ∈ λ

(uv)Λ = lim uλvλ = lim uλ lim vλ = uΛvΛ λ↑Λ λ↑Λ · λ↑Λ

4 - If u, v U 1, by (31), u, v U 0 and so, using the point 3, we have that ∈ ∈ ∗ ◦ ∗ ◦ ∗ ∗ ◦ Di (uv) = [∂i (uv)Λ] = [∂i (uΛvΛ)] = [∂i uΛvΛ + uΛ∂i vΛ] ∗ ◦ ∗ ◦ = [∂i uΛ] v + u [∂i vΛ] = Diuv + uDiv 

Since the U m’s have hyperfinite dimension then there exist finite dimensional m spaces Uλ ’s such that m m U = lim Uλ ; λ↑Λ the space of smooth ultrafunctions (or -regular ultrafunctions) is defined as follows: ∞

∞ m m U := U = lim Uλ . (46) λ↑Λ  N  m\≤α m≤|\λ∩ |   23 where α has been defined by (7). Clearly U ∞ = ∅, since 6 ϕ ϕ◦ U ∞. (47) ∈ D ⇒ ∈ For m N , we set ∈ ∪ {∞}

(U m)⊥ = u V ◦ ψ U m, uψ dx =0 ∈ | ∀ ∈  I  ⊥ and we denote by Πmu and Πmu the relative ”orthogonal” projection of u on U m and (U m)⊥ . Then every ultrafunction u can be split as follows

⊥ u = Πmu + Πmu (48)

Definition 3.7. Given the splitting (48), m N , Πmu will be called ⊥ ∀ ∈ ∪ {∞} the m-reguar part of u and Πmu the m-singular part of u. Remark 3.8. It is possible to define different types of regular ultrafunction namely we can choose different subspaces of V ◦ that satisfy suitable conditions. For example, we can set m m CΛ = lim C0 λ λ↑Λ ∩ m m m We have that CΛ U and hence the functions in CΛ satisfy less properties. Similarly, we can choose⊂ more regular spaces such as

U m,p := u U m v U m, u p−2 u U 0 ; p 2. (49) ∈ | ∀ ∈ | | ∈ ≥ n o Of course the choice of a particular space depends on the problems that we would like to treat. We can make an analogy with the theory of distributions; in this case the spaces Cm’s and the Sobolev spaces Hm,p’s can be considered as subspaces of ′ which present different kinds of regularity. D 3.5 Time-dependent ultrafunctions In evolution problems the time variable plays a different role that the space vari- k 1 ables; then the functional spaces used in these problems (e.g. C ([0,T ] ,H0 (Ω)), p R k Lloc( ,H (Ω)) etc.) reflect this fact. The same is true in the frame of ultrafunc- tion. This section is devoted in the description of the appropriate ultrafunction- spaces for evolution problems. First of all we need to recall some well known facts about free modules: Definition 3.9. Given a ring R and a module M over R, the set B M is a basis for M if: ⊂

• B is a generating set for M; that is to say, every element of M is a finite sum of elements of B multiplied by coefficients in R; • B is linearly independent.

24 Definition 3.10. A free module is a module with a basis. The following is a well known theorem: Theorem 3.11. If R is a commutative ring and M is a free R-module, then all the bases of E have the same cardinality. The cardinality of a basis is called rank of M. We will describe some free modules which will be used in the following part of this paper.

Examples: (i) If Γ is a finite set then Ck(R)Γ is a free module over Ck(R) of rank Γ and a basis is given by | | χ { a}a∈Γ If u Ck(R)Γ, then ∈ u(t, x)= c(t)χa(x) aX∈Γ Then we may think of u as a real function defined in R Γ or a function in × Ck(R, Γ) or in Ck(R) R F (Γ) . (ii) Let W F RN⊗ be a vector space of finite dimension, then ⊂ k
k N+1 C (R, W ) := C (R) R W F R ⊗ ⊂ is a free Ck-module of rank equal to dim W , namely every function
f Ck(R, W ) can be written as follows: ∈

dim W f(t, x) := ck(t)ek(x) kX=1 k where ck C (R) and ek is any basis in W . ∈ { } N ∗ (iii) If W = limλ↑Λ Wλ F R is an internal vector space of hyperfinite dimension, then by Ck(E, W⊂) we denote the internal Ck(R)∗-module defined by
k k C (E, W ) = lim C (R, Wλ). λ↑Λ

Now we are ready to define the time-dependent ultrafunctions: Definition 3.12. The space of time dependent ultrafunctions of order k N is the free Ck(R)∗-module given by ∈

Ck(E, V ◦).

Every time-dependent ultrafunction can be represented by the following hy- perfinite sum:

u(t, x)= c(t)χa(x) aX∈Γ

25 where c(t) Ck(R)∗ and χ (x) is the canonical basis of V ◦. If we set ∈ { a }a∈Γ k k C (E, VΛ) = lim C (R, Vλ), λ↑Λ the map ◦ k k ◦ ◦ ( ): C (E, V ) C (E, V ); w = w E Λ → | ×Γ is an isomorphism between free Ck(R)∗-modules: in fact, by using (20), we have that σ is a basis of Ck(E, V ) and we have that { a}a∈Γ Λ ◦

c(t)σa(x) = c(t)χa(x) (50) ! aX∈Γ aX∈Γ The restriction map (◦) can be extended to a Ck(R)∗-module homomorphism (◦): Ck(R, F RN )∗ Ck(E, V ◦) → by setting
◦ w (t, x)= w(t,a)χa(x) aX∈Γ In particular, if f Ck(R, F RN ), we have that ∈ ◦
∗ f (t, x)= f (t,a)χa(x) (51) aX∈Γ The notion of generalized derivative in the space variable is trivially defined by linearity:

Diu(t, x)= Di c(t)χa(x) = c(t)Diχa(x). (52) ! aX∈Γ aX∈Γ It is not necessary to introduce a generalized time-derivative, since the natural derivative ∂∗ : Ck+1(E, V ◦) Ck(E, V ◦), k 0, t → ≥ is well defined by setting

∗ ∗ ∗ ∂t u(t, x)= ∂t c(t)χa(x) = ∂t c(t)χa(x). (53) ! aX∈Γ aX∈Γ In our applications, we do not need a generalized nor a weak time-derivative for the functions in C0(E, V ◦). Theorem 3.13. If f Ck(R, Cm) and m 0, then, ∂∗f ◦ Ck−1(E,U m) and ∈ c ≥ t ∈ ∗ ◦ ∗ ∗ ∂t f (t, x)= ∂t f (t,a)χa(x). aX∈Γ Moreover, for i =1, ..., N and m 1, D f ◦ Ck(E,U m−1), and ≥ i ∈ ◦ ∗ ∗ Dif (t, x)= ∂i f (t,a)χa(x). aX∈Γ

26 Proof. The first equality follows immediately from (51) and (53). m m−1 Let us prove the second statement. We have that UΛ UΛ VΛ are hy- k R m k ⊂R m−1⊂ perfinite dimensional spaces and hence C ( ,UΛ ) and C ( ,UΛ ) are hyperfi- nite dimensional free Ck(R)∗-modules. By the definition 3.5, if w Ck(R,U m), ∈ Λ D w = ∂∗w Ck(R,U m−1). i i ∈ Λ Moreover t E, D w◦(t, ) = [∂∗w(t, )] ◦ U m−1 and hence ∀ ∈ i · i · ∈ ∗ ◦ ∗ Diw(t, x) = [∂i w(t, x)] = ∂i w(t,a)χa(x) aX∈Γ k R m ∗ k E m ∗ ∗ In particular, by Th. 3.6.2, if f C ( , Cc ), f C ( ,UΛ ), ∂i f k E m−1 ∈ ∈ ∈ C ( ,UΛ ) and

◦ ∗ ∗ ◦ ∗ ∗ Dif (t, x) = [∂i f (t, x)] = ∂i f (t,a)χa(x). Xa∈Γ 

4 Basic properties of ultrafunctions

In this section we analyze some properties of the fine ultrafunction that seems interesting in themselves and/or relevant in the applications.

4.1 The pointwise integral of 1 ultrafunctions L As we have seen, if f / V , in general ∈ f ◦(x) dx = f(x) dx 6 I Z since f ◦(x)dx takes account of the value of f in any single point. However, the followingI theorem holds: Theorem 4.1. If f 1, then, ∈ L f ◦(x) dx f(x) dx. ∼ I Z We recall (see (22)) that f V, ∀ ∈ ∗ fΛ = f where the operator v vΛ has been defined by (23). However this relation does not hold if f / V. The7→ following lemma is needed to prove the theorem above: ∈

27 Lemma 4.2. If f 1, then, ∈ L ∗ f ∗ f dx 0. | − Λ| ∼ Z Proof. We set

B (f)= v V x Γ , v(x)= f(x) λ { ∈ λ | ∀ ∈ λ } N where Γλ is a net of finite sets in R such that

Γ = lim Γλ λ↑Λ

∞ Clearly Bλ (f) is a closed set in Vλ with respect to the L -topology (remember that Vλ is a finite dimensional space); then the functional

v f v dx 7→ | − | Z has a minimizer which we will call fλ. fλ converges pointwise to the function f, and all the fλ are uniformly bounded in a neighborhood of supp(f); then we have that

lim f fλ dx =0. (54) λ→Λ | − | Z Since for every λ Λ, f V , ∈ λ ∈ λ

limfλ = fΛ; λ↑Λ so, we have that ∗ ∗ lim f fλ dx = f fΛ dx (55) λ↑Λ | − | | − | Z Z and by (9) and (54) ∗ f ∗ f dx 0. | − Λ| ∼ Z  Proof of Th. 4.1: If f 1, then, by (26) and Lemma 4.2 ∈ L ∗ ∗ ∗ f ◦ dx = f dx = f ∗dx (f ∗ f ) dx Λ − − Λ I Z ∗ Z Z f ∗dx = f dx. ∼ Z Z 

28 4.2 Ultrafunctions and measures We have seen that if f 1 , in general ∈ Lloc f ◦ dx = f dx. 6 I Z Thus it is a natural question to ask if there exists an ultrafunction u some way related to f such that u dx = f dx. (56) I Z This question has an easy answer if we think of f as the density of a measure µf . In fact, the following definition appears quite natural: Definition 4.3. If µ is a Radon measure, we define an ultrafunction µ◦ as follows: for every v V ◦, we set ∈

◦ µ (x)v(x)dx = lim vλ(x)dµ λ↑Λ I Z Example: 1 - If µ is a measure whose density is f 1 , then, v V ◦ f ∈ Lloc ∀ ∈ ∗ ◦ ∗ µf v dx = lim f(x)vλ(x)dx = f vΛdx. λ↑Λ I Z Z then, taking f 1, and v =1◦ ∈ L

◦ µf dx = f dx; I Z ◦ then (56) holds with u = µf . ◦ ◦ Example: 2 - If f V , then µf = f . δ ∈ δ◦ Example: 3-If a is the Dirac measure, then a = δa where δa is the Dirac ultrafunction defined by (43). EN Example: 4-IfΩ is a set of finite measure, and µΩ := µχΩ is the measure whose density is⊂χ , then u V ◦, by (26), we have that Ω ∀ ∈

◦ µΩ(x)u(x)dx = lim uλ(x)dµΩ = lim uλ(x)dx = u(x) dx. (57) λ↑Λ λ↑Λ Ω I Z Z ΩI◦

In particular, if f C0, ∈

◦ ◦ ◦ f (x) dx = f (x) dx = µΩ(x)f (x)dx (58) Ω Z ΩI◦ I

◦ The next proposition shows a useful way to represent µf :

29 Proposition 4.4. If f 1 , then ∈ Lloc ∗ ◦ ∗ µf (x)= f (y)σa(y)dy δa(x) (59) aX∈Γ Z  where σ (x) is the σ-basis (see (20)). { a }a∈Γ Proof: Let σa,λ be the net such that σa = limλ↑Λ σa,λ. Then,

◦ ◦ 1 ◦ 1 µf (a) = µf (y)δa(y)dy = µ (y)χa(y)dy = lim f(y)σa,λ (y) dy d(a) d(a) λ↑Λ I I Z 1 ∗ = f ∗(y)σ (y)dy d(a) a Z Hence, 1 ∗ µ◦ (x) = µ◦ (a)χ (x)= f ∗(y)σ (y)dy χ (x) f f a d(a) a a a∈Γ a∈Γ  Z  X ∗ X ∗ = f (y)σa(y)dy δa(x) aX∈Γ Z  

∗ ◦ Prop. 4.4 suggests to generalize the operator f µf to an operator w ∗ 7→ 7→ µ defined w L1 by setting w ∀ ∈ loc
∗ µw(x) := w(y)σa(y)dy δa(x) aX∈Γ Z 

µw(x) can be considered as a sort of measure density defined on Γ. ◦ Proposition 4.5. µw(x) is an element of V characterized by the following identity: u V ◦, ∀ ∈ ∗ µw(x)u (x) dx = w(x)u (x) dx = w(x)µu (x) dx (60) I Z I Proof. We have that ∗ µw(x)u (x) dx = w(y)σa(y)dy δa(x)u (x) dx I I a∈Γ Z  X ∗ = w(y)σa(y)dy δa(x)u (x) dx aX∈Γ Z  I ∗ ∗ = w(y)σa(y)u (a) dy = w(y) σa(y)u (a) dy a∈Γ Z  Z "a∈Γ # X∗ X = w(y)u (y) dx. Z

30 The last equality follows by symmetry. 

Eq. (60) allows to extend the pointwise integral over any internal set E Γ. ⊂ In fact, setting µE(x) := µχ (x), we have that E

∗ ∗ µE(x)u (x) dx = χE(x)u (x) dx = u (x) dx = lim uλ (x) dx λ↑Λ I Z ZE ZEλ where Eλ is a net of measurable sets such that limλ↑Λ Eλ = E. Then, it makes sense to define

u(x)dx := µE (x)u(x)dx (61) IE I By the above equation, we get

u(x)dx = u(a) µE(x)χa(x)dx; IE aX∈Γ I then setting dE (a)= µE(x)χa(x)dx, I

u(x)dx = u(a)dE(a) IE aX∈Γ generalizing eq. (27). The intuitive meaning of dE(a) is obvious.

4.3 The vicinity of a set

N 1 Given an open set Ω R and f C , then the value of f(x0) in a point x ∂Ω depends ony⊂ on the values∈ which f takes in Ω since∇ 0 ∈

f(x0) = lim f(x0) ∇ x∈Ω, ∇ x→x0

◦ ◦ If f is not continuous, f is not defined, but Df makes sense; however Df (x0) ◦ ∇ in a point x0 ∂Ω might depend on the values which f takes near by x0. A similar consideration∈ can be done about the integral (see (62)), These considerations make convenient the following definitions: Definition 4.6. Given an internal set E Γ, we define the vicinity of E as follows ⊂ vic(E) := supp ( Dµ + µ ) | E| E The operator Ω vic(Ω) reminds the closure operator Ω Ω but it is not a closure operator7−→ in the topological sense; in fact, in gneral,7−→vic (vic(E)) vic(E) in a strict sense. However, this similarity with the closure operator⊃ suggests the following

31 Definition 4.7. For any internal set E Γ, we define ⊂ bd (E) := x Γ Dµ (x) =0 or µ (x) =1 { ∈ | E 6 E 6 } and we will call bd (E) the pointwise boundary of E and

int(E) := x Γ Dµ (x)=0 and µ (x)=1 = vic(E) bd(E) { ∈ | E E } \ and we will call int(E) the pointwise interior of E.

By virtue of definition 4.7, we have that

u(x)dx = µE(x)u(x)dx = u(x)d (x)+ u(x)dE (x) . int bd IE I x∈X(E) x∈X(E) In particular this equation shows that in general, even for a smooth set Ω RN , and f V ⊂ ∈ f(x) dx = f ◦(x) dx = f ∗(x)d (x) , (62) Ω 6 ◦ Z ΩI◦ xX∈Ω actually, if f 0, we have that ≥ f ◦(x)d (x) < f(x)dx < f ◦(x)d (x) int ◦ Ω vic ◦ x∈X(Ω ) Z x∈ X(Ω ) ◦ since the integral over bd(Ω ) does not vanish and dΩ◦ (x) d (x). If we take E = RN ◦ = Γ, then bd (Γ) is called the boundary≤ at infinity. If x bd (Γ), x is an infinite number. Since ∈ | |

v V ◦, v(x)dx = v(x)dx = µ (x)v(x)dx, ∀ ∈ Γ I IΓ I we have that ◦ µΓ =1 . Since, bd (Γ) = ∅, the derivative of a constant is not null in every point x Γ: it vanishes for6 x int(Γ), but not in some points of bd (Γ); we have that ∈ ∈ supp (D 1◦) bd (Γ) . (63) i ⊂ ◦ Exercise 4.8. Given f V , x0 Γ E, u0 E, solve the following Cauchy problem in V ◦ (Γ) ∈ ∈ ⊂ ∈ Du = f(x);

u(x0)= u0.

32 Solution: In general, this problem does not have any solution in all Γ since, taking w(x)= D−1f(x), we might have that w(x ) = u 0 6 0 However, the function u : w(x) w(x )+ u − 0 0 solves the Cauchy problem in any point x int(Γ).  ∈

4.4 The Gauss’ divergence theorem First, we will define the pointwise integral extended to the pointwise boundary bd (E) of any set E EN . If Ω RN has a smooth boundary, then, it is natural to set ⊂ ⊂ ◦ u(x)dS = lim uλdS = µ∂Ω(x)u(x)dx (64) λ↑Λ ∂Ω bdI(Ω) Z I where S = S∂Ω denotes the (N 1)-dimensional mesure over ∂Ω. So, for any f C0 and any− bounded open set Ω with regular boundary, we have that ∈ ◦ ◦ ◦ f (x)dS = µ∂Ω(x)f (x)dx = f(x)dS. (65) ∂Ω bdI(Ω) I Z In order to analize the situation when ∂Ω is not smooth, we need to recall the notion of Caccioppoli set:

Definition 4.9. A Caccioppoli set Ω is a Borel set such that χΩ BV, namely such that (χ ) (the distributional gradient of the characteristic functi∈ on of Ω) ∇ Ω is a finite Radon measure. If Ω is a Caccioppoli set, then the measure (χΩ) is defined as follows: f C1, f 0, |∇ | ∀ ∈ c ≥

1 N f d ( (χ ) ) := sup (fφ) dx φ C , φ ∞ 1 |∇ Ω | ∇ · | ∈ k kL ≤ Z ZΩ 
The number p(Ω) := d ( (χ ) ) |∇ Ω | Z is called Caccioppoli perimeter of Ω. If the reduced boundary of Ω coincides with ∂Ω, we have that (see [28, Section 5.7])

f(x) d ( (χ ) )= f(x) d N−1 (66) |∇ Ω | H Z Z∂Ω

33 where N−1 is the (N 1)-dimensional Hausdorff measure of ∂Ω. In particular, if Ω isH a bounded open− set with smooth boundary,

f(x) d ( (χ ) )= f(x) dS. (67) |∇ Ω | Z Z∂Ω These facts suggest the following definition which generalizes (64): Definition 4.10. If E Γ is an internal set and u V ◦ we put ⊆ ∈ u dS = u Dµ dx (68) | E| bdI(E) I where µE is defined by (57). In particular, (66) and (67) become

f(x) d N−1 = f(x) Dµ◦ (x) dx H | Ω | Z∂Ω I

f(x) dS = f(x) Dµ◦ (x) dx. | Ω | Z∂Ω I Remark 4.11. If Ω is not sufficiently smooth and u / U 0, ∈ u(x) Dµ◦ (x) dx = u(x)µ◦ (x)dx. | Ω | 6 ∂Ω I I ◦ ◦ In this theory the measure DµΩ(x) is more useful than µ∂Ω. For example | | ◦ if ∂Ω is a set with Hausdorff dimension d>N 1, µ∂Ω is not defined, while ◦ ◦ − DµΩ(x) is well defined. Moreover DµΩ(x) allows to give a sense to the Gauss divergence| | theorem even when ∂Ω is| very wild| (see Th. 4.12 below). The above definitions allows to state the Gauss divergence theorem in the framework of ultrafunctions: Theorem 4.12. Let φ : Γ (V ◦)N be a (ultrafunctions) vector field and let E Γ be an internal set; then→ ⊆ D φ dx = φ n dS · · E IE bdI(E) where nE(x) is defined as follows

DµE (x) if Dµ (x) =0 |DµE | E n (x)= − 6 E   0 if DµE(x)=0 

34 Proof: By (61), (68) and (34)

D φ dx = µ D φ dx = Dµ φ dx · E · − E · IE IE I

DµE = φ DµE dx = φ nE dS. − DµE · | | · I | | bdI(E) 

An immediate consequence of the Gauss divergence theorem is the Funda- mental Theorem of Calculus: Corollary 4.13. Let Γ E, u V ◦(Γ) and [a,b] R; then ⊂ ∈ ⊂

u(x)dx = u(x)dS u(x)dS − [a,bI ]◦ NI(b) NI(a) where (b)= bd ([a,b] ◦) mon (a) and (b)= bd ([a,b] ◦) mon (b) . N ∩ N ∩ By (65), if f C0, ∈

f ◦(x)dS = f(b) and f ◦(x)dS = f(a)

NI(b) NI(a)

However, if f is not regular (you may think of the Dirichlet function) Cor. 4.13 holds, but the value of f ◦(x)dS depends on the values which f ◦ takes in I N (a) vicinity of a,b and not only on f(a) and f(b). { } 4.5 Ultrafunctions and distributions One of the most important properties of the ultrafunctions is that they can be seen (in some sense that we will make precise in this section) as a generalizations of the distributions. Definition 4.14. We say that an ultrafunction is distribution-like (DS) if there exist a distribution T such that for any ϕ ∈ D u(x)ϕ◦(x)dx = T, ϕ h i I We say that an ultrafunction is almost distribution like (ADS) if there exist a distribution T such that for any ϕ ∈ D u(x)ϕ◦(x)dx T, ϕ ∼ h i I

35 ◦ ◦ Example: The mesures DµΩ(x) and µ∂Ω(x) are distribution-like since for any ϕ | | ∈ D Dµ◦ ϕ◦dx = µ◦ (x)ϕ◦dx = T , ϕ | Ω| ∂Ω h µ i I I where Tµ is the distribution related to the measure µ. It is easy to see that: Proposition 4.15. For any distribution T there is a distribution-like ultrafunc- tion uT . Proof: Let us consider any projection P : V V and set λ λ → λ ∩ D

uT (x)v(x)dx = lim T, Pλvλ λ↑Λ h i I Then, ϕ ∀ ∈ D ◦ uT (x)ϕ (x)dx = lim T, Pλϕ = lim T, ϕ = T, ϕ λ↑Λ h i λ↑Λ h i h i I 

Clearly uT is not univocally defined since, in the proof of Prop. 4.15, the projection Pλ can be defined arbitrarily. So it make sense to set

◦ [u] ′ = v V v ′ u D { ∈ | ≈D } where ◦ v ′ u : ϕ , (u v) ϕ dx =0 ≈D ⇔ ∀ ∈ D − I Then there is a bijective map

′ ◦ Ψ: V / ′ (69) D → DS ≈D ◦ where VDS is the set of distribution like ultrafunction and

Ψ(T )= u V ◦ ϕ , uϕ◦dx = T, ϕ ∈ | ∀ ∈ D h i  I  The linear map is Ψ consistent with the distributional derivative, namely:

Proposition 4.16. If Ψ(T ) = [u]D′ then Ψ(∂iT ) = [Diu]D′

Proof: If Ψ(T ) = [u]D′ , then

D uϕ◦ dx = uD ϕ◦dx i − i I I Since ϕ◦ U ∞, then by Th. 3.6.45, D ϕ◦ = (∂ ϕ) ◦ and so ∈ i − i Duϕ◦ dx = uDϕ◦dx = u (∂ ϕ) ◦dx − i I I I = T, ∂ ϕ = ∂ T, ϕ − h i i h i i

36 Hence [Diu]D′ = Ψ(∂iT ). 

At this point it is a natural question to ask if there exists a linear map

Φ: ′ V ◦ D → which selects in any equivalence class [u]D′ a distribution-like ultrafunction Φ(u) in a way consistent with the distributional derivative, namely

Φ (∂iT )= DiΦ (T ) (70)

Actually this goal can be achieved in several ways. We will describe one of them: Definition 4.17. For every T ′, we denote by T ◦ the only ultrafunction in U ∞ (see(46)) such that ψ U∈∞ D ∀ ∈

◦ ∗ ∗ T (x)ψ(x)dx = lim T, ψλ = T , ψΛ (71) λ↑Λ h i h i I Clearly, T ◦ is a DS-ultrafunction since ϕ , ϕ◦ U ∞ and ∀ ∈ D ∈ T ◦(x)ϕ◦dx = lim T, ϕ = T, ϕ . λ↑Λ h i h i I Theorem 4.18. The map T T ◦ defined by (71) satisfies (70), namely 7→ ◦ ◦ (∂iT ) = DiT

Proof: By (47) and Th. 3.6.45, we have that ψ U ∞, ∀ ∈ D T ◦ψ dx = T ◦D ψ dx = T ◦∂∗ψ dx i − i − i I I I = T ∗, ∂∗ψ = ∂∗T ∗, ψ − h i Λi h i Λi ◦ = (∂iT ) ψ dx I 

1 Every function f loc defines a distribution Tf ; then, given f, we can ∈ L ◦ ◦ ◦ define three ultrafunctions: f ,µf and Tf . What is the relation between them? ◦ ∗ ◦ By Prop. 4.4, we have that µf is a projection of f over V , namely

∗ ◦ ∗ µf (x)= f (x)σa(y)dy δa(x) Xa∈Γ Z  ◦ Similarly also Tf is a projection, but over a smaller space as the following proposition shows:

37 Proposition 4.19. If f 1 , then ∈ Lloc ◦ ◦ Tf = Π∞f . where Π∞ has been defined by (48). Proof: By (47), we have that ψ U ∞, ∈ ◦ ◦ Π∞f ψ dx = lim fψλ dx = lim T, ψλ = Tf ψ dx λ↑Λ λ↑Λ h i I Z I ◦ ◦ ∞ Since both Tf and Π∞f U , the conclusion follows.  ∈

1 ◦ ◦ So, if f loc,Tf , similarly to µf , destroys some information contained ∈ L◦ ◦ ◦ ◦ in f; namely Tf (resp. µf ) cannot be distiguished by Tg (resp. µf ) if f and g agree almost everywere. Similarly, if µ is any Radon measure and Tµ is the ◦ ◦ corresponding distribution, then Tµ destroys some information contained in µ since ◦ ◦ Tµ = Π∞µ

Example: The δa ultrafunction is distribution like since for every ϕ , we have ∈ D δ ϕ◦(x)dx = ϕ(a)= δ , ϕ ; a h a i I (here we have used the boldface to distinguish the ultrafunction δa from the distribution δa). However δ = δ◦ a 6 a Actually, according to Def. 3.7, δ◦ ⊥ δa = a + Π∞δa, δ◦ namely a is the smooth part of δa.

5 Some applications

In this section we will sketch how the theory of fine ultrafunctions can be used in the study of Partial Differential Equations. In the framework of ultrafunc- tions, a very large class of problems is well posed and has solutions. Very often, hard a priori estimates are not necessary in proving the existence, but only in understanding the properties of a solution (qualitative analysis). In particular, if you have a problem from Physics or from Geometry, it is interesting to inves- tigate whether the generalized solutions describe the Physical or the Geometric phenomenon. We refer to [6],[8],[7],[14],..,[22] where such kind of problems have been treated in the framework of ultrafunction. A fortiori, these problems can be treated using fine ultrafunctions. In this section, we limit ourselves to give some new examples just to illustrate the use of fine ultrafunctions with a par- ticular emphasis in the study of ill posed problems. Obviously, each example is treated superficially. A deep analysis of each case, probably, would deserve a full paper.

38 5.1 Second order equations in divergence form Let Ω RN be an open set with regular boundary and let us consider the following⊆ boundary value problem:

[k(x, u) u]+ f(x, u)=0 in Ω (72) − ∇ · ∇ u(x)=0 for x ∂Ω (73) ∈ where f is a function and k(x, u) is a function or a (N N)-matrix with C1- coefficients. × A function which satisfies (72) and (73) is called classical solution if u 2 0 ∈ C (Ω) C0 Ω . We might translate this problem in the world of ultrafunctions as follows:∩
D [k∗(x, u)Du]+ f ∗(x, u)=0 in int (Ω) − · u(x)=0 if x ∂Ω◦, ∈ where ∂Ω◦ and int (Ω) has been defined by (18) and Def. 4.7. This formulation is not precise, since it does not prescribe what happens in bd (Ω) ∂Ω◦. Therefore we need a better formulation which take account of the behaviour⊃ of the solution in the points infinitely close to ∂Ω. To this aim, we rewrite equation (72) in the weak form:

v C1 Ω , [k(x, u) u v + f(x, u)v] dx = 0; ∀ ∈ ∇ · ∇ ZΩ
and we translate it in the ultrafunctions framework:

v V ◦ (Ω◦) , [k∗(x, u)Du Dv + f ∗(x, u)v] dx =0 ∀ ∈ · IΩ◦ where, for every internal set E Γ, ⊂ V ◦ (E)= u V ◦ u(x)=0 for x/ E . { ∈ | ∈ } The above equation is equivalent to (see section 4.2)

v V ◦ (Ω◦) , µ◦ (x)[k∗(x, u)Du Dv + f ∗(x, u)v] dx =0 ∀ ∈ Ω · I ◦ where µΩ have been defined by (57). Then, the right translation of problem (72), (73) in the framework of ultrafunctions is the following:

D [µ◦ (x) k∗(x, u)Du]+ µ◦ (x) f ∗(x, u)=0 in Ω◦ (74) − · Ω Ω u V ◦ (Ω◦) (75) ∈ A solution of (74), (75) will be called ultrafunction solution of problem (72), (73). The formulation (74), (75) allows the following theorem:

39 Theorem 5.1. If w is a classical solution of (72), (73) then u = w◦ is a ultrafunction solution. Proof: By the transfer principle, w∗ satisfies the equation

∗ [k∗(x, w∗) ∗w∗]+ f ∗(x, w∗)=0 in Ω∗ −∇ · ∇ 2 0 ∗ Since if w C (Ω) C0 Ω and since ∂Ω is smooth, we have that w ∈ ∩∗ ∈ H1 (Ω)∗ and v C1 Ω 0 ∀ ∈ 0
∗
[k∗(x, w∗) ∗w∗ ∗v + f ∗(x, w∗)v] dx =0 ∇ ∇ ZΩ Setting v =0 for x / Ω, then by (58), and Def.3.2.3a, ∈

◦ ∗ ◦ ◦ ◦ ∗ ◦ ◦ µΩ (x)[k (x, w )Dw Dv + f (x, w )v ] dx =0 I and hence, Def.3.2.3b

D [µ◦ (x) k∗(x, w◦)Dw◦]+ µ◦ (x) f ∗(x, w◦)=0. − · Ω Ω Then u = w◦ satisfies (74), (75). 

If a problem does not have a classical solution, we can look for weak solutions in some Sobolev space or in a space of distribution. However if there are not weak solutions, we can find a ultrafunction solution exploiting the following theorem: Theorem 5.2. If the operator

: V ◦ (Ω◦) V ◦ (Ω◦) A → is ( )-hemicontinuous2 and coercive, namely ∗ M, R E+, u R (u) u dx M u ∃ ∈ k k≥ ⇒ A ≥ ·k k I then the equation (u)= g A has at least one solution for every g V ◦ (Ω◦). ∈ 0 Proof: It is an immediate consequence of the Brower fixed point theorem and the fact that V ◦ (Ω◦) has hyperfinite dimension. 

2We recall that an operator over a vector space X is called hemicontinuous if it is continuous on finite dimensional subspaces of X. An operator over X ∗ is called (∗)-hemicontinuous if it is the Λ-limit of hemicontinuous operators.

40 Example 1: Let us consider the following problem:

u C2 Ω : (76) ∈ [k(x, u) u]=
f(x) in Ω (77) − ∇ · ∇ u(x)=0, for x ∂Ω (78) ∈ where k C1(R). We set ∈ k := inf k(x, s) (x, s) Ω R 0 { | ∈ × } and k0 > 0, (79) then, if k does not depend on u, (80) A = [k (x) u] is a strictly monotone operator and it is immediate to −∇ · ∇ 1 4 check that eq. (77) has a unique weak solution in H0 (Ω) L (Ω) for every f H−1(Ω)+L4/3(Ω). Moreover, if f and ∂Ω are smooth, by the∩ usual regularity results,∈ problem (76), (77) has a classical solution. If (79) or (80) is not satisfied, this problem is more delicate. In particular, if for some (but not all) value of u

k(x, u) < 0, the problem is not well posed and, in general, it has no solution in any distri- bution space. Nevertheless we have the following result: Theorem 5.3. If min lim k(x, u) k∞ > 0 (81) |u|→∞ ≥ then problem (76), (77), (78) has at least a ultrafunction solution, namely, f V ◦, there exists u V ◦(Ω), such that ∀ ∈ ∈ D [µ◦ k∗(x, u)Du]= µ◦ f x Ω◦. (82) − · Ω Ω ∀ ∈ Moreover, if (79) and (80) hold this solution is unique. Proof: By Th. 5.2 we have to prove that the operator is (82) is coercive. We have that

◦ ∗ D [µ k (x, u)Du] u dx = µ ◦ [k(x, u)Du] Du dx − · Ω · Ω I I 2 2 = µ ◦ k(x, u) Du dx = k(x, u) Du dx Ω | | | | I IΩ◦ By (81) and the continuity of k, there exists a constant C > 0 such that C k(x, u) k ≥ ∞ − 1+ u 2 | |

41 Then, for any M 0, ≥ Du 2 k(x, u) Du 2 dx k Du 2 dx C | | dx | | ≥ ∞ | | − 1+ u 2 IΩ◦ ΩI◦ ΩI◦ | | 2 2 2 Du Du k Du ◦ C | | dx | | dx. ≥ ∞ k kΩ − 1+ u 2 − 1+ u 2 |u|IM | |

Now, we take M so large that u >M implies that | | 2C 1+ u 2 | | ≥ k∞ Then Du 2 k | | dx ∞ Du 2 1+ u 2 ≤ 2C | | |u|I>M | | |uI|>M

k∞ 2 k∞ 2 Du = Du ◦ ≤ 2C | | 2C k kΩ I Moreover, there exists a constant C2 > 0 such that

Du 2 C | | dx 2 1+ u 2 ≤ C |u|I

Then

2 2 k∞ 2 k∞ 2 k(u) Du dx k Du ◦ C Du ◦ Du ◦ C . | | ≥ ∞ k kΩ − 2 − 2 k kΩ ≥ 2 k kΩ − 2 ΩI◦ Now, remember that V ◦(Ω) has hyperfinite dimension and hence the norm

Du Ω◦ is equivalent to the norm u . Hence the operator (82) is coercive. kThek last statement follows trivially byk k the fact that (79) and (80) implies that ◦ ∗ D [µΩk (x)Du] is strictly monotone. − ·

If k is not positive, our problem might have infinitely many solutions and they can be quite wild. For example, if we consider the problem

u V ◦(Ω), D (u3 u)Du =0. (83) ∈ 0 − · − we can check directly that, for any intenal set E int (Ω◦) the function u(x)= ⊂ χE(x) is a legitimate solution in the frame of ultrafunction. If (79) and (80) hold, problem (76), (77), (78) has a classical solution pro- vided that f is sufficiently good. Nevertheless, our problem has a unique gen- eralized solution even if f is a wild function (e.g. f / H−1(Ω) + L4/3(Ω) or ∈

42 f not measurable or f is a distribution not in H−1). For example if N 2, −1 ≥ then δa / H (Ω); in this case the ultrafunction solution, for any delta-like f, concentates∈ in mon (a) .

Example 2: Let us consider the following problem: u C2 Ω : ∈ [k(u) u]+ u3 =
f(x) in Ω (84) − ∇ · ∇ u(x)=0, for x ∂Ω ∈ where k is a matrix such that k(u)ξ ξ k ξ 2 . (85) · ≥− 0 | | Theorem 5.4. If (85) holds, problem (76), (77), (78) has at least a ultrafunc- tion solution, namely, f V ◦, there exists u V ◦(Ω), such that ∀ ∈ ∈ 0 ∗ 3 ◦ D [µ ◦ k (u)Du]+ µ ◦ u = µ ◦ f x Ω . (86) − · Ω Ω Ω ∀ ∈ Moreover, if (79) holds, and f C0(Ω) such a solution is unique and it corre- sponds to the classical solution.∈ Proof: By Th. 5.2 we have to prove that the operator is (82) is coercive. ◦ By (42), all the norms on V0 (Ω) are equivalemt; hence, we have that

D [k(u)Du]+ u3 u dx = k(u)Du Du + u4 dx − · · I I◦
Ω   k Du 2 + u 4 ≥ − 0 k k k k k D 2 u 2 + u 4 . ≥ − 0 k k k k k k an hence the operator (82) is coercive. The last statement follows trivially by Th. 5.1. 

As a particular case, we can take 1 0 k = ; −0 c2   so equation (84) reduces to the nonlinear wave equation: u + u3 = f;  = ∂2 c2∂2 1 − 2 If we take Ω=[0,T ] [0, 1] and we impose periodic boundary contusions in x1 this problem reduces× to the classical problem relative to the existence of periodic solution of the nonlinear wave equation. In general, this problem does not have classical solutions because of the presence of small divisors which prevent the approximate solutions to converge. So this is a problem that can be studied in the framework of ultrafunctions where the existence is guaranteed.

43 Remark 5.5. If we want to consider equation (72) with non-homogeneous boundary condition, i.e.

u(x)= g(x) for x ∂Ω (87) ∈ we can adopt the standard trick to set

w(x)= u(x) g¯(x) − where g¯(x) is any function which extends g in Ω and to solve the resulting equation in w with the homogeneous boundary conditions.

5.2 The Neumann boundary conditions Now let us consider the equation (72) with the homogeneous Neumann boundary conditions, i.e. du (x)=0 for x ∂Ω (88) dn ∈ Now we want to formulate the problem (72) and (88) in the framework of ultrafunctions in such a way that any classical solution w (if it exists) coincide withe the ultrafunction solution in the sense of Th. 5.1. The appropriate formulation is the following:

D [µ◦ (x) k∗(x, u)Du]+ µ◦ (x) f ∗(x, u)=0 in Ω (89) − · Ω Ω u V ◦ (vic (Ω◦)) (90) ∈ where vic (Ω◦) has been defined by Def. 4.6. Notice that the difference with the Dirichlet problem relies in the choice of the function space. This fact is not surprising since the same phenomenon happens with the weak solutions in Sobolev spaces. Theorem 5.6. If w is a classical solution of (72), (88) then u = w◦ is a ultrafunction solution of . Proof: By the transfer principle, w∗ satisfies the equation

∗ [k∗(x, u) ∗w∗]= f ∗(x) in Ω∗ −∇ · ∇ ∗ Since if w C2 Ω and since ∂Ω is smooth, we have that v C1 Ω ∈ ∀ ∈
∗
[k∗(x) ∗w∗ ∗v f ∗(x)v] dx =0 ∇ ∇ − ZΩ If we extend v to EN so that v◦ V ◦, then by (58), and Def.3.2.3a, ∈

◦ ∗ ◦ ◦ ◦ ∗ ◦ µΩ (x)[k (x, w )Dw Dv + f (x)v ] dx =0 I

44 and hence, Def.3.2.3b

D [µ◦ (x) k∗(x, w◦)Dw◦]+ µ◦ (x) f ∗(x, w◦)=0. − · Ω Ω 

Now let see how to formulate the Neumann problem with non-homogeneous boundary conditions, namely du (x)= g(x) for x ∂Ω (91) dn ∈ The weak formulation ot the problem (72),(91) is given by

v C1 Ω , [k(x, u) u v + f(x, u)v] dx + gv dS =0, ∀ ∈ ∇ · ∇ ZΩ Z∂Ω
then the weak formulation in the world of ultrafunction, using Def. 4.10, is

v V ◦ (vic (Ω◦)) , µ◦ (x)[k∗(x, u)Du Dv + f ∗(x, u)v] dx+ Dµ◦ gv dx =0 ∀ ∈ Ω · | Ω| I I and the explicit ultrafunction formulation of problem (72),(91) is

u V ◦ (vic (Ω◦)) ∈ D [µ◦ (x) k∗(x, u)Du]+ µ◦ (x) f ∗(x, u)+ Dµ◦ g =0 in Ω◦ (92) − · Ω Ω | Ω| Using the same arguments, we have that a clasical solution of (72),(91) is a solution of (92); however equation (92) is well posed even when ∂Ω is quite wild, for exemple when the Hausdorff dimension of ∂Ω larger that N 1. − 5.3 Regular weak solutions The expression regular weak sounds like an oxymoron, nevertheless it well de- scribes the notion we are going to present now. For example, let us consider the equation (83). It is possible that the function of the form χE are not acceptable as solutions of a some physical model described by (83). Then we may ask if there exist ”approximate” solutions of equation (83) which exibit some form of regolarity. More in general, given the second order operator (72) and (73), we might be interested in regular solutions u U m for some m N (see section 3.4). So, we are lead to the following∈ problem: find ∈ ∪ {∞}

u U m(Ω◦) (93) ∈ such that v U m(Ω◦), ∀ ∈ [k∗(x, u)Du Dv + f ∗(x, u)v] dx = 0 (94) · ΩI◦

45 Arguing as in Th. 5.1, if w Cm Ω is a classical solution of (72), (73) then u = w◦ is a solution of (93),∈ (93). In general a solution of (93), (94) does not satisfy equation (74), but the equation

D [µ◦ (x) k∗(x, u)Du]+ µ◦ (x) f ∗(x, u)= ψ(x) in Ω◦ (95) − · Ω Ω where ψ(x) U m(Ω◦)⊥ ∈ can be considered as a sort of error. In many situation, the error ψ can be considered negligible since

v U m(Ω◦), ψv dx =0 ∀ ∈ I Arguing as in theorem 5.2, we have the following result: Theorem 5.7. Let the operator (72), resticted to

span U m , m N+ { λ } ∈ ∪ {∞} be continuous and coercive λ L, then the equation ∀ ∈ [k∗(x, u)Du Dv + f ∗(x, u)v] dx =0, v U m(Ω◦), (96) · ∈ ΩI◦ has at least a solution in u U m(Ω◦). ∈ A solution u of (96) has the following characterization:

◦ u = lim uλ λ↑Λ   where u is a solution of the following problem: u U m(Ω) and λ λ ∈ λ [k(x, u)Du Dv + f(x, u)v] dx =0, v U m(Ω). · ∀ ∈ λ IΩ For suitable choices of k and f, it is possible that the regular weak solutions coincide with the ultrafunctions solutions, however for ill posed problem in gen- eral this fact does not happens. Then, in dealing with a problem relative to a physical phenomenon, the choice of the spece in which to work is very relevant (see Remark 3.8).

5.4 Calculus of variations The notion of regular weak solutions is natural when we deal with the calculus of variations. Consider the functional

J(u)= F (x, u, u) dx (97) ∇ ZΩ

46 If we assume the Dirichlet boundary condition, the natural space where to work 1 0 is C (Ω) C0 (Ω). If this functional does not have a minimum, then a natural space of∩ ultrafunctions where to work is U 1(Ω); in fact, by Th. 3.6.1, the generalized gradient Du, coincides with the natural extension of the gradient ( ∗u) ◦ and the following theorem holds. ∇ Theorem 5.8. Assume that for λ sufficiently large the functional (97) has a minimizer u in U m(Ω), m 1. Then λ λ ≥ u¯ = lim uλ λ↑Λ is a minimizer of

◦ ∗ ∗ ∗ J (u) := F (x,u,Du) dx = F (x, uΛ, uΛ) dx Ω∗ ∇ ΩI◦ Z in U m(Ω). Proof: Trivial. 

m m 0 Notice that Uλ (Ω) is a finite dimensional subspace of C (Ω) C0 (Ω) and hence, if F (x, u, ξ) is continuous in u and ξ and coercive, the minimum∩ exists. However also the case of discontinuous functions can be easily analyzed (see e.g. [8]).

Example: Let us consider the functional defined in C1(0, 1) C0([0, 1]) ∩ 0 2 J(u)= ∂u 2 1 + u2 dx (98) Ω | | − Z    which presents the well known Lavrentiev phenomenon, namely every minimiz- ing sequence un converges uniformly to 0 but

0 = lim J(un) = J(0) = 1. n→∞ 6 1 If we use Th. 5.8, the minimizing sequence uλ Uλ(Ω) has an infinitesimal Λ- limitu ¯ U 1(Ω) and J ◦(¯u) is a positive infinitesimal∈ which satisfies the following Euler-Lagrange∈ equations: 1 D Du¯ 2 1 Du¯ u¯ = ψ | | − − 2 h  i where v U 1(Ω)⊥, ψv dx =0. ∀ ∈ I However, it is possible to minimize the functional

2 J ◦(u) := Du 2 1 + u2 dx Ω◦ | | − Z    47 in V ◦ (Ω◦) ; in this case we get a minimizeru ˆ such that J ◦(ˆu)

5.5 Evolution problems Let Ω RN be an open set and let ⊂ A(x, ∂ ): D (Ω) C (Ω) ; i A → be a differential operator. We consider the following Cauchy problem: given u (x) C0 Ω , find 0 ∈ u C1(I,D (Ω)) C0(I, C (Ω)), 0 I R :
(99) ∈ A ∩ ∈ ⊆

∂tu = A(x, ∂i)[u] (100)

u(0, x)= u0(x). (101) A function which satisfies (99), (100), (101) is called classical solution. We want to translate this problem in the world of ultrafunctions. Because of the nature of the Cauchy problem, it is not convenient to translate this problem in the space V ◦ (R Ω) . It is better to use the internal space of functions C1(E, V ◦) defined in× section 3.5. We assume that the boundary conditions which are ”contained” in the definition of the domain DA (Ω) can be translated in a ◦ ◦ domain DA (Ω) V . In sections 5.1 and 5.2 we have seen how this can be done for second order⊂ operators with Dirichlet and Neumann bondary conditions. Then, setting

C1(I∗,D◦ (Ω)) = u C1(E, V ◦) t I∗, u(t, x) D◦ (Ω) A ∈ | ∀ ∈ ∈ A The problem (99), (100), (101) translates in the following one:

u C1(I,D◦ (Ω)) (102) ∈ A ∂∗u = A◦(x, D )[u] , t I∗ (103) t i ∈ u(0, x)= u0(x) (104) where A◦(x, D )[u] is defined by tht following relation: u, v D◦ (Ω) , i ∀ ∈ A ∗ ◦ ◦ ◦ ∗ ∗ A (x, Di)[u ] v dx = A (x, ∂i )[u] v dx Ω∗ ΩI◦ Z

48 ∗ Remember that the time derivative ∂t is not the generalized derivative, but the natural extension of ∂t defined by (53). A solution of (102), (103), (104) will be called ultrafunction solution of the problem (99), (100), (101). Using this notation, we can state the following fact: Theorem 5.9. If w is a classical solution of (99), (100), (101) and we assume that w be extended continuously in a neighborhood (Ω); then u = w◦ is a ultrafunction solution of (102), (103), (104). N Proof: If w satisfies (99), (100), w∗ satisfies the following:

w∗ C1(I,D (Ω))∗ C0(I, C (Ω))∗, (105) ∈ A ∩ ∗ ∗ ∗ ∗ ∗ ∗ ∂t w v dx = A (x, Di)[w ] v dx, v VΛ, (106) ∗ ∗ ∀ ∈ ZΩ ZΩ w∗(0, x)= u∗(x); x Ω∗, (107) 0 ∈ Then u = w◦ satisfies the following:

u C1(I∗,D◦ (Ω)) ∈ A ∂∗ uv dx = A◦(x, D )[u] v dx, v V ◦, (108) t i ∀ ∈ IΩ◦ IΩ◦ Then, putting u(t, x)=0for x / vic (Ω◦) , we have that u◦ satisfies (102), (103) (104). ∈ 

Also in the evolution case, the conditions which guarantee the existence of a ultrafunction solution are very weak.

Theorem 5.10. Assume that A(x, ∂i)[u] restricted to Vλ (Ω) is locally Liptshitz continuous in u; then there exists TΛ such that the problem (102), (103), (104) has a unique ultrafunction solution for t [0,TΛ) . Moreover, if there is an a prori bound for such a solution , then there∈ exists a unique ultrafunction solution 1 ∗ ◦ in C (I ,DA (Ω)). Proof. For every λ consider the following system of ODE′s in F (Γ Ω) λ ∩ ∂ u (t,a )= A(x, D )[u ] (t,a ), a Γ (109) t λ λ iλ λ λ λ ∈ λ

Since F (Γλ) is a finite dimensional vector space and A(x, Diλ)[u] restricted to Vλ (Ω) is locally Liptshitz continuous the above system has a local solution for t [0,T ) . Then, setting ∈ λ TΛ = lim Tλ λ↑Λ and taking the Λ-limit in (109), t [0,T ) , we get ∀ ∈ Λ ∗ ∂t u(t,a)= A(x, Di)[u] (t,a)

49 Moreover, if there is an a priori bound for such a solution, then there is a bound for the approximate solution (109) in [0,Tλ) (which might also depend on λ); ∗ then, it is well known that [0,Tλ)= I and hence [0,TΛ)= I 

5.6 Some examples of evolution problems Example 1: Let Ω be a bounded open set and let us consider the following problem: u C1(I, C0 Ω ) C0(I, C2 (Ω)): (110) ∈ 0 ∩ ∂tu =
[k(u) u] (111) ∇ · ∇ u(0, x)= u0(x) (112) If k(u) satisfies (79), then (111) is a parabolic equation and the problem, under suitable condition, has a classical solution. If k(u) < 0 for some u R, then the problem is ill-posed, and in general classical solutions do not exist.∈ Let us translate this problem in the world of ultrafunctions; taking account of the results of section 5.1, we get:

u C1(I∗, V ◦ (Ω◦)) (113) ∈ ∂∗u = D [µ◦ (x) k∗(u)Du] for x Ω◦ (114) t · Ω ∈ ◦ u(0, x)= u0(x) (115) So we can apply the theorems 5.10 and we get the existence of a unique (local in time) solution. It is not difficult to get sufficient conditions which guarantee the existence of a solution for every t I∗. For example ∈ Theorem 5.11. If the set

B = r R k(r) < 0 (116) { ∈ | } is bonded, problem (113),(114),(115) has a global solution. Proof - We set k(r) = k+(r) k−(r) and with some abuse of notation we write, for r E, k∗(r)= k+(r) k−−(r). Then, we have that ∈ − ∂∗ u2dx = 2 u∂∗udx =2 uD [k∗(u)Du] dx t t · ΩI◦ ΩI◦ ΩI◦ = 2 k∗(u) Du 2 dx 2 k−(u) Du 2 dx − | | ≤ | | ΩI◦ ΩI◦ 2 D 2 k−(u) u 2 dx ≤ k k | | ΩI◦ If we set M = max k−(r) r 2 r∈R | |

50 then, we have that ∂∗ u2dx 2M D 2 dx t ≤ k k · IΩ IΩ◦ and this implies the existence of a global solution. 

In many applications of eq. (111) u represent a density and k(u) u its flow. Then, thanks to the generalized Gauss’ theorem 4.12, it is easy to∇ prove that the ”mass” u(x)dx is preserved up to the flow crossing ∂Ω: I Ω ∗ ∗ ∂t u(x)dx = ∂t u(x)dx IΩ IΩ = D [k∗(u)Du] dx · IΩ = k∗(u)Du n dS · Ω ∂I◦Ω If we want to model a situation where the flow of u cannot cross ∂Ω, Ω bounded, in a classical context, the Neumann boundary conditions are imposed: du x ∂Ω, (x)=0; (117) ∀ ∈ dn in the framework of ultrafunction this situation can be easily described as fol- lows: u C1(I∗, V ◦ (vic (Ω◦))) (118) ∈ 0 ∂∗u = D [k∗(u)µ◦ (x)Du] (119) t · Ω ◦ u(0, x)= u0(x) (120) ◦ ∗ ◦ In this case, we have the flow µΩ(x)k (u)Du vanishes out of vic (Ω ) and we have the conservation of the ”mass” u(x)dx. We can prove this fact I ◦ ∗ directly, in fact, since Ω is bounded, also supp (D [µΩ(x)k (u)Du]) is bounded and hence, by (63) ·

∂∗ u(x)dx = D [µ◦ k∗(u)Du] dx t · Ω I I = D [µ◦ k∗(u)Du]1◦dx · Ω I = k∗(u)µ◦ Du D1◦dx =0. Ω · I Remark 5.12. This problem has been studied with nonstandard methods by Bottazzi in the framework of grid functions [25]. One of the main differences is that in the context of [25], theorem 5.9 does not hold.

51 Example 2. Now let us consider the following ”conservation law”:

u C1(I, C0 (Ω)) C0(I, C1 (Ω)): (121) ∈ ∩ ∂ u = F (x, u) (122) t ∇ · u(0, x)= u0(x) (123) where F : RN+1 RN is a smooth function with support in Ω as well as the → initial data u0(x). This problem is not well posed and when N > 1, very little is known. Nevertheless this problem is well posed in the frame of ultrafunctions:

u C1(I, V ◦ (vic (Ω◦))) : (124) ∈ ∂∗u = D F ∗(x, u) (125) t · ◦ u(0, x)= u0(x) (126) Theorem 5.13. If ∂F c + c u , (127) ∂u ≤ 1 2 | |

problem (121),(122),(123) has a unique (global in time) ultrafunction solution u(t, x) which satisfies the following properties:

supp (u(t, x)) vic (Ω◦) (128) ⊂

∗ ∂t u dx =0 (129) I Proof: The existence of a unique solution follows from Th. 5.10. (128) follows from the fact that for every x / vic (Ω◦) ,D F (x, u)=0. (129) is an immediate consequence of the Gauss’ theorem∈ 4.12. · 

A particular case of eq. (125) is the Burger’s equation

∂ u = u∂ u t − x u(0, x)= u (x) 0. 0 ≥ It is well known that this equation has infinitely many weak solutions which preserve the mass. One of them, the entropy solution, describes the phenomena occurring in fluid mechanics. It is convenient to write the Burger’s equation in the framework of utrafunctions as follows:

u C1(E, V ◦) (130) ∈ u u ∂∗u = D | | (131) t − x 2   ◦ u(0, x)= u0(x) (132)

52 Since the right hand side of this equation does not stisfy (127), the formula- tion (131) grantees a priori bounds and hence the existence of a global solutions. In fact, by (34), 1 1 ∂∗ u 3 dx = (u u ) ∂ u dx = (u u ) D (u u ) dx =0 t | | 3 | | t −6 | | x | | I I I The solutions of eq. (131) are different from the entropy solution since they preserves also the quantity u 3 dx. The viscosity solution can be modelled in | | the frame of ultrafunctionsI by the equation

u2 ∂∗u = D + νD2u t − x 2 x   where ν is a suitable infinitesimal (see [20]). Finally we remark that in the frame of ultrafunctions the equation (131) is different from the equation ∂∗u = uD u (133) t − x 2 even if u 0; in fact, in the point where u is singular Dx u =2uDxu and a priori eq.≥ (133) is not a conservation law since it does not have6 the form (125).
Nevertheless the mass is preserved since, by (34),

∂ u dx = uD u dx =0 t − x I I It is is immediate to see that u◦(x) 0 implies that, t, u(t, x) 0 and hence, 0 ≥ ∀ ≥ by the fact that u dx is constant, we get an a priori bound and the existence | | of a global solution.I ◦ ◦ Moreover, if supp (u0) [a,b] then supp (u (t, )) ng ([a,b] ) since for ◦ ⊂ · ⊂ every x / ng ([a,b] ) , ∂tu(x) = 0 for every t E. So a new phenomenon occurs: the∈ mass concentrates in the front of the∈ shock waves. For example consider the initial condition x ifx [0, 1] ◦ u (x)= ∈ 0  0 if x Γ [0, 1] ◦  ∈ \ In this case the solution, for t 0, is ≥ x if x [0, 1] ◦ mon (1) t+1 ∈ \ u(t, x)=  0 if x 0 or x> 0.  ≤ and hence, as t , all the mass concentrates in mon (1) . In conclusion,→ the ∞ translation of the Burger’s equation in the frame of ul- trafuctions, leads to several different situation which might reproduce different physical models.

53 Example 3. Let us consider the following problem:

u C2(I, C0) C0(I, C2) ∈ ∩ u + u p−2 u =0 in RN ; p> 2 | |

u(0, x) = u0(x),

∂tu(0, x) = u1(x); where u = ∂2u ∆u. t − For simplicity we assume that u0(x) and u1(x) have compact support. It is well known that this problem has a unique weak solution in suitable Sobolev spaces, provided that, for N 3, ≥ 2N p 2∗ = ≤ N 2 − and, in this case, the energy 1 1 1 E(u(t, )) = ∂∗u 2 + u 2 + u p dx · 2 | t | 2 |∇ | p | | Z   is a constant of motion. If p > 2∗, the problem has a weak solution but it is an open question if it is unique and if the energy is preserved. This problem can be translated and generalized in the framework of ultrafunctions:

u C2(I, V ◦) (134) ∈ 0 ∂∗2u D2u + u p−2 u =0 for x Γ (135) t − | | ∈ ◦ ∗ ◦ u(0, x)= u0(x); ∂t u(0, x)= u1(x). (136) Theorem 5.14. The problem (134),(135),(136) has a unique solution. More- over the energy equality holds, namely 1 1 1 ∂∗E(u(t, )) = ∂∗ ∂∗u 2 + Du 2 + u p dx =0 t · t 2 | t | 2 | | p | | I   Proof: Although the solution of this problem could be easily proved directly, we will reduce this problem to the form (102), (103) (104) by setting

∗ ∂t u = w ∂∗w = D2u u p−2 u t − | | so problem (134),(135),(136) can be reformulated as follows:

u C1(I∗, (V ◦)2): ∈ 0

54 A◦(D)[u]=0

u(0, x)= u0(x); w(0, x)= u1(x) where u w u = and A◦(D)[u]= ; w D2u u p−2 u    − | |  Then the existence of a unique solution is guaranteed by Th. 5.10 provided that we get an a priori estimate. This is given by the conservation of the energy:

1 1 1 ∂∗E(u(t, )) = ∂∗ w 2 + Du 2 + u p dx t · t 2 | | 2 | | p | | I   = w∂∗w + Du ∂∗Du + u p−1 ∂∗u dx t · t | | t I   = w∂∗w + Du Dw + u p−1 w dx t · | | I   = ∂∗w D2u + u p−1 w dx =0 t − | | I   ∗ ∗ Notice that ∂t and D commute, since ∂t ”behaves” as an ordinary derivative and D ”behaves” as a matrix in a finite dimensional space with the coefficients independent of time. 

5.7 Linear problems Let us consider the following linear boundary value problem:

u C2 (Ω) C0 Ω (137) ∈ ∩ 0 [k(x) u]+ λu = f(x) in
Ω, f C0 Ω . (138) − ∇ · ∇ ∈ For what we have discussed in section 5.1, it is convenient
to translate it in the framework of ultrafunctions as follows:

u V ◦ (Ω◦) (139) ∈ 0 D [µ◦ (x)k◦(x)Du]+ λu = µ◦ (x)f(x) in int(Ω), f V ◦ (Ω◦) (140) − · Ω Ω ∈ Since the operator D [µ◦ (x)k(x)Du] is symmetric it has a hyperfinite spec- − · Ω trum Σ = λk k∈K with an orthonormal basis of eigenvalues ek k∈K . Then the Fredholm{ alternative} holds and if λ / Σ, problem (139), (140){ has} a unique solution given by ∈

fk ◦ u(x)= ek where µΩ(x)f(x)= fkek(x). λk + λ kX∈K kX∈K

55 The operator L◦u := D [µ◦ (x)k◦(x)Du] can be regarded as a sort of − · Ω selfadjoint realization of L with respect to the scalar product (u, v) uv dx. 7→ Ω Let examine the spectrum of L◦ in some cases. If k(x) is a strictlyI positive smooth function and Ω is bounded, then the classical operator

Lu = [k(x) u] −∇ · ∇ has a discrete spectrum of positive eigenvalues which correspond to an orthonor- ◦ mal basis ek k∈N of smooth functions. Then the spectrum of L u has an or- { } ¯ ◦ ◦ thonormal basis hk k≤dim(V ◦(Ω◦)) . For some infinite number k< dim(V (Ω )), { } ∗ hk coincides with the spectrum of L ; in particular, if k N, the eigenvalues { }◦ ◦ ∈ of L concide with the eigenvalues of L and hk = ek. If k(x) is negative in a subset of Ω of positive measure, then Lu has a continuum unbounded spectrum. In this case the eigenvalues of L◦ are infinitely close to each other.

Example: Let us consider the following ill posed problem relative to the Tricomi equation: ∂2u + x ∂2u =0 in Ω; 0 Ω. 1 1 2 ∈ u = g(x) for x ∂Ω ∈ In this case L◦u := D [µ◦ (x) (D u + x D u)] . − · Ω 1 1 2 It is not difficult to prove that for a ”generic” open set Ω, 0 is not in the spectrum of L◦u. In this case, using Remark (5.5), this problem has a unique ultrafunction solutions.

6 A model for ultrafunctions

This section is devoted to prove Th. 6.21 namely to the construction a space of fine ultrafunctions. The construction presented here is the simplest which we have been able to find. Nevertheless it is quite involved but we do not know if a substantially simpler one exists. The main difficulty relies in the fact that all the properties of Def. 3.2 need to be satisfied; in particular, properties Def. 3.2.2b, 3.2.3a, and 3.2.3a are quite difficult to be obtained simultaneously (see also Remark 6.22). Our model of fine ultrafunctions combines the theory of ultrafunctions with the techniques related to step functions. Roughly speaking we can say that, in this model, the fine ultrafunctions, as well as the standard continuous functions, can be well approximated by step functions and this fact is a cornerstone of our construction.

56 6.1 Hyperfinite step functions The step functions are easy to handle ad hence they are largely used many branches of mathematics and in particular in nonstandard analysis. Let us describe the kind of step functions that will be considered in this paper. Given an infinite hypercube Q = [ L,L)N EN an infinitesimal partition − ∗ ⊂ of Q is a hyperfinite set ℘(RN ) such that every element S has P P ⊂ ∈ P an infinitesimal diameter. Such ah partitioni will be denoted by S { a}a∈Γ N where a E is a point such that a Sa. An infinitesimal partition η is called regular∈if each element of Q ∈is a hypercube of center s andP side η 0. s ∈ Pη ∼ Moreover, we ask that the number of elements of η be even (the reason of this choice will be clear in the proof of Prop 6.4). P It is easy to get such a regular partition; it sufficient to take L η = (141) β with β N∗ so large that η 0 and to set ∈ ∼ η G := N ; := zη + z Z∗; β z<β . (142) η I I 2 | ∈ − ≤ n o Given s = (s , .., s , .., s ) G , we set 1 i N ∈ η η η Q = x = (x , .., x , .., x ) EN i, s x < s + (143) s 1 i N ∈ | ∀ i − 2 ≤ i i 2 n o So = Q consists of (2β)N hypercubes. The number η, will be called η s s∈Gη meshP of{ the} partition and 2L will be called length of the partition. Definition 6.1. A step function z : EN E is defined as follows: →

z(x)= csχQs (x). sX∈Gη where c : G E is an internal function. The space of regular step functions s η → will be denoted by Sη (Q) . The integral of a step function is given by ∗ u(x)dx = ηN u(a). (144) Z aX∈G Let us define the partial derivative of a step function. We denote by ei the canonical base in RN and we set u(x + ηe ) u (x ηe ) Dηu(x) := i − − i (145) i 2η η We will refer to Di as the step derivative. Let us describe some (obvious) properties of the step derivative:

57 Proposition 6.2. The operator

Dη : S (Q) S (Q) i → is antisymmetric with respect to the scalar product induced by the step integral (144), namely

∗ ∗ u, v S (Q) , Dηuv dx = uDηv dx ∀ ∈ i − i Z Z Proof: We have that ∗ η N η Di uv dx = η Di u(a)v(a) Z aX∈Γ u(a + ηe ) u (a ηe ) = ηN i − − i v(a) 2η aX∈Γ   ηN−1 ηN−1 = u(a + ηe )v(a) u (a ηe ) v(a) 2 i − 2 − i aX∈Γ aX∈Γ On the other hand

u(a ηe )v(a)= u (a) a(a + ηe ) − i i aX∈Γ aX∈Γ and hence ∗ ηN−1 Dηuv dx = [u(a + ηe )v(a) u (a) v(a + ηe )] (146) i 2 i − i Z aX∈Γ By this equality, the conclusion follows straightforward. 

Proposition 6.3. If y = x +2mηei, then

m−1 η u(y)= u(x)+ η Di u(x + (2k + 1) ηei) Xk=0 Proof. We have that

m−1 m−1 u(x)+ η Dηu(x + (2k + 1) ηe ) = u(x)+ [u(x + 2 (k + 1) ηe ) u(x +2kηe )] i i i − i kX=0 Xk=0 = u(x +2mηei)= u(y). 

Proposition 6.4. If u S (Q), then for every i =1, ..., N ∈ η u 2 2L D u 2 k kL ≤ k i kL

58 Proof. Set F := Q Q ∂Q = ∅ Q s | s ∩ 6 and take arbitrarily x Q. By[ (141) and (142), the number of elements of is 2β, and hence it is even.∈ Then there exists m Z such that x + (2m 1) ηeI ∈ − i ∈ FQ. In order to fix the idea let us assume that m> 0. Then x +2mηei / Q and hence ∈ u (x +2mηei)=0 Then, by Prop. 6.3,

m−1 η 0= u(x)+ η Di u(x + (2k + 1) ηei) Xk=0 and so m−1 m−1 η η u(x) = η Di u(x + (2k + 1) ηei) η Di u(x + (2k + 1) ηei) | | ≤ | | k=0 k=0 X X Now, for t , we set ∈ I Γ = x Γ x = (x , .., x ,t,x , .., x ) t { ∈ | 1 i−1 i+1 N } then, since Γ contains (2L) /η elements, we have that | t| 1 1 2 2L 2 u(x) η Dηu(y) η Dηu(y) 2 | | ≤ | i |≤ · η ·  | i |  yX∈Γt   yX∈Γt 1 2   1 = (2Lη) 2 Dηu(y) 2 ·  | i |  yX∈Γt   and hence, 2L u(x) 2 2Lη Dηu(y) 2 =2Lη Dηu(y) 2 | | ≤ | i | · η | i | xX∈Γt xX∈Γt yX∈Γt yX∈Γt = 4L2 Dηu(y) 2 | i | yX∈Γt Then, ∗ u(x) 2 dx = ηN u(x) 2 = ηN u(x) 2 | | | | | | Z x∈Γ t∈I x∈Γt X X X ∗ 4ηN L2 Dηu(y) 2 =4L2 Dηu(y) 2 dy ≤ | i | | i | Xt∈I yX∈Γt Z 

From Prop. 6.4, the following corollaries follows straightforwardly:

59 Corollary 6.5. (Poincar´einequality for the grid derivative) If u S (Q), then ∈ 2L η u 2 D u 2 k kL ≤ √N k kL

Corollary 6.6. If i such that Dηu =0, then u =0. ∃ i

Any internal function u F(EN ) can be easily appproximated by a step functionu ˘ S (Q) by setting∈ ∈ η

u˘(x)= u(a)χQa (x) (147) aX∈Gη η This extension allows to define the step derivative Di to every function w F(RN )∗ by setting ∈ η η Di w := Di w.˘ (148)

6.2 σ-bases Definition 6.7. Let W F RN be a function space of finite dimension; we say that a family of functions⊂ σ , K RN is a σ-basis for W if every
a a∈K function u in W can be written{ as} follows: ⊂

u(x)= u(a)σa(x). K aX∈ in a unique way.

N N Given W F R and a set of points K = a1, ..., ak R , we can define ”restriction”⊂ map { }⊂
Φ: W F (K) (149) → Φ(f) = (f (a1) , ..., f (ak)) If e ,,,,e is a basis of W, then Φ can be ”represented” by the matrix { 1 m} e (a ) (150) { n l }n≤m,l≤k Definition 6.8. Let W F RN be a function space of finite dimension; we say that a set K = a , ...,⊂ a RN is: { 1 k}⊂
• independent in W if the map Φ is surjective and hence for any k-ple of points (c , ..., c ) RN , there exists f W such that 1 k ∈ ∈

f (al)= cl; l =1, ...., k. (151)

in this case the matrix (150) has rank k.

60 • complete in W if the map Φ is bijective and hence there exists a unique f W which satisfies (151); in this case ∈ det [e (a )] =0 (152) k l 6 • redundant in W if the map Φ is injective and hence ( f W, a K, f (a)=0) (f = 0) ; ∀ ∈ ∀ ∈ ⇒ in this case the matrix (150) has rank m. Notice that a set of points is complete in W if and only if it is independent and redundant. Lemma 6.9. Let W F RN be a function space of finite dimension and let K = a , ..., a RN⊂. Then it is complete if and only if there exists a σ-basis 1 m
σ (x{) . }⊂ { a }a∈K N Proof: Let K = a1, ..., an R be a of complete set points and let τ be the ”canonical”{ basis} in ⊂ F (K) namely { a}a∈K τ a(b)= δab Then, Φ−1 (τ ) is a σ-basis. If σ (x) is a σ-basis, then the map a a∈K { a }a∈K defined by Φ (σa)= τ a is bijective.  

It is evident that every finite dimensional vector space has a redundant set of points. The existence of a complete set is a consequence of the following theorem. Theorem 6.10. Let R be redundant in W . Then there exists a set C R, complete in W . ⊆

Proof: The matrix (150) has rank m = dim (W ). Let Mm be a nonsingular m m submatrix of the matrix M and let C be the image of the operator defined × by Mm. So, the system of equations σ (b)= σ (b), a,b C (153) a a ∈ has a unique solutions. Then C is complete.  Corollary 6.11. Every finite or hyperfinite dimensional vector space W has a complete system of points. Proof: If W is finite dimensional, take any redundant set of points in W and apply Th. 6.10. If the dimension of W is hyperfinite, take the Λ-limit.  Corollary 6.12. Every finite or hyperfinite dimensional vector space W has a σ-basis. Proof: It is an immediate consequence of Corollary 6.11 and Lemma 6.9. 

61 W 1 W 6.3 The spaces Λ and Λ 0 Let V = Cc . This and the next sections are devoted to construct the space VΛ so that it is possible to define a pointwise integral and a step derivative that satisfy the requests of Def. 3.2. This will be done in four steps building spaces

W 1 W 0 W V . Λ ⊂ Λ ⊂ Λ ⊂ Λ First of all we fix Q = [L,L)N (154) where L E is an infinite number such that for every ∈

u = lim uλ, uλ V λ, (155) λ↑Λ ∈ ∩ we have that supp (u) ( L,L)N . (156) ⊂ − We set 1 1 1 WΛ := lim Cc λ = lim uλ, uλ Cc λ ; (157) λ↑Λ ∩ λ↑Λ | ∈ ∩   a
WΛ = lim V λ; λ↑Λ ∩ W b = span u, ∂∗u u W 1 ; (158) Λ i | ∈ Λ c a b WΛ = WΛ + WΛ; (159) W 0 = span u,uv u, v W c . (160) Λ { | ∈ Λ} 1 0 0 Clearly, WΛ and WΛ are hyperfinite dimensional subspace of WΛ and u, v W 1 ∂∗u, v∂∗u W 0. (161) ∈ Λ ⇒ i i ∈ Λ Lemma 6.13. There exists a hyperfinite dimensional vector space W W 0 Λ ⊇ Λ which admits a σ-basis τ a such that { }a∈ΓW RN Γ ; (162) ⊂ W and supp (τ ) B (s,a) (163) a ⊂ ̺/2 where B (x ) EN is a ball of radious ̺/2 0. ̺/2 j ⊂ ∼

Proof: Let Ek k≤r be a hyperfinite partition of Q such that, for every k, diam (E ) < {̺ }0. We set k 2 ∼ 1 ∗ x y Θ (x) := ρ − dy. k rN r ZEk   ∗ where ρ is defined by (1) and hence Θ C0 . We assume that u W 0, k ∈ c ∀ ∈ Λ r < dist(supp(u), E
N Q). \

62 Then Θ is a partition of the unity of { k}k≤r Q− := x Q dist(x, EN Q) < r r ∈ | \ namely  x Q−, Θ (x)=1; (164) ∀ ∈ r Ek kX≤r Then, if u W 0, ∀ ∈ Λ u(x)= u(x) Θk(x) kX≤r Now, we set W (E )= u Θ u W 0 k · k| ∈ Λ and  W = W (E ) .... W (E ) Λ 1 ⊕ ⊕ r 0 Clearly W WΛ. Now, for every k r, take a σ-basis τ a of W (Ek) Λ ⊂ ≤ { }a∈Γk which exists by Cor. 6.11. If RN E = r = ∅, we can take Γ such that ∩ k { k} 6 k rk Γk; if we set ∈ r ΓW := Γk k[=0 then τ a a∈Γ is a σ-basis of WΛ which satisfies (162), and (163). { } W

6.4 The partitions η and η P W P Let η be a regular partition of Q with hypercubes of size P W

L N∗ ηW = , β1 2β1 ∈ where β1 is an infinite number. β1 is chosen such that ηW be so small to satisfy the following request and the lemma 6.14 below. We require that for every a ΓW (ΓW has been defined in Lemma 6.13), there is a unque R in η such W that∈ a R. Such a hypercube, will be denoted by R and we set P ∈ a

ΩW = Ra. (165) a∈[ΓW Now, we fix once for ever an infinitesimal 1 γ < (166) max m(Q) L2, Γ { · | W |} where m(Q) is the measure of Q, namely m(Q)=(2L)N .

63 Lemma 6.14. If ηW is sufficiently small, x, y R , τ (x) τ (y) γ2 (167) ∀ ∈ a | a − a |≤ ∗ τ (x)dx 1 γ2 m (R ) (168) a ≥ − a ZRa ∗
τ (x) dx γm (R ) (169) | a | ≤ a ZΩW \Ra Proof: Since the τ a’s are a finite number of continuous functions with compact support, then (167) follows from Cantor’s theorem about uniform con- tinuity. Then ∗ ∗ τ (x) 1 dx = τ (x) τ (a) dx γ2m (R ) | a − | | a − a | ≤ a ZRa ZRa and, hence

∗ ∗ ∗ τ (x)dx dx τ (x) 1 dx a ≥ − | a − | ZRa ZRa ZRa m (R ) γ2m (R ) 1 γ2 m (R ) ≥ a − a ≥ − a Moreover, if b = a,
6 ∗ ∗ τ (x) dx = τ (x) τ (b) dx γ2m (R )= γ2m (R ) | a | | a − a | ≤ b a ZRb ZRb Then, by (166)

∗ ∗ 2 τ a(x) dx = τ a(x) dx m (Ra) γ ΩW \Ra | | Rb | | ≤ Z b∈ΓXW \{a} Z b∈ΓXW \{a} = m (R ) Γ γ2 γm (R ) a | W | ≤ a 

Now we take a new partition of Q with hypercubes of size η = η /β , Pη W 2 where β2 is an infinite number, so that, setting β = β1β2 we obtain a regular partition as the one described in section 6.1 with L η = . 2β

We denote by Gη the relative grid defined by (142) and as usual, we set

η = Qs P { }s∈Gη

We take η so small (namely β2 so large) that the following lemmas be sat- isfied:

64 Lemma 6.15. If η is sufficiently small, s G , x, y Q , ∀ ∈ η ∀ ∈ s τ (x) τ˘ (y) γ2m (R ) | a − a |≤ a where the operator (˘) has ben defined by (147), namely ·

τ˘a(x)= τ a(x)χQa (x) aX∈Gη

Proof: Since the τ a’s are a finite number of continuous functions with compact support, then by Cantor’s theorem, we can choose η so small that for x, y Qs ∈ τ (x) τ (y) γ2m (R ); | a − a |≤ a then, τ (x) τ˘ (y) = τ (x) τ (s) γ2m (R ) . | a − a | | a − a |≤ a 

Lemma 6.16. If η is sufficiently small, w W 1(EN ) ∀ ∈ η ∂ w(x) D w(x) 2 γ w 2 . k i − i kL ≤ k kL Proof: We set

1 N := u W (E ) u 2 =1 K ∈ |k kL 1 N Since W (E ) is hyperfinite-dimensional, is compact and the function ∂iu are uniformly equicontinuous in . So, we canK choose η such that K x y η ∂ u (x) ∂ u (y) γ2. | − |≤ ⇒ | i − i |≤ By the Lagrange intermediate value theorem, we have that

Dηu(a)= ∂ u(ξ); ξ = a + re , η

65 N ◦ 6.5 The spaces VΛ(E ) and V (Γ) In order to define the space V we need to construct a new partition S Λ { a}a∈Γ which combines η = Ra (which is a partition of the set ΩW defined P W { }a∈ΓW by (165)) and η = Qs (which is a regular partition of Q defined by P { }s∈Gη (154)) in a suitable way. We set

Ω = Q Ω , Z \ W Γ := s G s Ω ; Γ := Γ Γ Z { ∈ η | ∈ Z } W ∪ Z Sa = Ra Qs { }a∈Γ { }a∈ΓW ∪{ }s∈ΓZ So we have constructed a hyperfinite Γ grid based on the partition Sa a∈Γ , such that, by (162), { } RN Γ Γ. ⊂ W ⊂ Notice that S is consistent with Q in the sense that χ S (Q). a a∈Γ s s∈Gη Sa η Now we set { } { } ∈ Z := span θ a Γ . { a | ∈ Z } where, θa is defined by 1 ∗ x y θ (x) := ρ 2 χ (x)= ρ − dy. a η ∗ Qa η2N η2 ZQa  
0 ∗ where ρ is defined by (1) and hence θa Cc . Notice that θa is a smooth approximation of χ and in particular we∈ have that Qa
∗ ∗ N θa(x)dx = χQa (x)dx = η . Z Z Lemma 6.17. W Z = 0 Λ ∩ { } Proof: Let u W Z. Since u Z, ∈ Λ ∩ ∈ a Γ , u (a)=0. ∀ ∈ W ΓW is a complete set for WΛ (see Def. 6.8) and u WΛ, then u =0.  ∈

◦ Finally we can define the spaces VΛ and V : V (EN )= W Z (170) Λ Λ ⊕ and V ◦ (Γ) = w◦ w V EN ; w◦ := w . (171) | ∈ Λ |Γ ◦ We will show that V (Γ) is a space of fine ultrafunctions
provided that a suit- able generalized derivative is defined. Before doing this we will examine some ◦ properties of the spaces V and VΛ. First of all observe that, by virtue of Lemma 6.17, τ a θa is { }a∈ΓW ∪{ }a∈ΓZ a basis of VΛ; now we need to build a σ-basis.

66 Lemma 6.18. If we set

θ (x) if a Γ a ∈ Z σa(x)=  , (172)  τ a(x) τ a(b)θb(x) if a ΓW  − ∈ bX∈ΓZ  then σ , Γ := Γ  Γ , is a σ-basis of V such that { a}a∈Γ W ∪ Z Λ supp (σ ) B (a). (173) a ⊂ ̺ Proof: First of all let us check that

σa (c)= δac (174) for every c Γ. ∈ • if a Γ , then, σ (c)= θ (c)= δ . ∈ Z a a ac • if a Γ , and c Γ then ∈ W ∈ Z σ (c)= τ (c) τ (b)θ (c)= τ (c) τ (c)=0 a a − a b a − a bX∈ΓZ

• if a ΓW , and c ΓW , since τ a is a σ-basis, then ∈ ∈ { }a∈ΓW σ (c)= τ (c) τ (b)θ (a)= δ 0= δ a a − a b ac − ac bX∈ΓZ So the σ ’s are Γ + Γ elements independent in V and hence they span a | W | | Z | Λ all VΛ. So, they form a σ-basis. Now let us prove (173). If σa(b) = 0, then τ a(b)θb(x) = 0 and supp (τ a) ∅ 6 ̺ 6 ∩ supp (θb) = ; hence, by(163), dist(a,b) 2 +2η < ̺.  6 ≤

The next lemma shows the reason why it has been necessary to use the partition S rather that a regular partition. { a}a∈Γ Lemma 6.19. For every a Γ, ∈ ∗ σadx > 0. Z Proof: If a Γ , σ = θ and hence ∈ Z a a ∗ ∗ N σadx = θadx = η > 0 Z Z

67 If a Γ , ∈ W ∗ ∗ ∗

τ a(b)θb(x)dx = τ a(b) θb(x)dx = τ a(b) χQb (x)dx Z b∈ΓZ b∈ΓZ Z b∈ΓZ Z X X∗ X∗

= τ a(b)χQb (x)dx = τ˘a(x)dx ΩZ Z bX∈ΓZ Z then, by Lemma 6.14, we have that

∗ ∗ σ dx = τ (x) τ (b)θ (x) dx a a − a b Z Z b∈ΓZ ! ∗ X∗ = τ (x)dx τ (b)θ (x)dx a − a b Z Z b∈ΓZ ∗ ∗ X = τ a(x)dx τ˘a(x)dx − ΩZ Z ∗ Z ∗ ∗ = τ (x)dx + τ (x)dx + (τ (x) τ˘ (x)) dx a a a − a ZRa ZΩW \Ra ZΩZ 1 γ2 m(R ) γm(R ) γm(R ) ≥ − a − a − a (1 3γ) m(R ) > 0 ≥ −
a 

6.6 The pointwise integral Now, we can define the pointwise integral for the functions in V ◦ as follows:

∗ u dx = uΛdx, (175) I Z Since VΛ is a hyperfinite space, then

VΛ = lim Vλ λ↑Λ for a suitable net of finite dimensional spaces Vλ V and hence this definition agrees with (32). In particular we have that ⊂

∗ u dx = u(a)d(a); d(a) := σa(x)dx. I aX∈Γ Z

So, if w VΛ, then ∈ ∗ w◦ dx = w dx, I Z

68 6.7 Definition of the generalized derivative We set ⊥ W 1 := u V v W 1, uv dx =0 (176) Λ ∈ Λ | ∀ ∈ Λ  I  Then, every u V can
be split as follows ∈ Λ ⊥ u = u + u ; u W 1, u W 1 (177) 1 0 1 ∈ Λ 0 ∈ Λ This splitting allows to define the generalized derivative
in the following way: u, v V , we set ∀ ∈ Λ ∗ D u◦v◦ dx = (∂∗u v + ∂∗u v u ∂∗v + Dηu v ) dx. (178) i i 1 1 i 1 0 − 0 i 1 i 0 0 I Z Now we need the following Lemma 6.20. For every u, v V and every a Γ, ∈ Λ ∈ ∗ ◦ ◦ η D u v dx D uvdx 2γ u 2 v 2 . i − i ≤ k kL ·k kL I Z

Proof: We set

F = ∂∗u Dηu u i 1 − i 1 Then, by Cor. 6.16, since u W 1, we have that 1 ∈ Λ ∗ ∗ ∗ F (x) 2 dx γ u (x) 2 dx γ u(x) 2 dx (179) | u | ≤ | 1 | ≤ | | Z Z Z Furthermore, ∗ Dηu◦v◦ dx = (∂∗u v + ∂∗u v u ∂∗v + Dηu v ) dx i i 1 1 i 1 0 − 0 i 1 i 0 0 I Z ∗ = (∂∗u v u ∂∗v + Dηu v ) dx i 1 − 0 i 1 i 0 0 Z ∗ = [(Dηu + F ) v u (Dηv + F )+ Dηu v ] dx i 1 u − 0 i 1 v i 0 0 Z ∗ ∗ = [Dηu v u Dηv + Dηv u ] dx + (F v F u ) dx 1 − 0 i 1 i 0 0 u − v 0 Z ∗ Z ∗ = [Dηu v + Dηu v + Dηu v ] dx + (F v F u ) dx i 1 i 0 1 i 0 0 u − v 0 Z ∗ ∗ Z = Dηuv dx + (F v F u ) dx i u − v 0 Z Z Then, ∗ ∗ ∗ ∗ Dηu◦v◦ dx Dηu (x) vdx (F v F u ) dx F vdx + F u dx i − ≤ u − v 0 ≤ u v 0 I Z Z Z Z

Fu L2 v L2 + Fv L2 u L2 ≤ k k ·k k k k ·k k γ u 2 v 2 + γ v 2 u 2 =2γ u 2 v 2 ≤ k kL ·k kL k kL ·k kL k kL ·k kL

69 

Finally we are ready to prove the following theorem from which Th. 3.3 follows. Theorem 6.21. V ◦ (Γ) is a fine space of ultrafunctions. Proof: We will check that V ◦ (Γ) verifies the requests of Def. 3.2. 3.2.1 follows from the construction of V ◦ (Γ) ; (31) is satisfied just taking

0 c UΛ = WΛ

c where WΛ has been defined by (159). 3.2.2a follows from (175). 3.2.2b follows from Lemma 6.19. 1 1 3.2.3a - If u = limλ↑Λ uλ, uλ C λ, then, by (157), u WΛ. Then u = u , and u = 0; then, by (178),∈ v V∩ ∈ 1 0 ∀ ∈ Λ ∗ ◦ ◦ ∗ Diu1v dx = ∂i u1v dx. I Z c ∗ 0 In particular, if you take v WΛ (see (159) and (160)), ∂i u1v WΛ VΛ and hence, by (175) ∈ ∈ ⊂

∗ ∗ ∗ ◦ ∗ ◦ ◦ ∂i u1v dx = (∂i u1v) dx = (∂i u1) v dx; Z I I then, v W c ∀ ∈ Λ ◦ ◦ ∗ ◦ ◦ Diu1v dx = (∂i u1) v dx I I ∗ c and since, by (158) and (159), ∂i u1 WΛ, we get the conclusion. 3.2.3b follows directly from (178).∈ 3.2.3c - If Du =0, then by Lemma 6.20, we have that

∗ η D uvdx 2γ u 2 v 2 i ≤ k kL ·k kL Z

η Then, taking v = Di u, we have that

η 2 η D u 2 2γ u 2 D u 2 k i kL ≤ k kL ·k i kL and hence, by the above inequality, Prop. 6.4 and (166)

η η D u 2 2γ u 2 2γ 2L D u 2 k i kL ≤ k kL ≤ · k i kL 1 η 4 L D u 2 ≤ · m(Q) L2 · k i kL · 1 η D u 2 ≤ 2 ·k i kL

70 η Then Di u L2 = 0 and the conclusion follows from Corollary 6.6. k k ∗ η 3.2.3d follows from the definition (178) of Di since both ∂i and Di are local operators. The inequality Th. 3.3 follows from lemma 6.20. 

Remark 6.22. The definition of generalized derivative given by (178) might appear rather awkward. Actually, in this contest, other possible definitions which appear more natural do not satisfy the requests of Def. 3.2. For example, if we set a ∗ η Di u = ∂i u1 + Di u0 a it turns out that Di is not antisymmetric (i.e. it does not satisfy the property 3.2.3b). If we set Dau (Da)† Dbu = i − i i 2 † b (where A denotes the adjoint of A), then Di u is antisymmetric, but it violates c the request 3.2.3a and probably (35). If we define a generalized derivative Di by the identity ∗ c ∗ η Di uvdx = (∂i u1v1dx + Di u0v0) dx, I Z also in this case, 3.2.3a and (35) do not hold. On the other hand, the definition (178) satisfies all the request of Def. 3.2. We do not know if it is possible to find a better definition.

7 Conclusive remarks

The definition 3.2 of fine ultrafunctions extends the notion of real functions in such a way that:

• (a) ”almost” all the partial differential equations (and other functional problems) have a solution; • (b) some of the main properties of the smooth functions are preserved.

The assumptions included in definition 3.2 seems quite natural; however they do not define a unique model and there is a lot of room to require others properties that allow to prove more facts about the solutions of a given problem. Let’s illustrate this point with an example: let us consider equation (135); we know that it has a unique solution which preserves the energy; however there are a lot of questions that are relevant for the physical interpretation such as:

• If f(u) grows more than u (N+2)/(N−2) most likely, this solution has sin- gular regions S Γ where| | the density of energy is an infinite number; there is a precise⊂ theorem? What can we say about the properties of S?

71 • In general the momentum is not ”exactly” preserved since the ”space” Γ is not invariant for translation (and hence we cannot apply Noether theorem); then the natural question is to know the initial conditions for which the momentum is preserved and/or the initial conditions for which the momentum is preserved up to an infinitesimal. • Under which assumptions the solutions converge to 0 as t ? → ∞ • And so on....

Some of these questions have an answer in the context of ultrafunctions provided that you work enough on a single question. However there are questions which do not have a YES/NO answer since there are models of ultrafunctions in which the answer is YES and other in which the answer is NO. A possible development of the theory is to add to the definition of ultrafunctions other properties which allows a more detailed description of the physical phenomenon described by the mathematical model. For example, in Def. 3.2 we have not included all the properties of the generalized derivative which can be deduced by Def. (178) and by a suitable approximation by Dη (which can be more accurate than the one given in Lemma 6.20). The main difficulty, if you want to add a new property to the ultrafunctions, is the proof of its consistency, namely the construction of a model. For example it is easy to prove that the Leibniz rule is not consitent with an algebra of functions which includes idempotent functions (see the discussion in section 3.4) but in general, it is difficult to guess if a ”reasonable property” is consistent with all the others. At this points, it is interesting compare the ultrafunction theory (and more in general a theory which includes infinitesimals) and a traditional theory based on real valued function spaces. In both cases there exist questions which do not have a YES/NO answer because of the Goedel incompleteness theorem, and, in both cases, we can add new axioms which allow to solve the problem. The difficult issue is the proof of the consistency of these new axioms. However, this issue is much easier in the world of Non-Archimedean mathematics, since the mathematical universe is wider and there is more room to construct a model. Let us consider an example that clarifies this point. We do not know if the Navier-Stokes equations have a smooth solution; however they have a unique ultrafunction solution. Probably this fact is not relevant for an applied mathe- matician or for an engineer since it does not help to descover new fact relative to the motion of a real fluid. However, it is possible to add new axioms which allow to prove properties of the solutions consistent with the experiments and to end with a richer mathematical model. This model can be used to get new prat- ical results even if the problem of the existence of a smooth solutions remains unsolved. For this reason we think that it is worthwile to investogate the potentialities of the Non-Archimedean mathematics.

72 References

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