The Caccioppoli Ultrafunctions

Ultrafunctions are a particular class of functions defined on a hyperreal field $\mathbb{R}^{\ast}\supset\mathbb{R}$. They have been introduced and studied in some previous works. In this paper we introduce a particular space of ultrafunctions which has special properties, especially in term of localization of functions together with their derivatives. An appropriate notion of integral is then introduced which allows to extend in a consistent way the integration by parts formula, the Gauss theorem and the notion of perimeter. This new space we introduce, seems suitable for applications to PDE's and Calculus of Variations. This fact will be illustrated by a simple, but meaningful example.


Introduction
The Caccioppoli ultrafunctions can be considered as a kind generalized functions. In many circumstances, the notion of real function is not sufficient to the needs of a theory and it is necessary to extend it. Among people working in partial differential equations, the theory of distributions of Schwartz is the most commonly used, but other notions of generalized functions have been introduced by Colombeau [13] and Sato [18,19]. This paper deals with a new kind of generalized functions, called "ultrafunctions", which have been introduced recently in [2] and developed in [6][7][8][9][10]. They provide generalized solutions to certain equations which do not have any solution, not even among the distributions.
Actually, the ultrafunctions are pointwise defined on a subset of (ℝ * ) N , where ℝ * is the field of hyperreal numbes, namely the numerical field on which nonstandard analysis (NSA in the sequel) is based. We refer to Keisler [15] for a very clear exposition of NSA and in the following, starred quantities are the natural extensions of the corresponding classical quantities.
The main novelty of this paper is that we introduce the space of Caccioppoli ultrafunctions V Λ (Ω). They satisfy special properties which are very powerful in applications to Partial Differential Equations and Calculus of Variations. The construction of this space is rather technical, but contains some relevant improvements with respect to the previous notions present in the literature (see e.g. [2,[4][5][6][7][8][9][10]).
The main peculiarities of the ultrafunctions in V Λ (Ω) are the following: there exist a generalized partial derivative D i and a generalized integral ◻ ∫ (called pointwise integral) such that the following hold: (4) To any distribution T ∈ D (Ω) we can associate an equivalence class of ultrafunctions [u] such that, for all v ∈ [u] and all φ ∈ D(Ω), where st( ⋅ ) denotes the standard part of an hyperreal number. The most relevant point, which is not present in the previous approaches to ultrafunctions, is that we are able the extend the notion of partial derivative so that it is a local operator and it satisfies the usual formula valid when integrating by parts, at the price of a suitable extension of the integral as well. In the proof of this fact, the Caccioppoli sets play a fundamental role.
It is interesting to compare the result about the Caccioppoli ultrafunctions with the well-known Schwartz impossibility theorem: Theorem (Schwartz impossibility theorem). There does not exist a differential algebra (A, +, ⊗, D) in which the distributions can be embedded, where D is a linear operator that extends the distributional derivative and satisfies the Leibniz rule (i.e. D(u ⊗ v) = Du ⊗ v + u ⊗ Dv) and ⊗ is an extension of the pointwise product on C (ℝ).
The ultrafunctions extend the space of distributions; they do not violate the Schwartz theorem since the Leibniz rule, in general, does not hold (see Remark 4.9). Nevertheless, we can prove the integration by parts rule (1.1) and the Gauss' divergence theorem (with the appropriate extension ◻ ∫ of the usual integral), which are the main tools used in the applications. These results are a development of the theory previously introduced in [9] and [11]. The theory of ultrafunctions makes deep use of the techniques of NSA presented via the notion of Λ-limit. This presentation has the advantage that a reader, which does not know NSA, is able to follow most of the arguments.
In the last section we present some very simple examples to show that the ultrafunctions can be used to perform a precise mathematical analysis of problems which are not tractable via the distributions.

Plan of the paper
In Section 2, we present a summary of the theory of Λ-limits and their role in the development of the ultrafunctions using nonstandard methods, especially in the context of transferring as much as possible the language of classical analysis. In Section 3, we define the notion of ultrafunctions, with emphasis on the pointwise integral. In Section 4, we define the most relevant notion, namely the generalized derivative, and its connections with the pointwise integral, together with comparison with the classical and distributional derivative. In Section 5, we show how to construct a space satisfying all the properties of the generalized derivative and integrals. This section is the most technical and can be skipped in a first reading. Finally, in Section 6, we present a general result and two very simple variational problem. In particular, the second problem is very elementary but without solutions in the standard H 1 -setting. Nevertheless, it has a natural and explicit candidate as solution. We show how this can be described by means of the language of ultrafunctions.

Notations
Let X be a set and let Ω be a subset of ℝ N . • P(X) denotes the power set of X and P fin (X) denotes the family of finite subsets of X. • F(X, Y) denotes the set of all functions from X to Y and F(Ω) = F(Ω, ℝ). • C (Ω) denotes the set of continuous functions defined on Ω.
• C k (Ω) denotes the set of functions defined on Ω which have continuous derivatives up to the order k.
• H k,p (Ω) denotes the usual Sobolev space of functions defined on Ω.
• If W(Ω) is any function space, then W c (Ω) will denote the function space of functions in W(Ω) having compact support.
• C 0 (Ω ∪ Ξ), Ξ ⊆ ∂Ω, denotes the set of continuous functions in C (Ω ∪ Ξ) which vanish for x ∈ Ξ. • D(Ω) denotes the set of the infinitely differentiable functions with compact support defined on Ω, and D (Ω) denotes the topological dual of D(Ω), namely the set of distributions on Ω.
, where supp is the usual notion of support of a function or a distribution.
• If W is a generic function space, its topological dual will be denoted by W and the pairing by ⟨ ⋅ , ⋅ ⟩ W • We denote by χ X the indicator (or characteristic) function of X, namely, • |X| will denote the cardinality of X.

Λ-theory
In this section we present the basic notions of Non-Archimedean Mathematics and of Nonstandard Analysis, following a method inspired by [3] (see also [2] and [6]).

Non-Archimedean fields
Here, we recall the basic definitions and facts regarding non-Archimedean fields. In the following, 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 be an ordered field and ξ ∈ . 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.

Definition 2.2. An ordered field is called non-Archimedean if it contains an infinitesimal
It is easily seen that all infinitesimal are finite, that the inverse of an infinite number is a nonzero infinitesimal number, and that the inverse of a nonzero infinitesimal number is infinite. Definition 2.3. A superreal field is an ordered field that properly extends ℝ.
It is easy to show, due to the completeness of ℝ, that there are nonzero infinitesimal numbers and infinite numbers in any superreal field. Infinitesimal numbers can be used to formalize a new notion of closeness: Definition 2.4. We say that two numbers ξ, ζ ∈ are infinitely close if ξ − ζ is infinitesimal. In this case, we Clearly, the relation ∼ of infinite closeness is an equivalence relation and we have the following theorem. Theorem 2.5. If is a superreal field, every finite number ξ ∈ is infinitely close to a unique real number r ∼ ξ , called the standard part of ξ . Given a finite number ξ , we denote its standard part by st(ξ), and we put st(ξ) = ±∞ if ξ ∈ is a positive (negative) infinite number. Definition 2.6. Let be a superreal field, and ξ ∈ a number. The monad of ξ is the set of all numbers that are infinitely close to it, i.e. mon(ξ) = {ζ ∈ | ξ ∼ ζ}, and the galaxy of ξ is the set of all numbers that are finitely close to it, i.e. gal(ξ) = {ζ ∈ | 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).

The Λ-limit
In this subsection we introduce a particular non-Archimedean field by means of Λ-theory¹ (for complete proofs and further information the reader is referred to [1,2,6]). To recall the basics of Λ-theory we have to recall the notion of superstructure on a set (see also [15]): Definition 2.7. Let E be an infinite set. The superstructure on E is the set where the sets V n (E) are defined by induction setting V 0 (E) = E and, for every n ∈ ℕ, V n+1 (E) = V n (E) ∪ P(V n (E)).
Here P(E) denotes the power set of E. By identifying the couples with the Kuratowski pairs and the functions and the relations with their graphs, it follows that V ∞ (E) contains almost every usual mathematical object that can be constructed starting with E; in particular, V ∞ (ℝ), which is the superstructure that we will consider in the following, contains almost every usual mathematical object of analysis.
Throughout this paper we let L = P fin (V ∞ (ℝ)) and we order L via inclusion. Notice that (L, ⊆) is a directed set. We add to L a point at infinity Λ ∉ L, and we define the following family of neighborhoods of Λ : where U is a fine ultrafilter on L, namely, a filter such that the following hold: In particular, we will refer to the elements of U as qualified sets and we will write Λ = Λ(U) when we want to highlight the choice of the ultrafilter. A function φ : L → E will be called net (with values in E). If φ(λ) is a real net, we have that lim if and only if for all ε > 0 there exists Q ∈ U such that for all λ ∈ Q, |φ(λ) − L| < ε. 1 Readers expert in nonstandard analysis will recognize that Λ-theory is equivalent to the superstructure constructions of Keisler (see [15] for a presentation of the original constructions of Keisler). As usual, if a property P(λ) is satisfied by any λ in a neighborhood of Λ, we will say that it is eventually satisfied.
Notice that the Λ-topology satisfies these interesting properties: Proposition 2.8. If the net φ(λ) takes values in a compact set K, then it is a converging net.
Proof. Suppose that the net φ(λ) has a subnet converging to L ∈ ℝ. We fix ε > 0 arbitrarily and we have to We argue indirectly and we assume that Q ε ∉ U. Then, by the definition of ultrafilter, N = L\Q ε ∈ U and hence This contradicts the fact that φ(λ) has a subnet which converges to L. Proposition 2.9. Assume that φ : there exists a sequence {λ n } in L such that lim n→∞ φ(λ n ) = x 0 . We refer to the sequence φ n := φ(λ n ) as a subnet of φ(λ).
Proof. It follows easily from the definitions. is uniquely defined and there exists a sequence n → φ(λ n ) which converges to v. Definition 2.11. The set of the hyperreal numbers ℝ * ⊃ ℝ is a set equipped with a topology τ such that the following hold: • Every net φ : L → ℝ has a unique limit in ℝ * if L and ℝ * are equipped with the Λ and the τ topology, respectively. • ℝ * is the closure of ℝ with respect to the topology τ. • τ is the coarsest topology which satisfies the first property.
The existence of such an ℝ * is a well-known fact in NSA. The limit ξ ∈ ℝ * of a net φ : L → ℝ with respect to the τ topology, following [2], is called the Λ-limit of φ and the following notation will be used: namely, we shall use the up-arrow "↑" to remind that the target space is equipped with the topology τ.  13. We observe that the field of hyperreal numbers is defined as a sort of completion of the real numbers. In fact, ℝ * is isomorphic to the ultrapower ℝ L /I, where The isomorphism resembles the classical one between the real numbers and the equivalence classes of Cauchy sequences. This method is well known for the construction of real numbers starting from rationals.

Natural extension of sets and functions
For our purposes it is very important that the notion of Λ-limit can be extended to sets and functions (but also to differential and integral operators) in order to have a much wider set of objects to deal with, to enlarge the notion of variational problem and of variational solution. So we will define the Λ-limit of any bounded net of mathematical objects in V ∞ (ℝ) (a net φ : L → V ∞ (ℝ) is called bounded if there exists n ∈ ℕ such that, for all λ ∈ L, φ(λ) ∈ V n (ℝ)). To do this, let us consider a net φ : L → V n (ℝ). (2.4) We will define lim λ↑Λ φ(λ) by induction on n. Definition 2.14. For n = 0, lim λ↑Λ φ(λ) is defined by (2.1). By induction we may assume that the limit is defined for n − 1 and we define it for the net (2.4) as follows: A mathematical entity (number, set, function or relation) which is the Λ-limit of a net is called internal. Definition 2.15. If for all λ ∈ L, E λ = E ∈ V ∞ (ℝ), we set lim λ↑Λ E λ = E * , namely, The set E * is called the natural extension of E.
Notice that, while the Λ-limit of a constant sequence of numbers gives this number itself, a constant sequence of sets gives a larger set, namely E * . In general, the inclusion E ⊆ E * is proper. Given any set E, we can associate to it two sets: its natural extension E * and the set E σ , where Clearly, E σ is a copy of E, however it might be different as set since, in general, X * ̸ = X.
Remark 2.16. If φ : L → X is a net with values in a topological space, we have the usual limit which, by Proposition 2.8, always exists in the Alexandrov compactification X ∪ {∞}. Moreover, we have that the Λ-limit always exists and it is an element of X * . In addition, the Λ-limit of a net is in X σ if and only if φ is eventually constant. If X = ℝ and both limits exist, then lim λ→Λ φ(λ) = st(lim λ↑Λ φ(λ)).
The above equation suggests the following definition. Definition 2.17. If X is a topological space equipped with a Hausdorff topology, and ξ ∈ X * , we set where ψ(λ) is a net of numbers such that A function which is a Λ-limit is called internal. In particular, if, for all λ ∈ L, The function f * : E * → ℝ * is called the natural extension of f . If we identify f with its graph, then f * is the graph of its natural extension.

Caccioppoli spaces of ultrafunctions
Let Ω be an open bounded set in ℝ N , and let W(Ω) be a (real or complex) vector space such that It is possible to prove (see e.g. [2]) that every space of ultrafunctions has a σ-basis. Clearly, if a, b ∈ Γ, then σ a (b) = δ ab , where δ ab denotes the Kronecker delta. Now we will introduce a class of spaces of ultrafunctions suitable for most applications. To do this, we need to recall the notion of Caccioppoli set:

Definition 3.1. A space of ultrafunctions modeled over the space W(Ω) is given by
The number is called Caccioppoli perimeter of E. From now on, with some abuse of notation, the above expression will be written as follows: this expression makes sense since "|∇(χ E )| dx" is a measure.
If E ⊂ Ω is a measurable set, we define the density function of E as follows: where η is a fixed infinitesimal and m is the Lebesgue measure. Clearly, θ E (x) is a function whose value is 1 in int(E) and 0 in ℝ N \ E; moreover, it is easy to prove that θ E (x) is a measurable function and we have that Now we can define a space V(Ω) suitable for our aims: , and n is a number which depends on f . Such a function will be called Caccioppoli function.
Notice that V(Ω) is a module over the ring C (Ω) and that, for all f ∈ V(Ω), , is a norm (and not a seminorm).
, such that, for all a ∈ Γ, the support of σ a is contained in mon(a).
The existence of a Caccioppoli space of ultrafunctions will be proved in Section 5. Remark 3.7. Usually in the study of PDEs, the function space where to work depends on the problem or equation which we want to study. The same fact is true in the world of ultrafunctions. However, the Caccioppoli space V Λ (Ω) has a special position since it satisfies the properties required by a large class of problems. First of all V Λ (Ω) ⊂ (L 1 (Ω)) * . This fact allows to define the pointwise integral (see the next subsection) for all the ultrafunctions. This integral turns out to be a very good tool. However, the space L 1 is not a good space for modeling ultrafunctions, since they are defined pointwise while the functions in L 1 are defined a.e. Thus, we are lead to the space L 1 (Ω) ∩ C (Ω), but this space does not contain functions like f(x)θ E (x) which are important in many situations; for example, the Gauss' divergence theorem can be formulated as whenever the vector field F and E are sufficiently smooth. Thus the space V Λ (Ω) seems to be the right space for a large class of problems.

The pointwise integral
From now on we will denote by V Λ (Ω) a fixed Caccioppoli space of ultrafunctions and by {σ a (x)} a∈Γ a fixed σ-basis as in Definition 3.6. If u ∈ V Λ (Ω), we have that Equality (3.2) suggests the following definition: Definition 3.8. For any internal function g : Ω * → ℝ * , we set In the sequel we will refer to ◻ ∫ as to the pointwise integral.
From Definition 3.8, we have that and, in particular, But in general these equalities are not true for L 1 functions. For example if However, for any set E ∈ C(Ω) and any function f ∈ C (Ω), Then, if f(x) ≥ 0 and E is a bounded open set, we have that since As we will see in the following part of this paper, in many cases, it is more convenient to work with the pointwise integral ◻ ∫ rather than with the natural extension of the Lebesgue integral ∫ * .
and similarly of course, the term 1 The pointwise integral allows us to define the following scalar product: From now on, the norm of an ultrafunction will be given by Moreover, we have that and hence the result. By the above equality, taking b = a, we get (3.4).

The δ-bases
Next, we will define the delta ultrafunctions: Definition 3.11. Given a point q ∈ Ω * , we denote by δ q (x) an ultrafunction in V Λ (Ω) such that and δ q (x) is called delta (or the Dirac) ultrafunction concentrated in q.
Let us see the main properties of the delta ultrafunctions: Theorem 3.12. The delta ultrafunction satisfies the following properties: (1) For every q ∈ Ω * there exists a unique delta ultrafunction concentrated in q.
Let us prove that δ q (x) actually satisfies (3.5). So δ q (x) is a delta ultrafunction concentrated in q. It is unique: in fact, if γ q (x) is another delta ultrafunction concentrated in q, then for every y ∈ Ω * we have and hence δ q (y) = γ q (y) for every y ∈ Ω * (3) We have The proof is complete.
By the definition of Γ, for all a, b ∈ Γ, we have that From this it follows readily the following result. Proposition 3.13. The set {δ a (x)} a∈Γ (Γ ⊂ Ω * ) is the dual basis of the sigma-basis; it will be called the δ-basis Let us examine the main properties of the δ-basis. Proposition 3.14. The δ-basis satisfies the following properties: Proof. (i) This is an immediate consequence of the definition of δ-basis.
(ii) By Theorem 3.10, it follows that (iii) This is an immediate consequence of (ii).

The canonical extension of functions
We have seen that every function f : Ω → ℝ has a natural extension f * : Ω * → ℝ * . However, in general, f * is not an ultrafunction; in fact, it is not difficult to prove that the natural extension f * of a function f is an ultrafunction if and only if f ∈ V(Ω). So it is useful to define an ultrafunction f°∈ V Λ (Ω) which approximates f * . More general, for any internal function u : Ω * → ℝ * , we will define an ultrafunction u°as follows.
if f : Ω → ℝ, with some abuse of notation, we set Since Ω ⊂ Γ, for any internal function u, we have that defined by P°(u) = u°is noting else but the orthogonal projection of u ∈ F(Ω) * with respect to the semidefinite bilinear form If a function f is not defined on a set S := Ω \ Θ, by convention, we define Example 3.18. By the definition above, for all x ∈ Γ, we have that We have that (see Example 3.16) .

Canonical splitting of an ultrafunction
In many applications, it is useful to split an ultrafunction u into a part w°which is the canonical extension of a standard function w and a part ψ which is not directly related to any classical object.  We will refer to as to the singular set of the ultrafunction u.
Notice that w°, the functional part of u, may assume infinite values, but they are determined by the values of w, which is a standard function defined on Ω \ Ξ. Example 3.22. Take ε ∼ 0, and In this case We conclude this section with the following trivial propositions which, nevertheless, are very useful in applications: Proof. It is an immediate consequence of Proposition 2.9.
If we use the notation introduced in Definition 2.17, the above proposition can be reformulated as follows: Proposition 3.24. If u λ ∈ V λ is weakly convergent to w in W and u := lim λ↑Λ u λ , then there is a sequence u n ∈ V Λ (Ω) which converges strongly to w in L 1 (Ω). Now set u := lim λ↑Λ u |λ| .
By Proposition 3.23, we have that On the other hand,

Differential calculus for ultrafunctions
In this section, we will equip the Caccioppoli space of ultrafunctions V Λ (Ω) with a suitable notion of derivative which generalizes the distributional derivative. Moreover, we will extend the Gauss' divergence theorem to the environment of ultrafunctions and finally we will show the relationship between ultrafunctions and distributions.

The generalized derivative
If u ∈ V Λ (Ω) ∩ [C 1 (Ω)] * , then ∂ * i u is well defined and hence, using Definition 3.17, we can define an operator as follows: However, it would be useful to extend the operator D i to all the ultrafunctions in V Λ (Ω) to include in the theory of ultrafunctions also the weak derivative. Moreover, such an extension allows to compare ultrafunctions with distributions. In this section we will define the properties that a generalized derivative must have (Definition 4.1) and in Section 5, we will show that these properties are consistent; we will do that by a construction of the generalized derivative. Definition 4.1. The generalized derivative is an operator defined on a Caccioppoli ultrafunction space V Λ (Ω) which satisfies the following properties: (I) V Λ has σ-basis {σ a (x)} a∈Γ such that, for all a ∈ Γ, the support of D i σ a is contained in mon(a).
where n E is the measure theoretic unit outer normal, integrated on the reduced boundary of E with respect to the (n − 1)-Hausdorff measure dS (see e.g. [14,Section 5.7]) and (e 1 , . . . , e N ) is the canonical basis of ℝ N .
We remark that, in the framework of the theory of Caccioppoli sets, the classical formula corresponding to (IV) is the following: for all v ∈ C (Ω), The existence of a generalized derivative will be proved in Section 5. Now let us define some differential operators: • ∇ = (∂ 1 , . . . , ∂ N ) will denote the usual gradient of standard functions, • ∇ * = (∂ * 1 , . . . , ∂ * N ) will denote the natural extension of the gradient (in the sense of NSA), • D = (D 1 , . . . , D N ) will denote the canonical extension of the gradient in the sense of the ultrafunctions (Definition 4.1). Next let us consider the divergence: Finally, we can define the Laplace operator: • △°or D 2 will denote the Laplace operator defined by D ⋅ D.

The Gauss' divergence theorem
By Definition 4.1 (IV), for any set E ∈ C Λ (Ω) and v ∈ V Λ (Ω), and by Definition 4.1 (III), If we take a vector field φ = (v 1 , . . . , v N ) ∈ [V Λ (Ω)] N , by the above identity, we get Now, if φ ∈ C 1 (Ω) and ∂E is smooth, we get the Gauss' divergence theorem: Then (4.1) is a generalization of the Gauss' theorem which makes sense for any set E ∈ C Λ (Ω). Next, we want to generalize Gauss' theorem to any measurable subset A ⊂ Ω.
First of all we need to generalize the notion of Caccioppoli perimeter p(E) to any arbitrary set. As we have seen in Section 3.1, if E ∈ C(Ω) is a special Caccioppoli set, we have that and it is possible to define an (n − 1)-dimensional measure dS as follows: In particular, if the reduced boundary of E coincides with ∂E, we have that (see [14,Section 5.7 Then the following definition is a natural generalization: Definition 4.2. If A is a measurable subset of Ω, we set In fact, the left-hand term has been defined as in particular, if ∂A is smooth and v(x) is bounded, ∑ x∈Γ v(x)χ°∂ A (x) η x is an infinitesimal number.

Theorem 4.4. If A is an arbitrary measurable subset of Ω, we have that
Proof. By Definition 4.1 (III), Then, using the definition of n°A(x) and (4.2), the above formula can be written as follows: The proof is complete.

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 subsection) as generalizations of the distributions. Definition 4.6. The space of generalized distributions on Ω is defined as follows: The equivalence class of u in V Λ (Ω) will be denoted by [u] D . Definition 4.7. Let [u] D be a generalized distribution. We say that [u] D is a bounded generalized distribution if, for all φ ∈ D(Ω), ∫ uφ * dx is finite. We denote by D GB (Ω) the set of the bounded generalized distributions.
We have the following result.

Theorem 4.8. There is a linear isomorphism
For a proof see e.g. [9].
From now on we will identify the spaces D GB (Ω) and D (Ω); so, we will identify [u] D with Φ([u] D ) and we will write [u] D ∈ D (Ω) and Moreover, with some abuse of notation, we will write also that [u] D ∈ L 2 (Ω), [u] D ∈ V(Ω), etc., meaning that the distribution [u] D can be identified with a function f in L 2 (Ω), V(Ω), etc. By our construction, this is equivalent to saying that f * ∈ [u] D . So, in this case, we have that for all φ ∈ D(Ω),

Remark 4.9.
Since an ultrafunction u : Ω * → ℝ * is univocally determined by its value in Γ, we may think of ultrafunctions as being defined only on Γ and to denote them by V Λ (Γ); the set V Λ (Γ) is an algebra which extends the algebra of continuous functions C (Ω) if it is equipped with the pointwise product.
Moreover, we recall that, by a well-known theorem of Schwartz, any tempered distribution can be represented as ∂ α f , where α is a multi-index and f is a continuous function. If we identify T = ∂ α f with the ultrafunction D α f°, we have that the set of tempered distributions S is contained in V Λ (Γ). However, the Schwartz impossibility theorem (see introduction) is not violated since (V Λ (Γ), +, ⋅ , D) is not a differential algebra, because the Leibnitz rule does not hold for some couple of ultrafunctions.

Construction of the Caccioppoli space of ultrafunctions
In this section we will prove the existence of Caccioppoli spaces of ultrafunctions (see Definition 3.6) by an explicit construction.

Construction of the space V Λ (Ω)
In this subsection we will construct a space of ultrafunctions V Λ (Ω) and in the next subsection we will equip it with a σ-basis in such a way that V Λ (Ω) becomes a Caccioppoli space of ultrafunctions according to Definition 3.6.
• B is the smallest family of sets which satisfies the above properties.
We we will refer to the family R of all the open sets which can be written by the expression (5.1) as to the family generated by R 0 .
Let us verify that Lemma 5.2. For any finite family of special Caccioppoli sets C 0 , there exists a basis B whose elements are special Caccioppoli sets. Moreover, also the set C generated by C 0 consists of special Caccioppoli sets. Proof. For any x ∈ Ω, we set We claim that {E x } x∈Ω is a basis. Since C 0 is a finite family, we also have that {E x } x∈Ω is a finite family and hence there is a finite set of indices K such that B = {E k } k∈K . Now it is easy to prove that B is a basis and it consists of special Caccioppoli sets. Also the last statement is trivial.
and we denote by B λ (Ω) and C λ (Ω) the relative basis and the generated family which exist by the previous lemma.
Lemma 5.3. The following properties hold true: where K(E) ⊂ B Λ (Ω) is a hyperfinite set and θ Q is the natural extension to C Λ (Ω) * of the function Q → θ Q defined on C Λ (Ω) by (3.1). Proof. It follows trivially by the construction.
The next lemma is a basic step for the construction of the space V Λ (Ω).   Proof. We set and we denote by ρ a smooth bell shaped function having support in B 1 (0); then the functions ρ( x−a λ r(λ) ), a λ ∈ λ ∩ Ω, have disjoint support. We set Ξ := {lim λ↑Λ a λ a λ ∈ λ ∩ Ω}, so that Ω ⊂ Ξ ⊂ Ω * and we divide all points a ∈ Ξ, among sets Ξ(Q), Q ∈ B Λ , in such a way that • if a ∈ Q, then a ∈ Ξ(Q); • if a ∈ ∂Q 1 ∩ ⋅ ⋅ ⋅ ∩ ∂Q l , there exists a unique Q j (j ≤ l) such that a ∈ Ξ(Q j ). With this construction, claims (1) and (2) are trivially satisfied. Now, for any a ∈ Ξ(Q), set and It is easy to check that the functions ζ a satisfy (3)-(6).
We set Finally, we can define the V Λ (Ω) as follows:

The σ-basis
In this subsection, we will introduce a σ-basis in such a way that V Λ (Ω) becomes a Caccioppoli space of ultrafunctions, according to Definition 3.6. Theorem 5.5. There exists a σ-basis for V Λ (Ω), {σ a (x)} a∈Γ , such that the following hold: is a δ-basis for Z(Q) ⊂ V 0 Λ (Q) (with respect to the scalar product (5.6)). In fact, if u ∈ Z(Q), then and hence, by Lemma 5.4, it follows that Next, we want to complete this basis and to get a δ-basis for V 0 Λ (Q). To this end, we take an orthonormal basis notice that this definition is not in contradiction with (5.3) since in the latter a ∈ Ξ. For every v ∈ Z(Q) ⊥ , we have that in fact, It is not difficult to realize that {d a (x)} a∈Q\Ξ generates all Z(Q) ⊥ and hence we can select a set Ξ ⋆ (Q) ⊂ Q\Ξ such that {d a (x)} a∈Ξ ⋆ (Q) is a basis for Z(Q) ⊥ . Taking we have that {d a (x)} a∈Q Γ is a basis for V 0 Λ (Q). Now let {σ a (x)} a∈Q Γ denote the dual basis of {d a (x)} a∈Q Γ namely a basis such that, for all a, b ∈ Q Γ , Clearly, it is a σ-basis for V 0 Λ (Q). In fact, if u ∈ V 0 Λ (Q), we have that Notice that if a ∈ Ξ(Q), then σ a (x) = ζ a (x). The conclusion follows taking Γ := ⋃ Q∈B Λ (Ω) Q Γ .
Brought to you by | Universita di Pisa Authenticated Download Date | 2/27/19 4:59 PM By the above theorem, the following corollary follows straightforward. Corollary 5.6. The set V Λ (Ω) is a Caccioppoli space of ultrafunctions in the sense of Definition 3.6.
If E ∈ C Λ (Ω) (see (5.2)), we set If, for any internal set A, we define then we have the following result: Proof. We have that Since uθ E ∈ V Λ (Ω), it follows from (5.5) that we can write Then by (5.7), By this equation and the fact that
and hence for all Then, by (5.9), we have, for all v ∈ V Λ (Ω), The conclusion follows from the arbitrariness of v. Next let us prove property (III). First we prove this property if u, v ∈ U 1 Λ . By (5.10), we have that Next we will compute ◻ ∫ uD i v dx and we will show that it is equal to − ◻ ∫ D i uv dx. So we replace u with v, in the above equality and we get Now, we compute ∑ Q∈B Λ (Ω) ∫ * Q ∂ * i v Q u Q dx and the last term of the above expression separately. We have that Moreover, the last term in (5.12), changing the order on which the terms are added, becomes On the right-hand side we can change the name of the variables Q and R: In the last step we have used the fact that x ∈ ∂Q ∩ ∂R implies n R (x) = −n Q (x). Replacing (5.13) and (5.14) in (5.12), we get Comparing (5.11) and the above equation, we get that Let us prove property (III) in the general case. We have that Now, replacing u with v and applying property (5.15) for u 1 Comparing the above equation with (5.16), we get that The proof is complete.

Some examples
We present a general minimization result and two very basic examples which can be analyzed in the framework of ultrafunctions. We have chosen these examples for their simplicity and also because we can give explicit solutions.

A minimization result
In this subsection we will consider a minimization problem. Let Ω be an open bounded set in ℝ N and let Ξ ⊂ ∂Ω be any nonempty portion of the boundary. We consider the following problem: minimize in the set C 1 (Ω) ∩ C 0 (Ω ∪ Ξ). We make the following assumptions: (1) a(u) ≥ 0 and a(u) ≥ k > 0 for u sufficiently large.
(3) f(x, u) is a lower semicontinuous function in u, measurable in x, such that |f(x, u)| ≤ M|u| q with 0 < q < p and M ∈ ℝ + .
Brought to you by | Universita di Pisa Authenticated Download Date | 2/27/19 4:59 PM Clearly, the above assumptions are not sufficient to guarantee the existence of a solution, not even in a Sobolev space. We refer to [20] for a survey of this problem in the framework of Sobolev spaces. On the other hand, we have selected this problem since it can be solved in the framework of the ultrafunctions.
More exactly, this problem becomes: Problem. Find an ultrafunction u ∈ V Λ (Ω) which vanishes on Ξ * and minimizes We have the following result: Theorem 6.1. If assumptions (1)-(3) are satisfied, then the functional J°(u) has a minimizer in the space Proof. The proof is based on a standard approximation by finite-dimensional spaces. Let us observe that, for each finite-dimensional space V λ , we can consider the approximate problem: The above minimization problem has a solution, being the functional coercive, due to the hypotheses on a( ⋅ ) and the fact that p > q. If we take a minimizing sequence u n λ ∈ V λ , then we can extract a subsequence weakly converging to some u λ ∈ V λ . By observing that in finite-dimensional spaces all norms are equivalent, it follows also that u n λ → u λ pointwise. Then, by the lower-semicontinuity of a and f , it follows that the pointwise limit satisfies J°(u λ ) ≤ lim inf J°(u n λ ).
Next, we use the very general properties of Λ-limits, as introduced in Section 2.2. We set u := lim λ↑Λ u λ .
Clearly, under this generality, the solution u could be very wild; however, we can state a regularization result which allows the comparison with variational and classical solutions. Theorem 6.2. Let the assumptions of the above theorem hold. If H N−1 (Ξ) > 0 and there exists ν ∈ ℝ such that a(u) ≥ ν > 0, then the minimizer has the following form: where w ∈ H 1,p (Ω) and ψ is null in the sense of distributions, namely,  Proof. Under the above hypotheses the minimization problem has an additional a priori estimate in H 1,p (Ω), due to the fact that a( ⋅ ) is bounded away from zero. Moreover, the fact that the function vanishes on a non-trascurable (N − 1)-dimensional part of the boundary shows that the generalized Poincaré inequality holds true. Hence, by Proposition 3.23, the approximating net {u λ } has a subnet {u n } such that u n → u weakly in H 1,p (Ω).
This proves the first statement, since obviously ψ := u − w ∘ vanishes in the sense of distributions. In this case, in general the minimum is not achieved in V(Ω) and hence J°(w ∘ + ψ) < J(w).
Next, if a( ⋅ ) is bounded also from above, by classical results of semicontinuity of De Giorgi (see Boccardo [12, Section 9, Theorem 9.3]) J is weakly l.s.c. Thus u is a minimizer and, by well-known results, u n → u strongly in H 1,p (Ω). This implies, by Proposition 3.23, that u ∼ w ∘ , and hence ψ is infinitesimal in H 1,p (Ω), proving the second part.

The Poisson problem in ℝ 2
Now we consider the very classical problem and it can be characterized in several ways.
First of all, it is the only solution in the Schwartz space S of tempered distributions obtained via the equationû (ξ) =φ (ξ) |ξ| 2 , (6.2) whereT denotes the Fourier transform of T. Moreover, it is the minimizer of the Dirichlet integral in the space D 1,2 (ℝ N ) which is defined as the completion of C 1 (ℝ N ) with respect to the Dirichlet norm Each of these characterizations provides a different method to prove its existence. The situation is completely different when N = 2. In this case, it is well known that the fundamental class of solutions is given by 2π ⋅ φ(x) * log |x|; however, none of the previous characterization makes sense. In fact, we cannot use equation (6.2), since 1 |ξ| 2 ∉ L 1 loc (ℝ 2 ) and hence 1 |ξ| 2 does not define a tempered distribution. Also, the space D 1,2 (ℝ 2 ) is not an Hilbert space and the functional J(u) is not bounded from below in D 1,2 (ℝ 2 ).
On the contrary, using the theory of ultrafunctions, we can treat equation (6.1) independently of the dimension.
First of all, we recall that in equation (6.1) with N ≥ 3, the boundary conditions are replaced by the condition u ∈ D 1,2 (ℝ N ). This is a sort of Dirichlet boundary condition. In the theory of ultrafunctions it is not necessary to replace the Dirichlet boundary condition with such a trick. In fact, we can reformulate the problem in the following way.
where ∆°is the "generalized" Laplacian defined in Section 4.1 and R is an infinite number such that Clearly, the solutions of the above problem are the minimizers of the Dirichlet integral , with the Dirichlet boundary condition. Notice that, in the case of ultrafunctions, the problem has the same structure independently of N. In order to prove the existence, we can use Theorem 6.1.³ The fact that J°(u) may assume infinite values does not change the structure of the problem and shows the utility of the use of infinite quantities. The relation between the classical solution w and the ultrafunction u is given by Some people might be disappointed that u depends on R and it is not a standard function; if this is the case, it is sufficient to take w = St D u and call w the standard solution of the Poisson problem with Dirichlet boundary condition at ∞. In this way we get the usual fundamental class of solutions and they can be characterized in the usual way also in the case N = 2. Concluding, in the framework of ultrafunctions, the Poisson problem with Dirichlet boundary condition is the same problem independently of the space dimension and it is very similar to the same problem when R is finite. This fact proves that the use of infinite numbers is an advantage which people should not ignore.

An explicit example
If the assumptions of Theorem 6.2 do not hold true, the solution could not be related to any standard object. For example, if H N−1 (Ξ) = 0 and f(x, u) > k|u| s (p < N, k > 0, 0 < s < q), the generalized solution u(x) takes infinite values for every x ∈ Ω. However, there are cases in which u(x) can be identified with a standard and meaningful function, but the minimization problem makes no sense in the usual mathematics. In the example which we will present here, we deal with a functional which might very well represent a physical model, even if the explicit solution cannot be interpreted in a standard world, since it involves the square of a measure (namely δ 2 ). Let us consider, for γ > 0, the one-dimensional variational problem of finding the minimum of the functional among the functions such that u(0) = 0. In particular, we are interested in the case in which a is the following degenerate function: 2 Such an R exists by overspilling (see e.g. [15][16][17]); in fact, for any r ∈ ℝ, χ Br ∈ V Λ (ℝ N ). 3 The fact that Ω is a standard set while B R is an internal set does not change the proof.
We recall that, by standard arguments, Hence, if γ < 2, the solution is explicitly computed since it turns out that 0 ≤ u(x) < 1 for all x ∈ (0, 1) and then the degeneracy does not take place.
Proof. First, we show that the generalized solution has at most one discontinuity. In fact, for γ > 2 the solution satisfies u(ξ) = 1, for some 0 < ξ < 1, and at that point the classical Euler equations are not anymore valid. On the other hand, where u > 2, the solution satisfies a regular problem, hence we are in the situation of having at least the following possible candidate as solution with a jump at and a discontinuity of derivatives at some ξ < η < 1.
In the specific case, we have (see Figure 1) We now show that this is not possible because the functional takes a lower value on the solution with only a jump at x = ξ . In fact, if we consider the functionũ (x) defined as we observe that u =ũ in [0, ξ], while u =ũ = γ(1 − x) for all x ∉ [ξ, η] and, by explicit computations, we have where Φ(s) = − s 2 + s 2 2 − s 3 6 is strictly decreasing, since Φ (s) = − 1 2 (s − 1) 2 ≤ 0. Actually, the solution is the one shown in Figure 2. Next we show that there exists a unique point ξ such that the minimum is attained. We write the functional J(u), on a generic solution with a single jump from the value u = 1 to the value u = 2 at the point 0 < ξ < 1 and such that the Euler equation is satisfied before and after ξ . We obtain the following value for the functional (in terms of the point ξ ): We observe that, for all γ > 2, To study the behavior of F(ξ), one has to solve some fourth-order equations (this could be possible in an explicit but cumbersome way), so we prefer to make a qualitative study. We evaluate hence we have that F (ξ) < 0 if and only if 0 < ξ < √ 2 γ < 1.
If γ ≥ 2(3 + 2 √ 2), the above argument fails. In this case we can observe that and in particular to investigate the structure of the singular set of u, both in the general case and in some particular situations in which it is possible to find explicit solutions (e.g., Ω = B R (0)).