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Advances in Nonlinear Analysis

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The Caccioppoli ultrafunctions

Vieri Benci
  • Corresponding author
  • Dipartimento di Matematica, Università degli Studi di Pisa, Via F. Buonarroti 1/c, 56127 Pisa, Italy
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  • De Gruyter OnlineGoogle Scholar
/ Luigi Carlo Berselli / Carlo Romano GrisantiORCID iD: https://orcid.org/0000-0001-9479-7492
Published Online: 2017-12-05 | DOI: https://doi.org/10.1515/anona-2017-0225

Abstract

Ultrafunctions are a particular class of functions defined on a hyperreal field . They have been introduced and studied in some previous works [2, 6, 7]. 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 Partial Differential Equations and Calculus of Variations. This fact will be illustrated by a simple, but meaningful example.

Keywords: Ultrafunctions; non-Archimedean mathematics; nonstandard analysis; delta function,distributions

MSC 2010: 35D05; 46F30; 03H05; 26E30

1 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 ()N, where 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 VΛ(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, 6, 7, 8, 9, 10, 4, 5]).

The main peculiarities of the ultrafunctions in VΛ(Ω) are the following: there exist a generalized partial derivative Di and a generalized integral (called pointwise integral) such that the following hold:

  • (1)

    The generalized derivative is a local operator, namely, if supp(u)E* (where E is an open set), then supp(Diu)E*.

  • (2)

    For all u,vVΛ(Ω),

    Diuv𝑑x=-uDiv𝑑x.(1.1)

  • (3)

    The “generalized” Gauss theorem holds for any measurable set A (see Theorem 4.4)

    ADφ𝑑x=Aφ𝐧A𝑑S.

  • (4)

    To any distribution T𝒟(Ω) we can associate an equivalence class of ultrafunctions [u] such that, for all v[u] and all φ𝒟(Ω),

    st(vφ𝑑x)=T,φ,

    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(uv)=Duv+uDv) 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.

1.1 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 H1-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.

1.2 Notations

Let X be a set and let Ω be a subset of N.

  • 𝒫(X) denotes the power set of X and 𝒫fin(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,).

  • 𝒞(Ω) denotes the set of continuous functions defined on Ω.

  • 𝒞k(Ω) denotes the set of functions defined on Ω which have continuous derivatives up to the order k.

  • Hk,p(Ω) denotes the usual Sobolev space of functions defined on Ω.

  • If W(Ω) is any function space, then Wc(Ω) will denote the function space of functions in W(Ω) having compact support.

  • 𝒞0(ΩΞ), ΞΩ, denotes the set of continuous functions in 𝒞(ΩΞ) which vanish for xΞ.

  • 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.

  • For any ξ(N),ρ, we set 𝔅ρ(ξ)={x(N)|x-ξ|<ρ}.

  • supp(f)=(supp(f))*, where supp is the usual notion of support of a function or a distribution.

  • 𝔪𝔬𝔫(x)={y(N)xy}, where xy means that x-y is infinitesimal.

  • 𝔤𝔞𝔩(x)={y(N)x-y is finite}.

  • If W is a generic function space, its topological dual will be denoted by W and the pairing by ,W

  • We denote by χX the indicator (or characteristic) function of X, namely,

    χX(x)={1if xX,0if xX.

  • |X| will denote the cardinality of X.

2 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]).

2.1 Non-Archimedean fields

Here, we recall the basic definitions and 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 and ξ𝕂. We say that:

  • ξ is infinitesimal if, for all positive n, |ξ|<1n.

  • ξ is finite if there exists n such that |ξ|<n.

  • ξ is infinite if, for all n, |ξ|>n (equivalently, if ξ is not finite).

Definition 2.2.

An ordered field 𝕂 is called non-Archimedean if it contains an infinitesimal ξ0.

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 𝕂 that properly extends .

It is easy to show, due to the completeness of , 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,ζ𝕂 are infinitely close if ξ-ζ is infinitesimal. In this case, we write ξζ.

Clearly, the relation of infinite closeness is an equivalence relation and we have the following theorem.

Theorem 2.5.

If K is a superreal field, every finite number ξK is infinitely close to a unique real number rξ, 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 ξ𝕂 is a positive (negative) infinite number.

Definition 2.6.

Let 𝕂 be a superreal field, and ξ𝕂 a number. The monad of ξ is the set of all numbers that are infinitely close to it, i.e.

𝔪𝔬𝔫(ξ)={ζ𝕂ξζ},

and the galaxy of ξ is the set of all numbers that are finitely close to it, i.e.

𝔤𝔞𝔩(ξ)={ζ𝕂ξ-ζis finite}.

By definition, it follows that the set of infinitesimal numbers is 𝔪𝔬𝔫(0) and that the set of finite numbers is 𝔤𝔞𝔩(0).

2.2 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

V(E)=nVn(E),

where the sets Vn(E) are defined by induction setting

V0(E)=E

and, for every n,

Vn+1(E)=Vn(E)𝒫(Vn(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(), 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=𝒫fin(V())

and we order 𝔏 via inclusion. Notice that (𝔏,) is a directed set. We add to 𝔏 a point at infinity Λ𝔏, and we define the following family of neighborhoods of Λ:

{{Λ}QQ𝒰},

where 𝒰 is a fine ultrafilter on 𝔏, namely, a filter such that the following hold:

  • For every A,B𝔏, if AB=𝔏, then A𝒰 or B𝒰.

  • For every λ𝔏 the set Q(λ):={μ𝔏λμ}𝒰.

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:𝔏E will be called net (with values in E). If φ(λ) is a real net, we have that

limλΛφ(λ)=L

if and only if for all ε>0 there exists Q𝒰 such that for all λQ,

|φ(λ)-L|<ε.

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. We fix ε>0 arbitrarily and we have to prove that Qε𝒰 where

Qε={λ𝔏|φ(λ)-L|<ε}.

We argue indirectly and we assume that Qε𝒰. Then, by the definition of ultrafilter, N=𝔏\Qε𝒰 and hence

|φ(λ)-L|εfor all λN.

This contradicts the fact that φ(λ) has a subnet which converges to L. ∎

Proposition 2.9.

Assume that φ:LE, where E is a first countable topological space; then if

limλΛφ(λ)=x0,

there exists a sequence {λn} in L such that

limnφ(λn)=x0.

We refer to the sequence φn:=φ(λn) as a subnet of φ(λ).

Proof.

It follows easily from the definitions. ∎

Example 2.10.

Let φ:𝔏V be a net with values in a bounded set of a reflexive Banach space equipped with the weak topology; then

v:=limλΛφ(λ)

is uniquely defined and there exists a sequence nφ(λn) which converges to v.

Definition 2.11.

The set of the hyperreal numbers is a set equipped with a topology τ such that the following hold:

  • Every net φ:𝔏 has a unique limit in if 𝔏 and are equipped with the Λ and the τ topology, respectively.

  • is the closure of with respect to the topology τ.

  • τ is the coarsest topology which satisfies the first property.

The existence of such an is a well-known fact in NSA. The limit ξ of a net φ:𝔏 with respect to the τ topology, following [2], is called the Λ-limit of φ and the following notation will be used:

ξ=limλΛφ(λ),(2.1)

namely, we shall use the up-arrow “” to remind that the target space is equipped with the topology τ.

Given

ξ:=limλΛφ(λ)andη:=limλΛψ(λ),

we set

ξ+η:=limλΛ(φ(λ)+ψ(λ)),(2.2)

and

ξη:=limλΛ(φ(λ)ψ(λ)).(2.3)

Then the following well-known theorem holds:

Theorem 2.12.

The definitions in (2.2) and (2.3) are well posed and R*, equipped with these operations, is a non-Archimedean field.

Remark 2.13.

We observe that the field of hyperreal numbers is defined as a sort of completion of the real numbers. In fact, * is isomorphic to the ultrapower /𝔏, where

={φ:𝔏φ(λ)=0 eventually}.

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.

2.3 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() (a net φ:𝔏V() is called bounded if there exists n such that, for all λ𝔏, φ(λ)Vn()). To do this, let us consider a net

φ:𝔏Vn().(2.4)

We will define limλΛφ(λ) by induction on n.

Definition 2.14.

For n=0, limλΛφ(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:

limλΛφ(λ)={limλΛψ(λ)|ψ:𝔏Vn-1(),ψ(λ)φ(λ) for all λ𝔏}.

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 λ𝔏, Eλ=EV(), we set limλΛEλ=E, namely,

E:={limλΛψ(λ)|ψ(λ)E}.

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 EE is proper.

Given any set E, we can associate to it two sets: its natural extension E and the set Eσ, where

Eσ={XXE}.

Clearly, Eσ is a copy of E, however it might be different as set since, in general, XX.

Remark 2.16.

If φ:𝔏X is a net with values in a topological space, we have the usual limit

limλΛφ(λ),

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 Xσ if and only if φ is eventually constant. If X= and both limits exist, then

limλΛφ(λ)=st(limλΛφ(λ)).

The above equation suggests the following definition.

Definition 2.17.

If X is a topological space equipped with a Hausdorff topology, and ξX*, we set

StX(ξ)=limλΛφ(λ)

if there is a net φ:𝔏X converging in the topology of X and such that

ξ=limλΛφ(λ),

and

StX(ξ)=

otherwise.

By the above definition we have that

limλΛφ(λ)=StX(limλΛφ(λ)).

Definition 2.18.

Let

fλ:Eλ,λ𝔏,

be a net of functions. We define a function

f:(limλΛEλ)

as follows: for every ξ(limλΛEλ) we set

f(ξ):=limλΛfλ(ψ(λ)),

where ψ(λ) is a net of numbers such that

ψ(λ)EλandlimλΛψ(λ)=ξ.

A function which is a Λ-limit is called internal. In particular, if, for all λ𝔏,

fλ=f,f:E,

we set

f=limλΛfλ.

The function f:E is called the natural extension of f. If we identify f with its graph, then f is the graph of its natural extension.

2.4 Hyperfinite sets and hyperfinite sums

Definition 2.19.

An internal set is called hyperfinite if it is the Λ-limit of a net φ:𝔏𝔉, where 𝔉 is a family of finite sets.

For example, if EV(), the set

E~=limλΛ(λE)

is hyperfinite. Notice that

EσE~E*,

so we can say that every set is contained in a hyperfinite set.

It is possible to add the elements of an hyperfinite set of numbers (or vectors) as follows: let

A:=limλΛAλ

be an hyperfinite set of numbers (or vectors); then the hyperfinite sum of the elements of A is defined in the following way:

aAa=limλΛaAλa.

In particular, if Aλ={a1(λ),,aβ(λ)(λ)} with β(λ), then setting

β=limλΛβ(λ),

we use the notation

j=1βaj=limλΛj=1β(λ)aj(λ).

3 Ultrafunctions

3.1 Caccioppoli spaces of ultrafunctions

Let Ω be an open bounded set in N, and let W(Ω) be a (real or complex) vector space such that

𝒟(Ω¯)W(Ω)L1(Ω).

Definition 3.1.

A space of ultrafunctions modeled over the space W(Ω) is given by

WΛ(Ω):=limλΛWλ(Ω)={limλΛfλ|fλWλ(Ω)},

where Wλ(Ω)W(Ω) is an increasing net of finite-dimensional spaces such that

Wλ(Ω)Span(W(Ω)λ).

So, given any vector space of functions W(Ω), the space of ultrafunction generated by {Wλ(Ω)} is a vector space of hyperfinite dimension that includes W(Ω)σ, as well as other functions in W(Ω)*. Hence the ultrafunctions are particular internal functions

u:Ω¯*.

Definition 3.2.

Given a space of ultrafunctions WΛ(Ω), a σ-basis is an internal set of ultrafunctions {σa(x)}aΓ such that ΩΓΩ* and for all uWΛ(Ω), we can write

u(x)=aΓu(a)σa(x).

It is possible to prove (see e.g. [2]) that every space of ultrafunctions has a σ-basis. Clearly, if a,bΓ, then σa(b)=δab, where δab 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 3.3.

A Caccioppoli set E is a Borel set such that χEBV, namely, such that (χE) (the distributional gradient of the characteristic function of E) is a finite Radon measure concentrated on E.

The number

p(E):=|(χE)|,1

is called Caccioppoli perimeter of E. From now on, with some abuse of notation, the above expression will be written as follows:

|(χE)|𝑑x;

this expression makes sense since “|(χE)|dx” is a measure.

If EΩ¯ is a measurable set, we define the density function of E as follows:

θE(x)=st(m(Bη(x)E*)m(Bη(x)(Ω¯)*)),(3.1)

where η is a fixed infinitesimal and m is the Lebesgue measure.

Clearly, θE(x) is a function whose value is 1 in int(E) and 0 in NE¯; moreover, it is easy to prove that θE(x) is a measurable function and we have that

θE(x)𝑑x=m(E);

also, if E is a bounded Caccioppoli set,

|θE|𝑑x=p(E).

Definition 3.4.

A set E is called special Caccioppoli set if it is open, bounded and m(E)=0. The family of special Caccioppoli sets will be denoted by (Ω).

Now we can define a space V(Ω) suitable for our aims:

Definition 3.5.

A function fV(Ω) if and only if

f(x)=k=1nfk(x)θEk(x),

where fk𝒞(N), Ek(Ω), 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 fV(Ω),

(|f(x)|dx=0)(xN,f(x)=0).

Hence, in particular, (|f(x)|2𝑑x)12, is a norm (and not a seminorm).

Definition 3.6.

The space VΛ(Ω) is called Caccioppoli space of ultrafunctions if it satisfies the following properties:

  • (i)

    VΛ(Ω) is modeled on the space V(Ω).

  • (ii)

    VΛ(Ω) has a σ-basis {σa(x)}aΓ, Γ(N)*, such that, for all aΓ, the support of σa 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 VΛ(Ω) has a special position since it satisfies the properties required by a large class of problems. First of all VΛ(Ω)(L1(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 L1 is not a good space for modeling ultrafunctions, since they are defined pointwise while the functions in L1 are defined a.e. Thus, we are lead to the space L1(Ω)𝒞(Ω¯), but this space does not contain functions like f(x)θE(x) which are important in many situations; for example, the Gauss’ divergence theorem can be formulated as

F(x)θE(x)𝑑x=E𝐧F(x)𝑑S

whenever the vector field F and E are sufficiently smooth. Thus the space VΛ(Ω) seems to be the right space for a large class of problems.

3.2 The pointwise integral

From now on we will denote by VΛ(Ω) a fixed Caccioppoli space of ultrafunctions and by {σa(x)}aΓ a fixed σ-basis as in Definition 3.6. If uVΛ(Ω), we have that

*u(x)𝑑x=aΓu(a)ηa,(3.2)

where

ηa:=*σa(x)𝑑x.

Equality (3.2) suggests the following definition:

Definition 3.8.

For any internal function g:Ω**, we set

g(x)𝑑x:=qΓg(q)ηq.

In the sequel we will refer to as to the pointwise integral.

From Definition 3.8, we have that

*u(x)𝑑x=u(x)𝑑x,uVΛ(Ω),

and, in particular,

f(x)𝑑x=f*(x)𝑑x,fV(Ω).

But in general these equalities are not true for L1 functions. For example if

f(x)={1if x=x0Ω,0if xx0,

we have that

*f*(x)𝑑x=f(x)𝑑x=0,

while

f*(x)𝑑x=ηx0>0.

However, for any set E(Ω) and any function f𝒞(Ω¯),

f*(x)θE*(x)𝑑x=Ef(x)𝑑x;

in fact,

f*(x)θE(x)𝑑x=*f*(x)θE*(x)𝑑x=f(x)θE(x)𝑑x=Ef(x)𝑑x.

Then, if f(x)0 and E is a bounded open set, we have that

f*(x)χE*𝑑x<f*(x)θE*(x)𝑑x<f*(x)χE¯*𝑑x.

since

χE<θE<χE¯.

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 *.

Example 3.9.

If E is smooth, we have that, for all xE, θE(x)=12 and hence, if E is open,

f*(x)χE*(x)𝑑x=f*(x)θE*(x)𝑑x-12f*(x)χE*(x)𝑑x=Ef(x)𝑑x-12f*(x)χE*(x)𝑑x,

and similarly

f*(x)χE¯*(x)𝑑x=Ef(x)𝑑x+12f*(x)χE*(x)𝑑x;

of course, the term 12f*(x)χE*(x)𝑑x is an infinitesimal number and it is relevant only in some particular problems.

The pointwise integral allows us to define the following scalar product:

u(x)v(x)𝑑x=qΓu(q)v(q)ηq.(3.3)

From now on, the norm of an ultrafunction will be given by

u=(|u(x)|2𝑑x)12.

Notice that

u(x)v(x)𝑑x=*u(x)v(x)𝑑xuvVΛ(Ω).

Theorem 3.10.

If {σa(x)}aΓ is a σ-basis, then {σa(x)ηa}aΓ is a orthonormal basis with respect to the scalar product (3.3). Hence for every uVΛ(Ω),

u(x)=qΓ1ηq(u(ξ)σq(ξ)𝑑ξ)σq(x).

Moreover, we have that

σa2=ηafor all aΓ.(3.4)

Proof.

By (3.3), we have that

σa(x)σb(x)𝑑x=qΓσa(q)σb(q)ηq=qΓδaqδbqηq=δabηa,

and hence the result. By the above equality, taking b=a, we get (3.4). ∎

3.3 The δ-bases

Next, we will define the delta ultrafunctions:

Definition 3.11.

Given a point qΩ, we denote by δq(x) an ultrafunction in VΛ(Ω) such that

v(x)δq(x)𝑑x=v(q)for all vVΛ(Ω),(3.5)

and δq(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 qΩ¯ there exists a unique delta ultrafunction concentrated in q.

  • (2)

    For every a,bΩ¯*,δa(b)=δb(a).

  • (3)

    δq2=δq(q).

Proof.

(1) Let {ej}j=1β be an orthonormal real basis of VΛ(Ω), and set

δq(x)=j=1βej(q)ej(x).

Let us prove that δq(x) actually satisfies (3.5). Let v(x)=j=1βvjej(x) be any ultrafunction. Then

v(x)δq(x)𝑑x=(j=1βvjej(x))(k=1βek(q)ek(x))𝑑x=j=1βk=1βvjek(q)ej(x)ek(x)𝑑x=j=1βk=1βvjek(q)δjk=k=1βvkek(q)=v(q).

So δq(x) is a delta ultrafunction concentrated in q. It is unique: in fact, if γq(x) is another delta ultrafunction concentrated in q, then for every yΩ¯* we have

δq(y)-γq(y)=(δq(x)-γq(x))δy(x)𝑑x=δy(q)-δy(q)=0,

and hence δq(y)=γq(y) for every yΩ¯.

(2) We have

δa(b)=δa(x)δb(x)𝑑x=δb(a).

(3) We have

δq2=δq(x)δq(x)𝑑x=δq(q).

The proof is complete. ∎

By the definition of Γ, for all a,bΓ, we have that

δa(x)σb(x)𝑑x=σa(b)=δab.

From this it follows readily the following result.

Proposition 3.13.

The set {δa(x)}aΓ (ΓΩ) is the dual basis of the sigma-basis; it will be called the δ-basis of VΛ(Ω) .

Let us examine the main properties of the δ-basis.

Proposition 3.14.

The δ-basis satisfies the following properties:

  • (i)

    One has

    u(x)=qΓ[σq(ξ)u(ξ)𝑑ξ]δq(x).

  • (ii)

    For all a,bΓ, σa(x)=ηaδa(x).

  • (iii)

    For all aΓ,

    δa2=δa(x)2𝑑x=δa(a)=ηa-1.

Proof.

(i) This is an immediate consequence of the definition of δ-basis.

(ii) By Theorem 3.10, it follows that

δa(x)=qΓ1ηq(δa(ξ)σq(ξ)𝑑ξ)σq(x)=qΓ1ηqδaqσq(x)=1ηaσa(x).

(iii) This is an immediate consequence of (ii). ∎

3.4 The canonical extension of functions

We have seen that every function f:Ω has a natural extension f*:Ω**. 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 fV(Ω). So it is useful to define an ultrafunction fVΛ(Ω) which approximates f*. More general, for any internal function u:Ω**, we will define an ultrafunction u as follows.

Definition 3.15.

If u:Ω** is an internal function, we define uVΛ(Ω) by the formula

u(x)=qΓu(q)σq(x);

if f:Ω, with some abuse of notation, we set

f(x)=(f*)(x)=qΓf*(q)σq(x).

Since ΩΓ, for any internal function u, we have that

u(x)=u(x)for all xΩ,

and

u(x)=u(x) for all xΩ*uVΛ(Ω).

Notice that

P:𝔉(Ω)*VΛ(Ω)

defined by P(u)=u is noting else but the orthogonal projection of u𝔉(Ω)* with respect to the semidefinite bilinear form

u(x)h(x)𝑑x.

Example 3.16.

If f𝒞(N), and E(Ω), then fθEV(Ω) and hence

(fθE)=f*θE*.

Definition 3.17.

If a function f is not defined on a set S:=ΩΘ, by convention, we define

f(x)=qΓΘ*f*(q)σq(x).

Example 3.18.

By the definition above, for all xΓ, we have that

(1|x|)={1|x|if x0,0if x=0.

If f𝒞(Ω), then ff* unless fVΛ(Ω). Let us examine what f looks like.

Theorem 3.19.

Let f:ΩR be continuous in a bounded open set AΩ. Then, for all xA* with mon(x)A*, we have that

f(x)=f*(x).

Proof.

Fix x0A. Since A is bounded, there exists a set E(Ω) such that

𝔪𝔬𝔫(x0)E*A*.

We have that (see Example 3.16)

f(x)=aΓf*(a)σa(x)=aΓf*(a)θE*(a)σa(x)+aΓf*(a)(1-θE*(a))σa(x)=f*(x)θE*(x)+aΓE*f*(a)(1-θE*(a))σa(x).

Since x0E*, it follows that θE*(x0)=1; moreover, since 𝔪𝔬𝔫(x0)E*, by Definition 3.6 (ii),

σa(x0)=σx0(a)=0for all aΓE*.

Then f(x0)=f*(x0). ∎

Corollary 3.20.

If fC(Ω), then, for any xΩ* such that |x| is finite, we get

f(x)=f*(x).

3.5 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.

If uVΛ(Ω), we set

Ξ={xΩu(x) is infinite}

and

w¯(x)={st(u(x))if xΩΞ,0if xΞ.

Definition 3.21.

For every ultrafunction u consider the splitting

u=w+ψ,

where

  • w=w¯|ΩΞ and w, which is defined by Definition 3.17, is called the functional part of u,

  • ψ:=u-w is called the singular part of u.

We will refer to

S:={xΩ*ψ(x)0}

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 ΩΞ.

Example 3.22.

Take ε0, and

u(x)=1x2+ε2.

In this case

w(x)=1x2,ψ(x)={-ε2x2(x2+ε2)if x0,1ε2if x=0,S:={x*ψ(x)0}𝔪𝔬𝔫(0).

We conclude this section with the following trivial propositions which, nevertheless, are very useful in applications:

Proposition 3.23.

Let W be a Banach space such that D(Ω)WLloc1(Ω) and assume that uλVλ is weakly convergent in W. Then if

u=w+ψ

is the canonical splitting of u:=limλΛuλ, there exists a subnet un:=uλn such that

limnun=wweakly in W

and

ψv𝑑x0for all vW.

Moreover, if

limnun-wW=0,

then ψW0.

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 uλVλ is weakly convergent to w in W and u:=limλΛuλ, then

w=StWweak(u).

If uλ is strongly convergent to w in W, then

w=StW(u).

An immediate consequence of Proposition 3.23 is the following:

Corollary 3.25.

If wL1(Ω), then

w(x)𝑑xw(x)𝑑x.

Proof.

Since VΛ(Ω) is dense in L1(Ω), there is a sequence unVΛ(Ω) which converges strongly to w in L1(Ω). Now set

u:=limλΛu|λ|.

By Proposition 3.23, we have that

u=w+ψ

with ψL1*0. Since u and w are in VΛ(Ω), it also follows that ψVΛ(Ω), so that ψ𝑑x=*ψ𝑑x0. Then

u(x)𝑑xw(x)𝑑x.

On the other hand,

u(x)𝑑x=*u(x)𝑑x=limλΛu|λ|𝑑xlimλΛu|λ|𝑑x=limnΩun𝑑x=w(x)𝑑x.

4 Differential calculus for ultrafunctions

In this section, we will equip the Caccioppoli space of ultrafunctions VΛ(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.

4.1 The generalized derivative

If uVΛ(Ω)[𝒞1(Ω)]*, then i*u is well defined and hence, using Definition 3.17, we can define an operator

Di:VΛ(Ω)[𝒞1(Ω)]*VΛ(Ω)

as follows:

Diu=(i*u).

However, it would be useful to extend the operator Di to all the ultrafunctions in VΛ(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

Di:VΛ(Ω)VΛ(Ω)

is an operator defined on a Caccioppoli ultrafunction space VΛ(Ω) which satisfies the following properties:

  • (I)

    VΛ has σ-basis {σa(x)}aΓ such that, for all aΓ, the support of Diσa is contained in 𝔪𝔬𝔫(a).

  • (II)

    If uVΛ(Ω)[𝒞1(Ω)]*, then

    Diu=(i*u).

  • (III)

    For all u,vVΛ(Ω),

    Diuv𝑑x=-uDiv𝑑x.

  • (IV)

    If E(Ω), then for all vVΛ(Ω),

    DiθEv𝑑x=-*Ev(𝐞i𝐧E)dS,

    where 𝐧E 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 (𝐞1,,𝐞N) is the canonical basis of N.

We remark that, in the framework of the theory of Caccioppoli sets, the classical formula corresponding to (IV) is the following: for all v𝒞(Ω),

iθEvdx=-Ev(𝐞i𝐧E)𝑑S.

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=(D1,,DN) will denote the canonical extension of the gradient in the sense of the ultrafunctions (Definition 4.1).

Next let us consider the divergence:

  • φ=1φ1++NφN will denote the usual divergence of standard vector fields φ[𝒞1(Ω¯)]N,

  • *φ=1*φ1++N*φN will denote the divergence of internal vector fields φ[𝒞1(Ω¯)*]N,

  • Dφ will denote the unique ultrafunction DφVΛ(Ω) such that, for all vVΛ(Ω),

    Dφv𝑑x=-φ(x)Dv𝑑x.

Finally, we can define the Laplace operator:

  • or D2 will denote the Laplace operator defined by DD.

4.2 The Gauss’ divergence theorem

By Definition 4.1 (IV), for any set EΛ(Ω) and vVΛ(Ω),

DiθEv𝑑x=-*Ev(𝐞i𝐧E)dS,

and by Definition 4.1 (III),

DivθE𝑑x=*Ev(𝐞i𝐧E)dS.

If we take a vector field φ=(v1,,vN)[VΛ(Ω)]N, by the above identity, we get

DφθE𝑑x=*Eφ𝐧EdS.(4.1)

Now, if φ𝒞1(Ω¯) and E is smooth, we get the Gauss’ divergence theorem:

Eφ𝑑x=Eφ𝐧E𝑑S.

Then (4.1) is a generalization of the Gauss’ theorem which makes sense for any set EΛ(Ω). Next, we want to generalize Gauss’ theorem to any measurable subset AΩ.

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(Ω) is a special Caccioppoli set, we have that

p(E)=|θE|𝑑x,

and it is possible to define an (n-1)-dimensional measure dS as follows:

Ev(x)𝑑S:=|θE|v(x)𝑑x.

In particular, if the reduced boundary of E coincides with E, we have that (see [14, Section 5.7])

Ev(x)𝑑S=Ev(x)𝑑N-1.

Then the following definition is a natural generalization:

Definition 4.2.

If A is a measurable subset of Ω, we set

p(A):=|DθA|𝑑x

and for all vVΛ(Ω),

Av(x)𝑑S:=v(x)|DθA|𝑑x.(4.2)

Remark 4.3.

Notice that

Av(x)𝑑Sv(x)χA(x)𝑑x.

In fact, the left-hand term has been defined as

Av(x)𝑑S=xΓv(x)|DθA(x)|ηx

while the right-hand term is

v(x)χA(x)𝑑x=xΓv(x)χA(x)ηx;

in particular, if A is smooth and v(x) is bounded, xΓv(x)χA(x)ηx is an infinitesimal number.

Theorem 4.4.

If A is an arbitrary measurable subset of Ω, we have that

DφθA𝑑x=Aφ𝐧A(x)𝑑S,(4.3)

where

𝐧A(x)={-DθA(x)|DθA(x)|if DθA(x)0,0if DθA(x)=0.

Proof.

By Definition 4.1 (III),

DφθA𝑑x=-φDθA𝑑x.

Then, using the definition of 𝐧A(x) and (4.2), the above formula can be written as follows:

DφθA𝑑x=φ𝐧A|DθA|𝑑x=Aφ𝐧A𝑑S.

The proof is complete. ∎

Clearly, if EΛ(Ω), then

Eφ𝐧E𝑑S=Eφ𝐧E𝑑S.

Example 4.5.

If A is the Koch snowflake, then the usual Gauss’ theorem makes no sense since p(A)=+; on the other hand equation (4.3) holds true. Moreover, the perimeter in the sense of ultrafunction is an infinite number given by Definition 4.2. In general, if A is a d-dimensional fractal set, it is an interesting open problem to investigate the relation between its Hausdorff measure and the ultrafunction “measure” dS=|DθA|dx.

4.3 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:

𝒟G(Ω)=VΛ(Ω)/N,

where

N={τVΛ(Ω)|τφ𝑑x0 for all φ𝒟(Ω)}.

The equivalence class of u in VΛ(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 φ𝒟(Ω), uφ*𝑑x is finite. We denote by 𝒟GB(Ω) the set of the bounded generalized distributions.

We have the following result.

Theorem 4.8.

There is a linear isomorphism

Φ:𝒟GB(Ω)𝒟(Ω),Φ([u]𝒟),φ𝒟(Ω)=st(uφ𝑑x).

Proof.

For a proof see e.g. [9]. ∎

From now on we will identify the spaces 𝒟GB(Ω) and 𝒟(Ω); so, we will identify [u]𝒟 with Φ([u]𝒟) and we will write [u]𝒟𝒟(Ω) and

[u]𝒟,φ𝒟(Ω):=Φ[u]𝒟,φ𝒟(Ω)=st(uφ𝑑x).

Moreover, with some abuse of notation, we will write also that [u]𝒟L2(Ω), [u]𝒟V(Ω), etc., meaning that the distribution [u]𝒟 can be identified with a function f in L2(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 φ𝒟(Ω),

[u]𝒟,φ𝒟(Ω)=st(uφ𝑑x)=st(fφ𝑑x)=fφ𝑑x.

Remark 4.9.

Since an ultrafunction u:Ω** is univocally determined by its value in Γ, we may think of ultrafunctions as being defined only on Γ and to denote them by VΛ(Γ); the set VΛ(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 αf, where α is a multi-index and f is a continuous function. If we identify T=αf with the ultrafunction Dαf, we have that the set of tempered distributions 𝒮 is contained in VΛ(Γ). However, the Schwartz impossibility theorem (see introduction) is not violated since (VΛ(Γ),+,,D) is not a differential algebra, because the Leibnitz rule does not hold for some couple of ultrafunctions.

5 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.

5.1 Construction of the space VΛ(Ω)

In this subsection we will construct a space of ultrafunctions VΛ(Ω) and in the next subsection we will equip it with a σ-basis in such a way that VΛ(Ω) becomes a Caccioppoli space of ultrafunctions according to Definition 3.6.

Definition 5.1.

Given a family of open sets 0, we say that a family of open sets 𝔅={Ek}kK is a basis for 0 if

  • for all kh, EkEh=,

  • for all A0, there is a set of indices KEK such that

    A=int(kKEEk¯),(5.1)

  • 𝔅 is the smallest family of sets which satisfies the above properties.

We we will refer to the family of all the open sets which can be written by the expression (5.1) as to the family generated by 0.

Let us verify that

Lemma 5.2.

For any finite family of special Caccioppoli sets C0, there exists a basis B whose elements are special Caccioppoli sets. Moreover, also the set C generated by C0 consists of special Caccioppoli sets.

Proof.

For any xΩ, we set

Ex={A0xA}.

We claim that {Ex}xΩ is a basis. Since 0 is a finite family, we also have that {Ex}xΩ is a finite family and hence there is a finite set of indices K such that 𝔅={Ek}kK. Now it is easy to prove that 𝔅 is a basis and it consists of special Caccioppoli sets. Also the last statement is trivial. ∎

We set

0,λ(Ω):=λ(Ω),

and we denote by 𝔅λ(Ω) and λ(Ω) the relative basis and the generated family which exist by the previous lemma.

Now set

Λ(Ω)=limλΛλ(Ω),𝔅Λ(Ω)=limλΛ𝔅λ(Ω).(5.2)

Lemma 5.3.

The following properties hold true:

  • Λ(Ω) and 𝔅Λ(Ω) are hyperfinite.

  • (Ω)σΛ(Ω)(Ω)*.

  • If EΛ(Ω) , then

    θE=QK(E)θQ(x),

    where K(E)𝔅Λ(Ω) is a hyperfinite set and θQ is the natural extension to Λ(Ω)* of the function QθQ defined on 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 VΛ(Ω).

Lemma 5.4.

For any QBΛ(Ω) there exists a set Ξ(Q)Q¯Ω, and a family of functions {ζa}aΞ(Q)such that the following hold:

  • (1)

    Ξ:={Ξ(Q)Q𝔅Λ(Ω)} is a hyperfinite set, and ΩΞΩ*.

  • (2)

    If Q,R𝔅Λ(Ω) and QR , then Ξ(Q)Ξ(R)=.

  • (3)

    If aΞ(Q) , then there exists fa𝒞1(Ω)* such that ζa=faθQ.

  • (4)

    For any a,bΞ, ab implies supp(ζa)supp(ζb)=.

  • (5)

    ζa0.

  • (6)

    For any aΞ,

    ζa(a)=1.

Proof.

We set

r(λ)=13min{d(x,y)x,yλΩ},

and we denote by ρ a smooth bell shaped function having support in B1(0); then the functions ρ(x-aλr(λ)), aλλΩ, have disjoint support. We set

Ξ:={limλΛaλ|aλλΩ},

so that ΩΞΩ* and we divide all points aΞ, among sets Ξ(Q), Q𝔅Λ, in such a way that

  • if aQ, then aΞ(Q);

  • if aQ1Ql, there exists a unique Qj (jl) such that aΞ(Qj).

With this construction, claims (1) and (2) are trivially satisfied.

Now, for any aΞ(Q), set

ρa(x):=limλΛρ(x-aλr(λ)),

and

ζa(x):=ρa(x)θQ(x)ρa(a)θQ(a).(5.3)

It is easy to check that the functions ζa satisfy (3)–(6). ∎

We set

VΛ1(Ω)=Span({ζa}aΞ)+limλΛ(λ𝒞1(Ω¯)),

and

VΛ1(Q)={uθQuVΛ1(Ω)};

so we have that, for any aΞ(Q), ζaVΛ1(Q). Also, we set

VΛ0(Ω)=Span({f,if,fg,giff,gVΛ1(Ω),i=1,,N}+limλΛ(λ𝒞(Ω¯)))(5.4)

and

VΛ0(Q)={uθQuVΛ0(Ω)}.

Finally, we can define the VΛ(Ω) as follows:

VΛ(Ω)=Q𝔅Λ(Ω)VΛ0(Q).

Namely, if uVΛ(Ω), then

u(x)=Q𝔅Λ(Ω)uQ(x)θQ(x)(5.5)

with uQVΛ0(Ω).

5.2 The σ-basis

In this subsection, we will introduce a σ-basis in such a way that VΛ(Ω) becomes a Caccioppoli space of ultrafunctions, according to Definition 3.6.

Theorem 5.5.

There exists a σ-basis for VΛ(Ω), {σa(x)}aΓ, such that the following hold:

  • (1)

    ΩΓΩ*.

  • (2)

    Γ=Q𝔅Λ(Ω)QΓ , where QΓQΓQ¯Γ and QΓRΓ= for QR.

  • (3)

    {σa(x)}aQΓ is a σ -basis for VΛ0(Q).

Proof.

First we introduce in VΛ(Ω) the following scalar product:

u,v=*uv𝑑x.(5.6)

For any Q𝔅Λ(Ω) we set

Z(Q)={aΞ(Q)γaζa(x)|γa*},

where Ξ(Q) and the functions {ζa}aΞ are defined in Lemma 5.4. If we set

𝔡a(x)=ζa(x)*|ζa(x)|2𝑑x,

we have that

{𝔡a(x)}aΞ(Q)

is a δ-basis for Z(Q)VΛ0(Q) (with respect to the scalar product (5.6)). In fact, if uZ(Q), then

u(x)=bΞ(Q)u(b)ζb(x),

and hence, by Lemma 5.4, it follows that

*u(x)𝔡a(x)𝑑x=*bΞ(Q)u(b)ζb(x)𝔡a(x)dx=bΞ(Q)u(b)*ζb(x)𝔡a(x)𝑑x=bΞ(Q)u(b)*ζb(x)(ζa(x)*|ζa(x)|2𝑑x)𝑑x=bΞ(Q)u(b)δab=u(a).

Next, we want to complete this basis and to get a δ-basis for VΛ0(Q). To this end, we take an orthonormal basis {ek(x)} of Z(Q), where Z(Q) is the orthogonal complement of Z(Q) in VΛ0(Q) (with respect to the scalar product (5.6)). For every aQ\Ξ, set

𝔡a(x)=kek(a)ek(x);

notice that this definition is not in contradiction with (5.3) since in the latter aΞ.

For every vZ(Q), we have that

*v(x)𝔡a(x)𝑑x=v(a);

in fact,

*v(x)𝔡a(x)𝑑x=*(kvkek(x))(heh(a)eh(x))𝑑x=k,hvkeh(a)*ek(x)eh(x)𝑑x=k,hvkeh(a)δhk=kvkek(a)=v(a).

It is not difficult to realize that {𝔡a(x)}aQ\Ξ generates all Z(Q) and hence we can select a set Ξ(Q)Q\Ξ such that {𝔡a(x)}aΞ(Q) is a basis for Z(Q). Taking

QΓ=Ξ(Q)Ξ(Q),

we have that {𝔡a(x)}aQΓ is a basis for VΛ0(Q).

Now let {σa(x)}aQΓ denote the dual basis of {𝔡a(x)}aQΓ namely a basis such that, for all a,bQΓ,

*σa(x)𝔡b(x)𝑑x=δab.

Clearly, it is a σ-basis for VΛ0(Q). In fact, if uVΛ0(Q), we have that

u(x)=aQΓ[*u(t)𝔡a(t)𝑑t]σa(x)=aQΓu(a)σa(x).

Notice that if aΞ(Q), then σa(x)=ζa(x). The conclusion follows taking Γ:=Q𝔅Λ(Ω)QΓ. ∎

By the above theorem, the following corollary follows straightforward.

Corollary 5.6.

The set VΛ(Ω) is a Caccioppoli space of ultrafunctions in the sense of Definition 3.6.

If EΛ(Ω) (see (5.2)), we set

EΓ=Q𝔅Λ(Ω),QEQΓ.

If, for any internal set A, we define

Au(x)𝑑x=aΓAu(a)ηa,

then we have the following result:

Theorem 5.7.

If uθEVΛ(Ω) and ECΛ(Ω), then

EΓu(x)𝑑x=*Eu(x)dx=u(x)θE(x)𝑑x.

Proof.

We have that

EΓu(x)𝑑x=aEΓu(a)ηa=*aEΓu(a)σa(x)dx=*QEaQΓu(a)σa(x)dx.(5.7)

Since uθEVΛ(Ω), it follows from (5.5) that we can write

u(x)θE(x)=QEuQ(x)θQ(x).

By Theorem 5.5 (3),

uQ(x)θQ(x)=aQΓu(a)σa(x)VΛ0(Q).

Then by (5.7),

EΓu(x)𝑑x=*QEuQ(x)θQ(x)dx=QE*uQ(x)θQ(x)𝑑x.(5.8)

By this equation and the fact that

*uQ(x)θQ(x)𝑑x=*Qu(x)dx,

it follows that

EΓu(x)𝑑x=QE*Qu(x)dx=*Eu(x)dx.

Moreover, since uQθQVΛ0(Q)VΛ(Ω),

*uQ(x)θQ(x)𝑑x=uQ(x)θQ(x)𝑑x,

by (5.8) we have

EΓu(x)𝑑x=QEuQ(x)θQ(x)𝑑x=QEuQ(x)θQ(x)dx=u(x)θE(x)𝑑x.

5.3 Construction of the generalized derivative

Next we construct a generalized derivative on VΛ(Ω).

We set

UΛ1=Q𝔅Λ(Ω)VΛ1(Q),

and

UΛ0=(UΛ1)

will denote the orthogonal complement of UΛ1 in VΛ(Ω). According to this decomposition, VΛ(Ω)=UΛ1UΛ0 and we can define the following orthogonal projectors:

Pi:VΛ(Ω)UΛi,i=0,1,

hence, any ultrafunction uVΛ(Ω) has the following orthogonal splitting: u=u1+u0, where ui=Piu.

Now we are able to define the generalized partial derivative for uVΛ1(Ω).

Definition 5.8.

We define the generalized partial derivative

Di:UΛ1(Ω)VΛ(Ω)

as follows:

Diuv𝑑x=Q𝔅Λ(Ω)i*uQvQθQdx-12Q𝔅Λ(Ω)R𝔜(Q)*QR(uQ-uR)vQ(𝐞i𝐧Q)dS,(5.9)

where

𝔜(Q)={R𝔅Λ(Ω){Q}QR,QR}

with

Q=Ω*Q𝔅Λ(Ω)Q¯.

Moreover, if u=u1+u0UΛ1UΛ0=VΛ(Ω), we set

Diu=Diu1-(DiP1)u0,

where, for any linear operator L, L denotes the adjoint operator.

Remark 5.9.

Notice that if u,vUΛ1, by Theorem 5.4 (2), we have

Diuv𝑑x=Q𝔅Λ(Ω)*Qi*uQvQdx-12Q𝔅Λ(Ω)R𝔜(Q)*QR(uQ-uR)vQ(𝐞i𝐧Q)dS.(5.10)

In fact, if u,vUΛ1, then uQ,vQVΛ1(Ω), and hence, by (5.4), i*uQvQVΛ0(Ω) and so

i*uQvQθQdx=*Qi*uQvQdx.

Theorem 5.10.

The operator Di:VΛ(Ω)VΛ(Ω), given by Definition 5.8, satisfies the requests (I)–(III) of Definition 4.1.

Proof.

Let us prove property (I). If uθQ,vθRVΛ(Ω) and Q¯R¯=, by Definition 5.8,

Di(uθQ)vθR𝑑x=0.

Set

δ:=max{diam(Q)Q𝔅Λ(Ω)}.

If qQ and rR, then

|q-r|>2δQ¯R¯=,

so, if σqVΛ0(Q), and rR, then

|q-r|>2δQ¯R¯=,

and hence, if we set ε0>3δ,

{R𝔅Λ(Ω)Q¯R¯}Bε0(q).

Since σqVΛ0(Q),

supp(Diσq){R𝔅Λ(Ω)Q¯R¯}¯Bε0(q).

We prove property (II). If u[𝒞1(Ω)]*VΛ(Ω), then

u=Q𝔅Λ(Ω)uθQ,

and hence for all xQR, uQ(x)-uR(x)=u(x)-u(x)=0. Then, by (5.9), we have, for all vVΛ(Ω),

Diuv𝑑x=Q𝔅Λ(Ω)i*uvQθQdx=i*u(Q𝔅Λ(Ω)vQθQ)𝑑x=i*uvdx.

The conclusion follows from the arbitrariness of v.

Next let us prove property (III). First we prove this property if u,vUΛ1. By (5.10), we have that

Diuv𝑑x=Q𝔅Λ(Ω)*Qi*uvdx-12Q𝔅Λ(Ω)R𝔜(Q)*QRuQvQ(𝐞i𝐧Q)dS+12Q𝔅Λ(Ω)R𝔜(Q)*QRuRvQ(𝐞i𝐧Q)dS=Q𝔅Λ(Ω)*Qi*uvdx-12Q𝔅Λ(Ω)*QuQvQ(𝐞i𝐧Q)dS+12Q𝔅Λ(Ω)R𝔜(Q)*QRuRvQ(𝐞i𝐧Q)dS.(5.11)

Next we will compute uDiv𝑑x and we will show that it is equal to -Diuv𝑑x. So we replace u with v, in the above equality and we get

Q𝔅Λ(Ω)*QuDivdx=Q𝔅Λ(Ω)*Qi*vQuQdx-12Q𝔅Λ(Ω)*QuQvQ(𝐞i𝐧Q)dS+12Q𝔅Λ(Ω)R𝔜(Q)*QRvRuQ(𝐞i𝐧Q)dS.(5.12)

Now, we compute Q𝔅Λ(Ω)Q*i*vQuQdx and the last term of the above expression separately. We have that

Q𝔅Λ(Ω)*Qi*vQuQdx=-Q𝔅Λ(Ω)*Qi*uQvQdx+Q𝔅Λ(Ω)*QuQvQ(𝐞i𝐧Q)dS.(5.13)

Moreover, the last term in (5.12), changing the order on which the terms are added, becomes

Q𝔅Λ(Ω)R𝔜(Q)*QRvRuQ(𝐞i𝐧Q)dS=R𝔅Λ(Ω)Q𝔜(R)*QRvRuQ(𝐞i𝐧Q)dS.

On the right-hand side we can change the name of the variables Q and R:

Q𝔅Λ(Ω)R𝔜(Q)*QRvRuQ(𝐞i𝐧Q)dS=Q𝔅Λ(Ω)R𝔜(Q)*QRvQuR(𝐞i𝐧R)dS=-Q𝔅Λ(Ω)R𝔜(Q)*QRvQuR(𝐞i𝐧Q)dS.(5.14)

In the last step we have used the fact that xQR implies 𝐧R(x)=-𝐧Q(x). Replacing (5.13) and (5.14) in (5.12), we get

Q𝔅Λ(Ω)*QuDivdx=-Q𝔅Λ(Ω)*Qi*uQvQdx+12Q𝔅Λ(Ω)*QuQvQ(𝐞i𝐧Q)dS-12Q𝔅Λ(Ω)R𝔜(Q)*QRvQuR(𝐞i𝐧Q)dS.

Comparing (5.11) and the above equation, we get that

Diuv𝑑x=-uDiv𝑑xfor all u,vUΛ1.(5.15)

Let us prove property (III) in the general case. We have that

Diu=Diu1-(DiP1)u0,

hence

Diuv𝑑x=Diu1v𝑑x-(DiP1)u0v𝑑x=Diu1v1𝑑x+Diu1v0𝑑x-u0DiP1v𝑑x=Diu1v1𝑑x+Diu1v0𝑑x-u0Div1𝑑x.(5.16)

Now, replacing u with v and applying property (5.15) for u1,v1UΛ1, we get

uDiv𝑑x=u1Div1𝑑x-Diu1v0𝑑x+u0Div1𝑑x=-Diu1v1𝑑x-Diu1v0𝑑x+u0Div1𝑑x.

Comparing the above equation with (5.16), we get that

Diuv𝑑x=-uDiv𝑑x.

The proof is complete. ∎

Before proving property (IV), we need the following lemma:

Lemma 5.11.

The following identity holds true: for all ECΛ(Ω), all uVΛ(Ω)[C1(Ω)]* and all vVΛ(Ω),

Di(uθE)v𝑑x=iuvθEdx-*Euv(𝐞i𝐧E)dS.(5.17)

Proof.

We can write

uθE=Q𝔅Λ(Ω)hQuθQ,

where

hQ={1if QE,0if QE.

Then we have that

hQu-hRu=u

if and only if

R𝔜Q,E+:={R𝔅Λ(Ω){Q}RE,QE,RΩE}.

Moreover, we have that

hQu-hRu=-u

if and only if

R𝔜Q,E-:={R𝔅Λ(Ω){Q}RE,QΩE,RE}.

Otherwise, we have that

hQu-hRu=0orRE=.

Then, by Theorem 5.7,

Di(uEθE)v𝑑x=Q𝔅Λ(Ω)i*(hQu)vθQdx-12Q𝔅Λ(Ω)R𝔜(Q)*QR(hQu-hRu)v(𝐞i𝐧Q)dS=Q𝔅Λ(Ω)hQQΓi*uvdx-12Q𝔅Λ(Ω)R𝔜Q,E+*QRuv(𝐞i𝐧Q)dS+12Q𝔅Λ(Ω)R𝔜Q,E-*QRuv(𝐞i𝐧Q)dS=EΓi*uvdx-12Q𝔅Λ(Ω),QE*QEuv(𝐞i𝐧E)dS+12Q𝔅Λ(Ω),QΩE*QEuv(𝐞i(-𝐧E))dS=i*uvθEdx-12*Euv(𝐞i𝐧E)dS+12*Euv(𝐞i(-𝐧E))dS=i*uvθEdx-*Euv(𝐞i𝐧E)dS.

Then (5.17) holds true. ∎

Corollary 5.12.

The operator Di:VΛ(Ω)VΛ(Ω) given by Definition 5.8 satisfies the request (IV) of Definition 4.1.

Proof.

The result follows straightforward from (5.17) just taking u=1. ∎

6 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.

6.1 A minimization result

In this subsection we will consider a minimization problem. Let Ω be an open bounded set in N and let ΞΩ be any nonempty portion of the boundary. We consider the following problem: minimize

J(u)=Ω[12a(u)|u(x)|p+f(x,u)]𝑑x,p>1,

in the set 𝒞1(Ω)𝒞0(ΩΞ). We make the following assumptions:

  • (1)

    a(u)0 and a(u)k>0 for u sufficiently large.

  • (2)

    a(u) is lower semicontinuous.

  • (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+.

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 uVΛ(Ω) which vanishes on Ξ* and minimizes

J(u):=Ω[12a*(u)|Du(x)|p-f*(x,u)]𝑑x,p>1.

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

{vVΛ(Ω)v(x)=0 for all xΞ*}.

Moreover, if J(u) has a minimizer w in V(Ω), then u=w.

Proof.

The proof is based on a standard approximation by finite-dimensional spaces. Let us observe that, for each finite-dimensional space Vλ, we can consider the approximate problem:

Problem.

Find uλVλ such that

J(uλ)=minvλVλJ(vλ).

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 uλnVλ, then we can extract a subsequence weakly converging to some uλVλ. By observing that in finite-dimensional spaces all norms are equivalent, it follows also that uλnuλ pointwise. Then, by the lower-semicontinuity of a and f, it follows that the pointwise limit satisfies

J(uλ)lim infJ(uλn).

Next, we use the very general properties of Λ-limits, as introduced in Section 2.2. We set

u:=limλΛuλ.

Then, taking a generic v:=limλΛvλ, from the inequality J(uλ)J(vλ), we get

J(u)J(v)for all vVΛ(Ω).

The last statement is trivial. ∎

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 HN-1(Ξ)>0 and there exists νR such that a(u)ν>0, then the minimizer has the following form:

u=w+ψ,

where wH1,p(Ω) and ψ is null in the sense of distributions, namely,

ψφ*𝑑x0for all φ𝒟(Ω).

In this case

J(u)infvV(Ω)J(v)

with V(Ω) as in Definition 3.5. Moreover, if in addition a(u)<M, with MR, we have that

ψH1,p(Ω)0

and J(u)J(w). Finally, if J(u) has a minimizer in wH1,p(Ω)C(Ω), then u=w and J(u)=J(w).

Proof.

Under the above hypotheses the minimization problem has an additional a priori estimate in H1,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 {uλ} has a subnet {un} such that

unuweakly in H1,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, unu strongly in H1,p(Ω). This implies, by Proposition 3.23, that uw, and hence ψ is infinitesimal in H1,p(Ω), proving the second part.

Finally, if the minimizer is a function wH1,p(Ω)𝒞(Ω)VΛ(Ω), we have that uλ=w eventually; then ψH1,p(Ω)=0. ∎

6.2 The Poisson problem in 2

Now we consider the very classical problem

-u=φ(x),φ𝒟(N).(6.1)

If N3, the solution is given by

φ(x)|x|-N+2(N-2)ωN

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 equation

u^(ξ)=φ^(ξ)|ξ|2,(6.2)

where T^ denotes the Fourier transform of T. Moreover, it is the minimizer of the Dirichlet integral

J(u)=[12|u(x)|2-φ(x)u(x)]𝑑x

in the space 𝒟1,2(N) which is defined as the completion of 𝒞1(N) with respect to the Dirichlet norm

u=|u(x)|2𝑑x.

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

2πφ(x)log|x|;

however, none of the previous characterization makes sense. In fact, we cannot use equation (6.2), since 1|ξ|2Lloc1(2) and hence 1|ξ|2 does not define a tempered distribution. Also, the space 𝒟1,2(2) is not an Hilbert space and the functional J(u) is not bounded from below in 𝒟1,2(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 N3, the boundary conditions are replaced by the condition u𝒟1,2(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.

Problem.

Find uVΛ(BR) such that

-u=φ(x)in BR,u=0on BR,

where is the “generalized” Laplacian defined in Section 4.1 and R is an infinite number such that χBRVΛ(N).2

Clearly, the solutions of the above problem are the minimizers of the Dirichlet integral

J(u)=[12|Du(x)|2-φ(x)u(x)]𝑑x

in the space uVΛ(BR), 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

u=w+ψ

with

St𝒟ψ=0.

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=St𝒟u

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.

6.3 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 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 xΩ. 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 δ2).

Let us consider, for γ>0, the one-dimensional variational problem of finding the minimum of the functional

J(u)=0112a(u)|u(x)|2-γu(x)dx(6.3)

among the functions such that u(0)=0. In particular, we are interested in the case in which a is the following degenerate function:

a(s)={1if s(-,1)(2,+),0if s[1,2].

Formally, the Euler equation, if u[1,2], is

u′′(x)=-γ.

We recall that, by standard arguments,

u(1)1u(1)=0.

Hence, if γ<2, the solution is explicitly computed

u(x)=γ2(2x-x2),

since it turns out that 0u(x)<1 for all x(0,1) and then the degeneracy does not take place.

If γ>2, we see that the solution does not live in H1(0,1), hence the problem has not a “classical” weak solution. More exactly, we have the following result:

Theorem 6.3.

If γ>2, then the functional (6.3) has a unique minimizer given by

u(x)={12(2γx-γx2),0<x<ξ,12(-γx2+2γx+2),ξ<x<1,

where ξ(0,1) is a suitable real number which depends on γ (see Figure 1).

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

ξ=γ-γ2-2γγ=1-1-2γ

and a discontinuity of derivatives at some ξ<η<1.

The function u⁢(x){{u}(x)} for γ=4{\gamma=4}.
Figure 1

The function u(x) for γ=4.

In the specific case, we have (see Figure 1)

u(x)={12(2γx-γx2),0<x<ξ,2,ξ<x<η,γη22-γη-γx22+γx+2,η<x<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 function u~(x) defined as

u~(x)={12(2γx-γx2),0<x<ξ,12(-γx2+2γx+2),ξ<x<1,

we observe that u=u~ in [0,ξ], while u=u~=γ(1-x) for all x[ξ,η] and, by explicit computations, we have

J(u~)-J(u)=γ2[-η2+η22-η36+ξ2-ξ22+ξ36]=γ2(Φ(η)-Φ(ξ))<0,ξ<η,

where Φ(s)=-s2+s22-s36 is strictly decreasing, since Φ(s)=-12(s-1)20.

The function u~⁢(x){\tilde{u}(x)} for γ=4{\gamma=4}.
Figure 2

The function u~(x) for γ=4.

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):

J(u)=F(ξ)=γ2ξ38-γ2ξ22+γ2ξ2-γ26+3γξ2-2γ+12ξ.

We observe that, for all γ>2,

F(0+)=+andF(1)=-γ224-γ2+12<0.

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

F(ξ)=3γ2ξ28-γ2ξ+γ22+3γ2-12ξ2,F′′(ξ)=3γ2ξ4-γ2+1ξ3,F′′′(ξ)=3γ24-3ξ4,

hence we have that

F′′′(ξ)<0if and only if0<ξ<2γ<1.

Consequently, the function F′′(ξ), which nevertheless satisfies, for all γ>2,

F′′(0+)=+,F′′(1)=1-γ24<0,

has a unique negative minimum at the point 2/γ.

The function F′′⁢(ξ){F^{\prime\prime}(\xi)} for γ=4{\gamma=4}.
Figure 3

The function F′′(ξ) for γ=4.

The function F′⁢(ξ){F^{\prime}(\xi)} for γ=14{\gamma=14}.
Figure 4

The function F(ξ) for γ=14.

From this, we deduce that there exists one and only one point 0<ξ0<2/γ such that

F′′(ξ)>0,0<ξ<ξ0,F′′(ξ)<0,ξ0<ξ1.

From the sign of F′′ we get that F is strictly increasing in (0,ξ0) and decreasing in (ξ0,1) (see Figure 3). Next for all γ>2,

F(0+)=-,F(1)=-γ28+3γ2-12,

hence in the case that

-γ28+3γ2-12>0,that is, γ<2(3+22)11.656,

then F has a single zero ξ1(0,ξ0) and, being a change of sign, ξ1 is a point of absolute minimum for F(ξ).

If γ2(3+22), the above argument fails. In this case we can observe that

F(1γ)=γ2+38>0,

hence F(1), which is negative at ξ=1 and near ξ=0 vanishes exactly two times, at the point ξ1, which is a point of local minimum and at another point ξ2>ξ1, which is a point of local maximum (see Figure 4). Hence, to find the absolute minimum, we have to compare the value of F(ξ1) with that of F(1).

In particular, we have that ξ1<2/γ and hence we can show that the minimum is not at ξ=1 simply by observing that we can find at least a point where F(ξ)<F(1) and this point is 2/γ. In fact,

M(γ):=F(2γ)-F(1)=γ322-γ28-5γ2+22γ-120.

In particular, M(2)=0 and

M(γ)=14(-γ+32γ+42γ-10)<0.

This follows since replacing γ with χ, we have to control the sign of the cubic

M~(χ)=-χ3+32χ2-10χ+42,

which is negative for all χ1, since M~(1)=-11+72, while

M~(χ)=-3χ2+62χ-10

is a parabola with negative minimum. ∎

Remark 6.4.

It could be interesting to study this problem in dimension greater than one, namely, to minimize

J(u)=Ω(12a(u)|Du(x)|2-γu(x))𝑑x

in the set

{vVΛ(Ω):u(x)=0 for all xΞ*}

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=BR(0)).

References

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Footnotes

  • 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). 

  • 2

    Such an R exists by overspilling (see e.g. [17, 15, 16]); in fact, for any r, χBrVΛ(N). 

  • 3

    The fact that Ω is a standard set while BR is an internal set does not change the proof. 

About the article

Received: 2017-10-13

Accepted: 2017-10-20

Published Online: 2017-12-05


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 946–978, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0225.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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