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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Generalized solutions of variational problems and applications

Vieri Benci
  • Dipartimento di Matematica, Università degli Studi di Pisa, Via F. Buonarroti 1/c, 56127 Pisa; and Centro Linceo interdisciplinare Beniamino Segre, Palazzo Corsini – Via della Lungara 10, 00165 Roma, Italy
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/ Lorenzo Luperi BagliniORCID iD: https://orcid.org/0000-0002-0559-0770 / Marco Squassina
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  • Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121 Brescia, Italy
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Published Online: 2018-08-08 | DOI: https://doi.org/10.1515/anona-2018-0146

Abstract

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

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

MSC 2010: 03H05; 26E35; 28E05; 46S20

1 Introduction

It is nowadays very well known that, in many circumstances, the needs of a theory require the introduction of generalized functions. Among people working in partial differential equations, the theory of distributions of L. Schwartz is the most commonly used, but other notions of generalized functions have been introduced, e.g. by J. F. Colombeau [15] and M. Sato [21, 22]. Many notions of generalized functions are based on non-Archimedean mathematics, namely mathematics handling infinite and/or infinitesimal quantities. Such an approach presents several positive features, the main probably being the possibility of treating distributions as non-Archimedean set-theoretical functions (under the limitations imposed by Schwartz’ result). This allows to easily introduce nonlinear concepts, such as products, into distribution theory. Moreover, a theory which includes infinitesimals and infinite quantities makes it possible to easily construct new models, allowing in this way to study several problems which are difficult even to formalize in classical mathematics. This has led to applications in various field, including several topics in analysis, geometry and mathematical physic (see e.g. [17, 19] for an overview in the case of Colombeau functions and their recent extension, called generalized smooth functions).

In this paper we deal with ultrafunctions, which are a kind of generalized functions that have been introduced recently in [1] and developed in [4, 5, 6, 7, 8, 2, 11, 10, 9].

Ultrafunctions are a particular case of non-Archimedean generalized functions that are based on the hyperreal field *, namely the numerical field on which nonstandard analysis is based. (We refer to Keisler [16] for a very clear exposition of nonstandard analysis.) No prior knowledge of nonstandard analysis is requested to read this paper: we will introduce all the nonstandard notions that we need via a new notion of limit, called Λ-limit (see [10] for a complete introduction to this notion and its relationships with the usual nonstandard analysis). The main peculiarity of this notion of limit is that it allows us to make a very limited use of formal logic, in contrast with most usual nonstandard analysis introductions.

Apart from being framed in a non-Archimedean setting, ultrafunctions have other peculiar properties that will be introduced and used in the following:

  • every ultrafunction can be split uniquely (in a sense that will be precised in Section 3.2) as the sum of a classical function and a purely non-Archimedean part;

  • ultrafunctions extend distributions, in the sense that every distribution can be identified with an ultrafunction; in particular, this allows to perform nonlinear operations with distributions;

  • although being generalized functions, ultrafunctions share many properties of 𝒞1 functions, like e.g. Gauss’ divergence theorem.

Our goal is to introduce all the aforementioned properties of ultrafunctions, so to be able to explain how they can be used to solve certain classical problems that do not have classical solutions; in particular, we will concentrate on singular problems arising in calculus of variations and in relevant applications (see e.g. [17] and references therein for other approaches to these problems based on different notions of generalized functions).

The paper is organized as follows: in Section 2, we introduce the notion of Λ-limit, and we explain how to use it to construct all the non-Archimedean tools that are needed in the rest of the paper, in particular, how to construct the non-Archimedean field extension * of and what the notion of “hyperfinite” means. In Section 3 we define ultrafunctions, and we explain how to extend derivatives and integrals to them. All the properties of ultrafunctions needed later on are introduced in this section: we show how to split an ultrafunction as the sum of a standard and a purely non-Archimedean part, how to extend Gauss’ divergence theorem and how to identify distributions with certain ultrafunctions. In Section 4, we present the main results of the paper, namely, we show that a very large class of classical problems admits generalized ultrafunction solutions. We study the main properties of these generalized solutions, concentrating in particular on the relationships between ultrafunction solutions and classical minimizing sequences for variational problems. Finally, in Section 5, we present two examples of applications of our methods: the first is the study of a variational problem related to the sign-perturbation of potentials, the second is a singular variation problem related to sign-changing boundary conditions.

The first part of this paper contains some overlap with other papers on ultrafunctions, but this fact is necessary to make it self-contained and to make the reader comfortable with it.

1.1 Notations

If X is a set and Ω is a subset of N, then

  • 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 ΩN;

  • 𝒞k(Ω) denotes the set of functions defined on ΩN which have continuous derivatives up to the order k (sometimes we will use the notation 𝒞0(Ω) instead of 𝒞(Ω));

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

  • 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 ΩN; 𝒟(Ω) denotes the topological dual of 𝒟(Ω), namely the set of distributions on Ω;

  • if AX is a set, then χA denotes the characteristic function of A;

  • 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;

  • a.e.xX means “for almost every xX”;

  • if a,b*, then

    • [a,b]*={x*:axb},

    • (a,b)*={x*:a<x<b};

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

  • if E is any set, then |E| will denote its cardinality.

2 Λ-theory

In this section, we present the basic notions of Non-Archimedean Mathematics (sometimes abbreviated as NAM) and of Nonstandard Analysis (sometimes abbreviated as NSA) following a method inspired by [3] (see also [1] and [4]). When we talk about NSA, we mean the NSA in the sense of Robinson (see [20]), and not the internal set theory developed by Nelson in [18].

2.1 Non-Archimedean fields

Here, we recall the basic definitions and facts regarding non-Archimedean fields. In the following, 𝕂 will denote a totally ordered infinite field. We recall that such a field contains (a copy of) the rational numbers. Its elements will be called numbers.

Definition 2.1.

Let 𝕂 be an ordered field. Let 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 infinitesimal numbers are actually 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, according to the following:

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 2.5.

If K is a totally ordered 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. In Definition 2.16, we will see how the notion of standard part can be generalized to any Hausdorff topological space.

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,

𝔪𝔬𝔫(ξ):-{ζ𝕂:ξζ}.

2.2 The Λ-limit

Let U be an infinite set of cardinality bigger than the continuum, and let 𝔏=𝒫fin(U) be the family of finite subsets of U.

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}Q:Q𝒰},

where 𝒰 is a fine ultrafilter on 𝔏, namely a filter such that

  • for every A,B𝔏, if AB=𝔏, then A𝒰 or B𝒰;

  • for every λ𝔏, the set Q(λ):-{μ𝔏:λμ}𝒰.

We will refer to the elements of 𝒰 as qualified sets. A function φ:𝔏E, defined on a directed set E, is called net (with values in E). If φ(λ) is a real net, we have that

limλΛφ(λ)=L

if, and only if, for every ε>0, there exists Q𝒰 such that |φ(λ)-L|<ε for all λQ.

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.

Proposition 2.7.

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 converging subnet 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=𝔏Qε𝒰, and hence,

|φ(λ)-L|εfor allλN.

This contradicts the fact that φλ has a subnet which converges to L. ∎

Proposition 2.8.

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.

Let {An:n} be a countable basis of open neighborhoods of x0. For every n, the set

In:-{λ𝔏:φ(λ)An}

is qualified. Hence, Jn:-jnIj. Let λnJn. Then the sequence {λn}n has trivially the desired property: for every n, for every mn, we have that φ(λm)An. ∎

Example 2.9.

Let φ:𝔏V be a net with values in a bounded subset 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 2.10.

The set of the hyperreal numbers * is a set equipped with a topology τ such that

  • 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 * is a well-known fact in NSA. The limit ξ* of a net φ:𝔏 with respect to the τ topology, following [1], 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),η:-limλΛψ(λ),

we set

:-limλΛ(φ(λ)+ψ(λ)),(2.2)ξη:-limλΛ(φ(λ)ψ(λ)).(2.3)

Then the following well-known theorem holds:

Theorem 2.11.

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

Remark 2.12.

We observe that the field of hyperreal numbers is defined as a sort of completion of real numbers. In fact, * is isomorphic to the ultrapower 𝔏/, where ={φ:𝔏φ(λ)=0eventually}. The isomorphism resembles the classical one between real numbers and equivalence classes of Cauchy sequences. This method is surely known to the reader for the construction of the real numbers starting from the rationals.

2.3 Natural extension of sets and functions

To develop applications, we need to extend the notion of Λ-limit to sets and functions (but also to differential and integral operators). This will allow to enlarge the notions of variational problem and of variational solution.

Λ-limits of bounded nets of mathematical objects in V() can be defined by induction (a net φ:𝔏V() is called bounded, if there exists n such that, for all λ𝔏, φ(λ)Vn()). To do this, consider a net

φ:𝔏Vn().(2.4)

Definition 2.13.

For n=0, limλΛφ(λ) is defined by (2.1); so 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(),for allλ𝔏,ψ(λ)φ(λ)}.

A mathematical entity (number, set, function or relation) which is the Λ-limit of a net is called internal.

Definition 2.14.

If for all λ𝔏, Eλ=EV(), we set limλΛEλ=E*, namely

E*:-{limλΛψ(λ):ψ(λ)E};

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:-{X*:XE}.(2.5)

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

Remark 2.15.

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

limλΛφ(λ)

which, by Proposition 2.7, always exists in the Alexandrov compactification X{}. Moreover, we have the Λ-limit, that 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)).(2.6)

The above equation suggests the following definition.

Definition 2.16.

If X is 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 2.17.

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;

f*:E*R* is called the natural extension of f. As expected, the natural extension of functions is a particular case of the extension of sets: in fact, if we identify f with its graph, then f* is the graph of its natural extension.

2.4 Hyperfinite sets and hyperfinite sums

Definition 2.18.

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 standard set is contained in a hyperfinite set.

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

A:-limλΛAλ

be a 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 Definition of ultrafunctions

We start by introducing the notion of hyperfinite grid.

Definition 3.1.

A hyperfinite set Γ such that NΓ(N)* is called hyperfinite grid.

From now on, we assume that Γ has been fixed once forever. Notice that, by definition, NΓ, and the following two simple (but useful) properties of Γ can be easily proven via Λ-limits:

  • for every xN there exists yΓ𝔪𝔬𝔫(x) so that xr;

  • there exists a hyperreal number ρ0, ρ>0, such that d(x,y)ρ for every x,yΓ, xy.

Definition 3.2.

A space of grid functions is a family 𝔊(N) of internal functions

u:Γ*

defined on a hyperfinite grid Γ. If EN, then 𝔊(E) will denote the restriction of the grid functions to the set E*Γ.

Let E be any set in N. To every internal function u𝔉(E)*, it is possible to associate a grid function by the “restriction” map

:𝔉(E)*𝔊(E)(3.1)

defined as follows:

u(x):-u*(x)for allxE*Γ;

moreover, if f𝔉(E), for short, we use the notation

f(x):-(f*)(x).(3.2)

So every function f𝔉(E) can be uniquely extended to a grid function f𝔊(E).

In many problems, we have to deal with functions defined almost everywhere in Ω, such as 1/|x|. Thus, it is useful to give a “rule” which allows to define a grid function for every xΓ.

Definition 3.3.

If a function f is defined on a set EN, we put

f(x)=aΓE*f*(a)σa(x),

where, for all aΓ, the grid function σa is defined as follows: σa(x):-δax.

If EN is a measurable set, we define the “density function” of E as follows:

θE(x)=st(m(Bη(x)E*)m(Bη(x))),(3.3)

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)

whenever m(E)<; also, if E is a bounded open set with smooth boundary, we have that θE(x)=12 for every xE.

Now, let V(N) be a vector space such that 𝒟(N)V(N)1(N).

Definition 3.4.

A space of ultrafunctions V(N) modeled over the space V(N) is a space of grid functions such that there exists a vector space VΛ(N)V*(N) such that the map1

:VΛ(N)V*(N)

is an *-linear isomorphism. From now on, we assume that V(N) satisfies the following assumption: if Ω is a bounded open set such that mN-1(Ω)< and fC0(N), then

fθΩV(N).

Next, we want to equip V(N) with the two main operations of calculus, the integral and the derivative.

Definition 3.5.

The pointwise integral

:V(N)*

is a linear functional which satisfies the following properties:

  • (1)

    for all uVΛ(N)

    u(x)dx=u(x)dx;(3.4)

  • (2)

    there exists an ultrafunction d:Γ* such that, for all xΓ, d(x)>0, and for all uV(N),

    u(x)dx=aΓu(a)d(a).

If EN is any set, we use the obvious notation

Eu(x)dx:-aΓE*u(a)d(a).

A few words to discuss the above definition: Point (2) says that the pointwise integral is nothing else but a hyperfinite sum. Since d(x)>0, every non-null positive ultrafunction has a strictly positive integral. In particular, if we denote by σa(x) the ultrafunctions whose value is 1 for x=a and 0 otherwise, we have that

σa(x)dx=d(a).

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

u(x)v(x)dx=aΓu(a)v(a)d(a).(3.5)

From now on, the norm of an ultrafunction will be defined as

u=(|u(x)|2dx)12.

Now, let us examine point (1) of the above definition. If we take fCcomp0(N), we have that f*VΛ(N), and hence

f(x)dx=f(x)dx.

Thus, the pointwise integral is an extension of the Riemann integral defined on Ccomp0(N). However, if we take a bounded open set Ω such that m(Ω)=0, then we have that

Ωf(x)dx=Ω¯f(x)dx.

However, the pointwise integral cannot have this property; in fact,

Ω¯f(x)dx-Ωf(x)dx=Ωf(x)dx>0,

since Ω. In particular, if Ω is a bounded open set with smooth boundary and fC0(N), then

Ωf(x)dx=f(x)χΩ(x)dx=f(x)θΩ(x)dx-12f(x)χΩ(x)dx=Ωf(x)dx-12Ωf(x)dx,

and similarly

Ω¯f(x)dx=Ωf(x)dx+12Ωf(x)dx;

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

Definition 3.6.

The ultrafunction derivative

Di:V(N)V(N)

is a linear operator which satisfies the following properties:

  • (1)

    for all fC1(N) and for all x(N)*, x finite,

    Dif(x)=if*(x);(3.6)

  • (2)

    for all u,vV(N),

    Diuvdx=-uDivdx;

  • (3)

    if Ω is a bounded open set with smooth boundary, then for all vV,

    DiθΩvdx=-Ω*v(𝐞i𝐧E)dS,

    where 𝐧E is the unit outer normal, dS is the (n-1)-dimensional measure and (𝐞1,,𝐞N) is the canonical basis of N;

  • (4)

    the support2 of Diσa is contained in 𝔪𝔬𝔫(a)Γ.

Let us comment the above definition. Point (1) implies that, for all fC1(N) and for all xN,

Dif(x)=if(x);(3.7)

namely, the ultrafunction derivative coincides with the usual partial derivative whenever fC1(N). The meaning of point (2) is clear; we remark that this point is very important in comparing ultrafunctions with distributions. Point (3) says that DiθΩ is an ultrafunction whose support is contained in ΩΓ; it can be considered as a signed measure concentrated on Ω. Point (4) says that the ultrafunction derivative, as well as the usual derivative or the distributional derivative, is a local operator, namely if u is an ultrafunction whose support is contained in a compact set K with KΩ, then the support of Diu is contained in Ω*. Moreover, property (4) implies that the ultrafunction derivative is well defined in V(Ω) for any open set Ω by the following formula:

Diu(x)=aΓΩ*u(a)Diσa(x).

Remark 3.7.

If uV(Ω) and u¯ is an ultrafunction in V(Ω¯) such that, for all xΩ, u(x)=u¯(x), then, by point (3), for all xΩ* such that 𝔪𝔬𝔫(x)Ω*, we have that

Diu(x)=Diu¯(x);

however, this property fails for some xΩ*. In fact, the support of Diσa is contained in 𝔪𝔬𝔫(a)Γ, but not in {a}.

Theorem 3.8.

There exists an ultrafunction space V(RN) which admits a pointwise integral and an ultrafunction derivative as in Definitions 3.5 and 3.6.

Proof.

In [2], there is a construction of a space VΛ(N) which satisfies the desired properties. The conclusion follows, taking

V(N)={u:uVΛ}.

3.2 The splitting of an ultrafunction

In many applications, it is useful to split an ultrafunction u in 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 uV(Ω), we set

S={xΩ:u(x)is infinite}

and

w(x)={st(u(x))ifxΩS,0ifxS.

We will refer to S as to the singular set of the ultrafunction u.

Definition 3.9.

For every ultrafunction u, consider the splitting

u=w+ψ,

where

  • w=w¯|ΩS and w, which is defined by Definition 3.3, is called the functional part of u;

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

Notice that w, the functional part of u, may assume infinite values for some xΩ*S*, but they are determined by the values of w which is a standard function defined on Ω.

Example 3.10.

Take ε0, and

u(x)=log(x2+ε2).

In this case,

w(x)={log(x2)=2log(|x|)ifx0,0ifx=0;ψ(x)={log(x2+ε2)-log(x2)=log(1+ε2x2)ifx0,log(ε2)ifx=0;S:-{0}.

We conclude this section with the following trivial proposition which, nevertheless, is very useful in applications.

Proposition 3.11.

Take a Banach space W such that D(Ω)WL1(Ω). Assume that {un}V(Ω) is a sequence which converges weakly in W and pointwise to a function w; then, if we set

u:-((limλΛu|λ|)),

we have that

u=w+ψ,

where

ψvdx0for allvW;

moreover, if

limnun-wW=0,

then ψW0.

Proof.

As a consequence of the pointwise convergence of {un} to w, we have that, for all aΓ, u(a)w(a). In particular, for all aΓ, ψ(a)0. As Γ is hyperfinite, the set {|ψ(a)|:aΓ} has a maximum η0. Hence, for every vW, we have

|ψvdx||ψ||v|dxη|v|dxη|v|dx0,

as η0 and |v|dx. For the second statement, let us notice that

ψW=u-wW=(limλΛu|λ|-wW)0,

as limnun-wW=0. ∎

An immediate consequence of Proposition 3.11 is the following:

Corollary 3.12.

If wL1(Ω), then

w(x)dxw(x)dx.

Proof.

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

u:-(limλΛu|λ|).

By Proposition 3.11, we have that

u=w+ψ

with ψL10. Then

u(x)dxw(x)dx.

On the other hand, since uV(Ω), by Definition 3.5 (1),

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

3.3 The Gauss divergence theorem

First of all, we fix the notation for the main differential operators:

  • =(1,,N) will denote the usual gradient of standard functions;

  • *=(1*,,N*) will denote the natural extension of internal functions;

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

Next, let us consider the divergence:

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

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

  • Dφ=D1φ1++DNφN will denote the divergence of vector-valued ultrafunctions φ[V(N)*]N.

And finally, we can define the Laplace operator of an ultrafunction uV(Ω) as the only ultrafunction uV(Ω) such that

uvdx=-DuDvdxfor allvV0(Ω¯),

where

V0(Ω¯):-{vV(Ω¯):for allxΩΓ,v(x)=0}.

By Definition 3.6 (3), for any bounded open set Ω with smooth boundary,

DiθΩvdx=-Ω*v(𝐞i𝐧E)dS,

and by Definition 3.6 (2),

DiθΩvdx=-DivθΩdx,

so that

DivθΩ𝑑x=Ω*v(𝐞i𝐧Ω)dS.

Now, if we take a vector field φ=(v1,,vN)[V(N)]N, by the above identity, we get

DφθΩdx=Ω*φ𝐧ΩdS.(3.8)

Now, if φC1, by Definition 3.6 (1), we get the Gauss divergence theorem

Ωφdx=Ωφ𝐧EdS.

Then, (3.8) is a generalization of the Gauss theorem which makes sense for any bounded open set Ω with smooth boundary and every vectorial ultrafunction φ. Next, we want to generalize Gauss’ theorem to any subset of AN. It is well known, that, for any bounded open set Ω with smooth boundary, the distributional derivative θΩ is a vector-valued Radon measure, and we have that

|θΩ|,1=mN-1(Ω).

Then, the following definition is a natural generalization.

Definition 3.13.

If A is a measurable subset of N, we set

mN-1(Ω):-|DθA|dx,

and, for all vV(N),

Av(x)dS:-v(x)|DθA|dx.(3.9)

Remark 3.14.

Notice that

Av(x)dSAv(x)dx.

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

Av(x)dS=xΓv(x)|DθA(x)|d(x),

while the right-hand term is

Av(x)dx=xΓA*v(x)d(x);

in particular, if A is smooth and v(x) is bounded, Av(x)dx is an infinitesimal number.

Theorem 3.15.

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

DφθAdx=Aφ𝐧A(x)dS,(3.10)

where

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

Proof.

By Definition 3.6 (3),

DφθAdx=-φDθAdx,

then, using the definition of 𝐧A(x) and (3.9), the above formula can be written as follows:

DφθAdx=φ𝐧A|DθA|dx=Aφ𝐧AdS.

3.4 Ultrafunctions and distributions

One of the most important properties of the ultrafunctions is that they can be seen (in some sense that we will make precise in this section) as generalizations of the distributions.

Definition 3.16.

The space of generalized distributions on Ω is defined as follows:

𝒟G(Ω)=V(Ω)/N,

where

N={τV(Ω):for allφ𝒟(Ω),τφdx0}.

The equivalence class of u in V(Ω) will be denoted by [u]𝒟.

Definition 3.17.

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 will denote by 𝒟GB(Ω) the set of bounded generalized distributions.

We have the following result.

Theorem 3.18.

There is a linear isomorphism

Φ:𝒟GB(Ω)𝒟(Ω)

such that, for every [u]DGB(Ω) and for every φD(Ω),

Φ([u]𝒟),φ𝒟(Ω)=st(uφ*dx).

Proof.

For the proof, see e.g. [7]. ∎

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,φ=st(uφ*dx).

If fCcomp0(Ω) and f*[u]𝒟, then, for all φ𝒟(Ω),

[u]𝒟,φ𝒟(Ω)=st(*uφ*dx)=st(*f*φ*dx)=fφdx.

Remark 3.19.

The set V(Ω) is an algebra which extends the algebra of continuous functions C0(N). If we identify a tempered distribution3 T=mf with the ultrafunction Dmf, we have that the set of tempered distributions 𝒮 is contained in V(Ω). However, the Schwartz impossibility theorem is not violated as (V(Ω),+,,D) is not a differential algebra since the Leibnitz rule does not hold for some pairs of ultrafunctions. See also [7].

4 Properties of ultrafunction solutions

The problems that we want to study with ultrafunctions have the following form: minimize a given functional J on V(Ω) subjected to certain restrictions (e.g., some boundary constrictions, or a minimization on a proper vector subspace of V(Ω)). This kind of problems can be studied in ultrafunctions theory by means of a modification of the Faedo–Galerkin method, based on standard approximations by finite-dimensional spaces. The following is a (maybe even too) general formulation of this idea.

Theorem 4.1.

Let W(Ω) be a vector subspace of V(Ω). Let

={f:V(Ω)for allEfinite-dimensional vector subspaces ofW(Ω),there existsuEf(u)=minvEf(v)}.

Then every FF* has a minimizer in WΛ(Ω).

Proof.

Let F=limλΛfλ, with fλ for every λ𝔏. By hypothesis, for every λ𝔏, there exists

uλWλ:-Span(Wλ)

that minimizes fλ on Wλ. Then u=limλΛuλ minimizes F on limλΛWλ=WΛ as, if v=limλΛvλWλ(Ω), then, for every λ𝔏, we have that fλ(vλ)fλ(uλ), hence,

F(v)=limλΛfλ(vλ)limλΛfλ(uλ)=F(u).

For applications, the following particular case of Theorem 4.1 is particularly relevant.

Corollary 4.2.

Let f(ξ,u,x) be coercive in ξ on every finite-dimensional subspace of V(Ω) and for every xΩ. Let F(u):-f(u,u,x)dx. Then F has a minimum on VΛ.

Proof.

Just notice that F, in the notations of Theorem 4.1. ∎

Theorem 4.1 provides a general existence result. However, such a general result poses two questions: the first is how wild such generalized solutions can be; the second is if this method produces new generalized solutions for problems that already have classical ones.

The answer to these questions depends on the problem that is studied. However, regarding the second question, we have the following result, which strengthens Theorem 4.1:

Theorem 4.3.

Let F:VΛ(Ω)R*, F=limλΛFλ. For every λL, let

Mλ:-{uVλ(Ω):Fλ(u)=minvVλFλ(v)}.

Assume that limλΛMλ. Then

MΛ:-{uVΛ(Ω):F(u)=minvVΛF(v)}=limλΛMλ.

Proof.

MΛlimλΛMλ: Let v=limλΛvλMΛ, and let u=limλΛuλlimλΛMλ. As F(v)F(u), there is a qualified set Q such that, for every λQ, Fλ(vλ)F(uλ). But then vλMλ, for every λQ, hence v=limλΛvλlimλΛMλ.

MΛlimλΛMλ: Let u=limλΛuλlimλΛMλ. Let v=limλΛvλVΛ(Ω). Let

Q={λ𝔏:uλMλ}.

Then Q is qualified and, for every λQ, Fλ(uλ)Fλ(vλ). Therefore F(u)F(v), and so uMΛ. ∎

The following easy consequences of Theorem 4.3 hold:

Corollary 4.4.

In the same notations of Theorem 4.3, let us now assume that there exists kN such that |Mλ|k for every λL. Then |MΛ|k.

Proof.

This holds, as the hypothesis on |Mλ| trivially entails that |limλΛMλ|k. ∎

Corollary 4.5.

In the same notations of Theorem 4.3, let us now assume that F=J*, where J:V(Ω)R. Let

M:-{vV(Ω):v=minwV(Ω)J(w)}.

Assume that M. Then the following facts are equivalent:

  • (1)

    u is a minimizer of F:VΛ(Ω)*.

  • (2)

    uM*VΛ(Ω).

In particular, if uM, then u* minimizes F.

Proof.

(1)(2) Let uM. Let Q(u):-{λ𝔏:uλ}. Then, for every λQ(u),

vMλJ(v)=J(u)vM,

hence MλM for every λQ(u), which is qualified, and so limλΛMλM*VΛ, and we conclude by Theorem 4.3.

(2)(1) By definition,

uM*F(u)=minv[V(Ω)]*F(v),

hence, if uM*Vλ(Ω), it trivially holds that u minimizes F. ∎

Corollary 4.6.

In the same hypotheses and notations of Corollary 4.3, let us assume that M={u1,,un} is finite. Then v minimizes F in VΛ(Ω) if, and only if, there exists uM such that u*=v.

Proof.

Just remember that S={s*:sS} for every finite set S, and that

Mσ={u*:uM}V(Ω)VΛ(Ω).

In general, one might not have minima, but minimization sequences could still exist. In this case, we have the following result (in which, for every ρ*, we set st(ρ)=- if, and only if, ρ is a negative infinite number). Notice that in the following result we are not assuming the continuity of J with respect to any topology on V(Ω), in general.

Theorem 4.7.

Let V(Ω) be a Banach space, let J:V(Ω)R and let infuV(Ω)J(u)=mR{-}. The following facts hold:

  • (1)

    J*(v)m for every vVΛ(Ω).

  • (2)

    There exists vVΛ(Ω) such that st(J*(v))=m.

  • (3)

    If vVΛ(Ω) is a minimum of J*:VΛ(Ω)* then J*(v)st(J*(v))=m.

  • (4)

    Let {un}n be a minimizing sequence that converges to uV(Ω) in some topology τ . Then there exists vVΛ(Ω) such that stτ(v)=u and J*(v)st(J*(v))=m . Moreover, if w+ψ is the canonical splitting of v , then

    • if τ is the topology of pointwise convergence, then w=v and w(x)=u(x) for every xΩ ;

    • if τ is the topology of pointwise convergence a.e., then w=v and w(x)=u(x) a.e. in xΩ ;

    • if τ is the topology of weak convergence, then w(x)=u(x) for every xΩ and ψ,φ**0 for every φ in the dual of V(Ω) ;

    • if τ is the topology associated with a norm and, moreover, {un}n converges pointwise to u , then w=u and ψ*0.

  • (5)

    If all minimizing sequences of J converge to uV(Ω) in some topology τ and v is a minimum of the functional J*:VΛ(Ω)* , then stτ(v)=u and J*(v)st(J*(v))=m.

Proof.

(1) Let v=limλΛvλ. Since m=infuV(Ω)J(u), we have that J(vλ)m for every λΛ, hence J*(v)m.

(2) By (1) it suffices to show that st(J*(v))=m. Let {un}n be a minimizing sequence for J. For every λ𝔏, let vλ:-u|λ|. Let v:-limλΛvλ. We claim that v is the desired ultrafunction.

To prove that st(J*(v))=limn+J(un)=m, we just have to observe that, by our definition of the net {vλ}λ, it follows that4

limn+J(un)=st(limλΛJ(vλ)),

and we conclude as limλΛJ(vλ)=J*(v) by definition.

(3) Let v=limλΛvλ, and let wVΛ(Ω) be such that st(J*(w))=m. Then mJ*(v) by (1), whilst st(J*(v))st(J*(w))=m. Hence, st(J*(v))=m, as desired.

(4) Let v be given as in point (2). Let us show that stV(Ω)(v)=u; let Aτ be an open neighborhood of u. As {un}n converges to u, there exists N>0 such that, for every m>N, unA. Let μ𝔏 be such that |μ|>N. Then, for every

λQμ:-{λ𝔏:μλ},vλA,

and as Qμ is qualified, this entails that vA*. Since this holds for every A neighborhood of u, we deduce that stτ(v)=u, as desired.

Now, let u=w+ψ be the splitting of u.

If τ is the pointwise convergence, stτ(v)(x)=u(x) for every xΩ, hence, by Definition 3.9, we have that the singular set of u is empty and that w(x)=u(x) for every xΩ, as desired. A similar argument works in the case of the pointwise convergence a.e.

If τ is the weak convergence topology, then stτ(v)=u means that v,φ**u,φ for every φ in the dual of V(Ω). Now, let S be the singular set of u. We claim that S=. If not, let xS and let φ=δx. Then v,φ**=v(x) is infinite, whilst u,δx=u(x) is finite, which is absurd. Henceforth, for every xΩ, we have that ψ(x)=0. But

u,φv,φ**=w+ψ,φ**=w,φ**+ψ,φ**=w,φ+ψ,φ**,

hence, stτ(ψ)=u-w. As ψ(x)=0 for all xΩ, this means that u(x)=w(x) for every xΩ. Then

u,φ+ψ,φ**=w,φ+ψ,φ**=v,φu,φ,

and so ψ,φ**0.

Finally, if τ is the strong convergence with respect to a norm and {un}n converges pointwise to u, then, by what we proved above, we have that v(x)u(x) for every xΩ, hence u(x)w(x) for every xΩ, which means u=w as both u,wV(Ω). Then ψ=u-w=u-v+v-w0.

(5) Let v=limλΛvλ. By point (2), the only claim to prove is that stτ(v)=u. We distinguish two cases:

Case 1: J*(v)r. As we noticed in point (2), it must be r=m. By contrast, let us assume that stτ(v)u. In this case, there exists an open neighborhood A of u such that the set

Q:-{λ𝔏:vλA}

is qualified. For every n, let

Qn:-{λ𝔏:|J(vλ)-r|<1n}Q.

Every Qn is qualified, hence nonempty. For every n, let λnQn. Finally, set un:-vλn. By construction, limnJ(un)=m. This means that {un}n is a minimizing sequence, hence, it converges to u in the topology τ, and this is absurd as, for every n, by construction, unA. Henceforth, stτ(v)=u.

Case 2: J*(v)-. As we noticed in the proof of point (2), in this case m=-. Let us assume that stV(Ω)(v)u. Then there exists an open neighborhood A of u such that the set

Q:-{λ𝔏:vλA}

is qualified. For every n, let

Qn={λ𝔏:J(vλ)<-n}Q

and let λnQn. Finally, let un:-vλn. Then J(un)<-n for every n, hence {un}n is a minimizing sequence, and so it must converge to u. However, by construction, unA for every n, which is absurd. ∎

Example 4.8.

Let Ω=(0,1), let

V(Ω)={u:Ωuis the restriction toΩof a piecewise𝒞1([0,1])function}

and let J:V(Ω) be the functional

J(u):-Ωu2(x)dx+Ω((u)2-1)2dx.

It is easily seen that infuV(Ω)J(u)=0, and that the minimizing sequences of J converge pointwise and strongly in the L2 norm to 0, but J(0)=1.

Let vVΛ(Ω) be the minimum of J*:VΛ(Ω). From points (4) and (5) of Theorem 4.7, we deduce that 0<J*(v)0, that stV(Ω)(v)=0 and that the canonical decomposition of v is v=0+ψ, with ψ=0 for every xΩ and Ω**ψ2dx0. Moreover, as J*(ψ)=0, we also have that Ω**((ψ)2-1)2dx0.

5 Applications

5.1 Sign-perturbation of potentials

The first problem that we would like to tackle by means of ultrafunctions regards the sign-perturbation of potentials.

Let us start by recalling some results recently proved by Brasco and Squassina in [13] as a refinement and extension of some classical result by Brezis and Nirenberg [14].

Let Ω be a bounded domain of N with5 N>2. Consider the minimization problem

𝒮(a):-infu𝒟01,2(Ω){uL2(Ω)2+Ωa|u|2dx:uL2*(Ω)=1},(5.1)

where aLN/2(Ω) is given, 2*=2N/(N-2),

𝒟01,2(Ω):-{uL2*(Ω):uL2(Ω),u=0onΩ}.

By Lagrange multipliers rule, minimizers of the previous problem (provided they exist) are constant sign weak solutions of

{-Δu+au=μ|u|2*-2uinΩ,u=0onΩ,(5.2)

with μ=𝒮(a), namely

Ωuφdx+Ωauφdx=μΩ|u|2*-2uφdx,

for every φ𝒟01,2(Ω).

The main result in [13] is the following theorem, where the standard notations

a+=max{a,0},a-=max{-a,0},BR(x0)={xN:|x-x0|<R}

are used.

Theorem 5.1 (Brasco, Squassina).

Let ΩRN be an open bounded set. Then the following facts hold:

  • (1)

    If a0 , then 𝒮(a) does not admit a solution.

  • (2)

    Let N>4 . Assume that there exist σ>0, R>0 and x0Ω such that

    a-σ,a.e. onBR(x0)Ω.

    Then 𝒮(a) admits a solution.

  • (3)

    Let 2<N4 . For any x0Ω , for any R>0 s.t. BR(x0)Ω , there exists σ=σ(R,N)>0 such that if

    a-σ,a.e. onBR(x0),

    then 𝒮(a) admits a solution.

In [1], V. Benci studied, in the ultrafunctions setting, the following similar (simpler) problem: minimize

minu𝔐pJ(u),

where

J(u)=Ω|u|2dx

and

𝔐p={u𝒞02(Ω¯):Ω|u|pdx=1}.

Here Ω is a bounded set in N with smooth boundary, N3 and p>2. In the ultrafunctions setting introduced in [1] (and with the notations of [1]), the problem takes the following form:

minu𝔐~pJ(u),(5.3)

where

J(u)=Ω*|u|2dx

and

𝔐~p={uV2,0(Ω¯):Ω*|u|pdx=1}

with V2,0(Ω¯)=[𝒞02(Ω¯)].

For every p>2, problem (5.3) has an ultrafunction solution u~p and, by setting m~p=J(u~p), one can show that

  • (i)

    if 2<p<2*, then m~p=mp+ and there is at least one standard minimizer u~p, namely u~p𝒞02(Ω¯);

  • (ii)

    if p=2* (and ΩN), then m~2*=m2*+ε, where ε is a positive infinitesimal;

  • (iii)

    if p>2*, then m~p=εp, where εp is a positive infinitesimal.

Our goal is to show that a similar result can be obtained for problem (5.2).

In the present ultrafunctions setting, problem (5.2) takes the following form: find

𝒮~(a):-infuVΛ(Ω){(N)**|u|2dx+(N)**a|u|2dx:u[L2*(N)]*=1},(5.4)

where a*[LN/2(Ω)] is given, and VΛ(Ω)=[𝒟01,2(Ω)]Λ. With the above notations, we can prove the following:

Theorem 5.2.

Let ΩRN be an open bounded set. Then the following facts hold:

  • (1)

    For every a[LN/2(Ω)]* , there exists uVΛ(Ω) that minimizes 𝒮~(a).

  • (2)

    Let a[LN/2(Ω)] . If u𝒞1(Ω)𝒞0(Ω¯) is a minimizer of problem ( 5.1 ), then u* is a minimizer of 𝒮~(a*).

  • (3)

    If a=0 , then 𝒮~(0)=S+ε , where

    S:-infu𝒟0(N){0}uL22uL2*2

    and

    ε={0𝑖𝑓Ω=N,a strictly positive infinitesimal𝑖𝑓ΩN;

    moreover, if u is the minimizer in VΛ(Ω) , then the functional part w of u is 0.

  • (4)

    Let a0 have an isolated minimum xm , and let uVΛ(Ω) be the minimum of problem ( 5.4 ). If u=w+ψ is the canonical splitting of u , then w=0 and ψ concentrates in xm , in the sense that, for every xmon(xm), ψ(x)0 . Moreover, ψ,φ**0 for every φ in the dual of V(Ω).

Proof.

1 This follows from Theorem 4.3, as the functional uL2(Ω)2+Ωa|u|2dx admits a minimum on every finite-dimensional subspace of VΛ.

2 This follows from Corollary 4.5.

3 In [13, Lemma 3.1], it was proved that, if we consider problem (5.1), we have that 𝒮(0)=S, and 𝒮(0) is attained in 𝒟0(Ω) if, and only if, Ω=N. Therefore, if Ω=N, the result follows from point (2). If ΩN, the fact that 𝒮~(0)=S+ε follows from Theorem 4.7 (3). Moreover, all minimizing sequences {un}n converge weakly to 0 in H1, therefore they converge strongly in L2(Ω) and so they converge pointwise a.e., hence, by Theorem 4.7 (5), we deduce that, in the splitting u=w+ψ, we have that w=0, namely the ultrafunction solution coincides with its singular part.

4 We start by following the approach of [13]. We let U be a minimizer of

infu𝒟01,2(Ω)[u]𝒟1,22uL2*2

and, for every ε>0, let Uε(r):-ε2-N2U(rε). Let δ>0 be such that Bδ(xm)Ω, and let uδ,ε be defined as follows:

uδ,ε={Uε(r)ifrδ,Uε(δ)Uε(r)-Uε(δΘ)Uε(δ)-Uε(δΘ)ifδ<rδΘ,0ifr>δΘ,

where Θ is a constant given in [13, Lemma 2.4]. Moreover, if F(u):-uL2(Ω)2+Ωa|u|2dx, for δ1,δ2 small enough, we have that F(uδ1,ε)F(uδ2,ε). Then (uε,ε) is a minimizing net (for ε0), so we can use Theorem 4.7 (4). As (uε,ε) converges pointwise to 0, we obtain that w=0, whilst the definition of the net ensures the concentration of ψ in xm. The last statement is again a direct consequence of Theorem 4.7 (4). ∎

Let us notice that the above theorem shows a strong difference between the ultrafunctions and the classical case. The existence of solutions in VΛ(Ω) is ensured independently of the sign of a whilst, as discussed in [13, Section 4], the conditions on a for the existence of solutions in the approach of Brasco and Squassina are essentially optimal. Of course, ultrafunction solutions might be very wild in general; their particular structure can be described in some cases, depending on a.

5.2 The singular variational problem

5.2.1 Statement of the problem

Let W be a C1-function defined in {0} such that

limt0W(t)=+

and

lim¯t±W(t)t2=0.

We are interested in the singular problem (SP).

Naive formulation of problem SP.

Find a continuous function

u:Ω¯,

which satisfies the equation

-Δu+W(u)=0inΩ(5.5)

with the following boundary condition:

u(x)=g(x)forxΩ,(5.6)

where Ω is an open set such that Ω and gL1(Ω) is a function different from 0 for every x which changes sign, e.g. g(x)=±1. Clearly, this problem does not have any solution in C1. This problem could be reformulated as a kind of free boundary problem in the following way:

Classical formulation of problem SP.

Find two open sets Ω1 and Ω2 and two functions

ui:Ωi,i=1,2,

such that all the following conditions are fulfilled:

Ω=Ω1Ω2Ξ,whereΞ=Ω¯1Ω¯2Ω;-Δui+W(ui)=0inΩi,i=1,2;(5.7)ui(x)=g(x)forxΩΩi,i=1,2;limxΞui(x)=0;Ξis locally a minimal surface.(5.8)

Condition (5.8) is natural, since formally equation (5.5) is the Euler–Lagrange equation relative to the energy

E(u)=12Ω(|u|2+W(u))dx,(5.9)

and the density of this energy diverges as xΞ. In general this problem is quite involved since the set Ξ cannot be a smooth surface and hence, it is difficult to be characterized. However, this problem becomes relatively easy if formulated in the framework of ultrafunctions.

Let us recall that the Laplace operator of an ultrafunction uV(Ω¯) is defined as the only ultrafunction ΔuV(Ω) such that

ΩΔuvdx=-ΩDuDvdx,for allvV0(Ω¯),

where

V0(Ω¯):-{vV(Ω¯):for allxΩΓ,v(x)=0}.

Notice that, we can assert that Δu(x)=DD(x) only in xΩ*.

Ultrafunction formulation of problem SP6.

Find uV(Ω¯) such that

u(x)0for allx(Ω¯)*Γ,(5.10)-Δu+W(u)=0forxΩ*Γ,(5.11)u(x)=g(x)forx(Ω)*Γ.(5.12)

As we will see in the next section, the existence of this problem can be easily proven using variational methods.

5.2.2 The existence result

The easiest way to prove the existence of an ultrafunction solution of problem SP is achieved exploiting the variational structure of equation (5.11). Let us consider the extension

E(u)=Ω(12|Du|2+W(u))dx(5.13)

of the functional (5.9) to the space

Vg(Ω¯):-{uV(Ω¯):for allx(Ω)*Γ,u(x)=g(x)}.

Remark 5.3.

We remark that the integration is taken over Ω*, but u is defined in Ω¯*. This is important, in fact, for some xΩ*, xΩ*, the value of Du(x) depends on the value of u in some point yΩ*,yx. This is a remarkable difference between the usual derivative and the ultrafunction derivative.

Lemma 5.4.

Equation (5.11) is the Euler–Lagrange equation of the functional (5.13).

Proof.

We use the expression of as given in Definition 3.6. As

E(u)=Ω(12|Du|2+W(u))dx,

let us compute separately the variations given by 12|Du|2 and W(u). As

Ω*12|Du|2dx=aΓΩ*12|Du(a)|2da

is a quadratic form, for vV0(Ω¯), we have that

(ddu)*(Ω*12|Du|2dx)[v]=Ω*DuDvdx=Ω*(-Δuv)dx.

The variation given by W(u) for vV0(Ω¯) is

(ddu)*(Ω*(W(u)dx))[v]=(ddu)*aΓΩ*W(u(a))v(a)da=Ω*W(u(x))v(x)dx.

Therefore, the total variation of E is

dE(u)[v]=Ω*(-Δu+W(u))vdx,

which proves our thesis. ∎

The existence of an ultrafunction solution of problem SP follows from the following lemma.

Lemma 5.5.

The functional (5.13) has a minimizer.

Proof.

The functional E(u) is coercive in the sense that, for any c*,

Ec:-{uVg(Ω¯):E(u)c}

is hypercompact (in the sense of NSA), since Vg(Ω¯) is a hyperfinite-dimensional affine manifold. Then, since E is hypercontinuous (in the sense of NSA), the result follows. ∎

Regarding Ξ being a minimal surface, we can prove the following:

Proposition 5.6.

Let u be the ultrafunction minimizer of problem (5.13), as given by Lemma 5.5. Then the sets

Ω1={xΩ:for ally𝔪𝔬𝔫(x)Γ,u(y)>0},Ω2={xΩ:for ally𝔪𝔬𝔫(x)Γ,u(y)<0}

are open, hence

Ξ:-{xΩ¯:there existy1,y2𝔪𝔬𝔫(x)Γsuch thatu(y1)<0,u(y2)>0}

is closed.

Proof.

This follows from overspill7. Let us prove it for Ω1. Let xΩ1. By definition of Ω1, for every ε0, we have that u(y)<0 for every yBε(x)Γ. Hence, by overspill, there exists a real number r>0 such that u(y)<0 for every yBr*(x)Γ. As Brσ(x)Br*(x)Γ, we deduce that the open ball Br(x)Ω1. ∎

Notice that property (1) in Proposition 5.6 is a first step towards property (5.8) in the classical formulation of problem SP. It is our conjecture, in fact, that Ξ is a minimal surface, at least under some rather general hypothesis. We have not been able to prove this yet, however.

Let us conclude with a remark. When studying problems like problem (5.13) with ultrafunctions, one would like to be able to generalize certain properties of elliptic equations based on the maximum principle. For example, one would expect to have the following properties:

  • (1)

    Let Ω be a bounded connected open set with smooth boundary and let g be a bounded function. Then, if

    u=w+ψ

    is the canonical splitting of u, as given in Definition 3.9, we have that wL and ψ(x)0 for every xΩ.

  • (2)

    Let Ω1,Ω2 be the sets defined in Proposition 5.6. Then in Ω1Ω2, we have

    -Δw+W(w)=0,Δψ(x)0.

  • (3)

    If a=inf(g), b=sup(g) and W(t)0 for all t(a,b), we have that

    au(x)b.

However, in the spaces of ultrafunctions constructed in this paper, the maximum principle does not hold directly. This is due to the fact that the kernel of the derivative is, in principle, larger than the space of constants. This problem could be avoided by modifying the space of ultrafunctions. As this leads to some technical difficulties, we prefer to postpone this study to a future paper.

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Footnotes

  • 1

    We use V*(E) as a shorthand notation for [V(E)]*. 

  • 2

    If u is an ultrafunction, the support of u is the set {xΓ:u(x)0}. 

  • 3

    We recall that, by a well-known theorem of Schwartz, any tempered distribution can be represented as mf, where m is a multi-index and f is a continuous function. 

  • 4

    A proof of this simple claim is given in [11, Lemma 28]. 

  • 5

    In [13], the authors work more in general with a p(1,N), and consider also a fractional version of Problem 5.2; however, in this paper, we prefer to consider only the local case p=2. 

  • 6

    If u is an ultrafunction and W, W, etc. are functions, for short, we shall write W(u), W(u), etc. instead of W*(u), (W)*(u), etc. 

  • 7

    Overspill is a well-known and very useful property in nonstandard analysis. The idea behind the version that we use here is the following: if a certain property P(x) holds for every x0, then there must be a real number r>0 such that P(x) holds for every x<r. For a proper formulation of overspill, we refer to [16]. 

About the article

Received: 2018-06-27

Accepted: 2018-06-27

Published Online: 2018-08-08

Published in Print: 2019-03-01


Funding Source: Austrian Science Fund

Award identifier / Grant number: M1876-N35

Award identifier / Grant number: P26859-N25

Award identifier / Grant number: P30821-N35

Marco Squassina is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) and of Istituto Nazionale di Alta Matematica (INdAM). Lorenzo Luperi Baglini has been supported by grants M1876-N35, P26859-N25 and P30821-N35 of the Austrian Science Fund FWF.


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 124–147, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0146.

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