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  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).
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  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  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 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.  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.
If X is a set and Ω is a subset of , then
denotes the power set of X and denotes the family of finite subsets of X;
denotes the set of all functions from X to Y and ;
denotes the set of continuous functions defined on ;
denotes the set of functions defined on which have continuous derivatives up to the order k (sometimes we will use the notation instead of );
denotes the usual Sobolev space of functions defined on ;
if is any function space, then will denote the function space of functions in having compact support;
, denotes the set of continuous functions in which vanish for ;
denotes the set of the infinitely differentiable functions with compact support defined on ; denotes the topological dual of , namely the set of distributions on Ω;
if is a set, then denotes the characteristic function of A;
where is the usual notion of support of a function or a distribution;
, where means that is infinitesimal;
means “for almost every ”;
if , then
if W is a generic function space, its topological dual will be denoted by and the pairing by ;
if E is any set, then will denote its cardinality.
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  (see also  and ). When we talk about NSA, we mean the NSA in the sense of Robinson (see ), and not the internal set theory developed by Nelson in .
2.1 Non-Archimedean fields
Here, we recall the basic definitions and facts regarding non-Archimedean fields. In the following, 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.
Let be an ordered field. Let . We say that
ξ is infinitesimal if, for all positive , ;
ξ is finite if there exists such that ;
ξ is infinite if, for all , (equivalently, if ξ is not finite).
An ordered field is called non-Archimedean if it contains an infinitesimal .
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.
A superreal field is an ordered field 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:
We say that two numbers are infinitely close if is infinitesimal. In this case, we write .
Clearly, the relation of infinite closeness is an equivalence relation and we have the following:
If is a totally ordered superreal field, every finite number is infinitely close to a unique real number , called the standard part of ξ.
Given a finite number ξ, we denote its standard part by , and we put if 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.
Let be a superreal field and a number. The monad of ξ is the set of all numbers that are infinitely close to it,
2.2 The Λ-limit
Let U be an infinite set of cardinality bigger than the continuum, and let be the family of finite subsets of U.
Notice that is a directed set. We add to a point at infinity , and we define the following family of neighborhoods of Λ:
where is a fine ultrafilter on , namely a filter such that
for every , if , then or ;
for every , the set .
We will refer to the elements of as qualified sets. A function , defined on a directed set E, is called net (with values in E). If is a real net, we have that
if, and only if, for every , there exists such that for all .
As usual, if a property is satisfied by any λ in a neighborhood of Λ, we will say that it is eventually satisfied.
If the net takes values in a compact set K, then it is a converging net.
Suppose that the net has a converging subnet to . We fix arbitrarily, and we have to prove that , where
We argue indirectly, and we assume that . Then, by the definition of ultrafilter, , and hence,
This contradicts the fact that has a subnet which converges to L. ∎
Assume that , where E is a first countable topological space; then if
there exists a sequence in such that
We refer to the sequence as a subnet of .
Let be a countable basis of open neighborhoods of . For every , the set
is qualified. Hence, . Let . Then the sequence has trivially the desired property: for every , for every , we have that . ∎
Let be a net with values in a bounded subset of a reflexive Banach space equipped with the weak topology; then
is uniquely defined, and there exists a sequence which converges to v.
The set of the hyperreal numbers is a set equipped with a topology τ such that
every net has a unique limit in , if 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 of a net with respect to the τ topology, following , is called the Λ-limit of φ, and the following notation will be used:
namely, we shall use the up-arrow “” to remind that the target space is equipped with the topology τ. Given
Then the following well-known theorem holds:
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 , where . 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 can be defined by induction (a net is called bounded, if there exists such that, for all , ). To do this, consider a net
A mathematical entity (number, set, function or relation) which is the Λ-limit of a net is called internal.
If for all , , we set , namely
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 . In general, the inclusion is proper.
Given any set E, we can associate to it two sets: its natural extension and the set , where
Clearly is a copy of E; however, it might be different as set since, in general, .
If is a net with value in a topological space, we have the usual limit
which, by Proposition 2.7, always exists in the Alexandrov compactification . Moreover, we have the Λ-limit, that always exists and it is an element of . In addition, the Λ-limit of a net is in if, and only if, φ is eventually constant. If , and both limits exist, then
The above equation suggests the following definition.
If X is topological space equipped with a Hausdorff topology, and , we set
if there is a net converging in the topology of X, and such that
By the above definition, we have that
be a net of functions. We define a function
as follows: for every , we set
where is a net of numbers such that
A function which is a Λ-limit is called internal. In particular, if, for all ,
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 is the graph of its natural extension.
2.4 Hyperfinite sets and hyperfinite sums
An internal set is called hyperfinite, if it is the Λ-limit of a net , where is a family of finite sets.
For example, if , the set
is hyperfinite. Notice that . 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
be a hyperfinite set of numbers (or vectors); then the hyperfinite sum of the elements of A is defined in the following way:
In particular, if with , then setting
we use the notation
3.1 Definition of ultrafunctions
We start by introducing the notion of hyperfinite grid.
A hyperfinite set Γ such that is called hyperfinite grid.
From now on, we assume that Γ has been fixed once forever. Notice that, by definition, , and the following two simple (but useful) properties of Γ can be easily proven via Λ-limits:
for every there exists so that ;
there exists a hyperreal number , , such that for every , .
A space of grid functions is a family of internal functions
defined on a hyperfinite grid Γ. If , then will denote the restriction of the grid functions to the set .
Let E be any set in . To every internal function , it is possible to associate a grid function by the “restriction” map
defined as follows:
moreover, if , for short, we use the notation
So every function can be uniquely extended to a grid function .
In many problems, we have to deal with functions defined almost everywhere in Ω, such as . Thus, it is useful to give a “rule” which allows to define a grid function for every .
If a function f is defined on a set , we put
where, for all , the grid function is defined as follows: .
If is a measurable set, we define the “density function” of E as follows:
where η is a fixed infinitesimal and m is the Lebesgue measure. Clearly, is a function whose value is 1 in and 0 in ; moreover, it is easy to prove that is a measurable function, and we have that
whenever ; also, if E is a bounded open set with smooth boundary, we have that for every .
Now, let be a vector space such that .
A space of ultrafunctions modeled over the space is a space of grid functions such that there exists a vector space such that the map1
is an -linear isomorphism. From now on, we assume that satisfies the following assumption: if Ω is a bounded open set such that and , then
Next, we want to equip with the two main operations of calculus, the integral and the derivative.
The pointwise integral
is a linear functional which satisfies the following properties:
there exists an ultrafunction such that, for all , , and for all ,
If is any set, we use the obvious notation
A few words to discuss the above definition: Point (2) says that the pointwise integral is nothing else but a hyperfinite sum. Since , every non-null positive ultrafunction has a strictly positive integral. In particular, if we denote by the ultrafunctions whose value is 1 for and 0 otherwise, we have that
The pointwise integral allows us to define the following scalar product:
From now on, the norm of an ultrafunction will be defined as
Now, let us examine point (1) of the above definition. If we take , we have that , and hence
Thus, the pointwise integral is an extension of the Riemann integral defined on . However, if we take a bounded open set Ω such that , then we have that
However, the pointwise integral cannot have this property; in fact,
since . In particular, if Ω is a bounded open set with smooth boundary and , then
of course, the term is an infinitesimal number and it is relevant only in some particular problems.
The ultrafunction derivative
is a linear operator which satisfies the following properties:
for all and for all , x finite,
for all ,
if Ω is a bounded open set with smooth boundary, then for all ,
where is the unit outer normal, is the -dimensional measure and is the canonical basis of ;
the support2 of is contained in .
Let us comment the above definition. Point (1) implies that, for all and for all ,
namely, the ultrafunction derivative coincides with the usual partial derivative whenever . The meaning of point (2) is clear; we remark that this point is very important in comparing ultrafunctions with distributions. Point (3) says that is an ultrafunction whose support is contained in ; it can be considered as a signed measure concentrated on . 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 , then the support of is contained in . Moreover, property (4) implies that the ultrafunction derivative is well defined in for any open set Ω by the following formula:
If and is an ultrafunction in such that, for all , , then, by point (3), for all such that , we have that
however, this property fails for some . In fact, the support of is contained in , but not in .
In , there is a construction of a space which satisfies the desired properties. The conclusion follows, taking
3.2 The splitting of an ultrafunction
In many applications, it is useful to split an ultrafunction u in a part which is the canonical extension of a standard function w and a part ψ which is not directly related to any classical object. If , we set
We will refer to S as to the singular set of the ultrafunction u.
For every ultrafunction u, consider the splitting
and , which is defined by Definition 3.3, is called the functional part of u;
is called the singular part of u.
Notice that , the functional part of u, may assume infinite values for some , but they are determined by the values of w which is a standard function defined on Ω.
Take , and
In this case,
We conclude this section with the following trivial proposition which, nevertheless, is very useful in applications.
Take a Banach space W such that . Assume that is a sequence which converges weakly in W and pointwise to a function w; then, if we set
we have that
As a consequence of the pointwise convergence of to w, we have that, for all , . In particular, for all , . As Γ is hyperfinite, the set has a maximum . Hence, for every , we have
as and . For the second statement, let us notice that
as . ∎
An immediate consequence of Proposition 3.11 is the following:
If , then
3.3 The Gauss divergence theorem
First of all, we fix the notation for the main differential operators:
will denote the usual gradient of standard functions;
will denote the natural extension of internal functions;
will denote the canonical extension of the gradient in the sense of ultrafunctions.
Next, let us consider the divergence:
will denote the usual divergence of standard vector fields ;
will denote the divergence of internal vector fields ;
will denote the divergence of vector-valued ultrafunctions .
And finally, we can define the Laplace operator of an ultrafunction as the only ultrafunction such that
By Definition 3.6 (3), for any bounded open set Ω with smooth boundary,
and by Definition 3.6 (2),
Now, if we take a vector field , by the above identity, we get
Now, if , by Definition 3.6 (1), we get the Gauss divergence theorem
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 . It is well known, that, for any bounded open set Ω with smooth boundary, the distributional derivative is a vector-valued Radon measure, and we have that
Then, the following definition is a natural generalization.
If A is a measurable subset of , we set
and, for all ,
In fact, the left-hand term has been defined as follows:
while the right-hand term is
in particular, if is smooth and is bounded, is an infinitesimal number.
If A is an arbitrary measurable subset of , we have that
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.
The space of generalized distributions on Ω is defined as follows:
The equivalence class of u in will be denoted by .
Let be a generalized distribution. We say that is a bounded generalized distribution if, for all , is finite. We will denote by the set of bounded generalized distributions.
We have the following result.
There is a linear isomorphism
such that, for every and for every ,
For the proof, see e.g. . ∎
From now on, we will identify the spaces and ; so, we will identify with and we will write and
If and , then, for all ,
The set is an algebra which extends the algebra of continuous functions . If we identify a tempered distribution3 with the ultrafunction , we have that the set of tempered distributions is contained in . However, the Schwartz impossibility theorem is not violated as is not a differential algebra since the Leibnitz rule does not hold for some pairs of ultrafunctions. See also .
4 Properties of ultrafunction solutions
The problems that we want to study with ultrafunctions have the following form: minimize a given functional J on subjected to certain restrictions (e.g., some boundary constrictions, or a minimization on a proper vector subspace of ). 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.
Let be a vector subspace of . Let
Then every has a minimizer in .
Let , with for every . By hypothesis, for every , there exists
that minimizes on . Then minimizes F on as, if , then, for every , we have that , hence,
For applications, the following particular case of Theorem 4.1 is particularly relevant.
Let be coercive in ξ on every finite-dimensional subspace of and for every . Let . Then has a minimum on .
Just notice that , 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:
Let , . For every , let
Assume that . Then
: Let , and let . As , there is a qualified set Q such that, for every , . But then , for every , hence .
: Let . Let . Let
Then Q is qualified and, for every , . Therefore , and so . ∎
The following easy consequences of Theorem 4.3 hold:
In the same notations of Theorem 4.3, let us now assume that there exists such that for every . Then .
This holds, as the hypothesis on trivially entails that . ∎
In the same notations of Theorem 4.3, let us now assume that , where . Let
Assume that . Then the following facts are equivalent:
u is a minimizer of .
In particular, if , then minimizes F.
Let . Let . Then, for every ,
hence for every , which is qualified, and so , and we conclude by Theorem 4.3.
hence, if , it trivially holds that u minimizes F. ∎
In the same hypotheses and notations of Corollary 4.3, let us assume that is finite. Then v minimizes F in if, and only if, there exists such that .
Just remember that for every finite set S, and that
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 , we set 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 , in general.
Let be a Banach space, let and let . The following facts hold:
for every .
There exists such that .
If is a minimum of then .
Let be a minimizing sequence that converges to in some topology τ . Then there exists such that and . Moreover, if is the canonical splitting of v , then
if τ is the topology of pointwise convergence, then and for every ;
if τ is the topology of pointwise convergence a.e., then and a.e. in ;
if τ is the topology of weak convergence, then for every and for every φ in the dual of ;
if τ is the topology associated with a norm and, moreover, converges pointwise to u , then and .
If all minimizing sequences of J converge to in some topology τ and v is a minimum of the functional , then and .
(1) Let . Since , we have that for every , hence .
(2) By (1) it suffices to show that . Let be a minimizing sequence for J. For every , let . Let . We claim that v is the desired ultrafunction.
To prove that , we just have to observe that, by our definition of the net , it follows that4
and we conclude as by definition.
(3) Let , and let be such that . Then by (1), whilst . Hence, , as desired.
(4) Let v be given as in point (2). Let us show that ; let be an open neighborhood of u. As converges to u, there exists such that, for every , . Let be such that . Then, for every
and as is qualified, this entails that . Since this holds for every A neighborhood of u, we deduce that , as desired.
Now, let be the splitting of u.
If τ is the pointwise convergence, for every , hence, by Definition 3.9, we have that the singular set of u is empty and that for every , as desired. A similar argument works in the case of the pointwise convergence a.e.
If τ is the weak convergence topology, then means that for every φ in the dual of . Now, let S be the singular set of u. We claim that . If not, let and let . Then is infinite, whilst is finite, which is absurd. Henceforth, for every , we have that . But
hence, . As for all , this means that for every . Then
and so .
Finally, if τ is the strong convergence with respect to a norm and converges pointwise to u, then, by what we proved above, we have that for every , hence for every , which means as both . Then .
(5) Let . By point (2), the only claim to prove is that . We distinguish two cases:
Case 1: . As we noticed in point (2), it must be . By contrast, let us assume that . In this case, there exists an open neighborhood A of u such that the set
is qualified. For every , let
Every is qualified, hence nonempty. For every , let . Finally, set . By construction, . This means that is a minimizing sequence, hence, it converges to u in the topology τ, and this is absurd as, for every , by construction, . Henceforth, .
Case 2: . As we noticed in the proof of point (2), in this case . Let us assume that . Then there exists an open neighborhood A of u such that the set
is qualified. For every , let
and let . Finally, let . Then for every , hence is a minimizing sequence, and so it must converge to u. However, by construction, for every , which is absurd. ∎
Let , let
and let be the functional
It is easily seen that , and that the minimizing sequences of J converge pointwise and strongly in the norm to 0, but .
Let be the minimum of . From points (4) and (5) of Theorem 4.7, we deduce that , that and that the canonical decomposition of v is , with for every and . Moreover, as , we also have that .
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 Ω be a bounded domain of with5 . Consider the minimization problem
where is given, ,
By Lagrange multipliers rule, minimizers of the previous problem (provided they exist) are constant sign weak solutions of
with , namely
for every .
The main result in  is the following theorem, where the standard notations
Theorem 5.1 (Brasco, Squassina).
Let be an open bounded set. Then the following facts hold:
If , then does not admit a solution.
Let . Assume that there exist , and such that
Then admits a solution.
Let . For any , for any s.t. , there exists such that if
then admits a solution.
In , V. Benci studied, in the ultrafunctions setting, the following similar (simpler) problem: minimize
For every , problem (5.3) has an ultrafunction solution and, by setting , one can show that
if , then and there is at least one standard minimizer , namely ;
if (and , then , where ε is a positive infinitesimal;
if , then , where 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
where is given, and . With the above notations, we can prove the following:
Let be an open bounded set. Then the following facts hold:
For every , there exists that minimizes .
Let . If is a minimizer of problem ( 5.1 ), then is a minimizer of .
If , then , where
moreover, if u is the minimizer in , then the functional part w of u is 0.
Let have an isolated minimum , and let be the minimum of problem ( 5.4 ). If is the canonical splitting of u , then and ψ concentrates in , in the sense that, for every , . Moreover, for every φ in the dual of .
3 In [13, Lemma 3.1], it was proved that, if we consider problem (5.1), we have that , and is attained in if, and only if, . Therefore, if , the result follows from point (2). If , the fact that follows from Theorem 4.7 (3). Moreover, all minimizing sequences converge weakly to 0 in , therefore they converge strongly in and so they converge pointwise a.e., hence, by Theorem 4.7 (5), we deduce that, in the splitting , we have that , namely the ultrafunction solution coincides with its singular part.
and, for every , let . Let be such that , and let be defined as follows:
where Θ is a constant given in [13, Lemma 2.4]. Moreover, if , for small enough, we have that . Then is a minimizing net (for ), so we can use Theorem 4.7 (4). As converges pointwise to 0, we obtain that , whilst the definition of the net ensures the concentration of ψ in . 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 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 -function defined in such that
We are interested in the singular problem (SP).
Naive formulation of problem SP.
Find a continuous function
which satisfies the equation
with the following boundary condition:
where Ω is an open set such that and is a function different from 0 for every x which changes sign, e.g. . Clearly, this problem does not have any solution in . 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 and and two functions
such that all the following conditions are fulfilled:
and the density of this energy diverges as . 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 is defined as the only ultrafunction such that
Notice that, we can assert that only in .
Ultrafunction formulation of problem SP6.
Find such that
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
of the functional (5.9) to the space
We remark that the integration is taken over , but u is defined in . This is important, in fact, for some , , the value of depends on the value of u in some point . This is a remarkable difference between the usual derivative and the ultrafunction derivative.
We use the expression of as given in Definition 3.6. As
let us compute separately the variations given by and . As
is a quadratic form, for , we have that
The variation given by for is
Therefore, the total variation of is
which proves our thesis. ∎
The existence of an ultrafunction solution of problem SP follows from the following lemma.
The functional (5.13) has a minimizer.
The functional is coercive in the sense that, for any ,
is hypercompact (in the sense of NSA), since is a hyperfinite-dimensional affine manifold. Then, since is hypercontinuous (in the sense of NSA), the result follows. ∎
Regarding Ξ being a minimal surface, we can prove the following:
are open, hence
This follows from overspill7. Let us prove it for . Let . By definition of , for every , we have that for every . Hence, by overspill, there exists a real number such that for every . As , we deduce that the open ball . ∎
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:
Let Ω be a bounded connected open set with smooth boundary and let g be a bounded function. Then, if
is the canonical splitting of u, as given in Definition 3.9, we have that and for every .
Let be the sets defined in Proposition 5.6. Then in , we have
If , and for all , we have that
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A proof of this simple claim is given in [11, Lemma 28].
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 holds for every , then there must be a real number such that holds for every . For a proper formulation of overspill, we refer to .
About the article
Published Online: 2018-08-08
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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