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  and Sato [18, 19]. This paper deals with a new kind of generalized functions, called “ultrafunctions”, which have been introduced recently in  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 , 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  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 . 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 are the following: there exist a generalized partial derivative and a generalized integral (called pointwise integral) such that the following hold:
The generalized derivative is a local operator, namely, if (where E is an open set), then .
For all ,
The “generalized” Gauss theorem holds for any measurable set A (see Theorem 4.4)
To any distribution we can associate an equivalence class of ultrafunctions such that, for all and all ,
where 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 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. ) and is an extension of the pointwise product on .
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  and .
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 -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.
Let X be a set and let Ω be a subset of .
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 Ω.
denotes the set of functions defined on Ω which have continuous derivatives up to the order k.
denotes the usual Sobolev space of functions defined on Ω.
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 Ω, and denotes the topological dual of , namely the set of distributions on Ω.
For any , we set .
, where is the usual notion of support of a function or a distribution.
, where means that is infinitesimal.
If W is a generic function space, its topological dual will be denoted by and the pairing by
We denote by the indicator (or characteristic) function of X, namely,
will denote the cardinality of X.
2.1 Non-Archimedean fields
Here, we recall the basic definitions and facts regarding non-Archimedean fields. In the following, will denote an ordered field. We recall that such a field contains (a copy of) the rational numbers. Its elements will be called numbers.
Let be an ordered field and . 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 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.
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:
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 theorem.
If is a 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.
Let be a superreal field, and a number. The monad of ξ is the set of all numbers that are infinitely close to it, i.e.
and the galaxy of ξ is the set of all numbers that are finitely close to it, i.e.
By definition, it follows that the set of infinitesimal numbers is and that the set of finite numbers is .
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 ):
Let E be an infinite set. The superstructure on E is the set
where the sets are defined by induction setting
and, for every ,
Here 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 contains almost every usual mathematical object that can be constructed starting with E; in particular, , which is the superstructure that we will consider in the following, contains almost every usual mathematical object of analysis.
Throughout this paper we let
and we order via inclusion. Notice that is a directed set. We add to a point at infinity , and we define the following family of neighborhoods of
where is a fine ultrafilter on , namely, a filter such that the following hold:
For every , if , then or .
For every the set .
In particular, we will refer to the elements of as qualified sets and we will write when we want to highlight the choice of the ultrafilter. A function will be called net (with values in E). If is a real net, we have that
if and only if for all 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.
Notice that the Λ-topology satisfies these interesting properties:
If the net takes values in a compact set K, then it is a converging net.
Suppose that the net has a subnet converging 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 .
It follows easily from the definitions. ∎
Let be a net with values in a bounded set 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 the following hold:
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 an 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.
Then the following well-known theorem holds:
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 , where
The isomorphism resembles the classical one between the real numbers and the equivalence classes of Cauchy sequences. This method is well known for the construction of real numbers starting from rationals.
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 (a net is called bounded if there exists such that, for all , ). To do this, let us consider a net
We will define by induction on n.
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,
The set 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 values in a topological space, we have the usual limit
which, by Proposition 2.8, always exists in the Alexandrov compactification . Moreover, we have that the Λ-limit 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 a 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 ,
The function is called the natural extension of f. 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 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
be an 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 Caccioppoli spaces of ultrafunctions
Let Ω be an open bounded set in , and let be a (real or complex) vector space such that
A space of ultrafunctions modeled over the space is given by
where is an increasing net of finite-dimensional spaces such that
So, given any vector space of functions , the space of ultrafunction generated by is a vector space of hyperfinite dimension that includes , as well as other functions in . Hence the ultrafunctions are particular internal functions
Given a space of ultrafunctions , a σ-basis is an internal set of ultrafunctions such that and for all , we can write
It is possible to prove (see e.g. ) that every space of ultrafunctions has a σ-basis. Clearly, if , then , where 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:
A Caccioppoli set E is a Borel set such that , namely, such that (the distributional gradient of the characteristic function of E) is a finite Radon measure concentrated on .
is called Caccioppoli perimeter of E. From now on, with some abuse of notation, the above expression will be written as follows:
this expression makes sense since “” is a measure.
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
also, if E is a bounded Caccioppoli set,
A set E is called special Caccioppoli set if it is open, bounded and . The family of special Caccioppoli sets will be denoted by .
Now we can define a space suitable for our aims:
A function if and only if
where , , and n is a number which depends on f. Such a function will be called Caccioppoli function.
Notice that is a module over the ring and that, for all ,
Hence, in particular, , is a norm (and not a seminorm).
The space is called Caccioppoli space of ultrafunctions if it satisfies the following properties:
is modeled on the space .
has a σ-basis , , such that, for all , the support of is contained in .
The existence of a Caccioppoli space of ultrafunctions will be proved in Section 5.
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 has a special position since it satisfies the properties required by a large class of problems. First of all . 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 is not a good space for modeling ultrafunctions, since they are defined pointwise while the functions in are defined a.e. Thus, we are lead to the space , but this space does not contain functions like which are important in many situations; for example, the Gauss’ divergence theorem can be formulated as
whenever the vector field F and E are sufficiently smooth. Thus the space seems to be the right space for a large class of problems.
3.2 The pointwise integral
From now on we will denote by a fixed Caccioppoli space of ultrafunctions and by a fixed σ-basis as in Definition 3.6. If , we have that
Equality (3.2) suggests the following definition:
For any internal function , we set
In the sequel we will refer to as to the pointwise integral.
From Definition 3.8, we have that
and, in particular,
But in general these equalities are not true for functions. For example if
we have that
However, for any set and any function ,
Then, if and E is a bounded open set, we have that
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 .
If is smooth, we have that, for all , and hence, if E is open,
of course, the term is an infinitesimal number and it is relevant only in some particular problems.
The pointwise integral allows us to define the following scalar product:
From now on, the norm of an ultrafunction will be given by
If is a σ-basis, then is a orthonormal basis with respect to the scalar product (3.3). Hence for every ,
Moreover, we have that
3.3 The δ-bases
Next, we will define the delta ultrafunctions:
Given a point , we denote by an ultrafunction in such that
and is called delta (or the Dirac) ultrafunction concentrated in q.
Let us see the main properties of the delta ultrafunctions:
The delta ultrafunction satisfies the following properties:
For every there exists a unique delta ultrafunction concentrated in q.
For every .
(1) Let be an orthonormal real basis of , and set
Let us prove that actually satisfies (3.5). Let be any ultrafunction. Then
So is a delta ultrafunction concentrated in q. It is unique: in fact, if is another delta ultrafunction concentrated in q, then for every we have
and hence for every .
(2) We have
(3) We have
The proof is complete. ∎
By the definition of Γ, for all , we have that
From this it follows readily the following result.
The set is the dual basis of the sigma-basis; it will be called the δ-basis of .
Let us examine the main properties of the δ-basis.
The δ-basis satisfies the following properties:
For all , .
For all ,
(i) This is an immediate consequence of the definition of δ-basis.
(ii) By Theorem 3.10, it follows that
(iii) This is an immediate consequence of (ii). ∎
3.4 The canonical extension of functions
We have seen that every function has a natural extension . However, in general, is not an ultrafunction; in fact, it is not difficult to prove that the natural extension of a function f is an ultrafunction if and only if . So it is useful to define an ultrafunction which approximates . More general, for any internal function , we will define an ultrafunction as follows.
If is an internal function, we define by the formula
if , with some abuse of notation, we set
Since , for any internal function u, we have that
defined by is noting else but the orthogonal projection of with respect to the semidefinite bilinear form
If , and , then and hence
If a function f is not defined on a set , by convention, we define
By the definition above, for all , we have that
If , then unless . Let us examine what looks like.
Let be continuous in a bounded open set . Then, for all with , we have that
Fix . Since A is bounded, there exists a set such that
We have that (see Example 3.16)
Since , it follows that ; moreover, since , by Definition 3.6 (ii),
Then . ∎
If , then, for any such that is finite, we get
3.5 Canonical splitting of an ultrafunction
In many applications, it is useful to split an ultrafunction u into 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
For every ultrafunction u consider the splitting
and , which is defined by Definition 3.17, is called the functional part of u,
is called the singular part of u.
We will refer to
as to the singular set of the ultrafunction u.
Notice that , 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 .
Take , and
In this case
We conclude this section with the following trivial propositions which, nevertheless, are very useful in applications:
Let W be a Banach space such that and assume that is weakly convergent in W. Then if
is the canonical splitting of , there exists a subnet such that
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:
If is weakly convergent to w in W and , then
If is strongly convergent to w in W, then
An immediate consequence of Proposition 3.23 is the following:
If , then
Since is dense in , there is a sequence which converges strongly to w in . Now set
By Proposition 3.23, we have that
with . Since u and are in , it also follows that , so that . Then
On the other hand,
4 Differential calculus for ultrafunctions
In this section, we will equip the Caccioppoli space of ultrafunctions 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 , then is well defined and hence, using Definition 3.17, we can define an operator
However, it would be useful to extend the operator to all the ultrafunctions in 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.
The generalized derivative
is an operator defined on a Caccioppoli ultrafunction space which satisfies the following properties:
has σ-basis such that, for all , the support of is contained in .
If , then
For all ,
If , then for all ,
where is the measure theoretic unit outer normal, integrated on the reduced boundary of E with respect to the -Hausdorff measure dS (see e.g. [14, Section 5.7]) and is the canonical basis of .
We remark that, in the framework of the theory of Caccioppoli sets, the classical formula corresponding to (IV) is the following: for all ,
The existence of a generalized derivative will be proved in Section 5.
Now let us define some differential operators:
will denote the usual gradient of standard functions,
will denote the natural extension of the gradient (in the sense of NSA),
will denote the canonical extension of the gradient in the sense of the ultrafunctions (Definition 4.1).
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 unique ultrafunction such that, for all ,
Finally, we can define the Laplace operator:
or will denote the Laplace operator defined by .
4.2 The Gauss’ divergence theorem
By Definition 4.1 (IV), for any set and ,
and by Definition 4.1 (III),
If we take a vector field , by the above identity, we get
Now, if and is smooth, we get the Gauss’ divergence theorem:
Then (4.1) is a generalization of the Gauss’ theorem which makes sense for any set . Next, we want to generalize Gauss’ theorem to any measurable subset .
First of all we need to generalize the notion of Caccioppoli perimeter to any arbitrary set. As we have seen in Section 3.1, if is a special Caccioppoli set, we have that
and it is possible to define an -dimensional measure dS as follows:
In particular, if the reduced boundary of E coincides with , we have that (see [14, Section 5.7])
Then the following definition is a natural generalization:
If A is a measurable subset of Ω, we set
and for all ,
In fact, the left-hand term has been defined as
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 Ω, we have that
By Definition 4.1 (III),
Then, using the definition of and (4.2), the above formula can be written as follows:
The proof is complete. ∎
Clearly, if , then
If A is the Koch snowflake, then the usual Gauss’ theorem makes no sense since ; 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 is a d-dimensional fractal set, it is an interesting open problem to investigate the relation between its Hausdorff measure and the ultrafunction “measure” .
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.
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 denote by the set of the bounded generalized distributions.
We have the following result.
There is a linear isomorphism
For a proof see e.g. . ∎
From now on we will identify the spaces and so, we will identify with and we will write and
Moreover, with some abuse of notation, we will write also that , , etc., meaning that the distribution can be identified with a function f in , , etc. By our construction, this is equivalent to saying that . So, in this case, we have that for all ,
Since an ultrafunction is univocally determined by its value in Γ, we may think of ultrafunctions as being defined only on Γ and to denote them by ; the set is an algebra which extends the algebra of continuous functions 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 , where α is a multi-index and f is a continuous function. If we identify with the ultrafunction , we have that the set of tempered distributions is contained in . However, the Schwartz impossibility theorem (see introduction) is not violated since 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
In this subsection we will construct a space of ultrafunctions and in the next subsection we will equip it with a σ-basis in such a way that becomes a Caccioppoli space of ultrafunctions according to Definition 3.6.
Given a family of open sets , we say that a family of open sets is a basis for if
for all , ,
for all , there is a set of indices such that
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 .
Let us verify that
For any finite family of special Caccioppoli sets , there exists a basis whose elements are special Caccioppoli sets. Moreover, also the set generated by consists of special Caccioppoli sets.
For any , we set
We claim that is a basis. Since is a finite family, we also have that is a finite family and hence there is a finite set of indices K such that . Now it is easy to prove that is a basis and it consists of special Caccioppoli sets. Also the last statement is trivial. ∎
and we denote by and the relative basis and the generated family which exist by the previous lemma.
The following properties hold true:
and are hyperfinite.
If , then
where is a hyperfinite set and is the natural extension to of the function defined on by ( 3.1 ).
It follows trivially by the construction. ∎
The next lemma is a basic step for the construction of the space .
For any there exists a set , and a family of functions such that the following hold:
is a hyperfinite set, and .
If and , then .
If , then there exists such that .
For any , implies .
For any ,
and we denote by ρ a smooth bell shaped function having support in ; then the functions , , have disjoint support. We set
so that and we divide all points , among sets , , in such a way that
if , then ;
if , there exists a unique () such that .
With this construction, claims (1) and (2) are trivially satisfied.
Now, for any , set
It is easy to check that the functions satisfy (3)–(6). ∎
so we have that, for any , . Also, we set
Finally, we can define the as follows:
Namely, if , then
5.2 The σ-basis
In this subsection, we will introduce a σ-basis in such a way that becomes a Caccioppoli space of ultrafunctions, according to Definition 3.6.
There exists a σ-basis for , , such that the following hold:
, where and for .
is a σ -basis for .
First we introduce in the following scalar product:
For any we set
where and the functions are defined in Lemma 5.4. If we set
we have that
is a δ-basis for (with respect to the scalar product (5.6)). In fact, if , then
and hence, by Lemma 5.4, it follows that
Next, we want to complete this basis and to get a δ-basis for . To this end, we take an orthonormal basis of , where is the orthogonal complement of in (with respect to the scalar product (5.6)). For every , set
notice that this definition is not in contradiction with (5.3) since in the latter .
For every , we have that
It is not difficult to realize that generates all and hence we can select a set such that is a basis for . Taking
we have that is a basis for .
Now let denote the dual basis of namely a basis such that, for all ,
Clearly, it is a σ-basis for . In fact, if , we have that
Notice that if , then . The conclusion follows taking . ∎
By the above theorem, the following corollary follows straightforward.
The set is a Caccioppoli space of ultrafunctions in the sense of Definition 3.6.
If (see (5.2)), we set
If, for any internal set A, we define
then we have the following result:
If and , then
5.3 Construction of the generalized derivative
Next we construct a generalized derivative on .
will denote the orthogonal complement of in . According to this decomposition, and we can define the following orthogonal projectors:
hence, any ultrafunction has the following orthogonal splitting: , where .
Now we are able to define the generalized partial derivative for .
We define the generalized partial derivative
Moreover, if , we set
where, for any linear operator L, denotes the adjoint operator.
Notice that if , by Theorem 5.4 (2), we have
In fact, if , then , and hence, by (5.4), and so
Let us prove property (I). If and , by Definition 5.8,
If and , then
so, if , and , then
and hence, if we set ,
We prove property (II). If , then
and hence for all , . Then, by (5.9), we have, for all ,
The conclusion follows from the arbitrariness of v.
Next let us prove property (III). First we prove this property if . By (5.10), we have that
Next we will compute and we will show that it is equal to . So we replace u with v, in the above equality and we get
Now, we compute and the last term of the above expression separately. We have that
Moreover, the last term in (5.12), changing the order on which the terms are added, becomes
On the right-hand side we can change the name of the variables Q and R:
Comparing (5.11) and the above equation, we get that
Let us prove property (III) in the general case. We have that
Now, replacing u with v and applying property (5.15) for , we get
Comparing the above equation with (5.16), we get that
The proof is complete. ∎
Before proving property (IV), we need the following lemma:
The following identity holds true: for all , all and all ,
We can write
Then we have that
if and only if
Moreover, we have that
if and only if
Otherwise, we have that
Then, by Theorem 5.7,
Then (5.17) holds true. ∎
The result follows straightforward from (5.17) just taking . ∎
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 and let be any nonempty portion of the boundary. We consider the following problem: minimize
in the set . We make the following assumptions:
and for u sufficiently large.
is lower semicontinuous.
is a lower semicontinuous function in u, measurable in x, such that with and .
Clearly, the above assumptions are not sufficient to guarantee the existence of a solution, not even in a Sobolev space. We refer to  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:
Find an ultrafunction which vanishes on and minimizes
We have the following result:
If assumptions (1)–(3) are satisfied, then the functional has a minimizer in the space
Moreover, if has a minimizer w in , then .
The proof is based on a standard approximation by finite-dimensional spaces. Let us observe that, for each finite-dimensional space , we can consider the approximate problem:
The above minimization problem has a solution, being the functional coercive, due to the hypotheses on and the fact that . If we take a minimizing sequence , then we can extract a subsequence weakly converging to some . By observing that in finite-dimensional spaces all norms are equivalent, it follows also that pointwise. Then, by the lower-semicontinuity of a and f, it follows that the pointwise limit satisfies
Next, we use the very general properties of Λ-limits, as introduced in Section 2.2. We set
Then, taking a generic , from the inequality , we get
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.
Let the assumptions of the above theorem hold. If and there exists such that , then the minimizer has the following form:
where and ψ is null in the sense of distributions, namely,
In this case
with as in Definition 3.5. Moreover, if in addition , with , we have that
and . Finally, if has a minimizer in , then and .
Under the above hypotheses the minimization problem has an additional a priori estimate in , due to the fact that is bounded away from zero. Moreover, the fact that the function vanishes on a non-trascurable -dimensional part of the boundary shows that the generalized Poincaré inequality holds true. Hence, by Proposition 3.23, the approximating net has a subnet such that
This proves the first statement, since obviously vanishes in the sense of distributions. In this case, in general the minimum is not achieved in and hence .
Next, if 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, strongly in . This implies, by Proposition 3.23, that , and hence ψ is infinitesimal in , proving the second part.
Finally, if the minimizer is a function , we have that eventually; then . ∎
6.2 The Poisson problem in
Now we consider the very classical problem
If , the solution is given by
and it can be characterized in several ways.
First of all, it is the only solution in the Schwartz space of tempered distributions obtained via the equation
where denotes the Fourier transform of T. Moreover, it is the minimizer of the Dirichlet integral
in the space which is defined as the completion of with respect to the Dirichlet norm
Each of these characterizations provides a different method to prove its existence.
The situation is completely different when . In this case, it is well known that the fundamental class of solutions is given by
however, none of the previous characterization makes sense. In fact, we cannot use equation (6.2), since and hence does not define a tempered distribution. Also, the space is not an Hilbert space and the functional is not bounded from below in .
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 , the boundary conditions are replaced by the condition . 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.
Find such that
Clearly, the solutions of the above problem are the minimizers of the Dirichlet integral
in the space , 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 may assume infinite values does not change the structure of the problem and shows the utility of the use of infinite quantities. The relation between the classical solution w and the ultrafunction u is given by
Some people might be disappointed that u depends on R and it is not a standard function; if this is the case, it is sufficient to take
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 . 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 and (, , ), the generalized solution takes infinite values for every . However, there are cases in which 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 ).
Let us consider, for , the one-dimensional variational problem of finding the minimum of the functional
among the functions such that . In particular, we are interested in the case in which a is the following degenerate function:
Formally, the Euler equation, if , is
We recall that, by standard arguments,
Hence, if , the solution is explicitly computed
since it turns out that for all and then the degeneracy does not take place.
If , we see that the solution does not live in , hence the problem has not a “classical” weak solution. More exactly, we have the following result:
If , then the functional (6.3) has a unique minimizer given by
where is a suitable real number which depends on γ (see Figure 1).
First, we show that the generalized solution has at most one discontinuity. In fact, for the solution satisfies , for some , and at that point the classical Euler equations are not anymore valid. On the other hand, where , the solution satisfies a regular problem, hence we are in the situation of having at least the following possible candidate as solution with a jump at
and a discontinuity of derivatives at some .
In the specific case, we have (see Figure 1)
We now show that this is not possible because the functional takes a lower value on the solution with only a jump at . In fact, if we consider the function defined as
we observe that in , while for all and, by explicit computations, we have
where is strictly decreasing, since .
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 , on a generic solution with a single jump from the value to the value at the point and such that the Euler equation is satisfied before and after ξ. We obtain the following value for the functional (in terms of the point ξ):
We observe that, for all ,
To study the behavior of , one has to solve some fourth-order equations (this could be possible in an explicit but cumbersome way), so we prefer to make a qualitative study. We evaluate
hence we have that
Consequently, the function , which nevertheless satisfies, for all ,
has a unique negative minimum at the point .
From this, we deduce that there exists one and only one point such that
From the sign of we get that is strictly increasing in and decreasing in (see Figure 3). Next for all ,
hence in the case that
then has a single zero and, being a change of sign, is a point of absolute minimum for .
If , the above argument fails. In this case we can observe that
hence , which is negative at and near vanishes exactly two times, at the point , which is a point of local minimum and at another point , which is a point of local maximum (see Figure 4). Hence, to find the absolute minimum, we have to compare the value of with that of .
In particular, we have that and hence we can show that the minimum is not at simply by observing that we can find at least a point where and this point is . In fact,
In particular, and
This follows since replacing with χ, we have to control the sign of the cubic
which is negative for all , since , while
is a parabola with negative minimum. ∎
It could be interesting to study this problem in dimension greater than one, namely, to minimize
in the set
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., ).
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Readers expert in nonstandard analysis will recognize that Λ-theory is equivalent to the superstructure constructions of Keisler (see  for a presentation of the original constructions of Keisler).
About the article
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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