1 Introduction and motivations
Singular problems in the calculus of variations have longly been studied both in mathematics and in relevant applications (see, e.g., [18, 7, 43, 25] and references therein). In this paper, we introduce an approach to variational problems involving singularities that allows the extension of the classical theory with very natural statements and proofs. We are interested in extremizing functionals which are either distributional themselves or whose set of extremals includes generalized functions. Clearly, distribution theory, being a linear theory, has certain difficulties when nonlinear problems are in play.
To overcome this type of problems, we are going to use the category of generalized smooth functions, see [13, 12, 14, 15]. This theory seems to be a good candidate, since it is an extension of classical distribution theory, which allows the modeling of nonlinear singular problems, while at the same time sharing many nonlinear properties with ordinary smooth functions like the closure with respect to composition and several non-trivial classical theorems of calculus. One could describe generalized smooth functions as a methodological restoration of Cauchy–Dirac’s original conception of generalized function, see [8, 28, 24]. In essence, the idea of Cauchy and Dirac (but also of Poisson, Kirchhoff, Helmholtz, Kelvin and Heaviside) was to view generalized functions as suitable types of smooth set-theoretical maps obtained from ordinary smooth maps depending on suitable infinitesimal or infinite parameters. For example, the density of a Cauchy–Lorentz distribution with an infinitesimal scale parameter was used by Cauchy to obtain classical properties, which nowadays are attributed to the Dirac delta function, cf. .
In the present work, the foundation of the calculus of variations is set for functionals defined by arbitrary generalized functions. This in particular applies to any Schwartz distribution and any Colombeau generalized function (see, e.g., [5, 6]), and hence justifies the title of the present paper.
For example, during the last years, the study of low regular Riemannian and Lorentzian geometry was intensified and made a huge amount of progress (cf. [26, 27, 41, 33, 30, 40]). It was shown that the exponential map is a bi-Lipschitz homeomorphism when metrics are considered [34, 26], or that Hawking’s singularity theorem still holds when , see . However, the calculus of variations in the classical sense may cease to hold when metrics with regularity, or below, are considered [19, 29]. This motivates the search for an alternative. In fact, if p, and denotes the set of all Lipschitz continuous curves connecting p and q, the natural question about what curves realize the minimal g-length leads to the corresponding geodesic equation, but the Jacobi equation is not rigorously defined. To be more precise, the Riemannian curvature tensor exists only as an function on , and is evaluated along γ. However, the image of γ has Lebesgue-measure zero if . Thus, we cannot state the Jacobi equations properly.
In order to present a possible way out of the aforementioned problems, the singular metric g is embedded as a generalized smooth function. In this way, the embedding has derivatives of all orders, valued in a suitable non-Archimedean ring (i.e., a ring that contains infinitesimal and infinite numbers). Despite the total disconnectedness of the ground ring, the final class of smooth functions on behaves very closely to that of standard smooth functions; this is a typical step one can recognize in other topics such as analytic space theory [4, 37] and non-Archimedean analysis, see, e.g.,  and references therein. We then apply our extended calculus of variations to the generalized Riemannian space , and sketch a way to translate the given problem into the language of generalized smooth functions, solve it there, and translate it back to the standard Riemannian space . Clearly, the process of embedding the singular metric g using introduces infinitesimal differences. This is typical in a non-Archimedean setting, but the notion of standard part comes to help: if is infinitely close to a standard real number s, i.e., for all , then the standard part of x is exactly s. We then show that (assuming that is geodesically complete) the standard part of the minimal length in the sense of generalized smooth functions is the minimal length in the classical sense, and give a simple way to check if a given (classical) geodesic is a minimizer of the length functional or not. In this way, the framework of generalized smooth functions is presented as a method to solve standard problems rather than a proposal to switch into a new setting.
The structure of the present paper is as follows. We start with an introduction into the setting of generalized smooth functions and give basic notions concerning generalized smooth functions and their calculus that are needed for the calculus of variations (Section 2). The paper is self-contained in the sense that it contains all the statements required for the proofs of calculus of variations we are going to present. If proofs of preliminaries are omitted, we clearly give references to where they can be found. Therefore, to understand this paper, only a basic knowledge of distribution theory is needed.
In Section 3, we obtain some preliminary lemmas regarding the calculus of variations with generalized smooth functions. The first variation and the notion of critical point will be defined and studied in Section 4. We prove the fundamental lemma of calculus of variations and the full connection between critical points of a given functional and solutions of the corresponding Euler–Lagrange equation. In Section 5, we study the second variation and define the notion of local minimizer. We also extend to generalized functions classical necessary and sufficient conditions to have a minimizer, and we give a proof of the Legendre condition. In Section 6, we introduce the notion of Jacobi field and extend to generalized functions the definition of conjugate points, so as to prove the corresponding Jacobi theorem. In Section 7, we extend the classical Noether’s theorem. We close with an application to Riemannian geometry in Section 8.
Note that Konjik, Kunzinger and Oberguggenberger  already established the calculus of variations in the setting of Colombeau generalized functions, by using a comparable methodological approach. Indeed, generalized smooth functions are related to Colombeau generalized functions, and one could say that the former is a minimal extension of the latter so as to get more general domains for generalized functions, and hence the closure with respect to composition and a better behavior on unbounded sets. However, there are some conceptual advantages in our approach.
(i) Whereas generalized smooth functions are closed with respect to composition, Colombeau generalized functions are not. This forced the authors of  to consider only functionals defined using compactly supported Colombeau generalized functions, i.e., functions assuming only finite values, or tempered generalized functions.
(ii) The authors of  were forced to consider the so-called compactly supported points (i.e., finite points in ), where the setting of generalized smooth functions gives the possibility to consider more natural domains like the interval . This leads us to extend in a natural way the statements of classical results of calculus of variations. Moreover, all our results still hold when we take as two infinite numbers such that , or as boundary points two unbounded points .
(iii) The theory of generalized smooth functions was developed to be very user friendly, in the sense that one can avoid cumbersome “ε-wise” proofs quite often, whereas the proofs in  frequently use this technique. Thus, one could say that some of the proofs based on generalized smooth functions are more “intrinsic” and close to the classical proofs in a standard smooth setting. This allows a smoother approach to this new framework.
(iv) The setting of generalized smooth functions depends on a fixed infinitesimal net , whereas the Colombeau setting considers only . This added degree of freedom allows to solve singular differential equations that are unsolvable in the classical Colombeau setting and to prove a more general Jacobi theorem on conjugate points.
(vi) We obtain more classical results like the Legendre condition, and the classical results about Jacobi fields and conjugate points.
(vii) The Colombeau generalized functions can be embedded into generalized smooth functions. Thus, our approach is a natural extension of .
2 Basic notions
The new ring of scalars
In this work, I denotes the interval and we will always use the variable ε for elements of I; we also denote ε-dependent nets simply by . By , we denote the set of natural numbers, including zero.
We start by defining the new simple non-Archimedean ring of scalars that extends the real field . The entire theory is constructive to a high degree, e.g., no ultrafilter or non-standard method is used. For all the proofs in this section, see [15, 13, 14].
Let be a net such that .
is called the asymptotic gauge generated by ρ. The net ρ is called a gauge.
If is a property of , we use the notation to denote . We can read as for ε small.
We say that a net is ρ-moderate and write if as .
Let , . We say that if as . This is a congruence relation on the ring of moderate nets with respect to pointwise operations, and we can hence define
In the following, ρ will always denote a net as in Definition 2.1. The infinitesimal ρ can be chosen depending on the class of differential equations we need to solve for the generalized functions we are going to introduce, see . For motivations concerning the naturality of , see .
We can also define an order relation on by saying that if there exists such that (we then say that is ρ-negligible) and for ε small. Equivalently, we have that if and only if there exist representatives and such that for all ε. Clearly, is a partially ordered ring. The usual real numbers are embedded in by considering constant nets .
Even in the case where the order is not total, we still have the possibility to define the infimum , and analogously the supremum function and the absolute value . Note, e.g., that and imply . In the following, we will also use the customary notation for the set of invertible generalized numbers. Our notations for intervals are and , and analogously for segments and . Finally, we write to denote that is an infinitesimal number, i.e., for all . This is equivalent to for all representatives and .
On the -module , we can consider the natural extension of the Euclidean norm, i.e., , where . Even if this generalized norm takes values in , it shares several properties with usual norms, like the triangular inequality or the property . It is therefore natural to consider on topologies generated by balls defined by this generalized norm and a set of radii .
Let , and .
We write if .
for each .
For each , denotes an ordinary Euclidean ball in .
The relation has better topological properties compared to the usual strict order relation and (that we will never use) because for the set of balls is a base for a topology on . The topology generated in the case is called sharp topology, whereas the one with the set of radii is called Fermat topology. We will call sharply open set any open set in the sharp topology, and large open set any open set in the Fermat topology; clearly, the latter is coarser than the former. The existence of infinitesimal neighborhoods implies that the sharp topology induces the discrete topology on . This is a necessary result when one has to deal with continuous generalized functions which have infinite derivatives. In fact, if is infinite, we have only for , see [11, 12]. With an innocuous abuse of language, we write instead of , and instead of . For example, . We will simply write to denote an open ball in the sharp topology and for an open ball in the Fermat topology. Also open intervals are defined using the relation , i.e., .
The following result is useful to deal with positive and invertible generalized numbers (cf. ).
Let . Then the following are equivalent:
x is invertible and , i.e., .
For each representative of x , we have .
For each representative of x , we have
We will also need the following result.
Let such that . Then the interior in the sharp topology is dense in .
Internal and strongly internal sets
A natural way to obtain sharply open, closed and bounded sets in is by using a net of subsets . We have two ways of extending the membership relation to generalized points .
Let be a net of subsets of .
is called the internal set generated by the net . See  for an introduction and an in-depth study of this notion in the case .
Let be a net of points of . We say that , and we read it as strongly belongs to , if , and if , then also for ε small. Also, we set , and we call it the strongly internal set generated by the net .
We say that the internal set is sharply bounded if there exists such that . Analogously, a net is sharply bounded if the internal set is sharply bounded.
Therefore, if there exists a representative such that for ε small, whereas this membership is independent from the chosen representative in the case of strongly internal sets. Note explicitly that an internal set generated by a constant net is simply denoted by .
The following theorem shows that internal and strongly internal sets have dual topological properties:
For , let and let . Then the following hold:
if and only if . Thus, if and only if .
if and only if , where . Hence, if , then if and only if .
is sharply closed and is sharply open.
, where is the closure of . On the other hand, , where is the interior of .
Generalized smooth functions and their calculus
Using the ring , it is easy to consider a Gaussian with an infinitesimal standard deviation. If we denote this probability density by , and if we set , where , we obtain the net of smooth functions . This is the basic idea we are going to develop in the following.
Let and be arbitrary subsets of generalized points. We say that is a generalized smooth function if there exists a net defining f in the sense that , and for all and all . The space of generalized smooth functions (GSF) from X to Y is denoted by .
Let us note explicitly that this definition states minimal logical conditions to obtain a set-theoretical map from X into Y, defined by a net of smooth functions. In particular, the following theorem states that the equality is meaningful, i.e., that we have independence from the representatives for all derivatives , .
Let and be arbitrary subsets of generalized points. Let be a net of smooth functions that defines a generalized smooth map of the type . Then the following hold:
For all , the GSF is locally Lipschitz in the sharp topology, i.e., each possesses a sharp neighborhood U such that for all x, and some .
Each is continuous with respect to the sharp topologies induced on X, Y.
Assume that the GSF f is locally Lipschitz in the Fermat topology and that its Lipschitz constants are always finite, i.e., . Then f is continuous in the Fermat topology.
is a GSF if and only if there exists a net defining a generalized smooth map of type such that .
Subsets with the trace of the sharp topology, and generalized smooth maps as arrows form a subcategory of the category of topological spaces. We will call this category the category of GSF , and denote it by .
The differential calculus for GSF can be introduced by showing existence and uniqueness of another GSF serving as incremental ratio.
Theorem 2.9 (Fermat–Reyes theorem for GSF).
Let be a sharply open set, let , and let be a generalized smooth map generated by the net of smooth functions . Then the following hold:
There exists a sharp neighborhood T of and a generalized smooth map , called the generalized incremental ratio of f along v , such that for all .
Any two generalized incremental ratios coincide on a sharp neighborhood of .
We have for every and we can thus define , so that .
If U is a large open set, then an analogous statement holds by replacing sharp neighborhoods by large neighborhoods.
Note that this result permits the consideration of the partial derivative of f with respect to an arbitrary generalized vector which can be, e.g., infinitesimal or infinite. Using this result, we can also define subsequent differentials as j-multilinear maps, and we set
The set of all the j-multilinear maps over the ring will be denoted by . For , we set , the generalized number defined by the operator norms of the multilinear maps .
The following result follows from the analogous properties for the nets of smooth functions defining f and g.
Let be an open subset in the sharp topology, and let and be generalized smooth maps. Then the following hold:
for all .
For each , the map is -linear in .
Let and be open subsets in the sharp topology, and let and be generalized smooth maps. Then, for all and all , we have .
We also have a generalization of the Taylor formula.
Let be a generalized smooth function defined in the sharply open set . Let be such that the line segment belongs to U. Then, for all , we have
If we further assume that all the n components of are invertible, then there exists , , such that
Formula (2.1) corresponds to a direct generalization of Taylor formulas for ordinary smooth functions with Lagrange remainder. On the other hand, in (2.2) and (2.3), the possibility that the differential may be infinite at some point is considered, and the Taylor formulas are stated so as to have infinitesimal remainder.
The following local inverse function theorem will be used in the proof of Jacobi’s theorem (see  for a proof).
Let and , and suppose that for some in the sharp interior of X, is invertible in . Then there exists a sharp neighborhood of and a sharp neighborhood V of such that is invertible and .
We can define right and left derivatives as, e.g., , which always exist if . The one-dimensional integral calculus of GSF is based on the following.
Let be a generalized smooth function defined in the interval , where . Let . Then there exists one and only one generalized smooth function such that and for all . Moreover, if f is defined by the net and , then for all .
Under the assumptions of Theorem 2.13, we denote by the unique generalized smooth function such that
All the classical rules of integral calculus hold in this setting:
Let and be generalized smooth functions defined on sharply open domains in . Let , with and . Then
for all ,
for all ,
Let and be generalized smooth functions defined on sharply open domains in . Let , with , such that , and . Finally, assume that . Then
Embedding of Schwartz distributions and Colombeau functions
We finally recall two results that give a certain flexibility in constructing embeddings of Schwartz distributions. Note that both the infinitesimal ρ and the embedding of Schwartz distributions have to be chosen depending on the problem we aim to solve. A trivial example in this direction is the ODE , which cannot be solved for , but it has a solution for . As another simple example, if we need the property , where H is the Heaviside function, then we have to choose the embedding of distributions accordingly. See also [16, 32] for further details.
If , and , we use the notations for the function and for the function . These notations permit to highlight that is a free action of the multiplicative group on and is a free action of the additive group on . We also have the distributive property .
Let be a net such that . Let . There exists a net of with the following properties:
for all .
for all .
If , then the net can be chosen so that .
It is worth noting that the condition (iv) of null moments is well known in the study of convergence of numerical solutions of singular differential equations, see, e.g., [42, 9, 21] and references therein.
Concerning the embeddings of Schwartz distributions, we have the following result, where
is called the set of compactly supported points in .
uniquely extends to a sheaf morphism of real vector spaces
and satisfies the following properties:
If for some , then is a sheaf morphism of algebras.
If , then .
for all and all .
commutes with partial derivatives, i.e., for each and .
Concerning the embedding of Colombeau generalized functions, we recall that the special Colombeau algebra on Ω is defined as the quotient of moderate nets over negligible nets, where the former is
and the latter is
Using , we have the following compatibility result.
A Colombeau generalized function defines a generalized smooth map , which is locally Lipschitz on the same neighborhood of the Fermat topology for all derivatives. This assignment provides a bijection of onto for every open set .
2.1 Extreme value theorem and functionally compact sets
For GSF, suitable generalizations of many classical theorems of differential and integral calculus hold such as the intermediate value theorem, mean value theorems, a sheaf property for the Fermat topology, local and global inverse function theorems, the Banach fixed point theorem and a corresponding Picard–Lindelöf theorem, see [15, 14, 31, 13].
Even though the intervals , , are neither compact in the sharp nor in the Fermat topology (see [15, Theorem 25]), analogously to the case of smooth functions, a GSF satisfies an extreme value theorem on such sets. In fact, we have the following theorem.
Let be a generalized smooth function defined on the subset X of . Let be an internal set generated by a sharply bounded net of compact sets . Then
We shall use the assumptions on K and given in this theorem to introduce a notion of “compact subset” which behaves better than the usual classical notion of compactness in the sharp topology.
A subset K of is called functionally compact, denoted by , if there exists a net such that
is sharply bounded,
We motivate the name functionally compact subset by noting that on this type of subsets, GSF have properties very similar to those that ordinary smooth functions have on standard compact sets.
By [36, Proposition 2.3], any internal set is closed in the sharp topology. In particular, the open interval is not functionally compact since it is not closed.
If is a non-empty ordinary compact set, then is functionally compact. In particular, we have that is functionally compact.
For the empty set, we have .
is not functionally compact since it is not sharply bounded.
In the present paper, we need the following properties of functionally compact sets.
Let , . Then implies .
If and , then .
Let us note that can also be infinite, e.g., , or , with . Finally, in the following result we consider the product of functionally compact sets.
Let and , then . In particular, if for , then .
A theory of compactly supported GSF has been developed in , and it closely resembles the classical theory of LF-spaces of compactly supported smooth functions. It establishes that for suitable functionally compact subsets, the corresponding space of compactly supported GSF contains extensions of all Colombeau generalized functions, and hence also of all Schwartz distributions.
3 Preliminary results for calculus of variations with GSF
In this section, we study extremal values of generalized functions at sharply interior points of intervals . As in the classical calculus of variations, this will provide the basis for proving necessary and sufficient conditions for general variational problems. Since the new ring of scalars has zero divisors and is not totally ordered, the following extension requires a more refined analysis than in the classical case.
The following lemma shows that we can interchange integration and differentiation while working with generalized functions.
Let , with and . Let also and assume that and . Then, for all , we have
We first note that , by the closure of GSF with respect to composition. Therefore, , and the right-hand side of (3.1) is well defined as an integral of a GSF. In order to show that also the left-hand side of (3.1) is well defined, we need to prove that also is a GSF. Let f be defined by the net , with , and let . Then and the extreme value Theorem 2.20 applied to yields the existence of such that
This proves that also the left-hand side of (3.1) is well defined as a derivative of a GSF. From the classical derivation under the integral sign, the Fermat–Reyes Theorem 2.9, and Theorem 2.13 about definite integrals of GSF, we obtain
The next result will be used frequently.
Let be a directed set and let be a set-theoretical map such that for all , and in the sharp topology. Then .
If , then if and only if Indeed, it suffices to let in .
Assume that x, and
Then taking in , we get .
We call componentwise invertible if and only if for all , we have that is invertible.
Let , where , and is a sharply open subset. Then if and only if for all componentwise invertible .
By Lemma 2.3, it follows that for , the set of invertible points in i.e., is dense in V (with respect to the sharp topology). This implies that is dense. By Theorem 2.8 (iv), f is sharply continuous, so Lemma 3.2 yields . The other direction is obvious. ∎
Analogously to the classical case, we say that is a local minimum of if there exists a sharply open neighborhood (in the trace topology) of such that for all . A local maximum is defined accordingly. We will write , which is a short hand notation to denote that is a (local) minimum of f.
Let and . If is a sharply interior local minimum of f, then .
Without loss of generality, we can assume , because of the closure of GSF with respect to composition. Let be such that and over U. Take any such that , so that . Thus, if , by Taylor’s Theorem 2.11, there exists such that
Set and . Due to the fact that is minimal, we have
Let , with , and let be such that for all sharply interior points . Then on .
Now, we are able to prove the “second-derivative-test” for GSF.
Let , with , and let be such that for some sharply interior . Then . Vice versa, if and , then .
As above, we can assume that . Let be such that and over U. Take any such that , so that , and set and . By Taylor’s Theorem 2.11, for some , we obtain
By assumption, for all , we have
By Lemma 3.6, we know that . Thus, for all , we obtain
Therefore, also . In this inequality we can set , assuming that and . We get , and the conclusion follows from Lemma 3.2 as .
Now assume that and , so that for some , by Lemma 2.3. Since , for all , Taylor’s formula gives
where . Therefore, . Now
We can hence write for all , which proves that is a local minimum. ∎
4 First variation and critical points
In this section, we define the first variation of a functional and prove that some classical results have their counterparts in this generalized setting, for example, the fundamental lemma (Lemma 4.4) or the connection between critical points and the Euler–Lagrange equations (Theorem 4.5).
If and , we define
When the use of the points is clear from the context, we adopt the simplified notation . We also note here that is an -module.
One of the positive features of the use of GSF for the calculus of variations is their closure with respect to composition. For this reason, the next definition of functional is formally equal to the classical one, though it can be applied to arbitrary generalized functions F and u.
Let with . Let and , and define
Let also . Then
We call the first variation of I. In addition, we call a critical point of I if for all .
To prove the fundamental lemma of the calculus of variations, Lemma 4.4, we first show that every GSF can be approximated using generalized strict delta nets.
Let be such that and let . Let and be such that . Assume that has the following properties:
For t small, is zero outside every ball , , i.e.,
We only have to generalize the classical proof concerning limits of convolutions with strict delta nets. We first note that
and so these integrals exist because . Using (i), for t small, say for , we get
For each , the sharp continuity of f at x yields for all y such that , and we can take . By (ii), for , we have
Lemma 4.4 (Fundamental lemma of the calculus of variations).
Let be such that , and let . If
Let . Because of Theorem 2.8 (iv) and Lemma 2.4, without loss of generality, we can assume that x is a sharply interior point, so that for some . Let be such that . Set , where and , and for all . Then, for t sufficiently small, we have and (4.3) yields . For t small, we both have that on and the assumptions of Lemma 4.3 hold. Therefore,
and hence Lemma 4.3 yields . ∎
Thus, we obtain the following theorem.
Let be such that , and let . Then u solves the Euler–Lagrange equations
for I given by (4.1), if and only if for all , i.e., if and only if u is a critical point of I.
5 Second variation and minimizers
Let , with . Let and . Then
The following result permits the calculation of the (generalized) norm using any net that defines v.
Under the assumptions of Definition 5.1, let and be such that for all ε. Then the following hold:
If the net defines v , then
if and only if .
for all .
For all , we have and for some .
By the standard extreme value theorem applied ε-wise, we get the existence of , such that
But , , so
This proves both that is well defined, i.e., it does not depend on the particular choice of points , as in Definition 5.1, and claim (i). The remaining properties (ii)–(v) follows directly from (i) and the usual properties of standard -norms. ∎
Using these -valued norms, we can naturally define a topology on the space .
Let , with , , and .
We set .
If , then we say that U is a sharply open set if
As in [15, Theorem 2], one can easily prove that sharply open sets form a topology on . Using this topology, we can assess when a curve is a minimizer of the functional I. Note explicitly that there are no restrictions on the generalized numbers , , e.g., they can also both be infinite.
Let , with , and .
For all , we set
Note that . The subscript “bd” stands here for “boundary values”.
We say that u is a local minimizer of I in if and
We define the second variation of I in the direction as
Note also explicitly that the points p, can have infinite norm, e.g., as . By using the standard Einstein’s summation conventions, we calculate
which we abbreviate as
The following results establish classical necessary and sufficient conditions to decide if a function u is a minimizer for the given functional (4.1).
Let , with , , , and let u be a local minimizer of I in . Then
for all ,
for all .
Let be such that (5.1) holds. Since , the map is well defined and continuous with respect to the trace of the sharp topology in its codomain. Therefore, we can find such that for all . We hence have . This shows that the GSF has a local minimum at . Now, by employing Lemmas 3.6 and 3.8, the claims are proven. ∎
Let , with , and . Let be such that
for all ,
for all and all , where and .
Then u is a local minimizer of the functional I in .
Moreover, if for all such that and all , then for all such that .
for some . But , and hence . Finally, Lemma 3.2 yields
which is our conclusion. Note explicitly that if for all and all , then and hence .
Now, assume that for all such that and all , and take such that . As above, set for all , so that . We have and for all because . Using Taylor’s theorem, we get for some . Therefore, . ∎
Let , and be sequences in . Assume that both and in the sharp topology as . Let . Finally, let for all . Then
We can apply the integral mean value theorem for each ε and each defining net of f to get the existence of such that
We now derive the so-called necessary Legendre condition.
Let , with , and let be a minimizer of the functional I. Then is positive semi-definite for all , i.e.,
Let and let be arbitrary. Let . We can assume that t is a sharply interior point, because otherwise we can use sharp continuity of the left-hand side of (5.2) and Lemma 3.2. We can also assume that λ is componentwise invertible because of Lemma 3.5. We want to mimic the classical proof of [23, Theorem 1.3.2], but considering a “regularized” version of the triangular function used there (see Figure 1). In particular: (1) The smoothed triangle must have an infinitesimal height which is proportional to λ, and we will take as this infinitesimal. (2) In the proof we need that the derivative at t is equal to λ, and this justifies the drawing of the peak in Figure 1. (3) To regularize the singular points of the triangular function, we need a smaller infinitesimal, and we can take, e.g., . So, consider a net of smooth functions on such that the following properties hold:
for all x.
The net defines a GSF because t is a sharply interior point. Setting for simplicity , by assumption, we have
Now, setting and , by (v), we have
Note that there always exists such that . Therefore,
But (iii) yields , and this concludes the proof. ∎
6 Jacobi fields
for all and . Note that if u minimizes I, then
As usual, we note that is a minimizer of the functional Q, and we are interested to know if there are others. In order to solve this problem, we consider the Euler–Lagrange equations for Q, which are given by
In other words,
Since u is given, (6.2) is an -linear system of second order equations in the unknown GSF η and with time dependent coefficients in . We call (6.2) the Jacobi equations I with respect to u . As in the classical setting, we introduce the following definition.
A solution of the Jacobi equations (6.2) is called a Jacobi field along u.
The following result confirms that the intuitive interpretation of a Jacobi field as the tangent space of a smooth family of solutions of the Euler–Lagrange equation still holds in this generalized setting.
Let , where . We write for all . Assume that each satisfies the Euler–Lagrange equations (4.4):
is a Jacobi field along u.
A straight forward calculation gives
6.1 Conjugate points and Jacobi’s theorem
The classical key result concerning Jacobi fields relates conjugate points and minimizers. The main aim of the present section is to derive this theorem in our generalized framework, by extending the ideas of the proof of [23, Theorem 1.3.4].
A crucial notion is hence that of piecewise GSF.
We call piecewise GSF an n-tuple with the following properties:
For all there exist such that and . Note that implies and because the relation is antisymmetric. Therefore, the points , are uniquely determined by the set-theoretical function .
For all , we have .
Every pointwise GSF defines a set-theoretical function:
For all , we set if .
Since the order relation is not a total one, we do not have that .
If is a set-theoretical function originating from a piecewise GSF , then neither the GSF nor the points are uniquely determined by ν. For this reason, we prefer to stress our notations with symbols like .
Every GSF can be seen as a particular case of a piecewise GSF.
If and , then also and are piecewise GSF, and we hence have a structure of an -module.
If and , then we can define the composition
Piecewise GSF inherit from their defining components a well-behaved differential and integral calculus. The former is even more general and taken from .
Let . Then we set
It is worth noting that induces an ultrametric on that generates exactly the sharp topology, see, e.g., [2, 11] and references therein. However, we will not use this ultrametric structure in the present paper, and we only introduced it to get an invertible infinitesimal that goes to zero with x. It is in fact easy to show that
in the sharp topology. The following definition is based on [1, Definition 2.2].
Let and let be an arbitrary set-theoretical function. Let be a sharply interior point of T. Then we say that f is differentiable at if
In this case, using Landau little-oh notation, we can hence write
As in the classical case, (6.3) implies the uniqueness of , so that we can define , and the usual elementary rules of differential calculus. By the Fermat–Reyes theorem, this definition of derivative generalizes that given for GSF.
In particular, this notion of derivative applies to the set-theoretical function induced by a piecewise GSF . We therefore have that is differentiable at each , and that , but clearly there is no guarantee that is also differentiable at each point .
The notion of definite integral is naturally introduced in the following definition.
Let be a piecewise GSF. Then
Since our main aim in using piecewise GSF is to prove Jacobi’s theorem, we do not need to prove that the usual elementary rules of integration hold, since we will always reduce to integrals of GSF.
Having a notion of derivative and of definite integral also for piecewise GSF allows to study functionals of the form
This leads to the following natural definition: we say that a piecewise GSF v is a piecewise GSF (global) minimizer if for all . For the proof of Jacobi’s theorem, we will only need this particular notion of global minimizer. Note explicitly that in (6.4), we only need the existence of right and left derivatives of GSF, because of Definition 6.7, and of Definition 2.14 of a definite integral of a GSF.
Classically, several proofs of Jacobi’s theorem use both some form of implicit function theorem and of uniqueness of solution for linear ODE.
Theorem 6.8 (Implicit function theorem).
Let , be sharply open sets. Let and . If is invertible in , then there exists a sharply open neighborhood of such that
Moreover, the function for all is a GSF and satisfies
The usual deduction of the implicit function theorem from the inverse function theorem in Banach spaces can be easily adapted by using Theorem 2.12 and noting that is a GSF such that . ∎
In the next theorem, the dependence of the entire theory on the initial infinitesimal net plays an essential role. Indirectly, the same important role will reverberate in the final Jacobi’s theorem.
Theorem 6.9 (Solution of first order linear ODE).
Let , where , , and let and . Assume that
where . Then there exists one and only one such that
Moreover, this y is given by for all .
We first note that
where , and . This exponential matrix in is a GSF because for all , we have
Therefore, all values of are ρ-moderate. Analogously, one can prove that also are moderate for all and . Considering that derivatives can be calculated ε-wise, we have that this GSF y satisfies (6.6), and this proves the existence part.
To show uniqueness, we can proceed as in the smooth case. Assume that satisfies (6.6), and set for all . Since , we have
From the uniqueness of primitives of GSF and Theorem 2.13, we have that . Therefore, . ∎
If α, , we write to denote that there exists such that . Therefore, assumption (6.5) can be written as . Note that this assumption is weaker, in general, than
The following result is the key regularity property that is needed to prove Jacobi’s theorem.
Let , with , and let . Let, in addition, be a piecewise GSF which satisfies the Euler–Lagrange equation
Finally, assume that is invertible. Then
In particular, if , then .
Set for all and all l, v, . For simplicity, set
Our assumption on the invertibility of makes it possible to apply the implicit function Theorem 6.8 to conclude that there exists a neighborhood of such that
But we have . Moreover, the unique function ϕ defined by for all is a GSF that belongs to . Now, for all , we have
Therefore, the uniqueness in (6.8) yields
We now integrate the Euler–Lagrange equation (6.7) on , obtaining
This entails that we can write
But the function has equal limits on the left and on the right of because on and on it is a GSF. In fact, for , we have
and this goes to 0 as , . Analogously, we can proceed for using β. Therefore,
Applying this equality in (6.9), we get , as claimed. Finally, if , then . ∎
In the following definition and below, we use the complete notation (see Definition 4.1).
Jacobi’s theorem shows that we cannot have minimizers if there are interior points conjugate to a. In order to prove it in the present generalized context, we finally need the following lemma.
Let and . Let be a Jacobi field along , with . Then
Since ψ is -homogeneous of second order in , we have
Thus, by integration by parts, we calculate
where we used the fact that η is a Jacobi field. ∎
After these preparations, we can finally prove Jacobi’s theorem.
Theorem 6.13 (Jacobi).
Let , with , and . Assume that the following hold:
is conjugate to a.
is invertible for all .
For all ,
Then u cannot be a local minimizer of I. Therefore, for any , there exists and such that but .
By contradiction, assume that u is a local minimizer, and let be a Jacobi field along such that the conditions from Definition 6.11 hold for η. We want to prove that . Define , which is a piecewise GSF since . Since also , Lemma 6.12 and the homogeneity of ψ yield
Thus, Theorem 5.5 (necessary condition for u being a minimizer) gives for all . Therefore, ν is a minimizer of the functional Q. Since ν is only a piecewise GSF, we cannot directly apply Theorem 4.5 (Euler–Lagrange equations). But, for all and all , we have
By the fundamental Lemma 4.4, this implies that η satisfies the Euler–Lagrange equations for ψ in the interval . Therefore, ν satisfies the same equations in . Moreover, is invertible by assumption (ii). Thus, all the hypotheses of the regularity Lemma 6.10 hold, and we derive that .
For all , we define
so that we can re-write the Jacobi equations (6.2) for η on as a system of first order ODE:
Note that if one of the quantities in (iii) depends even only polynomially on ε, then we are forced to take, e.g., to fulfill this assumption. This underlines the importance of the parameter ρ, making the entire theory dependent on the parameter ρ, in order to avoid unnecessary constraints on the scope of the functionals we look upon.
7 Noether’s theorem
In this section, we state and prove Noether’s theorem by following the lines of . We first note that any , where , can also be considered as a family in GSF which smoothly depends on the parameter . In this case, we hence say that is a generalized smooth family in . In particular, we can reformulate in the language of GSF the classical definition of one-parameter group of generalized diffeomorphisms of X as follows:
is a generalized smooth family in .
For all , the map is invertible, and .
for all .
for all s, .
The proof of Noether’s theorem is classically anticipated by the following time-independent version, which the general case is subsequently reduced to.
Let , where L, are sharply open sets. Let be a solution of the Euler–Lagrange equation corresponding to K, i.e., for all ,
Suppose that 0 is a sharply interior point of and is a generalized smooth family in such that for all ,
K is invariant under along w , i.e.,
Then, the quantity
is constant in .
Since the Euler–Lagrange equations (7.1) for K are given by , we have
which is our conclusion by the uniqueness part of Theorem 2.13. ∎
We are now able to prove Noether’s theorem. For the convenience of the reader, in its statement and proof we use the variables t, T, l, L, v, V so as to recall tempus, locus, velocitas, respectively.
Theorem 7.2 (Noether).
Let , with , and . Let, in addition, be a solution of the Euler–Lagrange equation (4.4) corresponding to F. Suppose that 0 is a sharply interior point of and is a generalized smooth family in . We denote by and , for all , the two projections of on and , respectively. We assume that for all ,
for all .
Then, the quantity
is constant in .
Since (7.3) is a GSF in , by sharp continuity it suffices to prove the claim for all . Set , (we recall that denotes the set of all invertible generalized numbers in ). Define by
and by for all . We note that are sharply open subsets and that . The notations for partial derivatives used in the present work result from the symbolic writing , so that the variables used in (7.4) yield
From these, for all and all , it follows that
Therefore, since u satisfies the Euler–Lagrange equations for F, this entails that w is a solution of the analogous equations for K in . Now, (i) gives
Moreover, (ii) gives . Finally,
is constant in . ∎
8 Application to Riemannian metric
In the following, we apply what we did so far to the problem of length minimizers in , where is a Riemannian metric. Furthermore, we assume that is geodesically complete. Note that the seeming restriction of considering only as our manifold weighs not so heavy. Indeed, the question of length minimizers can be considered to be a local one, since it is not guaranteed that global minimizers exist at all, whereas local minimizers always exist. Additionally, note that it was shown that it suffices to consider smooth manifolds (cf. [20, Theorem 2.9]) instead of manifolds with . Therefore, there is no need to consider non-smooth charts.
In this section, we fix an embedding , where satisfies for some , and where is an arbitrary open set, see Theorem 2.18. Actually, the embedding also depends on the dimension , but to avoid cumbersome notations, we denote embeddings always with the symbol ι.
By [27, Remark 2.6.2], it follows that we can always find a net of smooth functions such that by setting , we have that is a Riemannian metric for all ε. By Theorem 2.18 (iii), it follows that in norm. Let be the Christoffel symbols of , and set . A curve , with J being a sharply open subset of , is said to be a geodesic of if
We say that is geodesically complete if every solution of the geodesic equation belongs to , i.e., if for all and all , there exists a geodesic of such that and .
This definition includes also the possibility that the point p or the vector v could be infinite. By Theorem 2.19, it follows that if we consider only finite p and v, then any geodesic induces a Colombeau generalized function . Therefore, the space is geodesically complete in the sense of . We recall that is the set of compactly supported (i.e., finite) generalized points in Ω (see Theorem 2.18).
The definition of length of a (non-singular) curve needs the following.
Lemma 2.3 readily implies that . Therefore, the square root is defined on every strictly positive infinitesimal, but it cannot be extended to .
Let , , then
Moreover, for , we set
Let . Then we set , if this limit exists. Note that in this case.
Note that (8.1) are the usual geodesic equations for the generalized metric , whose derivation is completely analogous to that in the smooth case. Thus, they are the Euler–Lagrange equations of .
We are interested only in global minimizers of the functional , i.e., curves such that for all .
Let and , be such that and . Let be such that there exists
such that in as . Then
By assumption, , so that there exists such that
We hence obtain the claim by the triangle inequality and the convergence of , and to , and , respectively. ∎
Now, we consider , with . Let
be a solution of the geodesic equation
Let . Obviously, u is also the unique solution of
Using these initial conditions, for each fixed ε, we can solve the following problem:
for a unique and some .
Let u and be as above. Then the following hold:
For ε sufficiently small, the solution can be extended to a solution of ( 8.3 ) such that .
The net defines a GSF, i.e., .
(iii) If , then we have to show that for all ε, all the derivatives of are moderate. This is obviously true for and . The claim follows now from the fact that
so that there exists a polynomial P such that
If for all , then, by (ii), we have that in . Furthermore, in , by assumption, and we know that for some , since u is a g-geodesic (cf. [22, Lemma 1.4.5]). Therefore, we obtain that for small enough. ∎
Finally, the standard part of the generalized length of y is the length of u.
Let u and be as above. We conclude (using Lemma 8.4) that .
Let be as above. In addition, assume that each is -minimizing. Then is minimal.
Let . We have that and that . By assumption, for all ε, we have Therefore, , as claimed. ∎
Let be a minimizer of and assume that for ε small, is -minimizing. Then .
Corollary 8.8 gives us a way to answer the question if a certain classical geodesic between two given classical points p and q is a length-minimizer.
Furthermore, we are able to prove the following theorem, relating generalized minimizers to classical minimizers.
Let and let such that is minimal. Assume that exists and that there exists such that . Then w is g-minimizing and g-geodesic.
Assume to the contrary that there exists a curve connecting p and q (without loss of generality, σ is a g-geodesic) such that Now we construct (as done above) and , and set . Then
But, by assumption, we have that , which implies
This is a contradiction. ∎
We can summarize the present work as follows.
(i) The setting of GSF allows to treat Schwartz distributions more closely to classical smooth functions. The framework is so flexible and the extensions of classical results are so natural in many ways that one may treat it like smooth functions.
(ii) One key step of the theory is the change of the ring of scalars into a non-Archimedean one, and the use of the strict order relation to deal with topological properties. So, the use of and of -valued norms allows a natural approach to topology, even of infinite dimensional spaces (cf. Definition 5.3). On the other hand, the use of a ring with zero divisors and a non-total order relation requires a more refined and careful analysis. However, as proved in the present work, very frequently classical proofs can be formally repeated in this context, but paying particular attention to using the relation and invertibility instead of being non-zero in , and avoiding the total order property.
(iii) Others crucial properties are the closure of GSF with respect to composition and the use of the gauge ρ, because they do not force to narrow the theory into particular cases.
(iv) The present extension of the classical theory of calculus of variations shows that the use of GSF is a powerful analytical technique. The final application shows how to use them as a method to address problems in an Archimedean setting based on the real field .
Concerning possible future developments, we note the following.
(v) A generalization of the whole construction to piecewise GSF seems possible.
(vi) A more elegant approach for the integration of piecewise GSF could use the existence of right and left limits of and hyperfinite Riemann-like sums, i.e.,
extended to , where is the integer part function.
The present work could lay the foundations for further works concerning the possibility to extend other results of the calculus of variations in this generalized setting.
We would like to thank M. Kunzinger for helpful discussions and several suggestions that have led to considerable improvements.
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About the article
Published Online: 2017-09-15
Funding Source: Austrian Science Fund
Award identifier / Grant number: M1876-N35
Award identifier / Grant number: P25311-N25
Award identifier / Grant number: P25116-N25
Alexander Lecke has been supported by the uni:doc fellowship programme of the University of Vienna. Lorenzo Luperi Baglini has been supported by grant M1876-N35 of the Austrian Science Fund FWF. Paolo Giordano has been supported by grants P25311-N25 and P25116-N25 of the Austrian Science Fund FWF.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 779–808, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0150.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0