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Advances in Nonlinear Analysis

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


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The classical theory of calculus of variations for generalized functions

Alexander Lecke / Lorenzo Luperi BagliniORCID iD: https://orcid.org/0000-0002-0559-0770 / Paolo GiordanoORCID iD: https://orcid.org/0000-0001-7653-1017
Published Online: 2017-09-15 | DOI: https://doi.org/10.1515/anona-2017-0150

Abstract

We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler–Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi’s theorem on conjugate points and Noether’s theorem. We close with an application to low regularity Riemannian geometry.

Keywords: Calculus of variations; Schwartz distributions; generalized functions for nonlinear analysis, low regular Riemannian geometry

MSC 2010: 49-XX; 46F30; 53B20

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. [24].

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 g𝒞1,1 are considered [34, 26], or that Hawking’s singularity theorem still holds when g𝒞1,1, see [27]. However, the calculus of variations in the classical sense may cease to hold when metrics with 𝒞1,1 regularity, or below, are considered [19, 29]. This motivates the search for an alternative. In fact, if p, qd and Ω(p,q) denotes the set of all Lipschitz continuous curves connecting p and q, the natural question about what curves γΩ(p,q) 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 Lloc function on d, and is evaluated along γ. However, the image Im(γ) of γ has Lebesgue-measure zero if d>1. 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 ι(g) has derivatives of all orders, valued in a suitable non-Archimedean ring ~ρ (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 ~ρ 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., [17] and references therein. We then apply our extended calculus of variations to the generalized Riemannian space (~dρ,ι(g)), 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 (d,g). Clearly, the process of embedding the singular metric g using ι(g) introduces infinitesimal differences. This is typical in a non-Archimedean setting, but the notion of standard part comes to help: if x~ρ is infinitely close to a standard real number s, i.e., |x-s|r for all r>0, then the standard part of x is exactly s. We then show that (assuming that (d,g) 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 𝒞1,1 Riemannian geometry in Section 8.

Note that Konjik, Kunzinger and Oberguggenberger [25] 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 [25] 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 [25] were forced to consider the so-called compactly supported points c(Ω) (i.e., finite points in Ωn), where the setting of generalized smooth functions gives the possibility to consider more natural domains like the interval [a,b]~ρ. 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 a,b~ρ two infinite numbers such that a<b, or as boundary points two unbounded points p,q~dρ.

(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 [25] 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 (ρε)ε(0,1]0, whereas the Colombeau setting considers only ρε=ε. 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.

(v) In [25], only the notion of global minimizer is defined, whereas we define the notion of local minimizer, as in [10], using a natural topology in space of generalized smooth curves.

(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 [25].

2 Basic notions

The new ring of scalars

In this work, I denotes the interval (0,1] and we will always use the variable ε for elements of I; we also denote ε-dependent nets xI simply by (xε). 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].

Definition 2.1.

Let ρ=(ρε)I be a net such that limε0ρε=0+.

  • (i)

    (ρ):={(ρε-a)a>0} is called the asymptotic gauge generated by ρ. The net ρ is called a gauge.

  • (ii)

    If 𝒫(ε) is a property of εI, we use the notation 0ε:𝒫(ε) to denote ε0I,ε(0,ε0]:𝒫(ε). We can read 0ε as for ε small.

  • (iii)

    We say that a net (xε)I is ρ-moderate and write (xε)ρ if (Jε)(ρ):xε=O(Jε) as ε0+.

  • (iv)

    Let (xε), (yε)I. We say that (xε)ρ(yε) if (Jε)(ρ):xε=yε+O(Jε-1) as ε0+. This is a congruence relation on the ring ρ of moderate nets with respect to pointwise operations, and we can hence define

    ~ρ:=ρ/ρ,

    which we call Robinson–Colombeau ring of generalized numbers, see [38, 5, 6]. We denote the equivalence class x~ρ simply by x:=[xε]:=[(xε)]~ρ.

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 [16]. For motivations concerning the naturality of ~ρ, see [14].

We can also define an order relation on ~ρ by saying that [xε][yε] if there exists (zε)I such that (zε)ρ0 (we then say that (zε) is ρ-negligible) and xεyε+zε for ε small. Equivalently, we have that xy if and only if there exist representatives [xε]=x and [yε]=y such that xεyε for all ε. Clearly, ~ρ is a partially ordered ring. The usual real numbers r are embedded in ~ρ by considering constant nets [r]~ρ.

Even in the case where the order is not total, we still have the possibility to define the infimum min([xε],[yε]):=[min(xε,yε)], and analogously the supremum function max([xε],[yε]):=[max(xε,yε)] and the absolute value |[xε]|:=[|xε|]~ρ. Note, e.g., that xz and -xz imply |x|z. In the following, we will also use the customary notation ~*ρ for the set of invertible generalized numbers. Our notations for intervals are [a,b]:={x~ρaxb} and [a,b]:=[a,b], and analogously for segments [x,y]:={x+r(y-x)r[0,1]}~nρ and [x,y]n=[x,y]n. Finally, we write xy to denote that |x-y| is an infinitesimal number, i.e., |x-y|r for all r>0. This is equivalent to limε0+|xε-yε|=0 for all representatives x=[xε] and y=[yε].

Topologies on ~nρ

On the ~ρ-module ~nρ, we can consider the natural extension of the Euclidean norm, i.e., |[xε]|:=[|xε|]~ρ, where [xε]~nρ. Even if this generalized norm takes values in ~ρ, it shares several properties with usual norms, like the triangular inequality or the property |yx|=|y||x|. It is therefore natural to consider on ~nρ topologies generated by balls defined by this generalized norm and a set of radii .

Definition 2.2.

Let {~0*ρ,>0}, c~nρ and x,y~ρ.

  • (i)

    We write x<y if r:ry-x.

  • (ii)

    Br(c):={x~nρ|x-c|<r} for each r.

  • (iii)

    For each r>0, BrE(c):={xn|x-c|<r} denotes an ordinary Euclidean ball in n.

The relation < has better topological properties compared to the usual strict order relation ab and ab (that we will never use) because for {~0*ρ,>0} the set of balls {Br(c)r,c~nρ} is a base for a topology on ~nρ. The topology generated in the case =~0*ρ is called sharp topology, whereas the one with the set of radii =>0 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 f(x0) is infinite, we have f(x)f(x0) only for xx0, see [11, 12]. With an innocuous abuse of language, we write x<y instead of x<~0*ρy, and x<y instead of x<>0y. For example, ~0*ρ=~>0ρ. We will simply write Br(c) to denote an open ball in the sharp topology and BrF(c) for an open ball in the Fermat topology. Also open intervals are defined using the relation <, i.e., (a,b):={x~ρa<x<b}.

The following result is useful to deal with positive and invertible generalized numbers (cf. [35]).

Lemma 2.3.

Let xR~ρ. Then the following are equivalent:

  • (i)

    x is invertible and x0 , i.e., x>0.

  • (ii)

    For each representative (xε)ρ of x , we have 0ε:xε>0.

  • (iii)

    For each representative (xε)ρ of x , we have m,0ε:xε>ρεm

We will also need the following result.

Lemma 2.4.

Let a,bR~ρ such that a<b. Then the interior int([a,b]) in the sharp topology is dense in [a,b].

Internal and strongly internal sets

A natural way to obtain sharply open, closed and bounded sets in ~nρ is by using a net (Aε) of subsets Aεn. We have two ways of extending the membership relation xεAε to generalized points [xε]~ρ.

Definition 2.5.

Let (Aε) be a net of subsets of n.

  • (i)

    [Aε]:={[xε]~nρ0ε:xεAε} is called the internal set generated by the net (Aε). See [36] for an introduction and an in-depth study of this notion in the case ρε=ε.

  • (ii)

    Let (xε) be a net of points of n. We say that xεεAε, and we read it as (xε) strongly belongs to (Aε) , if 0ε:xεAε, and if (xε)ρ(xε), then also xεAε for ε small. Also, we set Aε:={[xε]~nρxεεAε}, and we call it the strongly internal set generated by the net (Aε).

  • (iii)

    We say that the internal set K=[Aε] is sharply bounded if there exists r~>0ρ such that KBr(0). Analogously, a net (Aε) is sharply bounded if the internal set [Aε] is sharply bounded.

Therefore, x[Aε] if there exists a representative [xε]=x such that xεAε 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 Aε=An is simply denoted by [A].

The following theorem shows that internal and strongly internal sets have dual topological properties:

Theorem 2.6.

For εI, let AεRn and let xεRn. Then the following hold:

  • (i)

    [xε][Aε] if and only if q>0,0ε:d(xε,Aε)ρεq . Thus, [xε][Aε] if and only if [d(xε,Aε)]=0~ρ.

  • (ii)

    [xε]Aε if and only if q>0,0ε:d(xε,Aεc)>ρεq , where Aεc:=nAε . Hence, if (d(xε,Aεc))ρ , then [xε]Aε if and only if [d(xε,Aεc)]>0.

  • (iii)

    [Aε] is sharply closed and Aε is sharply open.

  • (iv)

    [Aε]=[cl(Aε)] , where cl(S) is the closure of Sn. On the other hand, Aε=int(Aε) , where int(S) is the interior of Sn.

Generalized smooth functions and their calculus

Using the ring ~ρ, it is easy to consider a Gaussian with an infinitesimal standard deviation. If we denote this probability density by f(x,σ), and if we set σ=[σε]~>0ρ, where σ0, we obtain the net of smooth functions (f(-,σε))εI. This is the basic idea we are going to develop in the following.

Definition 2.7.

Let X~nρ and Y~dρ be arbitrary subsets of generalized points. We say that f:XY is a generalized smooth function if there exists a net fε𝒞(Ωε,d) defining f in the sense that XΩε, f([xε])=[fε(xε)]Y and (αfε(xε))ρd for all x=[xε]X and all αn. The space of generalized smooth functions (GSF) from X to Y is denoted by 𝒢ρ𝒞(X,Y).

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 f([xε])=[fε(xε)] is meaningful, i.e., that we have independence from the representatives for all derivatives [xε]X[αfε(xε)]~dρ, αn.

Theorem 2.8.

Let XR~nρ and YR~dρ be arbitrary subsets of generalized points. Let fεC(Ωε,Rd) be a net of smooth functions that defines a generalized smooth map of the type XY. Then the following hold:

  • (i)

    αn,(xε),(xε)ρn:[xε]=[xε]X(αuε(xε))ρ(αuε(xε)).

  • (ii)

    [xε]X,αn,q>0,0ε:supyBεqE(xε)|αuε(y)|ε-q.

  • (iii)

    For all αn , the GSF g:[xε]X[αfε(xε)]~d is locally Lipschitz in the sharp topology, i.e., each xX possesses a sharp neighborhood U such that |g(x)-g(y)|L|x-y| for all x, yU and some L~ρ.

  • (iv)

    Each f𝒢ρ𝒞(X,Y) is continuous with respect to the sharp topologies induced on X, Y.

  • (v)

    Assume that the GSF f is locally Lipschitz in the Fermat topology and that its Lipschitz constants are always finite, i.e., L . Then f is continuous in the Fermat topology.

  • (vi)

    f:XY is a GSF if and only if there exists a net vε𝒞(n,d) defining a generalized smooth map of type XY such that f=[vε(-)]|X.

  • (vii)

    Subsets S~sρ 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 𝒢ρ𝒞.

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 UR~nρ be a sharply open set, let v=[vε]R~nρ, and let fGρC(U,R~ρ) be a generalized smooth map generated by the net of smooth functions fεC(Ωε,R). Then the following hold:

  • (i)

    There exists a sharp neighborhood T of U×{0} and a generalized smooth map r𝒢ρ𝒞(T,~ρ) , called the generalized incremental ratio of f along v , such that f(x+hv)=f(x)+hr(x,h) for all (x,h)T.

  • (ii)

    Any two generalized incremental ratios coincide on a sharp neighborhood of U×{0}.

  • (iii)

    We have r(x,0)=[fεvε(xε)] for every xU and we can thus define Df(x)v:=fv(x):=r(x,0) , so that fv𝒢ρ𝒞(U,~ρ).

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 v~nρ which can be, e.g., infinitesimal or infinite. Using this result, we can also define subsequent differentials Djf(x) as j-multilinear maps, and we set

Djf(x)hj:=Djf(x)(h,,hj).

The set of all the j-multilinear maps (~nρ)j~dρ over the ring ~ρ will be denoted by Lj(~nρ,~dρ). For A=[Aε(-)]Lj(~nρ,~dρ), we set |A|:=[|Aε|], the generalized number defined by the operator norms of the multilinear maps AεLj(n,d).

The following result follows from the analogous properties for the nets of smooth functions defining f and g.

Theorem 2.10.

Let UR~nρ be an open subset in the sharp topology, and let vR~nρ and f,g:UR~ρ be generalized smooth maps. Then the following hold:

  • (i)

    (f+g)v=fv+gv.

  • (ii)

    (rf)v=rfv for all r~ρ.

  • (iii)

    (fg)v=fvg+fgv.

  • (iv)

    For each xU , the map df(x)v:=fv(x)~ρ is ~ρ -linear in v~nρ.

  • (v)

    Let U~nρ and V~dρ be open subsets in the sharp topology, and let g𝒢ρ𝒞(V,U) and f𝒢ρ𝒞(U,~ρ) be generalized smooth maps. Then, for all xV and all v~dρ , we have (fg)v(x)=df(g(x))gv(x).

We also have a generalization of the Taylor formula.

Theorem 2.11.

Let fGρC(U,R~ρ) be a generalized smooth function defined in the sharply open set UR~nρ. Let a,xR~nρ be such that the line segment [a,x] belongs to U. Then, for all nN, we have

ξ[a,x]:f(x)=j=0nDjf(a)j!(x-a)j+Dn+1f(ξ)(n+1)!(x-a)n+1.(2.1)

If we further assume that all the n components (x-a)kR~ρ of x-aR~nρ are invertible, then there exists ρR~>0ρ, ρ|x-a|, such that

kBρ(0),ξ[a-k,a+k]:f(a+k)=j=0nDjf(a)j!kj+Dn+1f(ξ)(n+1)!kn+1,(2.2)Dn+1f(ξ)(n+1)!kn+10.(2.3)

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 Dn+1f 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 [13] for a proof).

Theorem 2.12.

Let XR~nρ and fGρC(X,R~nρ), and suppose that for some x0 in the sharp interior of X, Df(x0) is invertible in L(R~nρ,R~nρ). Then there exists a sharp neighborhood UX of x0 and a sharp neighborhood V of f(x0) such that f:UV is invertible and f-1GρC(V,U).

We can define right and left derivatives as, e.g., f(a):=f+(a):=limta,a<tf(t), which always exist if f𝒢ρ𝒞([a,b],~dρ). The one-dimensional integral calculus of GSF is based on the following.

Theorem 2.13.

Let fGρC([a,b],R~ρ) be a generalized smooth function defined in the interval [a,b]R~ρ, where a<b. Let c[a,b]. Then there exists one and only one generalized smooth function FGρC([a,b],R~ρ) such that F(c)=0 and F(x)=f(x) for all x[a,b]. Moreover, if f is defined by the net fεC(R,R) and c=[cε], then F(x)=[cεxεfε(s)ds] for all x=[xε][a,b].

Definition 2.14.

Under the assumptions of Theorem 2.13, we denote by c(-)f:=c(-)f(s)ds𝒢ρ𝒞([a,b],~ρ) the unique generalized smooth function such that

ccf=0  and  (u(-)f)(x)=ddxuxf(s)ds=f(x)for all x[a,b].

All the classical rules of integral calculus hold in this setting:

Theorem 2.15.

Let fGρC(U,R~ρ) and gGρC(V,R~ρ) be generalized smooth functions defined on sharply open domains in R~ρ. Let a,bR~ρ, with a<b, and c,d[a,b]UV. Then

  • (i)

    cd(f+g)=cdf+cdg,

  • (ii)

    cdλf=λcdf for all λ~ρ,

  • (iii)

    cdf=cef+edf for all e[a,b],

  • (iv)

    cdf=-dcf,

  • (v)

    cdf=f(d)-f(c),

  • (vi)

    cdfg=[fg]cd-cdfg.

Theorem 2.16.

Let fGρC(U,R~ρ) and ϕGρC(V,U) be generalized smooth functions defined on sharply open domains in R~ρ. Let a,bR~ρ, with a<b, such that [a,b]V, ϕ(a)<ϕ(b) and [ϕ(a),ϕ(b)]U. Finally, assume that ϕ([a,b])[ϕ(a),ϕ(b)]. Then

ϕ(a)ϕ(b)f(t)dt=abf[ϕ(s)]ϕ(s)ds.

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 y=y/dε, which cannot be solved for ρ=(ε), but it has a solution for ρ=(e-1/ε). As another simple example, if we need the property H(0)=1/2, where H is the Heaviside function, then we have to choose the embedding of distributions accordingly. See also [16, 32] for further details.

If ϕ𝒟(n), r>0 and xn, we use the notations rϕ for the function xn1rnϕ(xr) and xϕ for the function ynϕ(y-x). These notations permit to highlight that is a free action of the multiplicative group (>0,,1) on 𝒟(n) and is a free action of the additive group (>0,+,0) on 𝒟(n). We also have the distributive property r(xϕ)=rxrϕ.

Lemma 2.17.

Let bR~ρ be a net such that limε0+bε=+. Let d(0,1). There exists a net (ψε)εI of D(Rn) with the following properties:

  • (i)

    supp(ψε)B1(0) for all εI.

  • (ii)

    ψε=1 for all εI.

  • (iii)

    αn,p:supxn|αψε(x)|=O(bεp) as ε0+.

  • (iv)

    j,0ε:1|α|jxαψε(x)dx=0.

  • (v)

    η>0,0ε:|ψε|1+η.

  • (vi)

    If n=1 , then the net (ψε)εI can be chosen so that -0ψε=d.

In particular, ψεb:=bε-1ψε satisfies (ii)(v).

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

c(Ω):={[xε][Ω]KΩ,0ε:xεK}

is called the set of compactly supported points in Ωn.

Theorem 2.18.

Under the assumptions of Lemma 2.17, let ΩRn be an open set and let (ψεb) be the net defined in Lemma 2.17. Then the mapping

ιΩb:T(Ω)[(Tψεb)(-)]𝒢ρ𝒞(c(Ω),~ρ)

uniquely extends to a sheaf morphism of real vector spaces

ιb:𝒟𝒢ρ𝒞(c((-)),~ρ),

and satisfies the following properties:

  • (i)

    If bdρ-a for some a>0 , then ιb|𝒞(-):𝒞(-)𝒢ρ𝒞(c((-)),~ρ) is a sheaf morphism of algebras.

  • (ii)

    If T(Ω) , then supp(T)=supp(ιΩb(T)).

  • (iii)

    limε0+ΩιΩb(T)εϕ=T,ϕ for all ϕ𝒟(Ω) and all T𝒟(Ω).

  • (iv)

    ιb commutes with partial derivatives, i.e., α(ιΩb(T))=ιΩb(αT) for each T𝒟(Ω) and α.

Concerning the embedding of Colombeau generalized functions, we recall that the special Colombeau algebra on Ω is defined as the quotient 𝒢s(Ω):=M(Ω)/𝒩s(Ω) of moderate nets over negligible nets, where the former is

M(Ω):={(uε)𝒞(Ω)I|KΩ,αn,N:supxK|αuε(x)|=O(ε-N)}

and the latter is

𝒩s(Ω):={(uε)𝒞(Ω)I|KΩ,αn,m:supxK|αuε(x)|=O(εm)}.

Using ρ=(ε), we have the following compatibility result.

Theorem 2.19.

A Colombeau generalized function u=(uε)+Ns(Ω)dGs(Ω)d defines a generalized smooth map u:[xε]c(Ω)[uε(xε)]R~d, which is locally Lipschitz on the same neighborhood of the Fermat topology for all derivatives. This assignment provides a bijection of Gs(Ω)d onto GρC(c(Ω),R~dρ) for every open set ΩRn.

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 [a,b]~, a,b, 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.

Theorem 2.20.

Let fGC(X,R~) be a generalized smooth function defined on the subset X of R~n. Let K=[Kε]X be an internal set generated by a sharply bounded net (Kε) of compact sets KεRn. Then

m,MK,xK:f(m)f(x)f(M).(2.4)

We shall use the assumptions on K and (Kε) 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.

Definition 2.21.

A subset K of ~n is called functionally compact, denoted by Kf~n, if there exists a net (Kε) such that

  • (i)

    K=[Kε]~n,

  • (ii)

    (Kε) is sharply bounded,

  • (iii)

    εI:Kεn.

If, in addition, KU~n, then we write KfU. Finally, we write [Kε]fU if (ii), (iii) and [Kε]U hold.

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.

Remark 2.22.

  • (i)

    By [36, Proposition 2.3], any internal set K=[Kε] is closed in the sharp topology. In particular, the open interval (0,1)~ is not functionally compact since it is not closed.

  • (ii)

    If Hn is a non-empty ordinary compact set, then [H] is functionally compact. In particular, we have that [0,1]=[[0,1]] is functionally compact.

  • (iii)

    For the empty set, we have =~f~.

  • (iv)

    ~n is not functionally compact since it is not sharply bounded.

  • (v)

    The set of compactly supported points ~c is not functionally compact because the GSF f(x)=x does not satisfy the conclusion (2.4) of Proposition 2.20.

In the present paper, we need the following properties of functionally compact sets.

Theorem 2.23.

Let KXR~n, fGC(X,R~d). Then KfR~n implies f(K)fR~d.

As a corollary of this theorem and Remark 2.22(ii), we get the following.

Corollary 2.24.

If a,bR~ and ab, then [a,b]fR~.

Let us note that a,b~ can also be infinite, e.g., a=[-ε-N], b=[ε-M] or a=[ε-N], b=[ε-M] with M>N. Finally, in the following result we consider the product of functionally compact sets.

Theorem 2.25.

Let KfR~n and HfR~d, then K×HfR~n+d. In particular, if aibi for i=1,,n, then i=1n[ai,bi]fR~n.

A theory of compactly supported GSF has been developed in [13], 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 [a,b]~ρ. 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 ~ρ 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.

Lemma 3.1.

Let a,b,c,dR~ρ, with a<b and c<d. Let also fGρC(X,Y) and assume that [a,b]×[c,d]XR~2ρ and YR~dρ. Then, for all s[c,d], we have

ddsabf(τ,s)dτ=absf(τ,s)dτ.(3.1)

Proof.

We first note that f(,s)𝒢ρ𝒞([a,b],Y), by the closure of GSF with respect to composition. Therefore, sf(,s)𝒢ρ𝒞([a,b],~dρ), 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 σ[c,d]abf(τ,σ)dτ~dρ is a GSF. Let f be defined by the net fε𝒞(Ωε,d), with XΩε, and let [σε][c,d]. Then [a,b]×{[σε]}f~2ρ and the extreme value Theorem 2.20 applied to nfσn yields the existence of N>0 such that

|dndσnaεbεfε(τ,σε)dτ|aεbε|nσnfε(τ,σε)|dτρε-N(bε-aε).

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

ddsabf(τ,s)dτ=ddsab[fε(τ,s)]dτ=dds[aεbεfε(τ,s)dτ]=[ddsaεbεfε(τ,s)dτ]=ab[sfε(τ,s)]dτ=absf(τ,s)dτ.

The next result will be used frequently.

Lemma 3.2.

Let (D,) be a directed set and let f:DR~ρ be a set-theoretical map such that f(d)0 for all dD, and limdDf(d)R~ρ in the sharp topology. Then limdDf(d)0.

Proof.

Note that the internal set [0,+)=[[0,+)] is sharply closed by (iii) of Theorem 2.6. ∎

Remark 3.3.

  • (i)

    If x~ρ, then x0 if and only if A>0,a>A:x-dρa. Indeed, it suffices to let a+ in f(a)=x+dρa.

  • (ii)

    Assume that x, y~nρ and

    s0~>0ρ,s~>0ρ:ss0|x|s|y|.

    Then taking s0 in f(s)=s|y|-|x|, we get x=0.

Definition 3.4.

We call x=(x1,,xd)~dρ componentwise invertible if and only if for all k{1,,d}, we have that xk~ρ is invertible.

Lemma 3.5.

Let fGρC(U,Y), where YR~ρ, and UR~dρ is a sharply open subset. Then f0 if and only if f(x)0 for all componentwise invertible xU.

Proof.

By Lemma 2.3, it follows that for V~ρ, the set of invertible points in V, i.e., V~*ρV is dense in V (with respect to the sharp topology). This implies that U(~dρ)*U is dense. By Theorem 2.8(iv), f is sharply continuous, so Lemma 3.2 yields f(x)0. The other direction is obvious. ∎

Analogously to the classical case, we say that x0X is a local minimum of f𝒢ρ𝒞(X) if there exists a sharply open neighborhood (in the trace topology) YX of x0 such that f(x0)f(y) for all yY. A local maximum is defined accordingly. We will write f(x0)=min!, which is a short hand notation to denote that x0 is a (local) minimum of f.

Lemma 3.6.

Let XR~ρ and fGρC(X,R~ρ). If x0X is a sharply interior local minimum of f, then f(x0)=0.

Proof.

Without loss of generality, we can assume x0=0, because of the closure of GSF with respect to composition. Let r~>0ρ be such that B2r(0)=:UX and f(0)=min! over U. Take any x~ρ such that 0<|x|<r, so that [-|x|,|x|]U. Thus, if x>0, by Taylor’s Theorem 2.11, there exists ξ[0,x] such that

f(x)=f(0)+f(0)x+f′′(ξ)2x2.

Set K:=[Brε(0)]fB2r(0)U and M:=maxxK|f′′(x)|~0ρ. Due to the fact that f(0) is minimal, we have

f(0)x+f′′(ξ)2x2=f(x)-f(0)0.

Thus, -f(0)xM2x2 and -f(0)M2|x|, since x>0. Analogously, if we take x<0, we get f(0)-M2x=M2|x|. Therefore, |f(0)|M2|x|, and the conclusion follows by Remark 3.3(ii). ∎

As a consequence of Lemma 2.4 and Theorem 2.8(iv), we have the following lemma.

Lemma 3.7.

Let a,bR~ρ, with a<b, and let fGρC([a,b],R~dρ) be such that f(x)=0 for all sharply interior points x[a,b]. Then f=0 on [a,b].

Now, we are able to prove the “second-derivative-test” for GSF.

Lemma 3.8.

Let a,bR~ρ, with a<b, and let fGρC([a,b],R~ρ) be such that f(x0)=min! for some sharply interior x0[a,b]. Then f′′(x0)0. Vice versa, if f(x0)=0 and f′′(x0)>0, then f(x0)=min!.

Proof.

As above, we can assume that x0=0. Let r~>0ρ be such that B2r(0)=:UX and f(0)=min! over U. Take any x~ρ such that 0<x<r, so that [0,x]U, and set K:=[Brε(0)]fB2r(0)U and M:=maxxK|f′′′(x)|~0ρ. By Taylor’s Theorem 2.11, for some ξ[0,x], we obtain

f(x)=f(0)+f(0)x+12f′′(0)x2+16f′′′(ξ)x3.

By assumption, for all a>0, we have

0f(x)-f(0)+dρa.

By Lemma 3.6, we know that f(0)=0. Thus, for all a>0, we obtain

f(x)-f(0)=12f′′(0)x2+16f′′′(ξ)x3-dρa.

Therefore, also 12f′′(0)x2+16Mx3-dρa. In this inequality we can set x=dρa/3, assuming that a>A and dρA<r. We get f′′(0)-(2+M3)dρa/3, and the conclusion follows from Lemma 3.2 as a+.

Now assume that f(0)=0 and f′′(0)>0, so that f′′(0)>dρa for some a>0, by Lemma 2.3. Since f(0)=0, for all xBr(0), Taylor’s formula gives

f(x)-f(0)=12f′′(0)x2+16f′′′(ξx)x3,

where ξx[0,x]. Therefore, f(x)-f(0)>x2(12dρa+16f′′′(ξx)x). Now

|16f′′′(ξx)x|16M|x|0as x0.

Thus,

s~>0ρ:s<r,xBs(0):-14dρa<16f′′′(ξx)x<14dρa.

We can hence write f(x)-f(0)>x2(12dρa-14dρa)=x214dρa0 for all xBs(0), which proves that x=0 is a local minimum. ∎

For the generalization of Lemmas 3.6 and 3.8 to the multivariate case, one can proceed as above, using the ideas of [25]. Note, however, that we do not need this generalization in the present work.

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).

Definition 4.1.

If a,b~ρ and a<b, we define

𝒢ρ𝒞0(a,b):={η𝒢ρ𝒞(~ρ,~dρ):η(a)=0=η(b)}.

When the use of the points a,b is clear from the context, we adopt the simplified notation 𝒢ρ𝒞0. We also note here that 𝒢ρ𝒞0(a,b) is an ~ρ-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.

Theorem 4.2.

Let a,bR~ρ with a<b. Let uGρC([a,b],R~dρ) and FGρC([a,b]×R~dρ×R~dρ,R~ρ), and define

I(u):=abF(t,u,u˙)dt.(4.1)

Let also ηGρC0. Then

δI(u;η):=ddsI(u+sη)|s=0=abη(Fu(t,u,u˙)-ddtFu˙(t,u,u˙))dt.

Proof.

Using Theorems 2.10 and 2.15, and Lemma 3.1, we have

ddsI(u+sη)|s=0=ddsabF(t,u+sη,u˙+sη˙)dt|s=0=absF(t,u+sη,u˙+sη˙)|s=0dt=abηFu(t,u,u˙)+η˙Fu˙(t,u,u˙)dt=[ηFu˙(t,u,u˙)]ab+abη(Fu(t,u,u˙)-ddtFu˙(t,u,u˙))dt=abη(Fu(t,u,u˙)-ddtFu˙(t,u,u˙))dt.

We call δI(u;η) the first variation of I. In addition, we call u𝒢ρ𝒞([a,b],~dρ) a critical point of I if δI(u;η)=0 for all η𝒢ρ𝒞0.

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.

Lemma 4.3.

Let a,bR~ρ be such that a<b and let fGρC([a,b],R~ρ). Let x[a,b] and RR~>0ρ be such that BR(x)[a,b]. Assume that GtGρC(R~ρ,R~ρ) has the following properties:

  • (i)

    -RRGt=1 for t~>0ρ small.

  • (ii)

    For t small, (Gt)t~>0ρ is zero outside every ball Bδ(0), 0<δ<R , i.e.,

    δ~>0ρ,ρ~>0ρ,tBρ(0)~>0ρ,y[-R,-δ][δ,R]:Gt(y)=0.

  • (iii)

    M~>0ρ,ρ~ρ,tBρ(0):-RR|Gt(y)|dyM.

Then

limt0+-RRf(x-y)Gt(y)dy=f(x).

Proof.

We only have to generalize the classical proof concerning limits of convolutions with strict delta nets. We first note that

-RRf(x-y)Gt(y)dy=x-Rx+Rf(y)Gt(x-y)dy,

and so these integrals exist because (x-R,x+R)=BR(x)[a,b]. Using (i), for t small, say for 0<t<S~>0ρ, we get

|-RRf(x-y)Gt(y)dy-f(x)|=|-RR[f(x-y)-f(x)]Gt(y)dy|-RR|f(x-y)-f(x)||Gt(y)|dy.

For each r~>0ρ, the sharp continuity of f at x yields |f(x-y)-f(x)|<r for all y such that |y|<δ~>0ρ, and we can take δ<R. By (ii), for 0<|t|<min(ρ,S), we have

|-RRf(x-y)Gt(y)dy-f(x)|r-δ+δ|Gt(y)|dy.(4.2)

The right-hand side of (4.2) can be taken arbitrarily small in ~>0ρ, because of (iii), the fact that [-δ,δ]f~ρ and the application of the extreme value Theorem 2.20 to the GSF Gt. ∎

Lemma 4.4 (Fundamental lemma of the calculus of variations).

Let a,bR~ρ be such that a<b, and let fGρC([a,b],R~ρ). If

abf(t)η(t)dt=0 for all η𝒢ρ𝒞0,(4.3)

then f=0.

Proof.

Let x[a,b]. 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 BR(x)[a,b] for some R~>0ρ. Let ϕ𝒟[-1,1]() be such that ϕ=1. Set Gt,ε(x):=1tεϕ(xtε), where x and t~>0ρ, and Gt(x):=[Gt,ε(xε)] for all x~ρ. Then, for t sufficiently small, we have Gt(x-)𝒢ρ𝒞0 and (4.3) yields abf(y)Gt(x-y)dy=0. For t small, we both have that Gt(x-)=0 on [a,x-R][x+R,b] and the assumptions of Lemma 4.3 hold. Therefore,

0=abf(y)Gt(x-y)dy=x-Rx+Rf(y)Gt(x-y)dy=-RRf(x-y)Gt(y)dy,

and hence Lemma 4.3 yields f(x)=0. ∎

Thus, we obtain the following theorem.

Theorem 4.5.

Let a,bR~ρ be such that a<b, and let uGρC([a,b],R~dρ). Then u solves the Euler–Lagrange equations

Fu-ddtFu˙=0,(4.4)

for I given by (4.1), if and only if δI(u;η)=0 for all ηGρC0, i.e., if and only if u is a critical point of I.

5 Second variation and minimizers

As in the classical case (see, e.g., [10]), thanks to the extreme value Theorem 2.20 and the property of the interval [a,b] of being functionally compact, we can naturally define a topology on the space 𝒢ρ𝒞([a,b],~dρ).

Definition 5.1.

Let a,b~ρ, with a<b. Let m and v𝒢ρ𝒞([a,b],~dρ). Then

vm:=maxnm1idmax(|dndtnvi(Mni)|,|dndtnvi(mni)|)~ρ,

where Mni,mni[a,b] satisfy

dndtnvi(mni)dndtnvi(t)dndtnvi(Mni)for all t[a,b].

The following result permits the calculation of the (generalized) norm vm using any net (vε) that defines v.

Lemma 5.2.

Under the assumptions of Definition 5.1, let a=[aε] and b=[bε] be such that aε<bε for all ε. Then the following hold:

  • (i)

    If the net (vε) defines v , then

    vm=[maxnm1idmaxt[aε,bε]|dndtnvεi(t)|].

  • (ii)

    vm0.

  • (iii)

    vm=0 if and only if v=0.

  • (iv)

    cvm=|c|vm for all c~ρ.

  • (v)

    For all u𝒢ρ𝒞([a,b],~dρ) , we have u+vmum+vm and uvmcmumvm for some cm~>0ρ.

Proof.

By the standard extreme value theorem applied ε-wise, we get the existence of m¯niε, M¯niε[aε,bε] such that

dndtnvεi(m¯niε)dndtnvεi(t)dndtnvεi(M¯niε)for all t[aε,bε].

Hence,

|dndtnvεi(t)|max(|dndtnvεi(m¯niε)|,|dndtnvεi(M¯niε)|).

Thus,

maxnm1idmaxt[aε,bε]|dndtnvεi(t)|maxnm1idmax(|dndtnvεi(m¯niε)|,|dndtnvεi(M¯niε)|).

But m¯niε, M¯niε[aε,bε], so

[maxnm1idmaxt[aε,bε]|dndtnvεi(t)|]=[maxnm1idmax(|dndtnvεi(m¯niε)|,|dndtnvεi(M¯niε)|)]=maxnm1idmax(|dndtnvi(m¯ni)|,|dndtnvi(M¯ni)|).

This proves both that vm is well defined, i.e., it does not depend on the particular choice of points mni, Mni as in Definition 5.1, and claim (i). The remaining properties (ii)(v) follows directly from (i) and the usual properties of standard 𝒞m-norms. ∎

Using these ~ρ-valued norms, we can naturally define a topology on the space 𝒢ρ𝒞([a,b],~dρ).

Definition 5.3.

Let a,b~ρ, with a<b, m, u𝒢ρ𝒞([a,b],~dρ) and r~>0ρ.

  • (i)

    We set Brm(u):={v𝒢ρ𝒞([a,b],~dρ)v-um<r}.

  • (ii)

    If U𝒢ρ𝒞([a,b],~dρ), then we say that U is a sharply open set if

    uU,m,r~>0ρ:Brm(u)U.

As in [15, Theorem 2], one can easily prove that sharply open sets form a topology on 𝒢ρ𝒞([a,b],~dρ). 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 a,b~ρ, a<b, e.g., they can also both be infinite.

Definition 5.4.

Let a,b~ρ, with a<b, and u𝒢ρ𝒞([a,b],~dρ).

  • (i)

    For all p,q~dρ, we set

    𝒢ρ𝒞bd(p,q):={v𝒢ρ𝒞([a,b],~dρ)v(a)=p,v(b)=q}.

    Note that 𝒢ρ𝒞bd(0,0)=𝒢ρ𝒞0. The subscript “bd” stands here for “boundary values”.

  • (ii)

    We say that u is a local minimizer of I in 𝒢ρ𝒞bd(p,q) if u𝒢ρ𝒞bd(p,q) and

    r~>0ρ,m,vBrm(u)𝒢ρ𝒞bd(p,q):I(v)I(u).(5.1)

  • (iii)

    We define the second variation of I in the direction η𝒢ρ𝒞0 as

    δ2I(u;η):=d2ds2|0I(u+sη).

Note also explicitly that the points p, q~dρ can have infinite norm, e.g., |pε|+ as ε0. By using the standard Einstein’s summation conventions, we calculate

δ2I(u;η)=d2ds2|0abF(t,u+sη,u˙+sη˙)dt=ab2s2|0F(t,u+sη,u˙+sη˙)dt=abFuiuj(t,u,u˙)ηiηj+2Fuiu˙j(t,u,u˙)ηiη˙j+Fu˙iu˙j(t,u,u˙)η˙iη˙jdt,

which we abbreviate as

δ2I(u;η)=abFuu(t,u,u˙)ηη+2Fuu˙(t,u,u˙)ηη˙+Fu˙u˙(t,u,u˙)η˙η˙dt.

The following results establish classical necessary and sufficient conditions to decide if a function u is a minimizer for the given functional (4.1).

Theorem 5.5.

Let a,bR~ρ, with a<b, FGρC([a,b]×R~dρ×R~dρ,R~ρ), p,qR~dρ, and let u be a local minimizer of I in GρCbd(p,q). Then

  • (i)

    δI(u;η)=0 for all η𝒢ρ𝒞0,

  • (ii)

    δ2I(u;η)0 for all η𝒢ρ𝒞0.

Proof.

Let r~>0ρ be such that (5.1) holds. Since η𝒢ρ𝒞0, the map s~ρu+sη𝒢ρ𝒞bd(p,q) is well defined and continuous with respect to the trace of the sharp topology in its codomain. Therefore, we can find r¯~>0ρ such that u+sηBrm(u)𝒢ρ𝒞bd(p,q) for all sBr¯(0). We hence have I(u+sη)I(u). This shows that the GSF sBr¯(0)I(u+sη)~ρ has a local minimum at s=0. Now, by employing Lemmas 3.6 and 3.8, the claims are proven. ∎

Theorem 5.6.

Let a,bR~ρ, with a<b, and p,qR~dρ. Let uGρCbd(p,q) be such that

  • (i)

    δI(u;η)=0 for all η𝒢ρ𝒞0,

  • (ii)

    δ2I(v;η)0 for all η𝒢ρ𝒞0 and all vBrm(u)𝒢𝒞bd(p,q) , where r~>0ρ and m.

Then u is a local minimizer of the functional I in GρCbd(p,q).

Moreover, if δ2I(v;η)>0 for all ηGρC0 such that ηm>0 and all vB2rm(u)GCbd(p,q), then I(v)>I(u) for all vBrm(u)GCbd(p,q) such that v-um>0.

Proof.

For any vBrm(u)𝒢𝒞bd(p,q), we set ψ(s):=I(u+s(v-u))~ρ for all sB1(0), so that we have u+s(v-u)Brm(u). Since (v-u)(a)=0=(v-u)(b), we have v-u𝒢ρ𝒞0, and properties (i), (ii) yield ψ(0)=δI(u;v-u)=0 and ψ′′(s)=δ2I(u+s(v-u);v-u)0 for all sB1(0). We claim that s=0 is a minimum of ψ. In fact, for all sB1(0), by Taylor’s Theorem 2.11, we have

ψ(s)=ψ(0)+sψ(0)+s22ψ′′(ξ)

for some ξ[0,s]. But ψ(0)=0, and hence ψ(s)-ψ(0)=s22ψ′′(ξ)0. Finally, Lemma 3.2 yields

lims1-ψ(s)=I(v)ψ(0)=I(u),

which is our conclusion. Note explicitly that if δ2I(v;η)=0 for all η𝒢ρ𝒞0 and all vBrm(u)𝒢𝒞bd(p,q), then ψ′′(ξ)=0 and hence I(v)=I(u).

Now, assume that δ2I(v;η)>0 for all η𝒢ρ𝒞0 such that ηm>0 and all vB2rm(u)𝒢𝒞bd(p,q), and take vBrm(u)𝒢𝒞bd(p,q) such that v-um>0. As above, set ψ(s):=I(u+s(v-u))~ρ for all sB3/2(0), so that u+s(v-u)B2rm(u). We have ψ(0)=0 and ψ′′(s)=δ2I(u+s(v-u);v-u)>0 for all sB3/2(0) because v-um>0. Using Taylor’s theorem, we get ψ(1)=ψ(0)+12ψ′′(ξ) for some ξ[0,1]. Therefore, ψ(1)-ψ(0)=I(v)-I(u)=12ψ′′(ξ)>0. ∎

Lemma 5.7.

Let (ak)kN, (bk)kN and (ck)kN be sequences in R~>0ρ. Assume that both (ak)k,(bk)k0 and ckak+bk1 in the sharp topology as k+. Let fGρC([a1,b1],R~ρ). Finally, let ak<t<bk for all kN. Then

f(t)=limk1ckt-akt+bkf(s)ds.

Proof.

We can apply the integral mean value theorem for each ε and each defining net (fε) of f to get the existence of τk[t-ak,t+bk] such that

f(τk)=1bk+akt-akt+bkf(s)ds=ckbk+ak1ckt-akt+bkf(s)ds.

Now, we take the limit for k, and the claim follows by assumption and Theorem 2.8(iv), i.e., by the sharp continuity of f. ∎

We now derive the so-called necessary Legendre condition.

Theorem 5.8.

Let a,bR~ρ, with a<b, and let uGρC([a,b],R~dρ) be a minimizer of the functional I. Then Fu˙u˙(t,u(t),u˙(t)) is positive semi-definite for all t[a,b], i.e.,

Fu˙iu˙j(t,u(t),u˙(t))λiλj0for all λ=(λ1,,λd)~dρ.(5.2)

This figure illustrates the function ϑε{\vartheta_{\varepsilon}} we are considering (blue). The dotted green triangle symbolizes the function which is used in the classical proofs of the Legendre necessary condition (cf. [23, Theorem 1.3.2]).
Figure 1

This figure illustrates the function ϑε we are considering (blue). The dotted green triangle symbolizes the function which is used in the classical proofs of the Legendre necessary condition (cf. [23, Theorem 1.3.2]).

Proof.

Let λ=[λε]~dρ and let k,h be arbitrary. Let t=[tε][a,b]. 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 ρεk 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., ρε2k. So, consider a net of smooth functions ϑε on [aε,bε] such that the following properties hold:

  • (i)

    ϑε(x)=0 for xtε-ρεk-ρε2k.

  • (ii)

    ϑε(x)=0 for xtε+ρεk+ρε2k.

  • (iii)

    ϑε(x)=λ(x-tε)+ρεkλ for x[tε-ρεk+ρε2k,tε].

  • (iv)

    ϑε(x)=-λ(x-tε)+ρεkλ for x[tε+ρε2k,tε+ρεk-ρε2k].

  • (v)

    |ϑε(x)|ρεk|λ|+2ρε2k|λ|.

  • (vi)

    |ϑ˙ε(x)|2|λ| for all x.

The net (ϑε) defines a GSF ϑ:=[ϑε(-)]𝒢ρ𝒞0 because t is a sharply interior point. Setting for simplicity ak:=dρk+dρ2k, by assumption, we have

0δ2I(u,ϑ)=t-akt+akFuu(t,u,u˙)ϑϑ+2Fu˙u(t,u,u˙)ϑ˙ϑ+Fu˙u˙(t,u,u˙)ϑ˙ϑ˙dt.(5.3)

Now, setting M:=max[a,b]|Fuu(t,u,u˙)| and N:=max[a,b]|Fuu˙(t,u,u˙)|, by (v), we have

|t-akt+akFuu(t,u,u˙)ϑϑdt|M|ϑ(t)|22ak=O(dρ3k),

where we used the evident notation Gk=O(dρk) to denote that there exists some A~>0ρ such that GkAdρk for all k. Using (v) and (vi), we analogously have

|t-akt+akFu˙u(t,u,u˙)ϑ˙ϑdt|4N|ϑ(t)|ak|λ|=O(dρ2k).

Note that there always exists C~ρ such that |λ|Cdρk. Therefore,

limk+12dρkt-akt+akFuu(t,u,u˙)ϑϑ+2Fu˙u(t,u,u˙)ϑ˙ϑdt=0.(5.4)

Using Lemmas 5.7 and 3.2, (5.4) and (5.3), we obtain

Fu˙u˙(t,u(t)u˙(t))ϑ˙(t)ϑ˙(t)=limk+12dρkt-akt+akFu˙u˙(t,u,u˙)ϑ˙ϑ˙dt0.

But (iii) yields ϑ˙(t)=λ, and this concludes the proof. ∎

6 Jacobi fields

As in the classical case, Theorem 5.5(ii) motivates us to define the accessory integral

Q(η):=abψ(t,η,η˙)dtfor all η𝒢ρ𝒞0,

where

ψ(t,l,v):=Fuu(t,u,u˙)ll+2Fuu˙(t,u,u˙)lv+Fu˙u˙(t,u,u˙)vv(6.1)

for all t[a,b] and (l,v)~dρ×~dρ. Note that if u minimizes I, then

Q(η)0for all η𝒢ρ𝒞0.

As usual, we note that η=0 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

ddtψη˙(t,η,η˙)=ψη(t,η,η˙).

In other words,

ddt{Fu˙u˙(t,u,u˙)η˙+Fuu˙(t,u,u˙)η}=Fuu˙(t,u,u˙)η˙+Fuu(t,u,u˙)η.(6.2)

Since u is given, (6.2) is an ~ρ-linear system of second order equations in the unknown GSF η and with time dependent coefficients in ~ρ. We call (6.2) the Jacobi equations I with respect to u . As in the classical setting, we introduce the following definition.

Definition 6.1.

A solution η𝒢ρ𝒞0 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.

Lemma 6.2.

Let uGρC([-δ,δ]×[a,b],R~dρ), where δR~>0ρ. We write us:=u(s,-) for all s[-δ,δ]. Assume that each us satisfies the Euler–Lagrange equations (4.4):

ddtFu˙(t,us,u˙s)=Fu(t,us,u˙s)for all s[-δ,δ].

Then

η(t):=dds|0us(t)for all t[a,b]

is a Jacobi field along u.

Proof.

A straight forward calculation gives

0=dds|0(ddtFu˙(t,us,u˙s)-Fu(t,us,u˙s))=ddt(Fu˙u˙(t,u,u˙)η˙+Fuu˙(t,u,u˙)η)-Fuu˙(t,u,u˙)η˙-Fuu(t,u,u˙)η.

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.

Definition 6.3.

We call piecewise GSF an n-tuple (f1,,fn) with the following properties:

  • (i)

    For all i=1,,n there exist ai,ai+1~ρ such that ai<ai+1 and fi𝒢ρ𝒞([ai,ai+1],~dρ). Note that [a,b]=[a,b] implies a=a and b=b because the relation is antisymmetric. Therefore, the points ai, ai+1 are uniquely determined by the set-theoretical function fi.

  • (ii)

    For all i=1,,n, we have fi(ai+1)=fi+1(ai+1).

Every pointwise GSF (f1,,fn) defines a set-theoretical function:

  • resume

    For all ti=1n[ai,ai+1], we set (f1,,fn)(t):=fi(t) if t[ai,ai+1].

We also use the arrow notation (f1,,fn):i=1n[ai,ai+1]~dρ to say that both (i) and (ii) hold.

Remark 6.4.

  • (i)

    Clearly, t[ai,ai+1][ai+1,ai+2] implies t=ai+1, so that condition (ii) yields that the evaluation (i) is well defined.

  • (ii)

    Since the order relation is not a total one, we do not have that [ai,ai+1][ai+1,ai+2]=[ai,ai+2].

  • (iii)

    If ν:[a1,a2][a2,a3]~dρ is a set-theoretical function originating from a piecewise GSF (f1,f2), then neither the GSF fi nor the points ai are uniquely determined by ν. For this reason, we prefer to stress our notations with symbols like (f1,f2)(t)~dρ.

  • (iv)

    Every GSF f𝒢ρ𝒞([a1,a2],~dρ) can be seen as a particular case of a piecewise GSF.

  • (v)

    If (g1,,gn),(f1,,fn):i=1n[ai,ai+1]~dρ and r~ρ, then also (g1,,gn)+(f1,,fn):=(g1+f1,,gn+fn) and r(f1,,fn):=(rf1,,rfn) are piecewise GSF, and we hence have a structure of an ~ρ-module.

  • (vi)

    If (f1,,fn):i=1n[ai,ai+1]~dρ and F𝒢ρ𝒞(~dρ,~nρ), then we can define the composition

    F(f1,,fn):=(Ff1,,Ffn):i=1n[ai,ai+1]~nρ.

Piecewise GSF inherit from their defining components a well-behaved differential and integral calculus. The former is even more general and taken from [1].

Definition 6.5.

Let x=[xε]~ρ. Then we set

  • (i)

    ν(x):=sup{b|xε|=O(ρεb)}{+},

  • (ii)

    |x|e:=e-ν(x)0,

  • (iii)

    dρ(x):=dρ-log|x|e~>0ρ.

It is worth noting that |-|e:~ρ0 induces an ultrametric on ~ρ 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 dρ(x) that goes to zero with x. It is in fact easy to show that

limx0xdρ(x)=1

in the sharp topology. The following definition is based on [1, Definition 2.2].

Definition 6.6.

Let T~ρ and let f:T~dρ be an arbitrary set-theoretical function. Let t0T be a sharply interior point of T. Then we say that f is differentiable at t0 if

m~dρ:limh0f(t+h)-f(t0)-mhdρ(h)=0.

In this case, using Landau little-oh notation, we can hence write

f(t+h)=f(t0)+mh+o(dρ(h))as h0.(6.3)

As in the classical case, (6.3) implies the uniqueness of m~dρ, so that we can define f(t0):=f˙(t0):=m, 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 (f1,,fn). We therefore have that (f1,,fn)(-) is differentiable at each ai<t<ai+1, and that (f1,,fn)(t)=fi(t), but clearly there is no guarantee that (f1,,fn)(-) is also differentiable at each point ai.

The notion of definite integral is naturally introduced in the following definition.

Definition 6.7.

Let (f1,,fn):i=1n[ai,ai+1]~dρ be a piecewise GSF. Then

a1an+1(f1,,fn)(t)dt:=i=1naiai+1fi(t)dt.

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

ν:=(f1,,fn),a1=a,an=bI(ν):=abF(t,ν(t),ν˙(t))dt~ρ.(6.4)

This leads to the following natural definition: we say that a piecewise GSF v is a piecewise GSF (global) minimizer if I(v)I(v~) for all v~𝒢ρ𝒞0. 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 UR~nρ, VR~dρ be sharply open sets. Let FGρC(U×V,R~dρ) and (x0,y0)U×V. If 2F(x0,y0) is invertible in L(R~dρ,R~dρ), then there exists a sharply open neighborhood U1×V1U×V of (x0,y0) such that

xU1,!yxV1:F(x,yx)=F(x0,y0).

Moreover, the function f(x):=yx for all xU1 is a GSF fGρC(U1,V1) and satisfies

Df(x)=-(2F(x,f(x)))-11F(x,f(x)).

Proof.

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 det[2F(-,-)] is a GSF such that |det[2F(x0,y0)]|~>0ρ. ∎

In the next theorem, the dependence of the entire theory on the initial infinitesimal net ρ=(ρε)0 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 AGρC([a,b],R~d×dρ), where a,bR~ρ, a<b, and let t0[a,b] and y0R~dρ. Assume that

|t0tA(s)ds|-Clogdρfor all t[a,b],(6.5)

where CR>0. Then there exists one and only one yGρC([a,b],R~dρ) such that

{y(t)=A(t)y(t)if t[a,b],y(t0)=y0.(6.6)

Moreover, this y is given by y(t)=exp(t0tA(s)ds)y0 for all t[a,b].

Proof.

We first note that

exp(t0tA(s)ds)=[exp(t0εtεAε(s)ds)],

where t=[tε], t0=[t0ε] and A(s)=[Aε(sε)]~d×dρ. This exponential matrix in ~d×dρ is a GSF because for all t[a,b], we have

exp(t0tA(s)ds)e-Clogdρdρ-C.

Therefore, all values of y(t)=exp(t0tA(s)ds)y0 are ρ-moderate. Analogously, one can prove that also y(k)(t) are moderate for all k and t[a,b]. 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 z𝒢ρ𝒞([a,b],~dρ) satisfies (6.6), and set h(t):=exp(-t0tA(s)ds) for all t[a,b]. Since h=-Ah, we have

(hz)=hz+hz=-Ahz+hAz=-Ahz+Ahz=0.

From the uniqueness of primitives of GSF and Theorem 2.13, we have that hz=h(t0)z(t0)=y0. Therefore, z=h-1y0. ∎

If α, β~ρ, we write α=O(β) to denote that there exists C>0 such that |α|C|β|. Therefore, assumption (6.5) can be written as t0tA(s)ds=O(logdρ). Note that this assumption is weaker, in general, than

(b-a)maxt[a,b]|A(t)|=O(logdρ).

The following result is the key regularity property that is needed to prove Jacobi’s theorem.

Lemma 6.10.

Let a,a,bR~ρ, with a<a<b, and let KGρC([a,b]×R~dρ×R~dρ,R~ρ). Let, in addition, ν=(η,β):[a,a][a,b]R~dρ be a piecewise GSF which satisfies the Euler–Lagrange equation

Ku(t,ν(t),ν˙(t))-ddtKu˙(t,ν(t),ν˙(t))=0for all t[a,a)(a,b].(6.7)

Finally, assume that det(Ku˙iu˙j(a,η(a),η˙(a))i,j=i,,d)R~ρ is invertible. Then

limtat<aν˙(t)=limtaa<tν˙(t)=η˙(a).

In particular, if β0|[a,b], then η˙(a)=0.

Proof.

Set Φ(t,l,v,q):=Ku˙(t,l,v)-q for all t[a,b] and all l, v, q~dρ. For simplicity, set

(t0,l0,v0,q0):=(a,η(a),η˙(a),Ku˙(a,η(a),η˙(a))).

Our assumption on the invertibility of Ku˙u˙(a,η(a),η˙(a))=vΦ(t0,l0,v0,q0) makes it possible to apply the implicit function Theorem 6.8 to conclude that there exists a neighborhood T×L×V×Q of (t0,l0,v0,q0) such that

(t,l,q)T×L×Q,!vV:Φ(t,l,v,q)=Φ(t0,l0,v0,q0).(6.8)

But we have Φ(t0,l0,v0,q0)=Ku˙(a,η(a),η˙(a))-q0=0. Moreover, the unique function ϕ defined by Φ(t,l,ϕ(t,l,q),q)=0 for all (t,l,q)T×L×Q is a GSF that belongs to 𝒢ρ𝒞(T×L×Q,V). Now, for all t[a,a)(a,b], we have

Φ(t,ν(t),ν˙(t),Ku˙(t,ν(t),ν˙(t)))=Ku˙(t,ν(t),ν˙(t))-Ku˙(t,ν(t),ν˙(t))=0.

Therefore, the uniqueness in (6.8) yields

ν˙(t)=ϕ(t,ν(t),Ku˙(t,ν(t),ν˙(t)))for all t[a,a)(a,b].

We now integrate the Euler–Lagrange equation (6.7) on [a,t], obtaining

Ku˙(t,ν(t),ν˙(t))=atKu(s,ν(s),ν˙(s))ds+Ku˙(a,η(a),η˙(a))for all t[a,a)(a,b].

This entails that we can write

ν˙(t)=ϕ(t,ν(t),atKu(s,ν(s),ν˙(s))ds+Ku˙(a,η(a),η˙(a)))for all t[a,a)(a,b].(6.9)

But the function t[a,a)(a,b]atKu(s,ν(s),ν˙(s))ds~dρ has equal limits on the left and on the right of a because on [a,a) and on (a,b] it is a GSF. In fact, for t<a, we have

|atKu(s,ν(s),ν˙(s))ds-aaKu(s,ν(s),ν˙(s))ds|maxt[a,a]|Ku(s,η(s),η˙(s))||t-a|,

and this goes to 0 as ta, t<a. Analogously, we can proceed for t>a using β. Therefore,

limtat<aatKu(s,ν(s),ν˙(s))ds=limtat>aatKu(s,ν(s),ν˙(s))ds.

Applying this equality in (6.9), we get limta,t<aν˙(t)=η˙(a)=limta,a<tν˙(t), as claimed. Finally, if β0|[a,b], then limta,a<tν˙(t)=0. ∎

In the following definition and below, we use the complete notation 𝒢ρ𝒞0(a,a) (see Definition 4.1).

Definition 6.11.

Let a,a,b~ρ, where a<a<b. We call a conjugate to a with respect to the variational problem (4.1) if there exists a non-identically vanishing Jacobi field η𝒢ρ𝒞0(a,a) along u|[a,a] such that η(a)=0=η(a), where ψ is given by (6.1).

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.

Lemma 6.12.

Let uGρC([a,b],R~dρ) and a(a,b). Let ηGρC0(a,a) be a Jacobi field along u|[a,a], with η(a)=0=η(a). Then

aaψ(t,η,η˙)dt=0.

Proof.

Since ψ is ~ρ-homogeneous of second order in (η,η˙), we have

2ψ(t,η,η˙)=ψη(t,η,η˙)η+ψη˙(t,η,η˙)η˙.

Thus, by integration by parts, we calculate

2aaψ(t,η,η˙)dt=aaηψη(t,η,η˙)+η˙ψη˙(t,η,η˙)dt=aaη(ψη(t,η,η˙)-ddtψη˙(t,η,η˙))dt=0,

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 a,bR~ρ, with a<b, FGρC([a,b]×R~dρ×R~dρ,R~ρ) and uGρC([a,b],R~ρ). Assume that the following hold:

  • (i)

    a(a,b) is conjugate to a.

  • (ii)

    detFu˙u˙(t,u(t),u˙(t))~ρ is invertible for all t[a,b].

  • (iii)

    For all t[a,a],

    atFu˙u˙-1(s,u(s),u˙(s))[ddsFuu˙(s,u(s),u˙(s))-Fuu(s,u(s),u˙(s))]ds=O(logdρ),atFu˙u˙-1(s,u(s),u˙(s))ddsFu˙u˙(s,u(s),u˙(s))ds=O(logdρ).

Then u cannot be a local minimizer of I. Therefore, for any rR~>0ρ, there exists vGρCbd(u(a),u(b)) and mN such that v-um<r but I(u)I(v).

Proof.

By contradiction, assume that u is a local minimizer, and let η𝒢ρ𝒞0(a,a) be a Jacobi field along u|[a,a] such that the conditions from Definition 6.11 hold for η. We want to prove that η0. Define ν:=(η,0|[a,b]), which is a piecewise GSF since η(a)=0. Since also η(a)=0, Lemma 6.12 and the homogeneity of ψ yield

Q(ν)=abψ(t,ν(t),ν˙(t))dt=aaψ(t,η(t),η˙(t))dt+abψ(t,0,0)dt=0.

Thus, Theorem 5.5 (necessary condition for u being a minimizer) gives Q(ν~)0=Q(ν) for all ν~𝒢ρ𝒞0(a,b). 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 ϕ𝒢ρ𝒞0(a,b) and all s~ρ, we have

Q(ν+sϕ)=abψ(t,ν+sϕ,ν˙+sϕ˙)dt=aaψ(t,η+sϕ,η˙+sϕ˙)dt+abψ(t,sϕ,sϕ˙)dt.(6.10)

This shows that s~ρQ(ν+sϕ)~ρ is a GSF, and hence s=0 is a minimum for this function. By Lemma 3.6 and (6.10), we get

δQ(ν,ϕ)=0=ddsQ(ν+sϕ)|0=aa(ψη(t,η,η˙)-ddtψη˙(t,η,η˙))ϕdt+ab(ϕψη(t,0,0)+ϕ˙ψη˙(t,0,0))dt=aa(ψη(t,η,η˙)-ddtψη˙(t,η,η˙))ϕdt.

By the fundamental Lemma 4.4, this implies that η satisfies the Euler–Lagrange equations for ψ in the interval [a,a). Therefore, ν satisfies the same equations in [a,a)(a,b]. Moreover, ψη˙η˙(a,η(a),η˙(a))=Fu˙u˙(a,u(a),u˙(a)) is invertible by assumption (ii). Thus, all the hypotheses of the regularity Lemma 6.10 hold, and we derive that η˙(a)=0.

For all t[a,b], we define

ξ(t):=-Fu˙u˙-1[ddtFuu˙(t,u,u˙)-Fuu(t,u,u˙)],ϑ(t):=-Fu˙u˙-1ddtFu˙u˙(t,u,u˙),

so that we can re-write the Jacobi equations (6.2) for η on [a,a] as a system of first order ODE:

{y˙:=(η˙z˙)=(01ξϑ)(ηz)=:Ayfor all t[a,a],y(a)=(η(a)η˙(a))=0.

By assumptions (iii), we obtain atA(t)=O(logdρ) for all t[a,a], and we can hence apply Theorem 6.9, obtaining y0, and thus η0. ∎

Note that if one of the quantities in (iii) depends even only polynomially on ε, then we are forced to take, e.g., ρε=ε1/ε 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 [3]. We first note that any X𝒢ρ𝒞(J×X,Y), where J~ρ, can also be considered as a family in GSF which smoothly depends on the parameter sJ. In this case, we hence say that (Xs)sJ is a generalized smooth family in 𝒢ρ𝒞(X,Y). In particular, we can reformulate in the language of GSF the classical definition of one-parameter group of generalized diffeomorphisms of X as follows:

  • (i)

    (Xs)s~ρ is a generalized smooth family in 𝒢ρ𝒞(X,X).

  • (ii)

    For all s~ρ, the map Xs:XX is invertible, and Xs-1𝒢ρ𝒞(X,X).

  • (iii)

    X0(x)=x for all xX.

  • (iv)

    XsXt=Xs+t for all s, t~ρ.

In our proofs, we will in fact only use properties (i) and (iii).

The proof of Noether’s theorem is classically anticipated by the following time-independent version, which the general case is subsequently reduced to.

Theorem 7.1.

Let KGρC(L×V,R~ρ), where L, VR~nρ are sharply open sets. Let wGρC((a,b),L) be a solution of the Euler–Lagrange equation corresponding to K, i.e., for all t(a,b),

w˙(t)V,Ku(w(t),w˙(t))=ddtKu˙(w(t),w˙(t)).(7.1)

Suppose that 0 is a sharply interior point of JR~ρ and (Xs)sJ is a generalized smooth family in GρC(L,L) such that for all t(a,b),

  • (i)

    tXs(w(t))V,

  • (ii)

    X0(w(t))=w(t),

  • (iii)

    K is invariant under (Xs)sJ along w , i.e.,

    K(w(t),w˙(t))=K(Xs(w(t)),tXs(w(t)))for all sJ.(7.2)

Then, the quantity

Ku˙j(w(t),w˙(t))s|s=0Xsj(w(t))~ρ

is constant in t(a,b).

Proof.

We first note that both sides of (7.2) are in 𝒢ρ𝒞((a,b),~ρ). Let τ(a,b) be arbitrary but fixed. Since s=0J is a sharply interior point, we can consider dds|s=0. Then, using (7.2) and (ii), we obtain

0=s|s=0K(Xs(w),tXs(w))=aτKu(w,w˙)s|s=0Xs(w)+Ku˙(w,w˙)ts|s=0Xs(w)dt.

Since the Euler–Lagrange equations (7.1) for K are given by Ku(w,w˙)=ddtKu˙(w,w˙), we have

0=ddt(Ku˙(w,w˙))s|s=0Xs(w)+Ku˙(w,w˙)ts|s=0Xs(w)=ddt(Ku˙(w,w˙)s|s=0Xs(w)),

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 a,bR~dρ, with a<b, and FGρC([a,b]×R~dρ×R~dρ,R~ρ). Let, in addition, uGρC([a,b],R~dρ) be a solution of the Euler–Lagrange equation (4.4) corresponding to F. Suppose that 0 is a sharply interior point of JR~ρ and (Xs)sJ is a generalized smooth family in GρC((a,b)×R~dρ,(a,b)×R~dρ). We denote by Ts(t,l):=Xs1(t,l)(a,b) and Ls(t,l):=Xs2(t,l)R~dρ, for all (t,l)(a,b)×R~dρ, the two projections of Xs on (a,b) and R~dρ, respectively. We assume that for all t(a,b),

  • (i)

    tTs(t,u(t))~ρ is invertible,

  • (ii)

    T0(t,u(t))=t and L0(t,u(t))=u(t),

  • (iii)

    F(t,u(t),u˙(t))=F[Ts(t,u),Ls(t,u),tLs(t,u)tTs(t,u)]tTs(t,u) for all sJ.

Then, the quantity

Fu˙j(t,u(t),u˙(t))s|s=0Lsj(t,u(t))+[F(t,u(t),u˙(t))-Fu˙k(t,u(t),u˙(t))u˙k(t)]s|s=0Ts(t,u(t))(7.3)

is constant in t[a,b].

Proof.

Since (7.3) is a GSF in t[a,b], by sharp continuity it suffices to prove the claim for all t(a,b). Set L:=(a,b)×~dρ, V:=~*ρ×~dρ (we recall that ~*ρ denotes the set of all invertible generalized numbers in ~ρ). Define K𝒢ρ𝒞(L×V,~ρ) by

K(t,l;p,v):=F(t,l,vp)pfor all (t,l)L and all (p,v)V,(7.4)

and w𝒢ρ𝒞((a,b),L) by w(t):=(t,u(t)) for all t(a,b). We note that L,V~d+1ρ are sharply open subsets and that w˙(t)=(1,u˙(t))V. The notations for partial derivatives used in the present work result from the symbolic writing K(u1,,ud+1;u˙1,,u˙d+1), so that the variables used in (7.4) yield

Kuj(t,l;p,v)={Kt(t,l;p,v)=Ft(t,l,vp)pif j=1,Klj(t,l;p,v)=Fuj-1(t,l,vp)pif j=2,,d+1,(7.5)

and

Ku˙j(t,l;p,v)={Kp(t,l;p,v)=F(t,l,vp)-Fu˙k(t,l,vp)vkpif j=1,Kvj(t,l;p,v)=Fu˙j-1(t,l,vp)if j=2,,d+1.(7.6)

From these, for all t(a,b) and all j=2,,d+1, it follows that

Ku1(w,w˙)-ddtKu˙1(w,w˙)=[ddtFu˙k(t,u,u˙)-Fuk(t,u,u˙)]u˙k,Kuj(w,w˙)-ddtKu˙j(w,w˙)=Fuj-1(t,u,u˙)-ddtFu˙j-1(t,u,u˙).

Therefore, since u satisfies the Euler–Lagrange equations for F, this entails that w is a solution of the analogous equations for K in (a,b). Now, (i) gives

tXs(w(t))=(tTs(t,u(t)),tLs(t,u(t)))~*ρ×~dρ=V.

Moreover, (ii) gives X0(w(t))=(T0(t,u(t)),L0(t,u(t)))=w(t). Finally,

K(w,w˙)=F(t,u,u˙),K(Xs(w),tXs(w))=F[Ts(t,u),Ls(t,u),tLs(t,u)tTs(t,u)]tTs(t,u).

We can hence apply Theorem 7.1, and from (7.5) and (7.6), we get that

Ku˙j(w,w˙)s|0Xsj(w)=Fu˙j(t,u(t),u˙(t))s|s=0Lsj(t,u(t))+[F(t,u(t),u˙(t))-Fu˙k(t,u(t),u˙(t))u˙k(t)]s|s=0Ts(t,u(t))

is constant in t(a,b). ∎

8 Application to 𝒞1,1 Riemannian metric

In the following, we apply what we did so far to the problem of length minimizers in (d,g), where g𝒞1,1 is a Riemannian metric. Furthermore, we assume that (d,g) is geodesically complete. Note that the seeming restriction of considering only d 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 𝒞k manifolds with 1k<+. Therefore, there is no need to consider non-smooth charts.

In this section, we fix an embedding (ιΩb)Ω, where b~ρ satisfies bdρ-a for some a>0, and where Ωd is an arbitrary open set, see Theorem 2.18. Actually, the embedding also depends on the dimension d>0, 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 (gijε) such that by setting g~:=ι(g)=:[gijε(-)]𝒢ρ𝒞(~dρ×~dρ,~ρ), we have that gijε is a Riemannian metric for all ε. By Theorem 2.18(iii), it follows that gijεgij in 𝒞0 norm. Let Γijε be the Christoffel symbols of gε, and set Γ~ij:=[Γijε(-)]𝒢ρ𝒞(~dρ,~dρ). A curve γ𝒢ρ𝒞(J,~dρ), with J being a sharply open subset of ~ρ, is said to be a geodesic of (~dρ,g~) if

γ¨(t)+Γ~ij(γ(t))γ˙i(t)γ˙j(t)=0for all tJ.(8.1)

Definition 8.1.

We say that (~dρ,g~) is geodesically complete if every solution of the geodesic equation belongs to 𝒢ρ𝒞(~ρ,~dρ), i.e., if for all p~dρ and all v~dρ, there exists a geodesic γ𝒢ρ𝒞(~ρ,~dρ) of (~dρ,g~) such that γ(0)=p and γ˙(0)=v.

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 γ𝒢ρ𝒞(~ρ,~dρ) induces a Colombeau generalized function γ|c()𝒢s()d. Therefore, the space (c(d),g~|c(d)×c(d)) is geodesically complete in the sense of [39]. We recall that c(Ω) 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.

Remark 8.2.

We set

-=(-)1/2:x=[xε]~>0ρ[xε]~>0ρ.

Lemma 2.3 readily implies that -𝒢ρ𝒞(~>0ρ,~>0ρ). Therefore, the square root is defined on every strictly positive infinitesimal, but it cannot be extended to ~0ρ.

Definition 8.3.

  • (i)

    Let p~, q~~dρ, then

    𝒢ρ𝒞>0(p~,q~):={λ𝒢ρ𝒞([0,1],~dρ)λ(0)=p~,λ(1)=q~,|λ˙(t)|>0t[0,1]}.

    Moreover, for λ𝒢ρ𝒞>0(p~,q~), we set

    Lg~(λ):=01(g~ij(α(t))α˙i(t)α˙j(t))1/2dt~ρ.

  • (ii)

    Let x=[xε]~nρ. Then we set st(x):=limε0xεd, if this limit exists. Note that xst(x) in this case.

Note that (8.1) are the usual geodesic equations for the generalized metric g~, whose derivation is completely analogous to that in the smooth case. Thus, they are the Euler–Lagrange equations of Lg~.

We are interested only in global minimizers of the functional Lg~, i.e., curves λ0X(p~,q~) such that Lg~(λ0)Lg~(λ) for all λ𝒢ρ𝒞>0(p~,q~).

Lemma 8.4.

Let p,qRd and p~, q~R~dρ be such that st(p~)=p and st(q~)=q. Let λ=[λε(-)]GρC>0(p~,q~) be such that there exists

λ¯𝒞>01(p,q):={w𝒞1([0,1],d)w(0)=p,w(1)=q,|w˙(t)|>0t[0,1]}

such that λελ¯ in C1 as ε0. Then st(Lg~(λ))=Lg(λ¯).

Proof.

We calculate

|01(gijε(λε)λ˙εiλ˙εj)1/2-(gij(λ¯)λ¯˙iλ¯˙j)1/2dt|=|01gijε(λε)λ˙εiλ˙εj-gij(λ¯)λ¯˙iλ¯˙j(gijε(λε)λ˙εiλ˙εj)1/2+(gij(λ¯)λ¯˙iλ¯˙j)1/2dt|.

By assumption, (gijε(λε)λ˙εiλ˙εj)1/2(gij(λ¯)λ¯˙iλ¯˙j)1/2, so that there exists C>0 such that

|01(gijε(λε)λ˙εiλ˙εj)1/2-(gij(λ¯)λ¯˙iλ¯˙j)1/2dt|C01|(gijε(λε)-gij(λε)+gij(λε)-gij(λ¯))λ˙εiλ˙εj+gij(λ¯)(λ˙εiλ˙εj-λ¯˙iλ¯˙j)|dt.

We hence obtain the claim by the triangle inequality and the convergence of λε, λ˙ε and gijε to λ¯, λ¯˙ and gij, respectively. ∎

Now, we consider p,qd, with pq. Let

u{u𝒞2,1([0,1],d)u(0)=p,u(1)=q}

be a solution of the geodesic equation

{u¨=-Γij(u)u˙iu˙j,p=u(0),q=u(1).(8.2)

Let c0:=u˙(0). Obviously, u is also the unique solution of

{u¨=-Γij(u)u˙iu˙j,p=u(0),c0=u˙(0).

Using these initial conditions, for each fixed ε, we can solve the following problem:

{y¨=-Γijε(y)y˙iy˙j,p=y(0),c0=y˙(0)(8.3)

for a unique yε𝒞([-dε,dε],d) and some dε>0.

Lemma 8.5.

Let u and yε be as above. Then the following hold:

  • (i)

    For ε sufficiently small, the solution yε can be extended to a solution yε𝒞([0,1],d) of ( 8.3 ) such that yε(1)=q.

  • (ii)

    yεu in 𝒞2.

  • (iii)

    The net (yε) defines a GSF, i.e., y:=[yε(-)]𝒢ρ𝒞>0(p,q).

Proof.

(i)(ii) For all i, j, we have that ΓijεΓij locally uniformly. Thus, we obtain these claims by (8.2) and by continuous dependence on parameters in ODE, see, e.g., [26, Lemma 2.3].

(iii) If y:=[yε(-)]𝒢ρ𝒞([0,1],~dρ), then we have to show that for all ε, all the derivatives of yε are moderate. This is obviously true for yε,y˙ε and y¨ε. The claim follows now from the fact that

dn+2dtn+2yε=-dndtn(Γijεk(y)y˙εiy˙εj),

so that there exists a polynomial P such that

dndtn(Γijεk(y)y˙εiy˙εj)=P(yε,ddtyε,,dn+1dtn+1yε,Γijεk,DΓijεk,,DnΓijεk).

If |y˙(t)|>0 for all t[0,1], then, by (ii), we have that yεu in 𝒞2. Furthermore, gεg in 𝒞1, by assumption, and we know that g(u˙,u˙)=c>0 for some c>0, since u is a g-geodesic (cf. [22, Lemma 1.4.5]). Therefore, we obtain that gε(y˙ε,y˙ε)>c2>0 for ε>0 small enough. ∎

Finally, the standard part of the generalized length of y is the length of u.

Theorem 8.6.

Let u and yε be as above. We conclude (using Lemma 8.4) that st(Lg~(y))=Lg(u).

Proposition 8.7.

Let y=[yε(-)] be as above. In addition, assume that each yε is Lgε-minimizing. Then Lg~(y) is minimal.

Proof.

Let λ=[λε(-)]𝒢ρ𝒞>0(p,q). We have that Lg~(λ)=[Lgε(λε)] and that Lg~(y)=[Lgε(yε)]. By assumption, for all ε, we have Lgε(λε)Lgε(yε). Therefore, Lg~(λ)Lg~(y), as claimed. ∎

Corollary 8.8.

Let λGρC>0(p,q) be a minimizer of Lg~ and assume that for ε small, yε is Lgε-minimizing. Then Lg~(y)=Lg~(λ).

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.

Theorem 8.9.

Let p,qRd and let γGρC>0(p,q) such that Lg~(γ) is minimal. Assume that st(Lg~(γ)) exists and that there exists wC>01(p,q) such that Lg(w)=st(Lg~(γ)). Then w is g-minimizing and g-geodesic.

Proof.

Assume to the contrary that there exists a curve σC2 connecting p and q (without loss of generality, σ is a g-geodesic) such that Lg(σ)<Lg(w). Now we construct (as done above) gε and σε, and set σ~:=[σε]. Then

st(Lg~(σ~))=Lg(σ)<Lg(w)=st(Lg~(γ)).

But, by assumption, we have that Lg~(γ)Lg~(σ~), which implies

st(Lg~(γ))st(Lg~(σ~))<st(Lg~(γ)).

This is a contradiction. ∎

9 Conclusions

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 ~ρ-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 (f1,,fn)(-) and hyperfinite Riemann-like sums, i.e.,

i=1Nf(xi)(xi-xi-1):=[i=1Nεfε(xi,ε)(xi,ε-xi-1,ε)]~dρ,

extended to N~:={[int(xε)][xε]~ρ}, where int(-) 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.

Acknowledgements

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

Received: 2017-06-26

Accepted: 2017-07-29

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.

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

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