Variational approximation of functionals defined on 1-dimensional connected sets in \mathbb{R}^{n}

Variational approximation of functionals defined on 1-dimensional connected sets in

M. Bonafini, G. Orlandi, É. Oudet Dipartimento di Matematica, Università di Trento, Italy, e-mail: mauro.bonafini@unitn.itDipartimento di Informatica, Università di Verona, Italy, e-mail: giandomenico.orlandi@univr.itLaboratoire Jean Kuntzmann, Université de Grenoble Alpes, France, e-mail: edouard.oudet@imag.fr
July 24, 2019
Abstract

In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert–Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in . Following the the analysis for the planar case presented in [4], we provide a variational approximation through Ginzburg–Landau type energies proving a -convergence result for .

1 Introduction

Given distinct points in and , the (single sink) Gilbert–Steiner problem, or -irrigation problem [3, 17] requires to find an optimal network along which to flow unit masses located at the sources to the target point , where the cost of moving a mass along a path of length scales like . The transportation network can be viewed as , with a path connecting to (i.e., the trajectory of the unit mass located at ), and thus the problem translates into

where represents the mass density along the network. In particular, reduces to the optimization of the total length of the graph and corresponds to the classical Euclidean Steiner Tree Problem (STP), i.e., finding the shortest connected graph which contains the terminal points . For any a solution to is known to exist and any optimal network turns out to be a tree [3].

As pointed out in the companion paper [4], the Gilbert–Steiner problem represents the basic example of problems defined on -dimensional connected sets, and it has recently received a renewed attention in the Calculus of Variations community. In the last years available results focused on variational approximations of the problem mainly in the planar case [8, 9, 15, 7], while higher dimensional approximations have been recently proposed in [10, 6].

In this paper we extend to the higher dimensional context the two dimensional analysis developed in [4] and we propose a variational approximation for in the Euclidean space , . We prove a result in the spirit of -convergence (see Theorem 4.6 and Proposition 4.5) by considering integral functionals of Ginzburg–Landau type [1, 2] (see also [16]). This approach builds upon the interpretation of as a mass minimization problem in a cobordism class of integral currents with multiplicities in a suitable normed group (as studied in [13, 12]). Thus, the relevant energy turns out to be a convex positively -homogeneous functional (a norm), for which one can use calibration type arguments to prove minimality of certain given configurations [12, 14]. The proposed method is quite flexible and can be adapted to a variety of situations, including manifold type ambients where a suitable formulation in vector bundles can be used (this will be treated in a forthcoming work).

Eventually, we remark that another way to approach the problem is to investigate possible convex relaxations of the limiting functional, as already pointed out in [4] and then further extended in [5], so as to include more general irrigation-type problems (with multiple sources/sinks) and even problems for -d structures on manifolds.

The plan of the paper is as follows. In Section 2 we briefly review the main concepts needed in the subsequent sections and in Section 3 we recall the variational setting for relying on the concept of -mass. We then provide in Section 4 a variational approximation of the problem in any dimension by means of Ginzburg–Landau type energies.

2 Preliminaries and notations

In this section we fix the notation used in the rest of the paper and some basic facts. We will follow closely [1, 2], to which we refer for a more detailed treatment.

For any , we denote by the standard basis of , is the open ball in with centre the origin and radius , is the unit sphere in , and

where stands for the Lebesgue measure of the given set. For we denote by the -dimensional Hausdorff measure. Furthermore, we assume we are given distinct points in , for and , and we denote . We also assume, without loss of generality, that .

Ginzburg–Landau functionals. We consider a continuous potential which vanishes only on and is strictly positive elsewhere, and we require

Given , open and , we set

(2.1)

where is the Euclidean norm of the matrix .

Currents. Given , let be the space of -covectors on and the space of -vectors. The canonical basis of will be denoted as . For a -covector we define its comass as

For , a -form on is a map from into the space of -covectors and a -dimensional current is a distribution valued into the space of -vectors. We denote as the space of all smooth -forms with compact support and as the space of all -currents. In particular, the space can be identified with the dual of the space and equipped with the corresponding weak topology. Furthermore, for and an open subset , we define the mass of in as

and we denote the mass of as . The boundary of a -current is the -current characterized as for every , where is the exterior differential of the form . Let be a current with locally finite mass, then there exist a positive finite measure on and a Borel measurable map with -a.e., such that

(2.2)

We denote the variation of the measure , so that, given , one has . A -current is said to be normal whenever both and have finite mass, and we denote as such space.

Given a -rectifiable set oriented by and a real-valued function , we define the current as

and we refer to as the multiplicity of the current. A -current is called rectifiable if it can be represented as for a -rectifiable set and an integer valued multiplicity . If both and are rectifiable, we say is an integral current and denote as the corresponding group. A polyhedral current in is a finite sum of -dimensional oriented simplexes endowed with some constant integer multiplicities , and we generally assume that is either empty of consists of a common face of and . As it is done in [2], we introduce the following flat norm of a current :

(2.3)

and the infimum is taken to be if is not a boundary.

Jacobians of Sobolev maps and boundaries. Given open and , following [11], we define the -form

and we set the Jacobian of to be

in the sense of distributions. This means that for any

where is the formal adjoint of . By means of the operator we can identify such a form with a -current . In our specific context, the operator can be defined, at the level of vectors/covectors, as follows: given a -covector , the vector is defined by the identity

Jacobians turn out to be the main tool in our analysis due to their relation with boundaries. In order to highlight such a relation we need some additional notation: given any segment in and given , let us define the set

If we identify the line spanned by with , we can write each point as , so that

We can now recall the main result of [1] (rewritten in our specific context).

Theorem 2.1 (Theorem 5.10, [1]).

Let be the (polyhedral) boundary of a polyhedral current of dimension in , and let denote the union of the faces of of dimension . Then there exists such that , with locally Lipschitz in the complement of and constant outside a bounded neighbourhood of , and belongs to for every and satisfies . Moreover, there exist small enough such that, for each -simplex , one has

3 Gilbert–Steiner problems and currents

In this section we briefly review (this time in terms of currents) the approach used in [4, 5], which is to say the framework introduced by Marchese and Massaccesi in [13, 12], and describe Gilbert–Steiner problems in terms of a minimum mass problem for a given family of rectifiable -currents in .

The set of possible minimizers for can be reduced to the set of (connected) acyclic graphs that are described as the superposition of curves.

Definition 3.1.

We define to be the set of acyclic graphs of the form

where each is a simple rectifiable curve connecting to and oriented by an -measurable unit vector field , with for -a.e. , and we denote by the corresponding global orientation, i.e., for -a.e. .

It can be shown (see, e.g., [13, Lemma 2.1]), that is equivalent to

(3.1)

Given now , we identify each component with the corresponding -current and we consider .

Definition 3.2.

We define to be the set such that each component is of the form for some , and write to highlight the supporting graph.

Given and a function , with , one sets

and for a norm on , we define the -mass measure of as

(3.2)

for open, where is the dual norm to w.r.t. the scalar product on , and we let the -mass norm of to be

(3.3)

As described in [13, 4, 5], the problem defined in (3.1) is equivalent to

(3.4)

where is the norm on for , and the norm for . This means that any minimizer of (3.4) is of the form for a minimizer of (3.1), and given any minimizer of (3.1) then the corresponding minimizes (3.4).

Remark 3.3.

In [13, 12] problem (3.4) is introduced in the context of a mass minimization problem for integral currents with coefficients in a suitable normed group. In that case, the -mass defined above is simply the mass of the current deriving from the particular choice of the norm for the coefficients group.

Calibrations. One of the main advantages of formulation (3.4) is the possibility to introduce calibration-type arguments for proving minimality of a given candidate. For a fixed , a (generalized) calibration associated to is a linear and bounded functional such that

  • ,

  • for any ,

  • for any .

The existence of a calibration is a sufficient condition to prove minimality in (3.4). Indeed, let be a competitor in (3.4) and be a calibration for . Consider any , with . By assumption, for each , one has , so that there exists a -current such that . Hence,

which proves the minimality of in (3.4) (and, more generally, also minimality among normal currents). We also remark that once a calibration exists it must calibrate all minimizers.

A calibration-type argument. The general idea behind calibrations can be used to tackle minimality in suitable subclasses of currents, as long as the previous derivation can be proved to still hold true. Consider, as displayed in figure 1, the Steiner tree problem for four points in with , , and . Let us identify the two points and , and fix as norm the norm on the coefficients space .

Figure 1: We consider the Steiner tree problem for vertices obtained as “opposite” couples of vertices of a rectangular cuboid.

Given a list of points , we write as the polyhedral current connecting them and oriented from to . Our aim is to prove that

is a minimizer of the -mass among all currents , where is the family of currents satisfying the given boundary conditions , and such that there exist a positive finite measure on , a unit vector field and a function such that . Let us formally identify any such object as (loosely speaking, we consider only the family of normal rank one currents with a prescribed superposition pattern for different flows). It can be easily seen that and for any we have . For proving minimality of for the -mass among all competitors in we can use a calibration argument: let us consider defined as

where are fixed to be

One can show by direct computations that , so that given any other and such that , we have , for which

because for -a.e. , and

Hence, for any . Up to permutations, the class represents every possible acyclic graph with additional Steiner points and thus the support of is an optimal Steiner tree within that family of graphs. Remark that any minimal configuration cannot have or Steiner points because these configurations violate the angle condition, so that we can conclude that the support of is indeed an optimal Steiner tree. This extends for the first time to an higher dimensional context calibration-type arguments which up to now have been extensively used almost exclusively in the planar case, e.g. in [13, 12].

In the companion paper [4], we investigate a variational approximation of (3.4) in the two dimensional case, relying on a further reformulation of the problem within a suitable family of functions and then providing a variational approximation based on Modica–Mortola type energies. Here, instead, we work in dimension three and higher and address (3.4) directly by means of Ginzburg–Landau type energies.

4 Variational approximation of -masses

In this section we state and prove our main results, namely Proposition 4.5 and Theorem 4.6, concerning the approximation of minimizers of -masses functionals through Jacobians of minimizers of Ginzburg–Landau type functionals, much in the spirit of [2].

4.1 Ginzburg–Landau functionals with prescribed boundary data

In this section, following closely [2], we consider Ginzburg–Landau functionals for functions having a prescribed trace on the boundary of a given open Lipschitz domain.

Domain and boundary datum. Fix two points , with , and let be a simple acyclic polyhedral curve joining and , and oriented from to . Let be the segments composing and, for small enough define

(4.1)

Consider the boundary datum defined as

(4.2)

By construction one has

In this context, for only two points, the -mass reduces (up to a constant) to the usual mass, and thus we can directly rely on Corollary 1.2 of [2], which yields the following.

Theorem 4.1.

For small enough, consider the Lipschitz domain defined in (4.1) and let be the boundary datum defined in (4.2).

  • Consider a (countable) sequence with trace on such that . Then, up to subsequences, there exists a rectifiable -current supported in , with , such that the Jacobians converge in the flat norm to and

    (4.3)
  • Given a rectifiable -current supported in such that , for every we can find such that on , and

In particular, given a sequence of minimizers of with trace on , then and, possibly passing to a subsequence, the Jacobians converge in the flat norm to , where minimizes the mass among all rectifiable -currents supported on with boundary .

Point of the previous theorem corresponds to the derivation of Section 3.1 in [4], where we consider Modica–Mortola functionals for maps with prescribed jump, and here the prescribed jump is somehow replaced by the prescribed boundary datum “around” the drift . As it is done in [4], the idea is now to extend the previous (single-component) result to problems involving -masses for .

4.2 The approximating functionals

We now consider Ginzburg–Landau approximations for -masses whenever we are given points. Fix then a norm on , and consider the -mass defined in (3.3).

Construction of the domain. Fix a family of simple polyhedral curves each one connecting to and denote by the associated -current (oriented from to ). Suppose, without loss of generality, that for any , i.e., any two curves do not intersect each other. Every can then be viewed as the concatenation of (oriented) segments , for each of which we consider the neighbourhood

for . Define now and observe that, by finiteness, we can fix sufficiently small such that for any . The domain we are going to work with is

(4.4)

Boundary datum and approximating functionals. Following the same idea used in the previous section, fix functions such that

By construction “winds around” and is constant on the rest of the given boundary. As such, one sees that . As our functional space we consider

(4.5)

and for and , we define the approximating functionals

(4.6)

or equivalently, thanks to (3.2),

(4.7)

Lower-bound inequality Results on “compactness” and lower-bound inequality presented in the previous section extends to as follows.

Proposition 4.2.

Consider a (countable) sequence such that . Then, up to subsequences, there exists a family of rectifiable -currents supported in , with , such that the Jacobians converge in the flat norm to and

(4.8)
Proof.

For each , by definition of we have

and the first part of the statement follows applying Proposition 4.1 componentwise. Fix now with for any and for all . Then, thanks to (4.3), we have

which yields (4.8) taking the supremum over .

Upper-bound inequality and behaviour of minimizers. We now state and prove a version of an upper-bound inequality for the functionals which is tailored to investigate the behaviour of Jacobians of minimizers of .

Proposition 4.3 (Upper-bound inequality).

Let , with an acyclic graph supported in . Then there exists a sequence such that , and

(4.9)
Proof.

Step 1. We assume that is an acyclic polyhedral graph fully contained in , which is to say , and let be its global orientation. Such a graph can then be decomposed into a family of oriented segments , with orientation given by . For each segment consider the set , for parameters and , and choose small enough so that sets are pairwise disjoint. Define as the union of the covering , and let . Eventually, define vectors as if and otherwise. Collect these vectors in a function defined as for .

For the construction of the approximating sequence we relay on the following fact, which is a direct consequence of Theorem 2.1: for each there exists and a finite set of points such that:

  • , which is to say satisfies the given boundary conditions, and furthermore ;

  • is locally Lipschitz in and

  • within the set every function behaves like

In particular, we observe that for any , if and , then on by (iii). Thus, we can define a “global” function such that for any and, consequently,