We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where is a non-negative convex function vanishing only at . We show that this property is always satisfied in dimension , while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when ) in dimension . The validity of the quadratic rigidity, which we prove in dimension , implies the existence of the trace of a divergence-measure vector field ξ on an -rectifiable set S, as soon as its weak normal trace is maximal on S. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.
The structure and the properties of vector fields, whose distributional divergence either vanishes or is represented by a locally finite measure, are of great interest in Mathematics and in Physics. Such vector fields arise, for instance, in fluid mechanics, in electromagnetism, and in conservation laws. We shall not give a detailed account, however the interested reader is referred to the monographs [20, 21], as well as to the papers [6, 18, 23, 24, 25], and to the references found therein.
In this paper we shall consider a rigidity property à la Liouville for divergence-free vector fields in defined hereafter.
Let be a convex function such that if and only if . We say that the φ-rigidity property holds in if, for any vector field such that
on in distributional sense,
almost everywhere on ,
one has that almost everywhere on .
We state here two results that establish this rigidity property.
Theorem 2 (Linear rigidity).
Let . Then the φ-rigidity property holds in when for some constant .
Theorem 3 (φ-rigidity in the plane).
Let . Then the φ-rigidity holds in for any choice of φ as in Definition 1.
Ideally, one would like to prove Theorem 3 in any dimension. Yet, proving φ-rigidity in dimension when φ is not linear is a rather delicate issue and it would require some further hypotheses. Indeed, we have found counterexamples for the choice in any dimension , as proved in Theorem 2. At the current stage it is unclear to us if Theorem 3 may hold in dimension . Notice that the counterexamples found in dimension are cylindrically symmetric; however, we prove in Proposition 3 that no such vector field can be a counterexample for .
The specific choice is quite interesting as it is closely related with a trace property of certain divergence-measure vector fields. More precisely, it allows us to deduce the existence of the trace of a divergence-measure vector field ξ on an oriented, -rectifiable set , under a maximality assumption of the weak normal trace of ξ at S. In general, see Section 2.3, given a divergence-measure vector field ξ defined on a bounded open set , one is able to define its normal trace only in a distributional sense. More specifically, assuming that Ω is weakly-regular (that is, the perimeter of Ω is finite and coincides with the -dimensional Hausdorff measure of , see Definition 5) one can show [34, Section 3] that there exists a function such that the following Gauss–Green formula holds for all :
where we have denoted by the -dimensional Hausdorff measure and by the measure-theoretic outer normal to . Such a function is called weak normal trace of ξ on . For the sake of completeness we recall that a first, fundamental weak version of the Gauss–Green formula is the classical result by De Giorgi [16, 17] and Federer , which states that (1.1) holds true in the case , and Ω with finite perimeter. A further extension due to Vol’pert [36, 37] holds when ξ is weakly differentiable or BV. In the seminal works of Anzellotti [4, 3], the concepts of weak normal trace and of pairing between vector fields and (gradients of) functions are introduced for the first time. Since then, the class of divergence-measure, bounded vector fields has been widely studied in view of applications to hyperbolic systems of conservation laws [7, 9, 10], to continuum and fluid mechanics , and to minimal surfaces [28, 34] among many others. In particular, the weak normal trace has been studied in different directions, see for instance [1, 11, 12, 13, 15, 14].
Despite some explicit characterizations of the weak normal trace are available (see the discussion in Section 2.3), a crucial issue coming with this distributional notion is that it is not possible to recover the pointwise value of such a trace by a standard, measure-theoretic limit, see Example 7. However, assuming that and that the weak normal trace attains the maximal value 1 at some point , it would seem quite natural to expect that ξ cannot oscillate too much around x, to ensure a maximal outflow at x. Thus, one is led to conjecture the existence of the classical trace, i.e. the validity of the formula
Indeed, this is exactly what we are able to prove, limitedly to the case , by employing Theorem 3 with the specific choice .
Let and an oriented -rectifiable with locally bounded -measure. Then, for -a.e. such that one has
In the above theorem the approximate limit is “one-sided” according to Definition 2. Notice that the same conclusion of the theorem applies when the weak normal trace attains a local maximum for the modulus of the vector field, that is, when there exists an open set U containing such that for almost every . Of course, by a scaling argument one can always restrict to vector fields ξ with .
The statement of Theorem 4 holds for -a.e. . More precisely, one needs to be such that: the normal of S at is defined; it is a Lebesgue point for the weak normal trace; and does not concentrate around it. Asking to satisfy these hypotheses, makes us indeed discard an -negligible set of S.
As the proof heavily relies on Theorem 3, we are able to show it only in dimension 2. Despite there exist counterexamples to the quadratic rigidity in when , we cannot exclude that Theorem 4 might be true in any dimension. Were it false, one should be able to construct a vector field with maximal normal trace at some -submanifold S, whose blow-ups at most points of S are not unique. Indeed, the existence of the classical, one-sided trace of ξ at is equivalent to the uniqueness of the blow-up of ξ at (see Proposition 2). This essentially corresponds to a pointwise almost-everywhere convergence property. However, a maximality condition on the value of the weak normal trace (viewed as a weak-limit of measures induced by classical traces on approximating smooth surfaces) might enforce no more than a -type convergence, in analogy with what happens to a sequence of negative functions that weakly converge to zero. The fact that, in turn, -convergence does not imply almost everywhere convergence (unless one extracts suitable subsequences) explains why proving Theorem 4 is so delicate.
Finally, we mention that Theorem 4 allows us to strengthen the main result of our previous paper . In there we considered solutions to the prescribed mean curvature equation in a set Ω, in the so-called extremal case. The extremality condition for the prescribed mean curvature equation
is a critical situation for the existence of solutions, occurring when the following, necessary condition for existence
becomes an equality precisely at . When is smooth, Giusti proved in his celebrated paper  that the above necessary condition is also sufficient for existence, and moreover that the extremal case can be characterized by other properties, among which the uniqueness of the solution up to vertical translations and the vertical contact of its graph with the boundary of the domain. In the physically meaningful case, i.e. , the extremal case for the prescribed mean curvature equation corresponds to the capillarity phenomenon for perfectly wetting fluids within a cylindrical container put in a zero gravity environment.
In our previous work  we extended Giusti’s result to the wider class of weakly-regular domains, obtaining an analogous characterization that involves the weak normal trace of the vector field
on . In virtue of the result proved here, we obtain that the boundary behavior of the capillary solution in the cylinder actually improves to a vertical contact realized in the classical trace sense, rather than just in the sense of the weak normal trace. In other words, the normalized gradient Tu is shown to admit a classical trace (equal to the outer normal ) at -almost-every point of . We remark that, at present, no other technique, like the one based on regularity for almost-minimizers of the perimeter, seems to be applicable in the case of a generic weakly-regular domain. The reason is that the boundary of the cylinder is not smooth enough to let the approaching boundary of the subgraph of the solution u be uniformly regularized via the standard excess-decay mechanism for almost-minimizers of the perimeter.
Briefly, the paper is organized as follows. In Section 2 we lay the notation, recall some basic facts from Geometric Measure Theory and weak normal traces. In Section 3 we give the proofs of Theorem 2 and of Theorem 3. Section 4 presents the construction of a counterexample to the -rigidity for . Finally, Section 5 is devoted to the weak normal trace and to the proof of Theorem 4.
2 Preliminary notions and facts
We first introduce some basic notations. Given , denotes the Euclidean n-dimensional space, the upper half-space , and the boundary of . For any and , denotes the Euclidean open ball of center x and radius r. Given a unit vector we set
Given a Borel set , we denote by its characteristic function, by its n-dimensional Lebesgue measure, and by its Hausdorff d-dimensional measure. We set
where , , and . Given an open set, we write whenever the topological closure of E is a compact subset of Ω. Whenever a measurable function, or vector field, f is defined on Ω, we denote by its -norm on Ω. We denote by the space of bounded vector fields defined in Ω whose divergence is a Radon measure. For brevity we set . It is convenient to consider the restriction to of the weak- topology of : given a sequence , we say that converges to in the weak- topology of , and write in -, if for every one has
2.2 Basic definitions in Geometric Measure Theory
We now recall some basic definitions and facts from Geometric Measure Theory and, in particular, from the theory of sets of locally finite perimeter.
Definition 1 (Points of density α).
Let E be a Borel set in , . If the limit
exists, it is called the density of E at x. In general , hence, we define the set of points of density for E as
Definition 2 (One-sided approximate limit).
Let be an open set, and let be an oriented -rectifiable set. Take such that the exterior normal is defined, and choose a measurable function, or vector field, f defined on Ω. We write
if for every the set has density 0 at z.
Definition 3 (Perimeter).
Let E be a Borel set in . We define the perimeter of E in an open set as
We set . If , we say that E is a set of finite perimeter in Ω. In this case (see ) one has that the perimeter of E coincides with the total variation of the vector-valued Radon measure (the distributional gradient of ), which is defined for all Borel subsets of Ω thanks to Riesz Theorem. We also recall that when is Lipschitz.
Theorem 4 (De Giorgi Structure Theorem).
Let E be a set of finite perimeter and let be the reduced boundary of E defined as
is countably -rectifiable,
for all , in as , where denotes the half-space through 0 whose exterior normal is ,
for any Borel set A, , thus in particular ,
for any .
Finally, we recall the notion of weakly-regular set which is useful for the next section.
Definition 5 (Weakly-regular set).
An open, bounded set Ω of finite perimeter, such that , is said to be weakly-regular.
For the sake of completeness, we recall that examples of weakly-regular sets are, for instance, Lipschitz sets, or minimal Cheeger sets with (see [28, 33] for an account of these facts, [26, 31, 32] for an introduction to Cheeger sets, and [27, 29] for recent results and examples).
2.3 The weak normal trace and the Gauss–Green formula
The weak normal trace was first defined in  for a vector field with divergence in , when Ω is bounded and Lipschitz. When Ω is a generic open set, and denoting by the Lebesgue measure on , the weak normal trace can be defined as the distribution
Taking of class , and defining in the same way, one can show that the distribution is represented by an function defined on , that we still denote by with a slight abuse of notation, so that
Then, up to showing a locality property of the weak normal trace on domains (see ), it is possible to define for any given, oriented -rectifiable set S contained in Ω. We remark, however, that some particular care has to be taken when Ω is a bounded open set with finite perimeter (and ξ is a-priori defined only on Ω). In order to guarantee that the distributional weak normal trace is represented by an function defined on one has to additionally assume that Ω is weakly-regular.
Thus, when Ω is weakly-regular one obtains the Gauss–Green formula below (see [34, Section 3]) exploiting the pairing between vector fields in and functions in . Another version of the formula with a slightly different pairing can be found in .
Theorem 6 (Generalized Gauss–Green formula).
Let be a weakly-regular set. For any and one has
where is the so-called weak normal trace of ξ on .
As already observed by Anzellotti in  (see also ), the weak normal trace is a proper extension of the scalar product between the exterior normal and the trace of the vector field ξ, assuming that the latter exists in the classical sense. More generally, one expects that the weak normal trace is obtained as a weak-limit of scalar products between (suitable averages of) the vector field ξ and the normal field . This actually corresponds to [4, Proposition 2.1], which says that, whenever S is of class and , then
It is worth recalling that there exist also some pointwise characterizations of the weak normal trace. A first one is obtained by testing (2.1) with a function ψ that approximates the characteristic function of a ball for . Taking x in a suitable subset of S of full -measure one obtains
where, for small enough, denotes the part of which is on the side of S pointed by . A second characterization given in [4, Proposition 2.2], assuming S of class , is the following. Given and small enough, let and set
Note that is well-defined on as soon as are small enough. In other words, is a curvilinear rectangle foliated by translates of in the direction. It is proved in  that
for -almost-every . This result can be also obtained by testing (2.1) with a function ψ that satisfies on , on , and which is linear on any segment , .
Despite the characterizations described above, one can easily construct examples of vector fields like the one presented below (a variant of a piece-wise constant one defined in ) that illustrate possible wild behaviors of divergence-free vector fields for which the weak normal trace on S is well-defined. More precisely, the example below shows that, in general, the weak normal trace does not coincide with any classical, or measure-theoretic, limit of the scalar product of the vector field with the normal to S, even when S is of class and the vector field is divergence-free and smooth in a neighborhood of S (minus S itself).
Let us set , , and . For and we set , , . Then, for such i and j we take with compact support in , so that in particular , and define . Notice that by our choice of parameters, the balls are pairwise disjoint. Whenever , we set
while otherwise (see Figure 1). One can suitably choose so that for all . Moreover, on and thus for any , by the Gauss–Green formula and owing to the definition of ξ, one has
so that on S. At the same time, ξ twists in any neighborhood of any point , , and the second component of the average of ξ on half balls centered at has a strictly smaller than its as . In conclusion, the scalar product does not converge to 0 in any pointwise or measure-theoretic sense.
3 Proofs of the rigidity results
We start by proving that the convexity assumption on the function φ makes Definition 1 essentially equivalent to a weaker one, in which the smoothness and the global Lipschitz-continuity of the vector field η are required.
Let φ and η be as in Definition 1. Let be the standard mollifier supported in a ball of radius . Then the regularized vector field is globally Lipschitz and satisfies (i)–(iii) of Definition 1 up to an ε-translation in the variable .
We note that almost everywhere on by (iii), therefore Jensen’s inequality implies that for all and
Then we observe that is globally Lipschitz, as a consequence of the boundedness of η, and verifies on , that is, (ii). Moreover, for every such that one has . Finally, up to a translation in the variable x, we can assume for all , hence property (i) is also satisfied. ∎
Proof of Theorem 2.
Without loss of generality, by Lemma 1 we can assume that η is smooth and globally Lipschitz. Let us fix and consider the one-parameter flow associated with the vector field , defined for every and by the Cauchy problem
Note that in the notation above we dropped the dependence upon the parameter ε for the sake of simplicity. Due to the smooth dependence from the initial datum, the map is smooth, and the map is a diffeomorphism for every . Let us denote by the n-th component of . Since for all , we have that , hence the function is strictly monotone and surjective. Therefore, for every and there exists a unique such that . By the Implicit Function Theorem, is smooth and one has
Fix now an open, bounded and smooth set and . Let us set and define the map
where we have set . Before proceeding it is convenient to introduce some more notation. We write
We start by computing the partial derivative of Ψ with respect to h:
Note that and by definition. Owing to the smoothness of Ψ, we first compute the partial derivative with respect to h of , for :
Moreover, it is immediate to check that for all and that , so that in particular the matrix form of the previous computation is
Moreover, the determinant of coincides with that of . If we assume that is fixed, and define , we find by standard calculations that
with initial condition . This shows that the Jacobian matrix of Ψ is uniformly invertible on compact subsets of . Consequently, Ψ is a smooth diffeomorphism and the set , depicted in Figure 2, is Lipschitz. Notice moreover that
thanks to the properties of η. Notice now that given a bounded , there exists a constant depending only on A and , such that is contained in for every . Setting , by applying the Divergence Theorem on to the vector field we find the identity
since the integral on the “lateral” boundary of vanishes. This happens because the lateral boundary of the flow-tube consists of integral curves of the flow X, and thus on this lateral boundary one has . Using the fact that we can pass to the limit as in the identity above, obtaining that vanishes on . By the arbitrary choice of both A and , and by property (iii) of Definition 1, we conclude that on . ∎
We now proceed with the proof of Theorem 3. Here, instead of using the “flow-tube” method employed in the proof of Theorem 2, we take advantage of a special feature of 2-dimensional cylinders of the form , i.e. the fact that their “lateral” perimeter is constantly equal to (and in particular it does not blow up when ).
Proof of Theorem 3.
By Lemma 1 we can additionally assume that η is smooth and globally Lipschitz. Since , by the Divergence Theorem we find
Therefore, for all and
hence, owing to the properties of φ, we get for all ,
This implies that
as , for all . Therefore, we can take the limit in (3.1) as and, thanks to the Dominated Convergence Theorem, we obtain that
Hence, the -norm of is zero, thus , for all . By (3.3) and the properties of φ we get on . ∎
4 Counterexamples to quadratic rigidity in dimension
Given , we set and . We consider vector fields such that for one has
with . We have the following:
Let be as in (4.1). Define
Then η satisfies
if and only if satisfies
Moreover, one has
We show in full detail the only if part. Since
equation (4.5) is equivalent to
Inequality (4.6), up to a change of the constant c, is equivalent to
so that in particular
and inequality (4.13) implies
The quadratic rigidity property does not hold in dimension .
Let us consider a positive parameter γ (to be chosen later) and define the function
Our aim is to verify properties (4.7)–(4.9) up to a suitable choice of γ, and then to use the equivalence stated in Proposition 1. Of course (4.7) is true by definition of V. Let us set and write for simplicity. One has
Notice that V is of class and for all and . We obtain
where the second inequality follows from the maximization of the function for . Let us consider the function
As , we have , while as we have . Moreover, when , one has , hence we deduce that
when , and
when . Therefore by (4.16) and the last inequalities we find that
Assuming , we obtain (4.8).
We now show that (4.9) (with the constant ) holds up to taking a smaller γ. Indeed, the relation , after separation of variables, becomes
We argue as for the upper bound of (more precisely, we discuss the two cases and ; in the first case we use the bound , while in the second case we use the fact that , and the inequalities and ), and find
At the same time, by easy calculations we infer that the function is bounded from below by 1. Hence (4.9) is implied by the condition
In conclusion, by taking γ small enough, and thanks to Proposition 1, a divergence-free vector field providing a counterexample to the rigidity property in dimension is given by
We remark that the vector field η is of class , however one can obtain a counterexample by mollification of η. ∎
Concerning the 3-dimensional case, the construction of a counterexample with cylindrical symmetry, as done in Theorem 2, does not work. Indeed we are unable to get an estimate like (4.17), as the function tends to 1 as . More precisely we can prove the following result.
We set and write (4.9) as
Let us assume by contradiction that and are not trivial, hence there exist and such that , , and . Setting and , we have four cases according to the sign of a and . We discuss the first case and . We consider the differential inequality
with . Therefore, the function is increasing and by separation of variables and integration between and we get
We denote by the maximal right interval of existence of the solution , for which . We can exclude the case , as we would obtain by maximality that as , however this would contradict the fact that for all . Contrarily, if , one gets a contradiction with (4.18). The remaining three cases can be discussed in a similar way. ∎
As a consequence of Proposition 3 we infer that in dimension no counterexample to the rigidity property can be found in the class of vector fields of the form .
5 The trace of a vector field with locally maximal normal trace
The results we shall discuss in this section are stated for -almost-every point of S, being S an oriented, -rectifiable set with locally finite -measure. Therefore, given , without loss of generality (see [2, Theorem 2.56]) we shall assume to be such that
the normal vector is defined at ,
is a Lebesgue point for the weak normal trace of z on S, with respect to the measure ,
Let be vector fields in with . Let be oriented, closed -rectifiable sets with locally finite -measure, satisfying the following properties:
Then in .
Fix and set and . By the Divergence Theorem coupled with the formula (see [1, Proposition 3.2])
hence by (ii) and (iii)
This shows that
which proves the thesis. ∎
Let and S a closed, oriented -rectifiable set with locally finite -measure. Then, for -almost-every and for any decreasing and infinitesimal sequence , the sequence of vector fields defined by converges up to subsequences to a vector field in -, such that setting , we have on and on Σ.
We show that hypotheses (i)–(iii) of Lemma 1 are satisfied. Since for all k, thanks to Banach–Alaoglu Theorem (see also [5, Theorem 3.28]) we can extract a not relabeled subsequence converging to , which gives (i). We set , then thanks to (c) we have
for all , which gives (iii). Owing to the localization property proved in [1, Proposition 3.2], we can replace S with the boundary of an open set Ω of class such that and . Defining , the proof of (ii) is reduced to showing that weakly converge as measures to , where H is the tangent half-space to Ω at (so that ). This fact is a consequence of Theorem 4. Now we can apply Lemma 1 and obtain on . In order to prove the last part of the statement we have to show that for any one has
where . Since we have proved that on , we only have to show that
Note that by the convergence of to η, the -convergence of to H, and the convergence of the measures to , we have
Therefore by (5.1) and (b) we infer that
By relying on Proposition 2 and on Theorem 3, we are now able to prove the main result of the section, i.e. Theorem 4 which states the existence of the classical trace for a divergence-measure vector field having a maximal weak normal trace on a oriented -rectifiable set S.
Proof of Theorem 4.
Without loss of generality, up to a translation we can suppose and up to a rotation that . Moreover, up to rescaling ξ, we can suppose . Let
We then want to show that the set
has density zero at 0 for all . Argue by contradiction and suppose there exist and a sequence of radii decreasing to zero such that
Define and the sequence for . Since the second component of is , one easily sees that
On top of that, with . By Proposition 2 the sequence defined above converges in - (up to subsequences, we do not relabel) to a vector field η such that on and on . We aim to show that η satisfies hypotheses (i)–(iii) of Theorem 3. Were this the case, one would conclude in and this would yield a contradiction with (5.6). Indeed, taking as a test function, we get
On the one hand, we know that hypothesis (i) of Theorem 3 is satisfied as on . On the other hand, as on , we get that
so that hypothesis (ii) of Theorem 3 holds as well. We are left to show that hypothesis (iii) of Theorem 3 is satisfied. Since is equibounded, by (5.5) we infer as well that converges (up to subsequences, we do not relabel) to some function ζ in -. Clearly, one has from (5.5) and the weak- convergence of and of that almost everywhere. We want to prove that the same holds with in place of ζ so to retrieve hypothesis (iii) of Theorem 3 with the choice . Take a probability measure with . Then, by Jensen’s inequality,
As , by the weak- convergence we get
Thus, by letting toward the Dirac measure centered at x, (iii) follows at once for almost-every point x (more precisely, x must be a Lebesgue point for the functions ). A direct application of Theorem 3 yields the desired contradiction. ∎
Let be weakly-regular and let . Then, for -a.e. such that one has
Thanks to the Gauss–Green formula (2.1) in the special case , one deduces that the vector field defined as on Ω and on belongs to . The conclusion is achieved by applying Theorem 4 with and in place of, respectively, ξ and S. ∎
5.1 An application to capillarity in weakly-regular domains
The trace property that we have studied in the last section is motivated by the study of the boundary behavior of solutions to the prescribed mean curvature equation in domains with non-smooth boundary (see ). Let us consider the vector field
associated with any given . We say that u is a solution to the prescribed mean curvature equation if
in the distributional sense, where H is a prescribed function on Ω. One of the main results of  is the following theorem.
Theorem ([28, Theorem 4.1]).
Let be a weakly-regular domain and let H be a given Lipschitz function on Ω. Assume that the necessary condition for existence of solutions to (PMC) holds, that is,
Then the following properties are equivalent:
(Uniqueness) ((PMC)) admits a solution u which is unique up to vertical translations.
(Maximality) Ω is maximal for ((PMC)), i.e. no solution can exist in any domain strictly containing Ω.
(Weak verticality) There exists a solution u which is weakly-vertical at , i.e.
where is the weak normal trace of Tu on .
We remark that, in the relevant case of H a positive constant, the extremality property (E) is equivalent to Ω being a minimal Cheeger set (i.e. Ω is the unique minimizer of the ratio among all measurable with positive volume, see for instance [26, 27, 29, 31, 32]) and H equals the Cheeger constant of Ω. In dimension , this extremal case corresponds exactly to capillarity in zero gravity for a perfectly wetting fluid that partially fills a cylindrical container with cross-section Ω. We also stress that the uniqueness property (U) holds in this case without any prescribed boundary condition; this means that the capillary interface in only depends upon the geometry of Ω. Another important remark should be made on the verticality condition (V), which corresponds to the tangential contact property that characterizes perfectly wetting fluids. In  this condition is obtained under the weak-regularity assumption on Ω, which somehow justifies the presence of the weak normal trace in the statement (see for instance in Figure 3 an example of non-Lipschitz, weakly-regular domain built in  and covered by [28, Theorem 4.1]).
Nevertheless, in the physical 3-dimensional case (i.e. when ), the weak-verticality (V) improves to strong-verticality, that is, the trace of Tu exists and is equal to almost-everywhere on thanks to Theorem 4 and to Corollary 3. However, we remark that the strategy of proof strongly relies on the rigidity property, which we have been able to prove only in dimension . It is an open question whether the weak-verticality condition always improves to the strong-verticality given by the existence of the classical trace of Tu at -almost-every point of , and more generally if Theorem 4 holds in any dimension.
Funding source: Istituto Nazionale di Alta Matematica
Award Identifier / Grant number: U-UFMBAZ-2019-000473 11-03-2019
Funding statement: Gian Paolo Leonardi and Giorgio Saracco have been partially supported by the INdAM-GNAMPA 2019 project “Problemi isoperimetrici in spazi Euclidei e non” (n. prot. U-UFMBAZ-2019-000473 11-03-2019).
The authors would like to thank Carlo Mantegazza for fruitful discussions on the topic, and Guido De Philippis for suggesting the “flow-tube” strategy used to prove Theorem 2.
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