Haïm Brezis

Minimizers of the W 1,1 -energy of S 1 -valued maps with prescribed singularities. Do they exist?

Keywords: Sobolev spaces, circle-valued maps, singularities, Jacobian 2002 AMS classification subject. 46E35, 58D15

The paper is concerned with the least W 1,1 -energy required to produce maps from a domain Ω ⊂ R 2 with values into S 1 having prescribed singularities (a i ) 1≤i≤k . The value of the infimum has been known for a long time and corresponds to the length of minimal configurations connecting the points (a i ) between themselves and/or to the boundary. We tackle here the question whether the infimum of this W 1,1 -energy is achieved. This natural topic turns out to be delicate and we have a complete answer only when k = 1. The bottom line for k ≥ 1 is that the infimum is "rarely" achieved. As a "substitute", we give a full description of the asymptotic behavior of all minimizing sequences and show that they "concentrate" along "convex combinations" of minimal configurations.

Introduction

Throughout this paper, the domain Ω ⊂ R 2 is smooth, bounded and simply connected. Given u ∈ W 1,1 (Ω; S 1 ), we write u = (u 1 , u 2 ). To such u, we may associate various objects and quantities which play a fundamental role in the study of W 1,1 (Ω; S 1 ) (see e.g. [START_REF] Bourgain | H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation[END_REF], [START_REF] Brezis | Sobolev maps with values into the circle[END_REF], [START_REF] Brezis | Distances between classes in W 1,1 (Ω; S 1 )[END_REF]).

One of these objects is the Jacobian of u, Ju. This distribution is defined by

Ju := 1 2 ∂ ∂x 1 u 1 ∂u 2 ∂x 2 -u 2 ∂u 1 ∂x 2 - ∂ ∂x 2 u 1 ∂u 2 ∂x 1 -u 2 ∂u 1 ∂x 1 . (1.1)
It is well-known (see e.g. [START_REF] Bourgain | H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation[END_REF], [START_REF] Brezis | W 1,1 -maps with values into S 1 . In Geometric analysis of PDE and several complex variables[END_REF] and [START_REF] Brezis | Sobolev maps with values into the circle[END_REF]) that

J(uv) = Ju + Jv, ∀ u, v ∈ W 1,1 (Ω; S 1 ), (1.2)
J u = -Ju, ∀ u ∈ W 1,1 (Ω; S 1 ), (1.3) and Ju = 0 if and only if u = e ıϕ for some ϕ ∈ W 1,1 (Ω; R). (1.4) If, in addition, u ∈ W 1,1 (Ω; S 1 ) ∩ C(Ω \ {a 1 , . . . , a k }; S 1 ),

where a 1 , . . . , a k ∈ Ω, then (see e.g. [4], [START_REF] Brezis | Harmonic maps with defects[END_REF], [START_REF] Bourgain | H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation[END_REF], [START_REF] Brezis | W 1,1 -maps with values into S 1 . In Geometric analysis of PDE and several complex variables[END_REF] and [START_REF] Brezis | Sobolev maps with values into the circle[END_REF])

Ju = π k j=1
M j δ a j in D (Ω).

(1.5)

Here, the integer M j := deg (u, a j ) is the degree of u restricted to any small circle centered at a j . (This integer does not depend on the small circle.) Roughly speaking, when u has a finite number of singularities, the Jacobian Ju detects the location and the topological strength of the singularities of u.

Given k points a 1 , . . . , a k in Ω and k integers M 1 , . . . , M k ∈ Z, we will use the notation a := (a 1 , . . . , a k ) and M := (M 1 , . . . , M k ). When k = 1, we let a := a 1 , M := M 1 and write a, M instead of a and M.

The quantity L(a, M) defined below appears in many questions involving maps in W 1,1 (Ω; S 1 ) (see e.g. [START_REF] Brezis | Harmonic maps with defects[END_REF], [START_REF] Bourgain | H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation[END_REF], [START_REF] Brezis | W 1,1 -maps with values into S 1 . In Geometric analysis of PDE and several complex variables[END_REF] and [START_REF] Brezis | Sobolev maps with values into the circle[END_REF]). .

L(a, M) := min

(1.6)

Here, the points P , N need not be all distinct. With no loss of generality we may assume that P = N , ∀ , and then the integer n satisfies n ≤ k j=1 |M j |. Note that the measure n =1 (δ P -δ N ) does not "see" points in ∂Ω, since equality in (1.6) holds in the sense of D (Ω). However, even in cases where the equality n =1 (δ P -δ N ) = n j=1 M j δ a j can be achieved using only points P , N ∈ {a 1 , . . . , a k }, it is sometimes advantageous to introduce artificial boundary points because they may lower the quantity n =1 |P -N | (see Example 3 below). If (P , N ) n =1 is a minimizer in (1.6), we say that the collection of segments (P , N ), = 1, . . . , n, is a minimal configuration associated with (a, M).

Here are a few examples.

Example 1. When a ∈ Ω and M = 1, we have L(a, M) = dist(a, ∂Ω). For any N ∈ ∂Ω such that |a -N| = dist(a, ∂Ω), the segment (a, N) is a minimal configuration.

Example 2. Let a = (a 1 , a 2 ) and M = (+1, +1). Then L(a, M) = dist(a 1 , ∂Ω) + dist(a 2 , ∂Ω). If N ∈ ∂Ω satisfy |a -N | = dist(a , ∂Ω), = 1, 2, then ((a 1 , N 1 ), (a 2 , N 2 )) is a minimal configuration.

Example 3. Let a = (a 1 , a 2 ) and M = (+1, -1). Then L(a, M) = min{|a 1 -a 2 |, dist(a 1 , ∂Ω) + dist(a 2 , ∂Ω)}. When |a 1 -a 2 | ≤ dist(a 1 , ∂Ω) + dist(a 2 , ∂Ω), a minimal configuration is (a 1 , a 2 ). When |a 1 -a 2 | ≥ dist(a 1 , ∂Ω)+dist(a 2 , ∂Ω), a minimal configuration is ((a 1 , N 1 ), (P 2 , a 2 )), with N 1 , P 2 ∈ ∂Ω satisfying |a 1 -N 1 | = dist(a 1 , ∂Ω), respectively |P 2 -a 2 | = dist(a 2 , ∂Ω).

To each u ∈ W 1,1 (Ω; S 1 ) we associate the number Σ(u) ≥ 0 defined by Σ(u) = inf ˆΩ |∇v|; v ∈ W 1,1 (Ω; S 1 ), Jv = Ju in D (Ω) .

(1.7) Roughly speaking, Σ(u) measures the least W 1,1 -energy required to produce S 1 -valued maps having the same singularities as u. The quantity Σ(u) is ubiquitous in the analysis of the space W 1,1 (Ω; S 1 ) (see e.g. [START_REF] Bourgain | H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation[END_REF], [START_REF] Brezis | Sobolev maps with values into the circle[END_REF], [START_REF] Brezis | W 1,1 -maps with values into S 1 . In Geometric analysis of PDE and several complex variables[END_REF], [START_REF] Brezis | Distances between classes in W 1,1 (Ω; S 1 )[END_REF]). It is well-known (see [START_REF] Brezis | Harmonic maps with defects[END_REF] and [START_REF] Brezis | Sobolev maps with values into the circle[END_REF]) that if u ∈ W 1,1 (Ω; S 1 ) satisfies

Ju = π k j=1
M j δ a j in D (Ω) (1.8) for some pair (a, D), then Σ(u) = 2πL(a, M).

(1.9)

Here are some natural questions related to (1.7):

Question 1. Is the infimum in (1.7) achieved? Question 2. Characterize the minimizers if they exist. Question 3. Analyze the asymptotic behavior of minimizing sequences for (1.7).

Questions 1 and 2 turn out to be much more delicate than expected. We have a full answer only in the case where u admits one singularity. In the case of multiple singularities, our results are incomplete. The bottom line is that the infimum in (1.7) is "rarely" achieved. On the other hand, we have a satisfactory answer to Question 3 in the most general situation.

Remark 1. Another natural approach consists of minimizing Σ(u) in the space BV (Ω; S 1 ), which is larger than W 1,1 (Ω; S 1 ). The main advantage of BV is that minimization problems are more likely to have solutions in BV than in W 1,1 . Concerning our specific minimization problem, a first difficulty to overcome is the definition of Ju when u is merely BV . Clearly, the right-hand side of (1.1) is meaningless for such u. In [START_REF] Ignat | The space BV(S 2 , S 1 ): minimal connection and optimal lifting[END_REF], Ignat proposed several definitions of Ju with u ∈ BV (Ω; S 1 ). They all coincide with (1.1) when u ∈ W 1,1 , but do not preserve the algebraic properties (1.2)-(1.3), and in addition they are not stable under strong BV convergence. More significant for our problem, relaxing the functional setting to BV is not of much help: the analogous infimum in (1.7) is still not achieved, in general. We will illustrate this in Appendix B. We therefore restrict ourselves to the W 1,1 setting, technically simpler.

We first present our main results for one singularity, say a = 0 ∈ Ω with degree M ≥ 1, so that Ju = πM δ 0 . In Theorems 1 and 2, we assume that dist(0, ∂Ω) = 1 and denote by D the open unit disc centered at 0. Set

X := {x ∈ ∂Ω; |x| = 1}, Y := {θ ∈ [0, 2π]; e ıθ ∈ X }.
(1.10)

Theorem 1. The infimum is attained in (1.7) if and only if |X | > 0. If |X | > 0, then v is a minimizer in (1.7) if and only if v is locally constant in Ω \ D and, in D, we have v(re ıθ ) = e ıh(θ) . Here, h ∈ W 1,1 ((0, 2π); R) satisfies h is nondecreasing, (1.11) h(2π) = h(0) + 2πM, (1.12)
and supp h ⊂ Y .

(1.13)

In the next result we discuss the asymptotic behavior of minimizing sequences in (1.7).

Theorem 2. Let (v n ) be a minimizing sequence in (1.7). Then, up to a subsequence, we have, for some probability measure µ on S 1 supported in X ,

ˆΩ |∇v n | ζ → 2πM ˆS1 ˆ1 0 ζ(rx) dr dµ(x), ∀ ζ ∈ C(Ω). (1.14)
And conversely.

Remark 2. Let us assume, for simplicity, that M = 1 and that X contains only two points, b and c. Let (v n ) be a minimizing sequence in (1.7). Then, up to a subsequence,

|∇v n | dx 2π λH 1 [0, b] + 2π (1 -λ)H 1 [0, c],
where λ could be any number in [0,[START_REF] Alberti | Functions with prescribed singularities[END_REF]. This is in sharp contrast with the situation analyzed by Brezis, Coron and Lieb in [START_REF] Brezis | Harmonic maps with defects[END_REF]. The minimization problem studied there is inf ˆΩ |∇v| 2 ; v :

Ω → S 2 ,
with Ω ⊂ R 3 . The inf is taken over all maps v which are smooth outside the origin and have degree one around the origin. Assuming that the set

X = {x ∈ ∂Ω; |x| = dist(0, ∂Ω)}
is finite, it is proved in [START_REF] Brezis | Harmonic maps with defects[END_REF] that, along a minimizing subsequence (v n ), we have

|∇v n | 2 dx 8πH 1 [0, b],
for some b ∈ X . The lack of quantization phenomenon for the W 1,1 -norm was originally pointed out in [START_REF] Brezis | Harmonic maps with defects[END_REF] (see the comments following Theorem 8.2 and at the end of Appendix E).

The proofs of Theorems 1 and 2 are presented in Section 2.

We now turn to multiple singularities. We start with a simple case of non-existence.

Theorem 3. Assume that there are only finitely or countably many minimal configurations associated with (a, M). Then the infimum in (1.7)-(1.8) is not attained. More generally, assume that there exists a null set (for the Lebesgue measure) Z ⊂ Ω such that for every minimal configuration (P , N ), = 1, . . . , n, associated with (a, M) we have

∪ n =1 (P , N ) ⊂ Z. Then the infimum in (1.7)-(1.8) is not attained.
Remark 3. We return to the case of a single singularity, as in Theorem 1. Assume e.g. that a = 0, M = 1 and dist (0, ∂Ω) = 1. Then the minimal configurations are (0, x), with x ∈ X (here, X is as in (1.10)), and thus the smallest set Z containing all minimal configurations is Z := ∪ x∈X (0, x). We may then rephrase the conclusion of Theorem 1 as follows. There exists a minimizer v in (1.7) if and only if Z has positive Lebesgue measure. In view of Theorem 3, it is tempting to conjecture that a statement similar to Theorem 1 holds in general. Roughly speaking, Theorem 3 asserts that, when the collection of all minimal configurations is a null set in Ω, there is no minimizer in (1.7). However, the converse is wrong; see Remark 5.

Here are some of our results concerning Question 3, i.e., the asymptotic behavior of minimizing sequences. Theorem 4. Assume that there are only finitely many minimal configurations U 1 , . . . ,U m associated with (a, M). Let (v n ) be a minimizing sequence in (1.7)- (1.8). Then, up to a subsequence,

|∇v n | dx 2π m j=1 λ j H 1 U j , with λ j ∈ [0, 1] and m j=1 λ j = 1.

And conversely.

The notation H 1 U j requires an explanation. Recall that a minimal configuration is a finite collection of segments (P , N ). Then we define

H 1 U j := H 1 (P , N ).
(1.15)

The proofs of Theorem 3 and 4 are given in Section 3. In fact, we will present in Section 3 far-reaching extensions of these results. Here is a typical example. Given a minimal configuration U associated with (a, M), we define H 1 U via (1.15). We set L min := {Q = H 1 U; U is a minimal configuration associated with (a, M)}.

(1.16)

Clearly, L min ⊂ M (Ω) = [C 0 (Ω)] * and Q M = L(a, M), ∀Q ∈ L min .
Therefore, L min is a bounded subset of M (Ω). A basic property is Lemma 1. L min is weakly- * closed in M (Ω), and thus L min equipped with the weak- * topology σ(M (Ω), C 0 (Ω)) is a compact metrizable space.

We have the following Theorem 5. Let (v n ) be a minimizing sequence in (1.7)- (1.8). Then there exists a probability measure µ on L min such that up to a subsequence we have

|∇v n | dx 2π κ in M (Ω), (1.17)
where

〈κ, ζ〉 := ˆLmin 〈Q, ζ〉 dµ(Q), ∀ ζ ∈ C 0 (Ω). (1.18)
Note that the right-hand side of (1.18) is well-defined, since Q → 〈Q, ζ〉 is continuous on the compact L min .

We call the attention of the reader to the following Open Problem. Find a tractable necessary and sufficient condition on (a, M) and Ω for the existence of a minimizer in (1.7)- (1.8).

In Section 4 we present some regularity results, e.g. continuity away from the points a i , for the minimizers of (1.7)-(1.8), assuming they exist.

Proofs of Theorems 1 and 2

Proof of Theorem 1. We know that Σ(u) = 2πL(0, M) = 2πM as in Example 1. The infimum in (1.7) is still 2πM if we replace Ω by D. Assume that v is a minimizer in (1.7). Then

2πM = ˆΩ |∇v| = ˆD |∇v| + ˆΩ\D |∇v| ≥ 2πM + ˆΩ\D |∇v|.
Consequently,

ˆD |∇v| = 2πM (2.1)
and

ˆΩ\D |∇v| = 0, (2.2)
so that ∇v = 0 in Ω \ D and thus

v is locally constant in Ω \ D. (2.3)
Set, in polar coordinates, (2.4)

Here, C(0, r) denotes the circle of radius r centered at 0. On the other hand, if w ∈ W 1,1 (S Assume temporarily that v ∈ C(D\{0}). By (1.5), we find that deg (v, C(0, r)) = M for every r ∈ (0, 1). If we remove the continuity assumption, then we have the following result, whose proof is postponed to Section 4. Lemma 2. Let v ∈ W 1,1 (D; S 1 ) be such that Ju = π M δ 0 . Then deg (v, C(0, r)) = M for a.e. r ∈ (0, 1). so that, on D, v does not depend on r. Therefore, we may write v(re ıθ ) = e ıh(θ) on D, with h ∈ W 1,1 ((0, 2π); R) nondecreasing and satisfying h(2π) = h(0) + 2πM. Since v is locally constant in Ω \ D, we deduce that h is locally constant in (0, 2π) \ Y , so that supp h ⊂ Y . Note that a function h with all the required properties exists if only if |Y | > 0. Conversely, it is easy to see that any function v locally constant in Ω \ D, which can be written as v(re ıθ ) = e ıh(θ) , where h ∈ W 1,1 ((0, 2π); R) satisfies (1.11)-(1.13), is a minimizer for (1.7).

Proof of Theorem 2. Theorem 2 is a special case of Theorems 5 and 7 (below). However, since it requires some additional arguments to see that the three statements coincide in the specific setting of Theorem 2, we present a direct proof.

We start with some observations. A measure µ on S 1 defines a (-1)-homogeneous measure µ in Ω, having in polar coordinates the form

µ(ζ) = ˆS1 ˆ1 0 ζ(rx) drdµ(x), ∀ ζ ∈ C 0 (Ω).
Here, "(-1)-homogeneous" comes from the fact that, if µ has density f , then µ has, in The following facts are straightforward: Fact 1. We have µ = 0 in

Ω \ D and µ(D) = µ(S 1 ). Fact 2. If µ n µ in M (S 1 ), then µ n µ in M (Ω). Fact 3. If w ∈ W 1,1 (D) is 0-homogeneous, then |∇w| is (-1)-homogenous in D.
The main ingredient of the proof of Theorem 2 is the following Lemma 3. Let (v n ) be a minimizing sequence in (1.7). Let ν be such that, up to a subsequence, |∇v n | dx ν in M (Ω). Then ν = ξ for some measure ξ on S 1 with supp ξ ⊂ X .

Proof of Theorem 2 completed. Let (v n ) be a minimizing sequence in (1.7). As in the proof of Theorem 1, we have ´Ω |∇v n | → 2πM, and also ´D |∇v n | ≥ 2πM. Therefore, we must have ´Ω\D |∇v n | → 0 and ´D |∇v n | → 2πM. It follows that any ν as in the above lemma must satisfy ν(D) = 2πM. Using Fact 1, we thus have ν = 2πM µ, with µ ∈ P (S 1 ) supported in X .

Conversely, let µ ∈ P (S 1 ) be supported in X . We may identify µ with an element in P (X ). We want to construct a sequence such that (1.14) holds. It suffices to consider the case where µ = k j=1 λ j δ b j , where

λ j ∈ [0, 1], b j ∈ X , ∀ j = 1, . . . , k, k j=1 λ j = 1.
Indeed, assuming this achieved, the case of a general µ ∈ P (X ) is settled as follows. Let

Q := {ν ∈ M (Ω); ∃ a sequence (v n ) minimizing in (1.7) such that |∇v n | dx ν}.
By definition, Q is clearly weakly sequentially closed. Using Fact 2, if µ n µ in M (S 1 ) and 2π µ n ∈ Q, ∀ n, then 2π µ ∈ Q. The above assumption is that 2π µ ∈ Q when µ is a discrete probability on X . Since such probabilities are weakly sequentially dense in P (X ), we find that Q ⊃ {2π µ; µ ∈ P (X )}, which is the desired converse.

We now return to the construction of (v n ) for a discrete measure µ = k j=1 λ j δ b j with

λ j ∈ (0, 1], ∀ j = 1, . . . , k, b j = b , ∀ j = and k j=1 λ j = 1. (2.7)
We may always take M = 1; the general case follows by choosing (v M n ). Assuming e.g. that b 1 = 1, we order the points b j in such a way that b j = e ıθ j with 0 = θ 1 < θ 2 < . . . < θ k < 2π.

We will use a variant of the dipole construction (see [START_REF] Brezis | Harmonic maps with defects[END_REF], [START_REF] Brezis | Sobolev maps with values into the circle[END_REF]). Given ε > 0 (sufficiently small) consider the cone in R 2 ,

Q ε := {(x 1 , x 2 ); |x 1 | < 2ε x 2 , 0 < x 2 < 1/2},
with height 1/2 and base of length 2ε.

Next we consider the symmetry S with respect to the line {(x 1 , 1/2); x 1 ∈ R}, and set

Q ε := Q ε ∪ S( Q ε ).
The set Q ε will be used in the construction of a dipole with vertices 0 and ı, as explained below. Given 0 < λ ≤ 1, consider the functions w ε :

Q ε → S 1 , defined by w ε (x 1 , x 2 ) := e -ıπ λ x 1 /(2ε x 2 )
and w ε :

Q ε → S 1 defined by w ε (x) := w ε (x), if x ∈ Q ε w ε (Sx), if x ∈ S( Q ε ) .
The standard dipole construction corresponds to λ = 1. Set

∂ + Q ε := (∂Q ε ) ∩ {(x 1 , x 2 ) ∈ R 2 ; x 1 > 0} and ∂ -Q ε := (∂Q ε ) ∩ {(x 1 , x 2 ) ∈ R 2 ; x 1 < 0}.
Note that

w ε = e -ıπ λ , on ∂ + Q ε e ıπ λ , on ∂ -Q ε , so that w ε is not constant on ∂Q ε unless λ = 1.
This is in sharp contrast with the standard dipole construction (where we can extend w ε by a constant outside Q ε ). Here, w ε jumps from e -ıπ λ to e ıπ λ when crossing

Q ε from ∂ + Q ε to π -Q ε (
which corresponds to the standard orientation on a small circle centered at 0). A straightforward calculation (see e.g. [START_REF] Brezis | Harmonic maps with defects[END_REF], [START_REF] Brezis | Sobolev maps with values into the circle[END_REF]) yields

ˆQε |∇w ε | ≤ 2π λ (1 + ε). (2.8)
We place a first dipole w ε on Q 1,ε with vortices at 0 and b 1 (instead of 0 and ı) and with λ := λ 1 . We multiply the corresponding function w ε by a factor ξ 1 := e ıπ λ 1 , so that the new function w 1,ε := ξ 1 w ε jumps from 1 to e 2ıπ λ 1 when crossing Q 1,ε . Next we place a second dipole Q 2,ε with vertices at 0 and b 2 and with λ := λ 2 . We multiply the corresponding function by a factor ξ 2 := e ıπ (2λ 1 +λ 2 ) , so that we obtain a function w 2,ε defined in Q 2,ε and satisfying

w 2,ε |∂ + Q 2,ε = w 1,ε |∂ -Q 1,ε = e 2ıπ λ 1 .
We proceed in the same manner until we reach Q k,ε . Choosing

ξ k := e ıπ (2λ 1 +2λ 2 +•••+2λ k-1 +λ k ) , we see that the new function w k,ε jumps when crossing Q k,ε from e 2ıπ (λ 1 +λ 2 +•••+λ k-1 ) to e 2ıπ (λ 1 +λ 2 +•••+λ k ) = e 2ıπ = 1 (by (2.7)),
which matches well with w 1,ε since

w k,ε |∂ -Q k,ε = w 1,ε |∂ + Q 1,ε = 1.
Next, we glue the functions w j,ε by taking appropriate constants on the components of D\({0}∪∪ k j=1 Q j,ε ), in order to obtain a map v ε continuous on D\{0}. Clearly, this map extends by continuity to D \ ({0} ∪ ∪ k j=1 {b j }). In addition, it satisfies

v ε ∈ Lip loc (D \ ({0} ∪ ∪ k j=1 {b j })), deg (v ε , C(0, r)) = 1 for every r < 1 and, by (2.8), ˆD |∇v ε | ≤ 2π (1 + ε).
Finally, we extend v ε to Ω as follows. Let ω be a connected component of Ω \ D. Then (∂ω) ∩ S 1 consists of an arc A on S 1 whose interior is disjoint from X , and which connects two points in X . (Here, we use in an essential way the assumption that Ω is simply connected.) Therefore, v ε is constant on A and we set v ε := v ε|A on ω.

Clearly, v n := v 1/n , n ≥ n 0 , has all the required properties.

Proof of Lemma 3. We start from the following version of (2.4):

2π M + o(1) = ˆD |∇v n | = ˆD ∂v n ∂r 2 + ∂v n ∂τ 2 ≥ ˆD ∂v n ∂τ = ˆD v n ∧ ∂v n ∂τ = ˆ1 0 ˆC(0,r) v n ∧ ∂v n ∂τ d dr ≥ ˆ1 0 ˆC(0,r) v n ∧ ∂v n ∂τ d dr ≥ 2π M.
(2.9)

We find that the following hold:

ˆD ∂v n ∂r → 0, ˆΩ\D |∇v n | → 0 (2.10)
and

ˆD v n ∧ ∂v n ∂τ - → 0. (2.11) Here, t -:= 0, if t ≥ 0 -t, if t < 0 .
In particular, up to a subsequence we have, for a.e. r ∈ (0, 1), ˆC(0,r)

v n ∧ ∂v n ∂τ - → 0. (2.12)
On the other hand, let us note the inequality ˆC(0,r)

|∇v n | d ≥ 2π
M, for a.e. r ∈ (0, 1); (2.13) this follows from Lemma 2 combined with the degree formula (2.6). By (2.9) and (2.13), we find that lim n→∞ ˆ1 0 ˆC(0,r)

|∇v n | d -2π M dr = 0. (2.14)
By (2.10) and (2.14), possibly up to a subsequence we have (0, ∞) r → ˆC(0,r)∩Ω

|∇v n | is dominated in L 1 ((0, ∞)).
(2.15)

By similar calculations, we obtain

ˆD(0,r) |∇v n | ≤ ˆΩ |∇v n | - ˆD\D(0,r) |∇v n | ≤ 2π M r + o(1), ∀ r ∈ (0, 1), (2.16)
and ˆ{x∈Ω;|x|≥1-ε}

|∇v n | ≤ 2π M ε + o(1), ∀ ε ∈ (0, 1). (2.17)
In view of (2.16)-(2.17), it suffices to prove the existence of a measure ξ on S 1 such that

ˆΩ |∇v n | ζ → ˆS1 ˆ1 0 ζ(rx) dr dξ(x), ∀ ζ ∈ C ∞ c (D \ {0}) (2.18) and supp ξ ⊂ X . (2.19)
Using a partition of unity in D \ {0}, it suffices to establish (2.18) when ζ is supported in a set of the form

A := re ıθ ; r 1 < r < r 2 , θ 1 < θ < θ 2 , 0 < r 1 < r 2 < 1, θ 2 -θ 1 < 2π.
Since A is simply connected and Jv n = 0 in A, we may write, in A, v n = e ıϕ n with ϕ n ∈ W 1,1 . In terms of ϕ n , (2.18) amounts to

ˆΩ ∇ϕ n ζ → ˆS1 ˆr2 r 1 ζ(rx) dr dξ(x), ∀ ζ ∈ C ∞ c (A), (2.20)
and it suffices to define ξ on the set B := e ıθ ; θ 1 < θ < θ 2 . (Assuming that this has been done, by covering S 1 with a finite number of B's we obtain a global object such that (2.18) holds.)

Consider some R n ∈ (r 1 , r 2 ) such that:

1. The trace y n of v n on C(0, R n ) belongs to W 1,1 and the sequence (y n (R n •)) is bounded in W 1,1 (S 1 ).

The trace

ψ n of ϕ n on C(0, R n ) ∩ A belongs to W 1,1 and e ıψ n = y n in C(0, R n ) ∩ A.
3. (2.12) holds with r = R n .

Let ξ be a measure on B such that, possibly up to a subsequence, we have

ˆS1 ∂ y n ∂τ (R n ω) g(ω) d (ω) = ˆS1 ∂ψ n ∂τ (R n ω) g(ω) d (ω) → ˆS1 g dξ, ∀ g ∈ C c (B; R). (2.
|∇ϕ n | is dominated in L 1 ((r 1 , r 2 )).
(2.24) By (2.12) and (2.21), we also have

ˆS1 ∂ψ n ∂τ (R n ω) g(ω) d (ω) → ˆS1 g dξ, ∀ g ∈ C c (B; R). (2.25)
Using (2.22) and (2.24), for a.e. r in (r 1 , r 2 ) we have

ϕ n (r•) -ψ n (R n •) → 0 in L 1 (B), uniformly in r ∈ (r 1 , r 2 ) as n → ∞ (2.26)
and thus (using (2.25) and (2.26))

-ˆA ϕ n (x) ∂ζ ∂τ (x) dx = -ˆA ψ n (R n x/|x|) ∂ζ ∂τ (x) dx + o(1) → ˆB ˆr2 r 1 ζ(rx) dr dξ(x) as n → ∞, ∀ ζ ∈ C ∞ c (A).
(2.27)

Equivalently, we have

ˆA ∂ϕ n ∂τ ζ → ˆS1 ˆr2 r 1 ζ(rx) dr dξ(x), ∀ ζ ∈ C ∞ c (A), (2.28)
We obtain (2.20) from (2.22), (2.23), (2.24) and (2.28).

It remains to prove (2.19). In view of (2.28), this amounts to proving that, if the above B satisfies B ∩ X = , then

ˆA ∂ϕ n ∂τ ζ → 0, ∀ ζ ∈ C ∞ c (A). (2.29)
In turn, (2.29) is obtained via a slight modification of the above arguments. Let ρ > 1 be such that {rx; x ∈ B, 0 ≤ r < ρ} ⊂ Ω. Existence of such ρ follows from the assumption B ∩ X = . Consider now some R n ∈ (1, ρ) (instead of R n ∈ (r 1 , r 2 )) such that the above properties 1-3 and (2.21) hold. Define ξ starting from these R n 's. This ξ will satisfy (2.27). By (2.10), we find that ξ = 0.

Proofs of Theorems 3, 4 and 5

Since we will present generalizations of these theorems we need some preliminary material. We denote by E the following class of distributions:

E := T ∈ D (Ω); T = (δ P j -δ N j ), with P j , N j ∈ Ω, |P j -N j | < ∞ . (3.1)
Up to a factor π, the class E characterizes all possible Jacobians of maps u ∈ W 1,1 (Ω; S 1 ) ( [START_REF] Alberti | Functions with prescribed singularities[END_REF], [START_REF] Brezis | W 1,1 -maps with values into S 1 . In Geometric analysis of PDE and several complex variables[END_REF], [START_REF] Brezis | Sobolev maps with values into the circle[END_REF]) as explained below.

1. Let u ∈ W 1,1 (Ω; S 1 ). Then there exist points P j , N j ∈ Ω such that

|P j -N j | < ∞ (3.2)
and

Ju = π (δ P j -δ N j ) in (W 1,∞ 0 (Ω)) * (and thus in D (Ω)). (3.3) 2.
Conversely, given T ∈ E , there exists u ∈ W 1,1 (Ω; S 1 ) such that Ju = π T.

3. In addition, we have

Σ(u) = 2π inf | P j -N j |; P j , N j satisfy (3.2) -(3.3) . (3.4)
4. An equivalent formulation of (3.4) is the following. Define, for T ∈ E , the Wasserstein norm

T W := sup 〈T, ζ〉; ζ ∈ W 1,∞ 0 (Ω; R), ∇ζ L ∞ ≤ 1 < ∞. (3.5)
(Alternatively, we could take, in (3.5), the sup over functions

ζ ∈ C ∞ c (Ω; R).) Then Σ(u) = 2 Ju W . (3.6)
For such a general T, (generalized) minimal configurations need not exist ( [START_REF] Ponce | On the distributions of the form i (δ p i -δ n i )[END_REF]), i.e., the inf in (3.4) need not be achieved. In this context, it is convenient to work with vectorvalued measures. Following [START_REF] Brezis | Sobolev maps with values into the circle[END_REF], we will consider connections and minimal connections. A connection C associated with T ∈ E is an R 2 -valued measure C on Ω satisfying:

C := ν i H 1 (S i ∩ Ω), (3.7)
where (S i ) is a countable collection of Borel subsets S i of C 1 oriented curves (3.9)

γ i ⊂ R 2 with normals ν i , such that H 1 (S i ∩ Ω) < ∞, and curl C = T in D (Ω). ( 3
6. The infimum in (3.9) is always achieved. A minimal connection is a minimizer in (3.9).

7. Let us return to the situation considered in the previous sections, where T := 1 π Ju is a finite sum of Dirac masses. Then T W = L(a, M) (by (1.9) and (3.5)). Consider a minimal configuration ((P , N )) n =1 associated with (a, M). Then C := ν H 1 (N , P ) (with (N , P ) the oriented segment from N to P ) is a minimal connection ( [START_REF] Brezis | Sobolev maps with values into the circle[END_REF]). And conversely.

In view of these considerations, Theorem 3 is an immediate consequence of the following. Theorem 6. Let T ∈ E \ {0}. Assume that there are only finitely or countably many minimal connections associated with T.

Then the infimum

inf ˆΩ |∇v|; v ∈ W 1,1 (Ω; S 1 ), Jv = πT (3.10)
is not attained. More generally, the same conclusion holds if we assume that there exists a null set (for the Lebesgue measure) Z ⊂ Ω such that supp C ⊂ Z H 1 -a.e., for every minimal connection C .

The proof of Theorem 6 relies on the coarea formula for Sobolev maps (see e.g. [START_REF] Hajlasz | Sobolev mappings, co-area formula and related topics[END_REF], [START_REF] Malý | The co-area formula for Sobolev mappings[END_REF]) and a consequence of this formula. This key ingredient, Lemma 4 below, is a delicate result due to Alberti, Baldo and Orlandi ([1, Theorem 3.8], with roots in [2, Appendices A.5-A.8] and [11, 4.3]).

We first recall the Sobolev version of the coarea formula, following the presentation in [1, Sections 7.4, 7.5]. Let v : Ω → S 1 be a Borel function such that v ∈ W 1,1 . Let E be the set of points where v is not approximatively differentiable (for the definition of the approximate differential, see e.g. [3, Definition 3.70]). Then E is a null set (for the Lebesgue measure). For α ∈ S 1 , we set

D α = [v = α] := {x ∈ Ω \ E; v(x) = α}.
Then there exists a full measure set A ⊂ S 1 such that D α is H 1 -rectifiable, and v ∧ ∇v = 0 H 1 -a.e. on D α , ∀ α ∈ A.

(3.11)

In addition, the following "coarea formula" holds

ˆΩ |∇v| = ˆS1 H 1 (D α ) dα. (3.12)
More generally, we have the following

ˆΩ g|∇v| = ˆS1 ˆDα g dH 1 D α dα. (3.13)
Here, equality is valid for any non-negative Borel function g or for any Borel function g such that g|∇v| ∈ L 1 (Ω).

The next result is a reformulation of [1, Theorem 3.8]. (The latter is stated in terms of rectifiable currents. One "translates" it in terms of connections using the "dictionary" (3.18)-(3.20) below.)

Lemma 4. Let v ∈ W 1,1 (Ω; S 1 ). Set T := 1 π Jv and D α := [v = α], ∀ α ∈ S 1 .
Then, for a.e. α,

C α := v ∧ ∇v |v ∧ ∇v| H 1 D α
is a connection associated with T.

Note that, in view of (3.11), the definition of C α makes sense for a.e. α ∈ S 1 .

Proof of Theorem 6. We may assume that Z is a Borel set. Argue by contradiction and assume that v is a minimizer in (3.10), so that

ˆΩ |∇v| = 2π T W . (3.14)
By Lemma 4 and (3.9), we have

H 1 (D α ) = C α M ≥ T W , for a.e. α ∈ S 1 . (3.15)
By (3.14), (3.15) and the coarea formula (3.12), we find that C α is a minimal connection, for a.e. α ∈ S 1 . For any such α, we have

D α = supp C α ⊂ Z H 1 -a.e. ( 3.16)
Let now g := 1 Z , so that for a.e. α ∈ S 1 , g = 1 H 1 -a.e. on D α .

(3.17)

Since g = 0 a.e., we find, using (3.17) and the coarea formula (3.13), that

0 = ˆΩ g |∇v| = ˆS1 ˆDα g dH 1 D α dα = ˆS1 H 1 (D α ) dα = 2π T W > 0.
This contradiction completes the proof of Theorem 6.

Remark 4. The idea of using the coarea formula to prove the inequality ´Ω |∇v| ≥ 2π T W when v is smooth except at a finite number of singularities goes back to Almgren, Browder and Lieb [START_REF] Almgren | Co-area, liquid crystals, and minimal surfaces[END_REF].

Remark 5. The converse to Theorem 6 is wrong, even when T is a finite sum of Dirac masses. Actually, while the converse is true for one singularity (see Theorem 1), it is already wrong for two singularities. Indeed, consider a domain Ω and points a 1 , a 2 ∈ Ω such that the following hold:

1. The set

X 1 := {x ∈ ∂Ω; |a 1 -x| = dist (a 1 , ∂Ω)} has positive length. 2. The set X 2 := {x ∈ ∂Ω; |a 2 -x| = dist (a 2 , ∂Ω)} consists of a single point, say N 2 .
Let a := (a 1 , a 2 ) and M := (+1, +1), so that we have T = δ a 1 + δ a 2 . Then the minimal configurations are ((a 1 , N 1 ), (a 2 , N 2 )), with N 1 ∈ X 1 . Therefore, any set Z as in Theorem 6 must contain ∪ N 1 ∈X 1 (a 1 , N 1 ), and thus has positive Lebesgue measure.

We claim that, for the above T, there exists no minimizer v in (3.10). Indeed, argue by contradiction. As in the proof of Theorem 6, for a.e. α ∈ S 1 the set D α is the support of a minimal connection. In our case, this means that, for such α, D α equals, H 1 -a.e., a set of the form S N 1 := (a 1 , N 1 ) ∪ (a 2 , N 2 ). However, this is impossible since by definition D α ∩ D β = , for a.e. α, β ∈ S 1 such that α = β, while H

1 (S N 1 ∩ S N 1 ) > 0, ∀ N 1 , N 1 ∈ X 1 .
Some preliminaries are needed in order to extend Theorems 4 and 5 to the case of a general T. These considerations may also be understood using tools from geometric measure theory. This would require working with rectifiable currents instead of connections. The two are related through the following straightforward equivalence:

C is a connection associated with T ⇐⇒ C := C ⊥ satisfies (3.19) below:

(3.18)

C is a finite mass integer multiplicity 1-rectifiable current such that ∂C = T. (3.19)

(If C = (C x , C y ), then C ⊥ := (-C y , C x ).)
In addition, we have

|C | = |C|. (3.20)
We will take advantage of our specific situation and work only with connections and BV functions. This is reminiscent of the well-known observation of Hardt and Pitts [START_REF] Hardt | Solving Plateau's problem for hypersurfaces without the compactness theorem for integral currents[END_REF] that in codimension one the theory of rectifiable currents is essentially equivalent to the theory of BV functions.

Fix some T ∈ E , and define K := {C ; C is a connection associated with T}.

(3.21)

Given m ≥ 0, set

K m := {C ∈ K ; C M ≤ m}. (3.22)
(In view of (3.9), K m is non-empty if and only if m ≥ T W .) By definition, K m is a subset of the closed ball B(0, m) of M (Ω; R 2 ) = [C 0 (Ω; R 2 )] * , and thus has a metrizable topology inherited from the weak- * topology σ(M (Ω; R 2 ), C 0 (Ω; R 2 )) on B(0, m). We have the following vectorial analogue of Lemma 1 mentioned in the introduction.

Lemma 5. K m equipped with the weak- * σ(M (Ω; R 2 ), C 0 (Ω; R 2 )) topology is a compact metrizable space.

Although it is possible to give a direct proof of Lemma 5 (see Remark 7 below), it will be more pleasant to work with a distance, δ, defined globally on K , such that the following holds. Lemma 6. The metric δ induces on K m the σ(M (Ω; R 2 ), C 0 (Ω; R 2 )) topology, ∀ m ≥ 0.

The definition of δ we present below is reminiscent of the one of the flat norm [START_REF] Federer | Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften[END_REF]Section 4.1.19,p. 377]. Let C 1 , C 2 ∈ K . Then curl (C 2 -C 1 ) = 0, and therefore there exists some ψ ∈ BV (Ω) such that C 2 = C 1 + Dψ. We define the distance

δ(C 1 , C 2 ) := inf ψ L 1 ; ψ ∈ BV (Ω), C 2 = C 1 + Dψ . (3.23)
To start with, let us note the following simple Lemma 7. We have

〈C 2 , χ〉 -〈C 1 , χ〉 ≤ δ(C 1 , C 2 ) div χ L ∞ , ∀ C 1 , C 2 ∈ K , ∀ χ ∈ C 1 c (Ω; R 2 ).
Proof. This follows from the fact that, if

C 2 = C 1 + Dψ, then 〈C 2 , χ〉 -〈C 1 , χ〉 = |〈Dψ, χ〉| = ˆΩ ψ div χ ≤ ψ L 1 div χ L ∞ .
Proof of Lemma 6. We have to prove that, for C n , C ∈ K m , we have

C n C in M (Ω; R 2 ) ⇐⇒ δ(C n , C ) → 0. (3.24)
Implication " ⇐= " is clear, in view of Lemma 7. Conversely, write C n = C + Dψ n , with ´Ω ψ n = 0. Then ψ n is bounded in BV (Ω). Let ψ ∈ BV (Ω) be such that ´Ω ψ = 0 and, possibly up to a subsequence, ψ n → ψ in L 1 (Ω). Then

ˆΩ ψ div χ = lim n→∞ ˆΩ ψ n div χ = lim n→∞ 〈C n -C , χ〉 = 0, ∀ χ ∈ C 1 c (Ω; R 2 ). (3.25)
It follows that Dψ = 0, and thus ψ = 0. Therefore, for the full original sequence (ψ n ), we have

ψ n → 0 in L 1 (Ω), so that δ(C n , C ) → 0.
In what follows, we endow K with the distance δ. Lemma The proof of Lemma 8 relies on the following Lemma 9. Fix some C 0 ∈ K . For any vector-valued distribution F ∈ D (Ω; R 2 ), we have

F ∈ K ⇐⇒ [F = C 0 + Dψ for some ψ ∈ BV (Ω; Z)]. (3.26)
Before proceeding to the proof of Lemma 9, let us recall a few facts concerning the fine structure of the distributional gradient Dψ with ψ ∈ BV (Ω); see e.g. [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]Chapter 3]. If ψ ∈ BV (Ω; R), then the measure Dψ can be (uniquely) written as a sum of an absolutely continuous part with respect to the Lebesgue measure, D a ψ, whose density is denoted ∇ψ, a Cantor part D c ψ and a jump part D j ψ ([3, Definition 3.91]). With an abuse of notation, we write this decomposition as:

Dψ = ∇ψ + D c ψ + D j ψ.
(3.27)

Assuming that ψ equals, outside its jump set ([3, Definition 3.67]), its approximate limit ([3, Definition 3.63]), we may write

D j ψ = ∞ i=0 (ψ + -ψ -) ν i H 1 S i .
(3.28)

Here, S i are disjoint Borel subsets of C 1 curves γ i , ν i is a normal vector orienting γ i , u ± are the approximate side limits of ψ on γ i ([3, Definition 3.67]) -the "+" and "-" sides are determined by ν i -and i ´Si |ψ + -ψ -| dH 1 < ∞. With a more compact notation, if we set S := ∪ i S i and ν := ν i on S i , then D j ψ = (ψ + -ψ -) ν H 1 S, and S is the jump set of ψ.

We next recall Volpert's chain rule. If f is C 1 and Lipschitz and ψ is BV , then we have

D( f • ψ) = f (ψ)∇ψ + f (ψ)D c ψ + f (ψ + ) -f (ψ -) ψ + -ψ -D j ψ. (3.29)
Proof of Lemma 9. " =⇒ " Let F ∈ K and ψ 0 ∈ BV (Ω; R) such that F = C 0 + Dψ 0 . Then ∇ψ 0 = 0, D c ψ 0 = 0 and D j ψ 0 = F -C 0 . Taking the specific form of the elements in K into account, we have ψ + 0 -ψ - 0 ∈ Z H 1 -a.e. on the jump set of ψ 0 . Volpert's chain rule implies that D [sin(2π ψ 0 )] = 0 and D [cos(2π ψ 0 )] = 0, and thus e ı2π ψ 0 is constant. It follows that, for some appropriate C ∈ R, ψ := ψ 0 +C satisfies e ı2π ψ = 1, and this ψ satisfies the requirements of (3.26).

" ⇐= " Let ψ ∈ BV (Ω; Z) be such that S = C 0 + Dψ. Since sin(2πψ) = 0, we have ∇ψ = 0 and D c ψ = 0 (by (3.29)). On the other hand, we have (possibly after performing a suitable decomposition of the S i 's and by changing, if necessary their orientations)

D j ψ = ∞ i=1 i ν i H 1 S i , with ∞ i=1 i H 1 (S i ) < ∞.
By repeating i times each S i , we may rewrite this as

D j ψ = ∞ =1 ν H 1 Σ , with ∞ =1 H 1 (Σ ) < ∞.
In view of the specific form of C 0 , we find that

F = D ψ -C 0 ∈ K .
Proof of Lemma 8. We may assume that K m is non-empty. Let (C n ) ⊂ K m . Fix some C 0 ∈ K , and write (using Lemma 9) C n = C 0 + D ψ n for some ψ n ∈ BV (Ω; Z). Without loss of generality, we may assume that ffl Ω ψ n ∈ [0, 1). Thus (ψ n ) is bounded in BV (Ω). Let ψ ∈ BV (Ω) be such that an appropriate subsequence satisfies ψ n k → ψ in L 1 (Ω) and a.e. Then ψ is Z-valued. Set C := C 0 +D ψ, so that (again by Lemma 9) C ∈ K . By construction, we have δ(C n k , C ) → 0 as k → ∞. The fact that C ∈ K m follows from Corollary 1.

Corollary 2. Set

K min := {C ∈ K ; C M = T W }.
(3.30)

Then K min is a non-empty set, and (K min , δ) is compact.

Proof. Compactness follows from Lemma 8, non-emptiness from (3.9) and Corollary 1.

Consider next a finite Borel measure µ on K m . By Lemma 6, the integral

ν(χ) := ˆK m 〈C , χ〉 dµ (3.31) makes sense for any χ ∈ C 0 (Ω; R 2 ).
With a slight abuse of notation, we still denote B(0, m) a closed ball of M (Ω; R). We have the following Lemma 10. Set m 0 := T W . Then the map

(K min , δ) C → |C | ∈ (B(0, m 0 ), σ(M (Ω; R), C 0 (Ω; R)) is continuous.
The proof of Lemma 10 uses the following standard result. In view of Lemma 10, if µ is a finite Borel measure on K min , then ´Kmin 〈|C |, ζ〉 dµ makes sense for any ζ ∈ C 0 (Ω; R). This allows us to give a meaning to (3.34) below.

Theorem 4 is a consequence of the following

Theorem 7. Let T ∈ E . Let (v n ) be a minimizing sequence in inf ˆΩ |∇v|; v ∈ W 1,1 (Ω; S 1 ), Jv = πT . (3.32)
Then there exists a probability measure µ on K min such that, up to a subsequence, we have

ˆΩ(v n ∧ ∇v n ) • χ → 2π ˆKmin 〈C , χ〉 dµ, ∀ χ ∈ C 0 (Ω; R 2 ). (3.33)
In addition, if we set for some λ 1 , . . . , λ m ∈ [0, 1] such that m j=1 λ j = 1. Therefore, Theorem 4 is a special case of Theorem 7.

〈κ, ζ〉 := ˆKmin 〈|C |, ζ〉 dµ, ∀ ζ ∈ C 0 (Ω; R), (3.34
Proof of Theorem 7 and Remark 6. " =⇒ " Given any minimizing sequence (v n ) in (3.32) and any ε > 0, we will construct a subsequence (v n k ) and a discrete probability measure µ on K min , depending on the sequence (v n ) and on ε, such that

ˆΩ(v n k ∧ ∇v n k ) • χ -2π ˆKmin 〈C , χ〉 dµ ≤ ε χ W 1,∞ , ∀ χ ∈ C 1 c (Ω; R 2 ). (3.38)
Assuming this achieved, by a diagonal process we may construct probability measures µ j , µ on K min and a subsequence (v n k ) such that

ˆΩ(v n k ∧ ∇v n k ) • χ -2π ˆKmin 〈C , χ〉 dµ j ≤ 1 j χ W 1,∞ , ∀ k ≥ j ≥ 1, ∀ χ ∈ C 1 c (Ω; R 2 ) (3.39)
and µ j µ. We find that this µ satisfies (3.33).

It remains to construct (v n k ) and µ as in (3.38). Let (v n ) be a minimizing sequence in (3.32). We let C α,n (respectively D α,n ) denote the connections (respectively the level sets) associated with v n as in Lemma 4.

By (3.9) and Lemma 4, for every n there exists a full measure set A n ⊂ S 1 such that

H 1 (D α,n ) = C α,n M ≥ T W , ∀ α ∈ A n . (3.40)
On the other hand, using the coarea formula (3.12) and the fact that (v n ) is a minimizing sequence in (3.32), we find that

ˆΩ |∇v n | = ˆS1 H 1 (D α,n ) dα → 2π T W . (3.41)
By (3.40) and (3.41), possibly up to a subsequence there exists some f ∈ L 1 (S 1 ; [0, ∞)) and a full measure set A ⊂ S 1 such that

H 1 (D α,n ) ≤ f (α), ∀n, and H 1 (D α,n ) → T W , ∀ α ∈ A. (3.42) By (3.40) and (3.42)
, we have

C α,n M ≤ f (α) and C α,n M → T W , ∀ α ∈ A. (3.43)
Consider a finite family (C i ) i∈I ⊂ K min , depending on ε, such that the family of balls (B(C i , ε/(2π))) i∈I (for the distance δ) covers K min . Such a family exists, by Corollary 2. By Corollary 1, there exists some t > 0 such that

C M ≥ T W + t, ∀ C ∈ K \ ∪ i∈I B(C i , ε/(2π)). (3.44) Set, for i ∈ I, A n,i,ε := {α ∈ S 1 ; δ(C α,n , C i ) < ε/(2π)}.
In view of Lemma 8 and of (3.43)-(3.44), we have

S 1 \ ∪ i∈I A n,i,ε → 0 as n → ∞.
Possibly after passing to a subsequence, for simplicity still denoted (v n ), we may assume that

ˆS1 \∪ i∈I A n,i,ε C α,n M ≤ ε 2 , ∀ n, (3.45)
|A n,i,ε | → 2πλ i , ∀ i ∈ I, with λ i ∈ [0, 1] and i∈I λ i = 1, (3.46)
and i∈I ||A n,i,ε | -2πλ i | ≤ ε 2 T W . ( 3
(v n ∧ ∇v n ) • χ |∇v n | : ˆΩ(v n ∧ ∇v n ) • χ -2π ˆKmin 〈C , χ〉 dµ = ˆΩ(v n ∧ ∇v n ) • χ -2π i∈I λ i 〈C i , χ〉 = ˆS1 〈C α , χ〉 dα -2π i∈I λ i 〈C i , χ〉 ≤ ˆS1 \∪ i∈I A n,i,ε C α,n M χ L ∞ + i∈I ˆAn,i,ε |〈C α,n -C i , χ〉| + i∈I ||A n,i,ε | -2πλ i | C i M χ L ∞ ≤ ε χ W 1,∞ . (3.48)
This implies the validity of (3.38), and thus of (3.33).

We next derive (3.35) from (3.33) as follows. Set

〈ν, χ〉 := ˆKmin 〈C , χ〉, ∀ χ ∈ C 0 (Ω; R 2 ) (3.49)
(as in the right-hand side of (3.33)).

Since Jv n = πT and v n ∧ ∇v n 2π ν, we find that ν satisfies curl ν = T. We next invoke the following improvement of (3.9) ( [START_REF] Brezis | Sobolev maps with values into the circle[END_REF]):

inf{ ξ M ; curl ξ = T} = T W ; (3.50)
this implies that the quantity T W could be computed not only starting from connections, but also from general Borel measures satisfying curl ξ = T.

Using (3.50) and the fact that (v n ) is a minimizing sequence in (3.32), we find that We first prove that discrete measures belong to S . Equivalently, if C j ∈ K min and λ j ∈ [0, 1], ∀ j = 1, . . . , k, with j λ j = 1, we have to prove that

lim n→∞ ˆΩ |v n ∧ ∇v n | = lim n→∞ ˆΩ |∇v n | = 2π T W ≤ 2π ν M . ( 3
j λ j C j = j λ j |C j |.
(3.56)

Inequality "≤" in (3.56) is clear. In order to obtain the opposite inequality, it suffices to that

j λ j C j M ≥ j λ j C j M . (3.57)
In turn, (3.57) follows from (3.50), since curl ( j λ j C j ) = T.

In order to complete the proof of (3.55), it suffices to prove that S is weakly- * sequentially closed. Let µ n , µ ∈ P (K min ) be such that µ n µ and µ n ∈ S , ∀ n. We clearly have ν(µ n ) ν(µ) and κ(µ n ) κ(µ). Assume that, possibly up to a subsequence, |ν(µ n )| ξ. Then ξ ≥ κ(µ). On the other hand, we have ξ M = κ(µ n ) M = T W = κ(µ) M , and then Lemma 11 implies that ξ = κ(µ) and thus µ ∈ S .

Proof of Theorem 7 completed. During the proof of Lemma 12, we have established not only (3.35), but also (3.37) (while (3.36) is clear). In particular, we have justified Remark 6. Therefore, it remains to establish " ⇐= " As explained in the proof of the " ⇐= " part of Theorem 2, it suffices to consider the case where µ is discrete, that is of the form µ = k j=1 λ j δ C j , with λ j ≥ 0, k j=1 λ j = 1, C j ∈ K min , ∀ j = 1, . . . , k. We construct a sequence (v n ) satisfying (3.33), (3.35), Jv n = πT and ˆΩ |∇v n | → 2π T W (3.58) in the case of two minimal connections, C 1 and C 2 . The construction is exactly the same in case of more than two minimal connections.

Let u ∈ W 1,1 (Ω; S 1 ) be such that Ju = πT. Then curlG = 0, where We set v n := u e -ıϕ n and claim that the sequence (v n ) has all the required properties. Indeed, on the one hand we have Jv n = J(u e -ıϕ n ) = Ju = πT. By (3.60), we have

G := u ∧ ∇u -2π(λ 1 C 1 + λ 2 C 2 ) ∈ M (Ω; R 2 ).
ˆΩ |u ∧ ∇u -∇ϕ n | → 2π λ 1 C 1 + λ 2 C 2 M = 2π T W ; (3.61)
for the latter equality, see (3.56).

We obtain (3.33), (3.35), (3.58) using (3.59), (3.61) and the identity v n ∧ ∇v n = u ∧ ∇u -∇ϕ n .

Remark 7. We present here a proof of Lemma 5 which does not rely on the distance δ. What we have to prove is that, if C n ∈ K and C n µ, then µ ∈ K . In order to prove this, we rely on the following characterization of connections ( [START_REF] Brezis | Sobolev maps with values into the circle[END_REF]). Fix some u ∈ W 1,1 (Ω; S 1 ) such that Ju = π T. Then

[C ∈ K ] ⇐⇒ [∃ ψ ∈ BV (Ω; R) such that u = e ıψ and Dψ = u ∧ ∇u -2π C ].
(3.62)

With C n as above, consider ψ n corresponding to C n as in (3.62). We may assume that ´Ω ψ n ∈ [0, 2π), and then (ψ n ) is bounded in BV . Up to a subsequence, there exists some ψ ∈ BV such that ψ n → ψ in L 1 and a.e. Thus u = e ıψ . On the other hand, we have Dψ = u ∧ ∇u -2π µ. In view of (3.62), we find that µ ∈ C .

Proof of Lemma 1. In view of Corollary 2 and Lemma 10, it suffices to prove the following.

If U is a minimal configuration and C is the corresponding minimal connection, then

|C | = H 1 U. (3.63)
Inequality "≤" is clear. On the other hand, we have

H 1 U M = L(a, M) = C M ,
whence the equality in (3.63).

Proof of Theorem 5. We will actually prove that the existence of µ as in (3.34)-(3.35) is equivalent to the existence of µ as in (1.17)- (1.18). This is obtained as follows. For ζ ∈ C(Ω), set

ϕ ζ : L min → R, ϕ ζ (Q) := 〈Q, ζ〉, ∀Q ∈ L min ,
and

ψ ζ : K min → R, ψ ζ (C ) := 〈|C |, ζ〉, ∀ C ∈ K min .
Note that

[ϕ ζ = ϕ ζ ] ⇐⇒ [ψ ζ = ψ ζ ] (3.64)
and that

ϕ ζ L ∞ = ψ ζ L ∞ . ( 3
|T (ϕ ζ )| ≤ ψ ζ L ∞ = ϕ ζ L ∞ .
We find that T extends to a finite Borel measure µ on L min , with µ M (L min ) ≤ 1. On the other hand, if we let ζ = 1, we see that

L(a, M) µ(L min ) = L(a, M),
and thus µ is a probability measure on L min such that κ(µ) = κ(µ).

The construction of µ starting from µ follows the same lines.

Where continuity enters

In this section we consider two distinct directions where continuity enters.

Minimizing over continuous maps

Given a pair (a, M), we set W(a, M) = v ∈ W 1,1 (Ω; S 1 ) ∩ C(Ω \ {a 1 , . . . , a k }); deg (v, a j ) = M j , ∀ j = 1, . . . , k , and

W ∞ (a, M) = v ∈ W 1,1 (Ω; S 1 ) ∩ C ∞ (Ω \ {a 1 , . . . , a k }); v(z) = z -a j |z -a j | M j
near each a j , ∀ j = 1, . . . , k .

Clearly,

W ∞ (a, M) ⊂ W(a, M) ⊂ v ∈ W 1,1 (Ω; S 1 ); Jv = π k j=1 M j δ a j . Set Λ(a, M) = inf v∈W(a,M) ˆΩ |∇v| and Λ ∞ (a, M) = inf v∈W ∞ (a,M)
ˆΩ |∇v|, so that (with L(a, M) defined in (1.6))

L(a, M) ≤ Λ(a, M) ≤ Λ ∞ (a, M).

Theorem 8. We have

L(a, M) = Λ(a, M) = Λ ∞ (a, M). (4.1)
Clearly, Theorem 8 is an immediate consequence of the following Lemma 13. Given v ∈ W 1,1 (Ω; S 1 ) satisfying Jv = π k j=1 M j δ a j , there exists a sequence

(v n ) ⊂ W ∞ (a, M) such that v n → v in W 1,1 .
Proof. Consider a reference map v 0 ∈ W ∞ (a, M). Since J (v v 0 ) = 0, we may write v = v 0 e ıϕ , where ϕ ∈ W 1,1 (Ω; R). Consider a sequence (ϕ n ) ⊂ C ∞ (Ω; R) such that ϕ n → ϕ in W 1,1 and ϕ n = 0 near each a j ; this is possible since a point in the plane has zero W 1,1 capacity ([10, Section 4.7, Theorem 2]). Letting v n = v 0 e ıϕ n , we have

v n ∈ W ∞ (a, M) and v n → v in W 1,1 . Proof of Lemma 2. With Ω = D, let (v n ) ⊂ W ∞ (0, M) be such that v n → v in W 1,1 .
Possibly after passing to a subsequence, we may also assume that v n (r•) → v(r•) in W 1,1 (S 1 ) (and thus uniformly on S 1 ) for a.e. r ∈ (0, 1). The conclusion of Lemma 2 follows from the stability of the degree under uniform convergence and the fact that deg (v n , C(0, r)) = M, ∀ n, ∀ r ∈ (0, 1).

Continuity of minimizers

Given a pair (a, M), consider the minimization problem inf ˆΩ |∇v|; v ∈ W 1,1 (Ω; S 1 ), Jv = π j M j δ a j .

(4.2)

Theorem 9. Assume that w is a minimizer in (4.2). (Warning: such a minimizer need not exist; see Theorems 1 and 3.) Then w ∈ C(Ω \ {a 1 , . . . , a k }).

Proof. Let ω = D(b, R) be a disc in Ω \ {a 1 , . . . , a k }. Since Jw = 0 in D (ω), we may write, in ω, w = e ıϕ for some ϕ ∈ W 1,1 (ω; R).

where the subscript c stands for compactly supported. In turn, (4.5) follows from the fact that ϕ minimizes (4.3) combined with the fact that for every ψ as in (4.5) there exists a sequence (ψ n ) ⊂ C ∞ c (D(b, r)) such that ψ n → ψ a.e. and ˆD(b,r)

|∇(ϕ + ψ n )| → ∇ϕ + Dψ M
(see the proof of Lemma 14 in Appendix A below). We conclude by invoking the fact that, in a disc, a minimizer of (4.4) subject to a continuous Dirichlet boundary condition is continuous; this follows by combining [17, Theorems 3.5 and 3.7] with [START_REF] Sternberg | The Dirichlet problem for functions of least gradient[END_REF]Theorem 3.6].

Assuming that u equals its approximate limit outside its jump set, define

〈J β u, ζ〉 := - 1 2 ˆΩ ∇ ⊥ ζ•[u ∧(∇u + D c u)]- 1 2 ˆS ρ β (u + , u -) ν•∇ ⊥ ζ dH 1 , ∀ ζ ∈ C 1 c (Ω; R), (B.1)
and set J := J 0 . Here, ∇ ⊥ ζ := (∇ζ) ⊥ = (-∂ζ/∂x 2 , ∂ζ/∂x 1 ). Since the definition (1.1) is equivalent to

〈Ju, ζ〉 = - 1 2
ˆΩ ∇ ⊥ ζ • [u ∧ ∇u], ∀ ζ ∈ C 1 c (Ω; R), (B.2)
we see that J β u coincides with Ju when u ∈ W 1,1 (Ω; S 1 ).

We next investigate in more details the properties of J := J 0 (following [START_REF] Ignat | The space BV(S 2 , S 1 ): minimal connection and optimal lifting[END_REF]); the case of an arbitrary β is similar.

1. The main result in [START_REF] Ignat | The space BV(S 2 , S 1 ): minimal connection and optimal lifting[END_REF] is that 1 π Ju ∈ E , ∀ u ∈ BV (Ω; S 1 ); here, E is the class defined in (3.1).

2. In connection with the study of J, the following "adapted energy" is natural:

u → ˆΩ |Du| S 1 , with |Du| S 1 := |∇u| + |D c u| + dist S 1 (u + , u -) H 1 S. (B.3)
Here, dist S 1 is the geodesic distance on S 1 . By analogy with (1.7), we consider the quantity

Clearly, ´Ω

Σ S 1 (u) := inf ˆΩ |Dv| S 1 ; v ∈ BV (Ω; S 1 ), Jv = Ju in D (Ω) (B.7)
and address the question of the existence of a minimizer in (B.7). In general, a minimizer does not exist. Indeed, assume that 0 ∈ Ω and that dist(0, ∂Ω) = 1. Consider, as in (1.10), the non-empty set X := {x ∈ ∂Ω; |x| = 1}. Then we have the following partial analogue of Theorem 1.

Theorem 10. Let M ≥ 1 be an integer. Then " ⇐= " Assume that #X ≥ 2M + 1. Consider b 1 , . . . , b 2M+1 ∈ X mutually distinct points such that b j+1 is "after" b j for the natural (counterclockwise) orientation of S 1 . Let R j := [0, b j ], ∀ j = 1, . . . , 2M + 1, and set R 2d+2 := R 1 . For j = 1, . . . 2M + 1, let U j denote the domain delimited by R j , R j+1 and ∂Ω. Set v := e ı( j-1)p in U j , with p := 2π M 2M + 1 . If we orientate R j with the unit tangent vector to S 1 at R j (for the natural orientation on S 1 ), then the jump of v across R j is p. Using the fact that p ∈ [0, π), we find that ρ(v + , v -) ν = pν on R j . Since v has only pure jumps, formula (B.1) leads to 5. If θ is a jump point of h, then h(θ+) -h(θ-) ≤ π.

inf ˆΩ |Dv| S 1 ; v ∈ BV (Ω; S 1 ), Jv = π Mδ 0 in D (Ω) = 2π M, (
6. If, in addition h(θ+) -h(θ-) = π, then ρ(e ıh(θ+) , e ıh(θ-) ) = π.

The situation is even worse if we consider the standard energy ´Ω |Dv|. Indeed, with the same assumptions on Ω as above, we have the following result. Proof. We first establish "≤" in (B.19). We assume, with no loss of generality, that 1 ∈ X . For sufficiently small ε and for j = 0, . . . , 2M, let b j = b j,ε = t j e ı jε be the first intersection point with ∂Ω of the half-line {t e ı jε ; t > 0}. Then t 0 = 1 and t j → 1 as ε → 0. Let R j := [0, b j ], j = 0, . . . , 2M, R 2M+1 := R 0 , b 2M+1 := b 0 = 1. Let U j be the domain delimited by R j , R j+1 and the arc of ∂Ω from b j to b j+1 (with the natural orientation on ∂Ω), ∀ j = 0, . . . , 2M. Let α j = α j,ε := e ı j (π-ε) , j = 0, . . . , 2M, and α 2M+1 := α 0 = 1. Define v = v ε by v := α j in U j , j = 0, . . . , 2M. When and equality requires that ∇v = 0, D c v = 0 and that v is a minimizer in (B.8). In particular, this implies "≥" in (B.19). By the above, the proof of Theorem 10 and (B.18), we find that, if v is a minimizer in (B.19), then in D we have v(x) = w(x/|x|), where w ∈ BV (S 1 ; S 1 ) has only pure jumps part (and thus is a step function). Moreover, equality in (B.22) requires that at any jump point x ∈ S 1 of w we have w(x+) = -w(x-), and therefore ρ(w(x+), w(x-)) = ±π . In addition, since v is a minimizer in (B.8), we have ρ(w(x+), w(x-)) > 0 at any jump point x (see (B.15) and (B.16)). However, this is impossible (see the argument after (B.17)). This implies that there exists no minimizer for (B.19).

  | P , N ∈ {a 1 , . . . , a k } ∂Ω, = 1, . . . , n,and n =1 (δ P -δ N ) = n j=1 M j δ a j in D (Ω)

D, density 1

 1 |x| f (x/|x|), which is (-1)-homogenous. In the special case where µ = δ b for some point b ∈ S 1 , we have µ = H 1 [0, b].

. 8 ) 5 .

 85 It turns out that Σ(u) can also be computed starting from connections. More specifically, let u ∈ W 1,1 (Ω; S 1 ) and set T := 1 π Ju. Then we have ([7]) inf { C M ; C is a connection associated with T} = T W = 1 2π Σ(u).

Lemma 11 .

 11 Let ν n , ν be R N -valued finite Borel measures on a locally compact separable metric space X . If ν n ν and lim sup n→∞ ν n M ≤ ν M , then |ν n | |ν|. Proof of Lemma 11. Let µ be such that, possibly up to a subsequence still denoted (ν n ), we have |ν n | µ. Then µ ≥ |ν| ([3, Proposition 1.62 (b)]. Combining this with µ(X ) = lim n→∞ |ν n |(X ) ≤ |ν|(X ), we find that µ = |ν|. Proof of Lemma 10. Let C n , C ∈ K min be such that C n C . Let κ be such that, possibly up to a subsequence, we have |C n | κ. We want prove that κ = |C |. As in the proof of Lemma 11, we have κ ≥ |C |. On the other hand, we clearly have C n M = T W , and thus κ M = T W = C M . We conclude via Lemma 11.

) then |∇v n | dx 2π κ. ( 3 Remark 6 .

 36 As we will see in the proof below, in the special case where there are only finitely many minimal connections, C 1 , . . . , C m , (3.33) and (3.35) become v n ∧ ∇v n dx 2π

  .47) Define µ := i∈I λ i δ C i . In view of Lemma 7 and (3.45)-(3.47), we obtain, using the coarea formula (3.13) with g :=

Lemma 12 .

 12 .51) Combining (3.33), Lemma 11 and (3.51), we obtain |∇v n | dx 2π |ν|. (3.52) If we compare (3.35) with (3.52), formula (3.35) is a consequence of (3.52) and of the following Let µ be a finite positive Borel measure on K min . Set 〈ν, χ〉 = 〈ν(µ), χ〉 := ˆKmin 〈C , χ〉 dµ, ∀ χ ∈ C 0 (Ω; R 2 ). Then 〈|ν|, ζ〉 = ˆKmin 〈|C |, ζ〉 dµ, ∀ ζ ∈ C 0 (Ω; R). (3.53) Proof of Lemma 12. We may assume that µ is a probability. Let κ = κ(µ) denote the measure defined by right-hand side of (3.53). Set S := {µ ∈ P (K min ); |ν(µ)| = κ(µ)}. (3.54) Then (3.53) amounts to S = P (K min ). (3.55)

  Consider ϕ ∈ BV (Ω; R) such that G = Dϕ. By Lemma 14 in Appendix A below, we may find a sequence (ϕn ) ⊂ C ∞ (Ω) ∩ W 1,1 (Ω) such that ∇ϕ n Dϕ in M (Ω) (3.59) and lim n→∞ ˆΩ |u ∧ ∇u -∇ϕ n | = u ∧ ∇u -Dϕ M .(3.60)

  .65) We continue as follows. Given µ ∈ P (K min ) and κ = κ(µ) as in (3.34), set T (ϕ ζ ) := 〈κ, ζ〉 = ˆKmin ψ ζ dµ. By (3.64)-(3.65), T is well-defined, and clearly

  For a.e. r ∈ (R/2, R), we have ϕ |C(b,r) ∈ W 1,1 (C(b, r)), and thus ϕ is continuous on C(b, r). On the other hand, ϕ is a minimizer of the problem inf ˆD(b,r)|∇ψ|; ψ ∈ W 1,1 (D(b, r); R) and ψ = ϕ on C(b, r) . (4.3)This follows from the fact that for every ψ as in (4.3), the mapv = u, in Ω \ D(b,r) e ıψ , in D(b, r) is a competitor in (4.2). We claim that actually ϕ is a minimizer of inf Dψ M ; ψ ∈ BV (D(b, r); R) and ψ = ϕ on C(b, r) . (4.4) Indeed, by a result of Sternberg and Ziemer [18, Theorem 2.2], the minimality of ϕ in (4.4) is equivalent to ˆD(b,r) |∇ϕ| ≤ ∇ϕ + Dψ M , ∀ψ ∈ BV c (D(b, r); R),

〈Jv, ζ〉 = 1 2 ( 2 2 Lemma 16 .

 22216 2M + 1) p ζ(0) = 〈π Mδ 0 , ζ〉, ∀ ζ ∈ C 1 c (Ω).Thus v is a competitor in (B.8). On the other hand, clearlyˆΩ |Dv| S 1 = (2M + 1) p = 2π M,and thus v achieves the minimum in (B.8)." =⇒ " Let ζ 0 (x) = (1 -|x|) + , and let Υ(x) := x ⊥ |x|, so that -∇ ⊥ ζ 0 = 1 D Υ. We claim that the following formula holds for any T ∈ E and v ∈ BV (Ω; S 1 ) such that Jv = πT:π〈T, ζ 0 〉 = 〈Jv, ζ 0 〉 = 1 ˆD v ∧ (Υ • ∇v + Υ • D c v) + 1 ˆS∩D ρ(v + , v -) Υ • ν dH 1 . (B.9)Let us note that, although the quantity 〈Jv, ζ〉 was defined only forζ ∈ C 1 c (Ω), it makes sense more generally for ζ ∈ W 1,∞ 0 (Ω), since T ∈ E . Equality (B.9) is obtaining by a straightforward limiting procedure starting from the formula of 〈Jv, ζ〉, with ζ(x) := f (|x|) and f ∈ C 1 c ([0, 1)) such that f = 1 near 0. If, in addition v is a competitor in (B.8), then T = π M δ 0 , and thus (B.9) readsˆD v ∧ (Υ • ∇v + Υ • D c v) + ˆS∩D ρ(v + , v -) Υ • ν dH 1 = 2π M. (B.10)Assume now that v is a minimizer in (B.8). ThenˆΩ[|∇v| + |D c v|] + ˆS |ρ(v + , v -)| dH 1 = 2π M.(B.11) Comparing (B.10) and (B.11), we find that v is locally constant in Ω \ D and H 1 (S \ D) = 0. (B.12) We next invoke the following straightforward consequence of the polar decomposition (see the proof of Lemma 15). Let µ ∈ M (Ω ; R N ). If G, H : Ω → R N are Borel vector fields such that |G| = 1 |µ|-a.e., G • H = 0 |µ|-a.e. and if ´Ω G • µ = ´Ω |µ|, then H • µ = 0. Proof of Lemma 16. Let µ = F |µ| be the polar decomposition of µ. Then 0 = ˆΩ[|µ| -G • µ] ≥ ˆΩ[|µ| -|G • µ|] = ˆΩ(1 -|F • G|) |µ| ≥ 0. It follows that F ∥ G |µ|-a.e., and thus F ⊥ H |µ|-a.e., whence the conclusion.

  3. h is non-decreasing. 4. h is locally constant in {θ ∈ (β, β + 2π); e ıθ ∈ X }.

Theorem 12 .

 12 Let M ≥ 1 be an integer. Theninf ˆΩ |Dv|; v ∈ BV (Ω; S 1 ), Jv = π Mδ 0 in D (Ω) = 4 M, (B.19)and the infimum in (B.19) is never achieved.

2 π

 2 j+1 , α j ) δ 0 = (2 M (πε) + 2 M ε) δ 0 = 2π M δ 0 ,and thus v is a competitor in (B.19). On the other hand, we haveˆΩ |Dv| = 2M j=0 |α j+1 -α j | = 2 M |1 + e -ıε | + |1 -e -ı2 M ε | → 4 M as ε → 0, whence "≤" in (B.19).For the reverse inequality, consider an arbitrary competitor v in (B.19). Using|z 1 -z 2 | ≥ 2 π ρ(z 1 , z 2 ), ∀ z 1 , z 2 ∈ S 1 ,with equality iff z 1 = ±z 2 , (B.20) we find that |Dv| ≥ 2 π |Dv| S 1 , with equality iff ∇v = 0, D c v = 0 and v + = -v -H 1 -a.e. (B.21) By (B.21), we find that ˆΩ |Dv| ≥ ˆΩ |Dv| S 1 ≥ 4 M, (B.22)

1 ; S 1 ), then we have

	ˆS1	w ∧	∂w ∂τ	d = 2π deg w;	(2.5)
	this is just another way of writing the classical Cauchy formula
	deg w =	1 2ıπ ˆS1	1 w	∂w ∂τ	d .
	Rescaling (2.5) and returning to v we have
	ˆC(0,r)	v ∧	∂v ∂τ	d = 2π deg (v, C(0, r)) for a.e. r ∈ (0, 1).	(2.6)

  6 implies the following Corollary 1. The map K C → C M is lower semi-continuous.

	The next statement is a variant of Federer's compactness theorem [11, Theorem 4.2.17,
	p. 414].
	Lemma 8. (K

m , δ) is compact.

  |Du| S 1 ≥ ´Ω |Du|, since |Du| = |∇u| + |D c u| + |u + -u -| H 1 S.

		(B.4)
	4. The interest of this new energy stems from the straightforward inequality
	|〈Ju, ζ〉| ≤	1 2

∇ζ L ∞ ˆΩ |Du| S 1 , ∀ ζ ∈ C 1 c (Ω; R); (B.5)

this uses the identity |ρ(z 1 , z 2 )| = dist S 1 (z 1 , z 2 ).

5. If we set

T := 1 π Ju, then (B.5) implies ˆΩ |Du| S 1 ≥ 2π T W , ∀ u ∈ BV (Ω; S 1

) such that Ju = πT.

(B.6)

Acknowledgments

We thank Peter Sternberg for pointing out to us the references [17], [18]. HB was partially supported by NSF grant DMS-1207793. Part of this work was completed while PM was invited professor at the Simion Stoilow Institute of Mathematics of the Romanian Academy. He thanks the Institute and the Centre Francophone de Mathématiques in Bucharest for their support.

Appendix A. An approximation lemma

The main result in this appendix is standard when F = 0; see e.g. [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]Theorem 3.9]. Lemma 14. Let F ∈ L 1 (Ω; R N ) and ϕ ∈ BV (Ω). Then there exists a sequence 1 (Ω) such that (A.3)

Therefore, it suffices to find, for each ε > 0, a map

Then ϕ n := ψ 1/n has all the required properties.

Consider an exhaustion Ω = ∪ j≥0 Ω j of Ω, with each Ω j open and Ω j Ω j+1 . Let U 0 := Ω 0 and, for j ≥ 1, U j := Ω j+1 \Ω j-1 . Thus Ω = ∪ j≥0 U j and each x ∈ Ω belongs to at most two U j 's. Let (ξ j ) be a partition of unity subordinated to the covering (U j ) of Ω. Let ρ ∈ C ∞ c (B 1 (0)) be a mollifier. With the sequence (ε j ) to be fixed later, we let ψ := j ψ j , where ψ j := (ϕξ j ) * ρ ε j .

We claim that, for an appropriate choice of ε j , ψ has the required properties. To start with, if ε j < dist(supp ξ j , ∂U j ), then ψ j ∈ C ∞ c (U j ), and thus ψ ∈ C ∞ (Ω). Next, if ε j is sufficiently small, then ψ j -ϕξ j L 1 < ε 2 j+1 , and thus ψϕ L 1 < ε.

We continue by noting that

, provided the ε j 's are sufficiently small.

Similarly, we may write

where S L 1 < ε 2 , provided the ε j 's are sufficiently small. Here, we use the fact that F belongs to L 1 , and is not merely a measure. Finally, for small ε j we have

in the above, the next to the last equality is a consequence of the following simple result.

In particular, if ξ j are non-negative Borel functions, then

Appendix B. Minimization in BV

When u ∈ BV (Ω; S 1 ), there no obvious definition of Ju. We recall here the definition(s) proposed by Ignat [START_REF] Ignat | The space BV(S 2 , S 1 ): minimal connection and optimal lifting[END_REF]. In [START_REF] Ignat | The space BV(S 2 , S 1 ): minimal connection and optimal lifting[END_REF], one considers maps in BV (S 2 ; S 1 ) instead of BV (Ω; S 1 ), but the considerations there may be easily adapted to our setting.

Let, for β ∈ R, Arg β : S 1 → (βπ, β + π] be the determination of the argument given by Arg β (e ıθ ) := θ, ∀ θ ∈ (βπ, β + π], and set Arg := Arg 0 .

Define

and set ρ := ρ 0 .

Proof of Theorem 10 continued. Set

Let us note that Using the fact that |ρ(z 1 , z 2 )| ≤ π, we find from (B.16) that k ≥ 2M. We actually claim that we cannot have k = 2M (whence the desired implication). Indeed, if k = 2M and (B.15) and (B.16) hold, then ρ(α j , α j+1 ) = π, ∀ j = 1, . . . , 2M. This is equivalent to α j+1 = -α j , Im α j+1 ≥ 0 and α j = -1.

(B.17)

However, (B.17) cannot hold simultaneously for j and j+1. This contradiction completes the proof of the theorem. Remark 8. Let us take a closer look at what we have proved. Without any assumption on X , we have obtained that any minimizer v (if it exists) satisfies: With more work, we may prove the following exact counterpart of Theorem 1, that we state here without proof. Theorem 11. v is a minimizer in (B.8) if and only if v is as in (B. [START_REF] Sternberg | The Dirichlet problem for functions of least gradient[END_REF], where w has the following form: w(e ıθ ) = e ıh(θ) , ∀ θ ∈ [β, β + 2π], for some β ∈ R and some h ∈ BV ([β, β + 2π]) such that:

1. h is continuous at its endpoints β and β + 2π.

h(β