Radu Ignat
email: radu.ignat@math.u-psud.fr

Two-dimensional unit-length vector fields of vanishing divergence

Keywords: AMS classification: 35B65, 46E35 Sobolev spaces, regularity, approximation, vortices, kinetic formulation, entropy

We prove the following regularity result: any two-dimensional unit-length divergence-free vector field belonging to H 1/2 (or W 1,1 ) is locally Lipschitz except at a locally finite number of vortices. We also prove approximation results for such vector fields: the dense sets are formed either by unit-length divergence-free vector fields that are smooth except at a finite number of points and the approximation result holds in the W 1,p loc -topology (1 ≤ p < 2), or by everywhere smooth unit-length vector fields (not necessarily divergence-free) and the approximation result holds in a weaker topology.

Introduction

Let Ω ⊂ R 2 be an open bounded set. We will focus on measurable vector fields m : Ω → R 2 that satisfy |m| = 1 a.e. in Ω and ∇ • m = 0 in D ′ (Ω).

(

) 1
One can equivalently consider measurable vector fields v : Ω → R 2 such that |v| = 1 a.e. in Ω and

∇ × v = 0 in D ′ (Ω). (2)
(The passage from (1) to ( 2) is done via v = m ⊥ = (-m 2 , m 1 ).) Locally, m (resp. v) can be written in terms of a stream function ψ, i.e., m = ∇ ⊥ ψ (resp. v = ∇ψ) so that we get to the eikonal equation through ψ:

|∇ψ| = 1. (3)
Typically, one can construct such vector fields by considering stream functions of the form ψ = dist (•, K) for some closed set K ⊂ R 2 ; these vector fields are called Landau states in micromagnetic jargon (see Figure 1). However, not every stream function can be written as a distance function (up to a sign ±1 and an additive constant); for example, if ψ(x) = max{dist (x, P 1 ), dist (x, P 2 )} for two different points P 1 , P 2 ∈ R 2 , then (3) holds even if ψ is not a distance function.

1 Figure 1: Landau states in a rectangle and a disk.

Main results

For p ≥ 1 and s > 0, we denote by W s,p div (Ω, S 1 ) = {m ∈ W s,p (Ω, R 2 ) : m satisfies (1)}.

Regularity results

The first goal is to prove the following regularity result:

Theorem 1 If m ∈ W 1,1 div (Ω, S 1 ) (or m ∈ H 1/2 div (Ω, S 1
)) then m is locally Lipschitz continuous inside Ω except at a locally finite number of singular points. Moreover, every singular point P of m corresponds to a vortex singularity of degree 1 of m, i.e., there exists a sign α = ±1 such that m(x) = α (x -P ) ⊥ |x -P | for every x = P in any convex neighborhood of P in Ω.

In particular, if m ∈ H 1 div (Ω, S 1 ) then m is locally Lipschitz.

Remark 1

The above result was proved by Jabin-Otto-Perthame [START_REF] Jabin | Line-energy Ginzburg-Landau models: zero-energy states[END_REF] in the particular case of zero-energy states of a line-energy Ginzburg-Landau model. More precisely, for ε > 0, one defines the functional E ε :

H 1 (Ω, R 2 ) → R + by E ε (m ε ) = ε Ω |∇m ε | 2 dx + 1 ε Ω (1 -|m| 2 ) 2 dx + 1 ε ∇ • m ε 2 H -1 (Ω) , m ε ∈ H 1 (Ω, R 2 )
(we refer to [START_REF] Ambrosio | Line energies for gradient vector fields in the plane[END_REF][START_REF] Aviles | On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg-Landau type energy for gradient fields[END_REF][START_REF] Jin | Singular perturbation and the energy of folds[END_REF][START_REF] Desimone | A compactness result in the gradient theory of phase transitions[END_REF][START_REF] Jabin | Compactness in Ginzburg-Landau energy by kinetic averaging[END_REF][START_REF] Rivière | Limiting domain wall energy for a problem related to micromagnetics[END_REF][START_REF] Jabin | Line-energy Ginzburg-Landau models: zero-energy states[END_REF] for the analysis of this model). A vector field m : Ω → R 2 is called zero-energy state if there exists a family {m ε ∈ H 1 (Ω, R 2 )} ε→0 satisfying

m ε → m in L 1 (Ω) and E ε (m ε ) → 0 as ε → 0.
Then m satisfies (1) and shares the structure stated in Theorem 1 (see [START_REF] Jabin | Line-energy Ginzburg-Landau models: zero-energy states[END_REF]).

The hypothesis m ∈ W 1,1 (or m ∈ H 1/2 ) in Theorem 1 is a critical regularity assumption in order to avoid line-singularities for vector fields m satisfying (1) (see Proposition 8 in Section 7). As consequence of Theorem 1, one has {m ∈ W 1,1 loc (Ω, R 2 ) : m satisfies (1)} = {m ∈ H Let us now discuss the optimality of the result in Theorem 1: Firstly, observe that Lipschitz regularity cannot be improved.

Proposition 2

There exist an open simply-connected nonconvex piecewise Lipschitz domain Ω and a vector field m ∈ W 1,p div (Ω, R 2 ) for every p ∈ [1, 2) that has infinitely many vortices {P 1 , P 2 , . . . }.

Observe that W 1,p loc (Ω, S 1 ) ⊂ H 1/2 loc (Ω, S 1 ) for p > 1, and the embedding fails for p = 1 (see Proposition 9 in Section 7). Also notice that configurations with infinitely many (interior) vortices can occur only in a non-Lipschitz domain Ω; indeed, if ∂Ω is Lipschitz, then a configuration m ∈ W 1,1 div (Ω, S 1 ) or m ∈ H 1/2 div (Ω, S 1 ) has only a finite number of interior vortex singularities (see Proposition 7).

The main ingredient of the proof of Theorem 1 resides in the following kinetic formulation. It is a generalization to the case of W 1,1 div (Ω, S 1 ) or H 1/2 div (Ω, S 1 ) vector fields of the result in [START_REF] Jabin | Line-energy Ginzburg-Landau models: zero-energy states[END_REF] for zero-energy states of E ε in Remark 1: Proposition 3 (Kinetic formulation) Let m ∈ W 1,1 div (Ω, S 1 ) (or m ∈ H 1/2 div (Ω, S 1 )). For every direction ξ ∈ S 1 , we define χ(•, ξ) : Ω → {0, 1} (resp. χ(•, ξ) : S 1 → {0, 1}) by χ(x, ξ) = χ(m(x), ξ) = 1 for m(x) • ξ > 0, 0 for m(x) • ξ ≤ 0.

Then the following kinetic equation holds for every ξ ∈ S 1 :

ξ • ∇χ(•, ξ) = 0 in D ′ (Ω). (4)
Here, χ corresponds to the concept of characteristic of a weak solution m satisfying [START_REF] Ambrosio | Line energies for gradient vector fields in the plane[END_REF]. Indeed, if m is smooth around a point x ∈ Ω, then the characteristic of m at x (by means of the eikonal equation [START_REF] Aviles | On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg-Landau type energy for gradient fields[END_REF] with m = ∇ ⊥ ψ around x) is given by Ẋ(t, x) = m ⊥ (X(t, x)) with the initial condition X(0, x) = x; then the orbit {X(t, x)} t is a straight line (i.e., X(t, x) = x + tm ⊥ (x) for t in some interval around 0) along which m is perpendicular and constant. Therefore, in the direction ξ := m ⊥ (x), either ∇χ(•, ξ) locally vanishes (if m is constant in a neighborhood of x), or it concentrates on {X(t, x)} t and is oriented by ξ ⊥ (see Figure 2). The knowledge of χ(•, ξ) in every direction ξ ∈ S 1 determines completely the vector field m due to the straightforward formula

m(x) = 1 2 S 1 ξχ(x, ξ) dξ for a.e. x ∈ Ω. ( 5
)
Remark 2 Classical kinetic averaging lemma (see e.g. Golse-Lions-Perthame-Sentis [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF]) shows that a measurable vector-field m : Ω → S 1 satisfying (4) belongs to H 1/2 loc (due to ( 5)). This property can be read as the inverse of Proposition 3 for the case m ∈ H 1/2 (Ω, S 1 ). Moreover, such a vector field has stronger regularity since it shares the structure described in Theorem 1.

Remark 3

The proof of Proposition 3 strongly relies on the structure of lifting of vector fields m ∈ W 1,1 (Ω, S 1 ) (resp. m ∈ H 1/2 (Ω, S 1 )) and an appropriate chain rule. More precisely, if m ∈ W 1,1 (Ω, S 1 ), then there exists a lifting Θ ∈ SBV (Ω, R) such that m = e iΘ a.e. in Ω (see e.g. [START_REF] Giaquinta | Cartesian currents in the calculus of variations. II, volume 38 of Ergebnisse der Mathematik und ihrer Grenzgebiete[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] Dávila | Lifting of BV functions with values in S 1[END_REF], [START_REF] Ignat | The space BV(S 2 , S 1 ): minimal connection and optimal lifting[END_REF]). While if m ∈ H 1/2 (Ω, S 1 ), then one can find a lifting Θ = Θ 1 + Θ 2 with Θ 1 ∈ H 1/2 , Θ 2 ∈ SBV and e iΘ2 ∈ H 1/2 ∩ W 1,1 (see [START_REF] Bourgain | H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation[END_REF]). Recall that SBV (Ω, R d ) is the subspace of vector fields m ∈ BV (Ω, R d ) whose differential Dm has vanishing Cantor part D c m (i.e., D c m ≡ 0 as a measure in Ω).

We address the following open problem:

Open Problem 1 Is it true that every m ∈ BV (Ω, R 2 ) with (1) satisfies m ∈ SBV ?

This question is related with a recent work of Bianchini-DeLellis-Robyr [START_REF] Bianchini | SBV regularity for Hamilton-Jacobi equations in R n[END_REF]: they show that the viscosity solution ψ of a Hamilton-Jacobi equation H(∇ψ) = 0 in Ω (with a uniformly convex hamiltonian H) satisfies ∇ψ ∈ SBV . Open Problem 1 asks whether for the particular case of the eikonal equation ( 3), the result in [START_REF] Bianchini | SBV regularity for Hamilton-Jacobi equations in R n[END_REF] still holds when replacing the assumption of viscosity solution with the hypothesis of a general solution ψ of (3) with ∇ψ ∈ BV .

Density results

The second goal of the paper is to present approximation results for the class of vector fields

W 1,1 div (Ω, S 1 ) (or H 1/2 div (Ω, S 1
)): our subsets are formed either by divergence-free vector fields that are smooth except at a finite number of points and the approximation result holds in the W 1,1 (or H 1/2 )-topology, or by everywhere smooth vector fields (not necessarily divergence-free) and the approximation result holds in a weaker topology. We start by extending Bethuel-Zheng's density result (see [START_REF] Bethuel | Density of smooth functions between two manifolds in Sobolev spaces[END_REF]) for W 1,1 (Ω, S 1 ) vector fields, respectively Riviere's density result (see [START_REF] Rivière | Dense subsets of H 1/2 (S 2 , S 1 )[END_REF]) for H 1/2 (Ω, S 1 ) vector fields to the case of divergence-free vector fields:

Theorem 2 Let Ω be a Lipschitz bounded simply-connected domain and m ∈ W 1,1 div (Ω, S 1 ) (or m ∈ H 1/2 div (Ω, S 1 )). Then m has a finite number k ≥ 0 of vortices {P 1 , . . . , P k } and m can be approximated in W 1,p loc (Ω) (for any p ∈ [1, 2)) by divergence-free vector fields m n ∈ C ∞ (Ω\{P 1 , . . . , P k }, S 1 ) that are smooth except at the k vortex points of m. In particular, if m ∈ H 1 div (Ω, S 1 ), the sequence {m n } can be chosen to be smooth everywhere in Ω and the approximation result holds in H 1 loc (Ω).

In various applications (see e.g. Remarks 1 and 4), we need to approximate vector fields m (with the structure given in Theorem 1) by H 1 (Ω, S 1 ) vector fields. But H 1 (Ω, S 1 )-vector fields cannot allow for vortices. Therefore, an approximation result by everywhere smooth vector fields is needed in some weak topology. What is the optimal weak topology where such a density result holds? The following result shows that L 1 -topology is too strong for having density of smooth vector fields of vanishing divergence and values in S 1 .

Proposition 4 Let m : B 2 → S 1 be the vortex configuration m(x) = x ⊥
|x| in the unit disk B 2 . Then there exists no sequence of vector fields

m n ∈ C ∞ (Ω, S 1 ) of vanishing divergence such that m n → m a.e. in B 2 .
We now generalize this property: the density result still fails if we relax the divergence-free constraint on the approximated smooth vector fields, but we impose this restriction in the limit in L 1 -topology (or H -s weak topology for some s ∈ [0, 1 2 )).

Proposition 5 Let m : B 2 → S 1 be the vortex configuration m(x) = x ⊥ |x| in B 2 . Then there exists no sequence of vector fields m n ∈ C ∞ (Ω, S 1 ) such that m n → m a.e. in B 2 and one of the following two conditions holds:

a) ∇ • m n → 0 in L 1 (B 2 ); b) ∇ • m n ⇀ 0 weakly in H -s (B 2 ) for some s ∈ [0, 1 2 ).
Finally, we prove an approximation result in L 1 -topology by smooth vector fields with values in S 1 (not necessary divergence-free), but the divergence-free constraint holds in the limit in the H -1/2 topology. This topology is optimal due Proposition 5 b).

Theorem 3 Let Ω be a Lipschitz bounded simply-connected domain and m

∈ W 1,1 div (Ω, S 1 ) (or m ∈ H 1/2 div (Ω, S 1 )).
Then there exists a sequence of vector fields

m n ∈ C ∞ (Ω, S 1 ) such that m n → m a.e. in Ω and (∇ • m n )1 Ω → 0 in Ḣ-1/2 (R 2 ).
Remark 4 The motivation of Theorem 3 comes from thin-film micromagnetics. The following 2D energy (see [START_REF] Desimone | A reduced theory for thin-film micromagnetics[END_REF]) is considered as an approximation of the full 3D micromagnetic model for thin films: for ε > 0, one defines the functional

F ε : H 1 (Ω, S 1 ) → R + by F ε (m ε ) = ε Ω |∇m ε | 2 dx + (∇ • m ε )1 Ω 2 Ḣ-1/2 (R 2 ) , m ε ∈ H 1 (Ω, S 1 ).
This model was analyzed in [START_REF] Desimone | 2-d stability of the Néel wall[END_REF], [START_REF] Ignat | A compactness result in thin-film micromagnetics and the optimality of the Néel wall[END_REF], [START_REF] Ignat | Vortex energy and 360 • Néel walls in thin-film micromagnetics[END_REF]. In particular, it is proved in [START_REF] Ignat | Vortex energy and 360 • Néel walls in thin-film micromagnetics[END_REF] that a vortex configuration m 0 (x) = x ⊥ |x| in B 2 is a zero-energy state, i.e., there exists a family {m ε ∈ H 1 (B 2 , S 1 )} such that m ε → m 0 a.e. in B 2 and F ε (m ε ) → 0 as ε → 0. The role of Theorem 3 is to generalize this approximation result for every vector field m ∈ W 1,1 (resp. m ∈ H 1/2 ) satisfying [START_REF] Ambrosio | Line energies for gradient vector fields in the plane[END_REF].

Most of the results of the paper have been announced in [START_REF] Ignat | Gradient vector fields with values into S 1[END_REF]. The outline of the paper is the following: in Section 3, we define a class of entropies that is used in the proof of Proposition 3 and Theorem 1. In Section 4, we present several examples of vector fields satisfying (1); in particular, we prove Propositions 1 and 2. Section 5 deals with the proof of non-density results in Propositions 5 and 4, while in Section 6, we present the proofs of density results in Theorems 2 and 3. We finish with Section 7 where we recall some properties of Sobolev spaces.

3 Entropies. Proof of Theorem 1.

The starting point consists in regarding the structure (1) of our configurations as a scalar conservation law. Indeed, writing m = (u, h(u)) for the flux h(u) = ± √ 1u 2 , the vanishing divergence condition in m turns into

∂ t u + ∂ s h(u) = 0, (6)
where (t, s) := (x 1 , x 2 ) correspond to (time, space) variables. Let us recall some definitions from the theory of scalar conservation laws. Since the flux h is nonlinear, there is in general no smooth solution of the Cauchy problem associated to [START_REF] Bourgain | H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation[END_REF]. Therefore, the solutions of (6) are to be understood in the sense of distributions and in general, there are infinitely many weak solutions for the Cauchy problem. The concept of entropy solution has been formulated in order to have uniqueness (see Kružkov [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]). To introduce this notion, the pair (entropy, entropy-flux) is defined as a couple of scalar functions (η, q) such that dq ds = dh ds dη ds and for every smooth solution u of (6), the entropy production vanishes, i.e.,

∂ t [η(u)] + ∂ s [q(u)] = 0.
A solution u of (6) (in the sense of distributions) is called entropy solution if for every convex entropy η, the entropy production ∂ t [η(u)] + ∂ s [q(u)] ≤ 0 is a nonpositive measure. Such solutions u have the property that for every pair (η, q), the entropy production concentrates on lines (corresponding to "shocks" of u). It suggests the interest of using "global" quantities (η, q) to detect "local" line-singularities of u. This idea has been used by Jin-Kohn [START_REF] Jin | Singular perturbation and the energy of folds[END_REF], Aviles-Giga [START_REF] Aviles | On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg-Landau type energy for gradient fields[END_REF] DeSimone-Kohn-Müller-Otto [START_REF] Desimone | A compactness result in the gradient theory of phase transitions[END_REF], Ambrosio-DeLellis-Mantegazza [START_REF] Ambrosio | Line energies for gradient vector fields in the plane[END_REF], Ignat-Merlet [START_REF] Ignat | Lower bound for the energy of Bloch walls in micromagnetics[END_REF][START_REF] Ignat | Entropy method for line energies[END_REF].

In the sequel we will always use the following notion of entropy introduced in [START_REF] Desimone | A compactness result in the gradient theory of phase transitions[END_REF] (see also [START_REF] De | Structure of entropy solutions to the eikonal equation[END_REF], [START_REF] Ignat | Entropy method for line energies[END_REF]). It corresponds to the pair (entropy, entropy-flux) from the scalar conservation laws, but the pair is defined in terms of the couple (u, h(u)).

Definition 1 ([13]) We will say that

Φ ∈ C ∞ (S 1 , R 2 ) is an entropy if d dθ Φ(z) • z = 0, for every z = e iθ = (cos θ, sin θ) ∈ S 1 . (7)
Here, d dθ Φ(z) = d dθ [Φ(e iθ )] stands for the angular derivative of Φ. The set of all entropies is denoted by EN T . This notion is coherent with the property that a smooth vector field m satisfying (1) induces vanishing entropy production ∇ • [Φ(m)] = 0. In fact, it is equivalent to Definition 1 as stated in the following property:

Proposition 6 Let Φ ∈ C ∞ (S 1 , R 2 ). Then Φ is an entropy if and only if for every m ∈ W 1,1 div (Ω, S 1 ) (or m ∈ H 1/2 div (Ω, S 1
)), the following identity holds:

∇ • [Φ(m)] = 0 in D ′ (Ω). ( 8
)
Proof. We divide the proof in several steps:

Step 1. If Φ ∈ EN T and m ∈ W 1,1 div (Ω, S 1
), then (8) holds. Indeed, let us consider an SBV lifting Θ of m in Ω, i.e., m = e iΘ in Ω (see e.g. [START_REF] Giaquinta | Cartesian currents in the calculus of variations. II, volume 38 of Ergebnisse der Mathematik und ihrer Grenzgebiete[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] Dávila | Lifting of BV functions with values in S 1[END_REF], [START_REF] Ignat | The space BV(S 2 , S 1 ): minimal connection and optimal lifting[END_REF]). For the SBV -function Θ, the measure DΘ splits into two terms

DΘ = D a Θ + (Θ + -Θ -)νH 1 J(Θ) (9)
where D a Θ = ∇ a ΘH 2 is the absolutely continuous part of the measure DΘ with respect to twodimensional Lebesgue measure H 2 and the last term stands for the jump part concentrated on the H 1 -rectifiable set J(Θ) oriented by the unit normal vector ν and the traces of Θ on J(Θ) with respect to ν are denoted by Θ ± (x) = lim ε↓0 Θ(x ± εν(x)) in L 1 loc (J(Θ)). (Recall that a SBV function Θ has vanishing Cantor part of the measure DΘ.) Since m ∈ W 1,1 , the chain rule applied to m = e iΘ implies that Θ + -Θ -∈ L 1 (J(Θ), 2πZ) and ∇ a Θ = m ∧ ∇m ∈ L 1 (Ω). Moreover, since ∇ • m = 0, the chain rule also yields m ⊥ • ∇ a Θ = 0 in L 1 (Ω). Therefore, there exists a function

λ := m • ∇ a Θ ∈ L 1 (Ω) such that ∇ a Θ = λ m. Applying now the chain rule for Φ(e iΘ ) ∈ W 1,1 , we deduce ∇ • [Φ(m)] = ∇ • [Φ(e iΘ )] = d dθ Φ(m) • ∇ a Θ = d dθ Φ(m) • m λ (7) = 0 in L 1 (Ω), (10)
i.e., (8) holds.

Step

2. If Φ ∈ EN T and m ∈ H 1/2 div (Ω, S 1
), then (8) holds. Indeed, let B ⊂ Ω be an arbitrary ball and let us consider a lifting

Θ = Θ 1 + Θ 2 of m in B with Θ 1 ∈ H 1/2 (B), Θ 2 ∈ SBV (B) and h = e iΘ2 ∈ W 1,1 ∩ H 1/2 (B, S 1 ) (see Brezis-Bourgain-Mironescu [6], Theorem 5). Therefore, the corresponding decomposition (9) of Θ 2 satisfies Θ + 2 -Θ - 2 ∈ L 1 (J(Θ 2 ), 2πZ) and ∇ a Θ 2 = h ∧ ∇h ∈ L 1 (B). Claim. For every Ψ ∈ C 2 (S 1 , R 2 ), then Ψ(m) ∈ H 1/2 (B) and ∇ • [Ψ(m)] = d dθ Ψ(m) • (∇Θ 1 + ∇ a Θ 2 ) in H -1/2 (B). (11)
Here, H -1/2 (B) is the dual space of H 1/2 00 (B) (see Section 7 for more details).

Proof of Claim. The fact that if

m ∈ H 1/2 (B) then Ψ(m) ∈ H 1/2 (B) is standard and follows from B B |Ψ(m(x)) -Ψ(m(y))| 2 |x -y| 3 dxdy ≤ ∇Ψ 2 L ∞ B B |m(x) -m(y)| 2 |x -y| 3 dxdy < ∞. Therefore, ∇ • [Ψ(m)] ∈ H -1/2 (B) (since the differential operator is continuous from H 1/2 (B) to H -1/2 (B)
, see e.g. Lions-Magenes [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]). Before proving [START_REF] Desimone | 2-d stability of the Néel wall[END_REF], let us observe that the RHS of ( 11) is a distribution (a-priori, it doesn't belong to H -1/2 (B)). Indeed, one has that

d dθ Ψ(m) ∈ H 1/2 ∩ L ∞ (B) (here we use that Ψ ∈ C 2 (S 1 ) so that d dθ Ψ ∈ C 1 (S 1 )); therefore, d dθ Ψ(m) • ∇ a Θ 2 ∈ D ′ (B) as a duality product between L ∞ and L 1 , while d dθ Ψ(m)•∇Θ 1 ∈ D ′ (B) since for every test function ζ ∈ C ∞ c (B), one has < d dθ Ψ(m) • ∇Θ 1 , ζ > (D ′ (B),C ∞ c (B)) =< ∇Θ 1 , ζ d dθ Ψ(m) > (H -1/2 (B),H 1/2 00 (B)) . (12)
In order to prove [START_REF] Desimone | 2-d stability of the Néel wall[END_REF], we consider an approximating sequence Θ

1,n ∈ C 1 ( B) such that Θ 1,n → Θ 1 in H 1/2 (B). We set m n = e i(Θ1,n+Θ2) ∈ W 1,1 ∩ H 1/2 (B, S 1
). Applying the clain rule as in [START_REF] De | Structure of entropy solutions to the eikonal equation[END_REF] for

Ψ(m n ) ∈ W 1,1 , we obtain ∇ • [Ψ(m n )] = d dθ Ψ(m n ) • (∇Θ 1,n + ∇ a Θ 2 ) in L 1 ∩ H -1/2 (B).
We want to pass to the limit n → ∞ in order to get to [START_REF] Desimone | 2-d stability of the Néel wall[END_REF]. For that, we have e iΘ1,n → e iΘ1 in H 1/2 (B) and we deduce that m n → m in H 1/2 (B) by Proposition 10 (see Section 7) and also, a.e. in B (up to a subsequence). The continuity of the differential operator from

H 1/2 to H -1/2 combined with Proposition 10 lead to ∇ • [Ψ(m n )] → ∇ • [Ψ(m)] in H -1/2 (B).
On the other hand, the same arguments lead to d dθ Ψ(m n ) → d dθ Ψ(m) in H 1/2 (B) and a.e. in B (up to a subsequence); thus, by duality as in [START_REF] Desimone | A reduced theory for thin-film micromagnetics[END_REF], one has

d dθ Ψ(m n ) • ∇Θ 1,n → d dθ Ψ(m) • ∇Θ 1 in D ′ (B)
and by dominated convergence theorem, one also deduces that

d dθ Ψ(m n ) • ∇ a Θ 2 → d dθ Ψ(m) • ∇ a Θ 2 in L 1 (B)
, which yields [START_REF] Desimone | 2-d stability of the Néel wall[END_REF].

Coming back to Step 2,[START_REF] Desimone | 2-d stability of the Néel wall[END_REF]

applied for Ψ(z) = z for z ∈ S 1 yields m ⊥ • (∇Θ 1 + ∇ a Θ 2 ) = 0 in H -1/2 (Ω) since ∇ • m = 0. Idem, defining Ψ(z) = -z ⊥ for z ∈ S 1 , (11) leads to λ := m•(∇Θ 1 +∇ a Θ 2 ) = ∇•[Ψ(m)] ∈ H -1/2 (B)
. Therefore, one writes (as at Step 1) ∇Θ 1 +∇ a Θ 2 = mλ which belongs to D ′ (B) by duality as in [START_REF] Desimone | A reduced theory for thin-film micromagnetics[END_REF]. As in [START_REF] De | Structure of entropy solutions to the eikonal equation[END_REF], we apply [START_REF] Desimone | 2-d stability of the Néel wall[END_REF] to obtain

∇ • [Φ(m)] = d dθ Φ(m) • m λ=0 in D ′ (B).
Since B is an arbitrary ball in Ω, using a partition of unity, we conclude that (8) holds in D ′ (Ω).

Step 3.

Conversely, let Φ ∈ C ∞ (S 1 , R 2 ) such that (8) holds for every m ∈ W 1,1 div (Ω, S 1 ) (or m ∈ H 1/2 div (Ω, S 1 )). Set z ∈ S 1 .
We prove that (7) holds for z using the same argument as in [START_REF] Ignat | Lower bound for the energy of Bloch walls in micromagnetics[END_REF][START_REF] Ignat | Entropy method for line energies[END_REF]. Up to translations, we may assume that Ω contains the origin 0. Motivated by [START_REF] De | Structure of entropy solutions to the eikonal equation[END_REF], we consider a map m given by the vortex structure centered at z ⊥ , i.e.,

m(x) := x -z ⊥ |x -z ⊥ | ⊥ for x ∈ Ω. Then m ∈ W 1,p div (Ω, S 1 ) for every p ∈ [1, 2) (in particular, m ∈ H 1/2 div (Ω, S 1
)) and m(0) = z. Moreover, since m is smooth around the origin 0, m has a smooth lifting Θ around 0 (unique up to a constant) that satisfies ∇Θ(0) = z. Then by [START_REF] Coddington | Theory of ordinary differential equations[END_REF] we know that ∇ • [Φ(m)](0) = 0. Therefore, as in [START_REF] De | Structure of entropy solutions to the eikonal equation[END_REF], we obtain

d dθ Φ(z) • z = d dθ Φ(m(0)) • ∇Θ(0) = ∇ • [Φ(m)](0) = 0.
Remark 5 A way to construct entropies is given by the following equivalent definition

: if Φ ∈ C ∞ (S 1 , R 2 ), then Φ ∈ EN T is an entropy if and only if there exists a (unique) 2π-periodic ϕ ∈ C ∞ (R) such that for every z = e iθ ∈ S 1 , Φ(z) = ϕ(θ)z + dϕ dθ (θ)z ⊥ (13)
(see details in [START_REF] Desimone | A compactness result in the gradient theory of phase transitions[END_REF][START_REF] Ignat | Entropy method for line energies[END_REF]).

As a consequence, we prove the kinetic formulation (4):

Proof of Proposition 3. For every ξ ∈ S 1 , we define "elementary entropies" Φ ξ :

S 1 → R 2 given by Φ ξ (z) := ξ χ(z, ξ) = ξ for z • ξ > 0, 0 for z • ξ ≤ 0.
Although Φ ξ is not a smooth entropy (in fact, Φ ξ has a jump at the points ±ξ ⊥ ∈ S 1 ), the equality [START_REF] Brezis | W 1,1 -maps with values into S 1 . In Geometric analysis of PDE and several complex variables[END_REF] trivially holds in D ′ (S 1 ). That's why Φ ξ is a generalized entropy. Moreover, as shown in [START_REF] Desimone | A compactness result in the gradient theory of phase transitions[END_REF], there exists a sequence of smooth entropies {Φ k } ⊂ EN T such that {Φ k } is uniformly bounded and lim k Φ k (z) = Φ ξ (z) for every z ∈ S 1 . Indeed, this smoothing result follows by [START_REF] Desimone | A compactness result in the gradient theory of phase transitions[END_REF]: if one writes ξ = e iθ0 with θ 0 ∈ (-π, π], then the unique 2π-periodic function ϕ ∈ C(R) satisfying [START_REF] Desimone | A compactness result in the gradient theory of phase transitions[END_REF] for Φ ξ is given by:

ϕ(θ) = ξ • z1 {z•ξ>0} = cos(θ -θ 0 )1 {θ-θ0∈(-π/2,π/2)} for z = e iθ , θ ∈ (-π + θ 0 , π + θ 0 ).
By [START_REF] Desimone | A compactness result in the gradient theory of phase transitions[END_REF] for Φ ξ , the choice of ϕ ′ is fixed at the jump points ±ξ ⊥ ∈ S 1 :

ϕ ′ (θ) = -sin(θ -θ 0 )1 {θ-θ0∈(-π/2,π/2)} for θ ∈ (-π + θ 0 , π + θ 0 ). Now, one regularizes ϕ by 2π-periodic functions ϕ k ∈ C ∞ (R) that are uniformly bounded in W 1,∞ (R) and lim k ϕ k (θ) = ϕ(θ) as well as lim k ϕ ′ k (θ) = ϕ ′ (θ)
for every θ ∈ R. Thus, the desired (smooth) approximating entropies Φ k are given by ϕ k via [START_REF] Desimone | A compactness result in the gradient theory of phase transitions[END_REF]. Therefore, Proposition 6 implies that for every

m ∈ W 1,1 div (Ω, S 1 ) (or m ∈ H 1/2 div (Ω, S 1 )), one has Ω Φ k (m) • ∇ζ dx = 0 for every ζ ∈ C ∞ c (Ω)
and by the dominated convergence theorem, one concludes that

0 = ∇ • [Φ ξ (m)] = ∇ • [ξ χ(m, ξ)] = ∇ • [ξχ(•, ξ)] = ξ • ∇χ(•, ξ).
Proof of Theorem 1. It is a consequence of Proposition 3 combined with the strategy of Jabin-Otto-Perthame (see Theorem 1.3 in [START_REF] Jabin | Line-energy Ginzburg-Landau models: zero-energy states[END_REF]). For completeness of the writing, let us recall the main steps of that argument: let m : Ω → S 1 be a measurable function that satisfies (4) for every ξ ∈ S 1 . Notice that the divergence-free condition is automatically satisfied (in D ′ (Ω)) because of [START_REF] Bianchini | SBV regularity for Hamilton-Jacobi equations in R n[END_REF]. The first step consists in defining a L ∞ -trace of m on each segment Σ ⊂ Ω. More precisely, if

Σ := {0} × [-1, 1] ⊂ Ω, then there exists a trace m ∈ L ∞ (Σ, S 1 ) such that lim r→0 1 r r -r 1 -1 |m(x 1 , x 2 ) -m(x 2 )| dx 2 dx 1 = 0 and for each Lebesgue point (0, x 2 ) ∈ Σ of m, one has m(0, x 2 ) = m(x 2 ).
Observe that this step is straightforward in the case of m ∈ W 1,1 div (Ω, S 1 ); however, it is essential for the case of m ∈ H 1/2 div (Ω, S 1 ). The second step is to prove that the trace m of m on Σ is almost everywhere orthogonal at Σ (which coincides with the classical principle of characteristics for smooth vector fields m). The key point for that resides in a relation of order of characteristics of m, i.e., for every two Lebesgue points x, y ∈ Ω of m with the segment [x, y] ⊂ Ω, the following implication holds:

m(x) • (y -x) > 0 ⇒ m(y) • (y -x) > 0.
The final step consists is proving that on any open convex subset ω ⊂ Ω with d = dist (ω, ∂Ω) > 0, only two situations may occur: either two characteristics of m intersect at P ∈ Ω with dist (P, ω) < d and m Proof of Proposition 1. Let Θ : (0, 1) → ( π 4 , π 3 ) be a a Lipschitz function that is not in C 1 (0, 1). On the "space" axis s, we define m as given by m(s, 0) = (cos Θ(s), sin Θ(s)) = e iΘ(s) ∈ S 1 for every s ∈ (0, 1).

(x) = ± (x-P ) ⊥ |x-P | for x ∈ ω \ {P }, or m is 1/d-Lipschitz in ω, i.e., |m (x)
Then m has a unique Lipschitz extension satisfying (1): The initial value (at "time" t = 0) of m determines the characteristics along which m remains constant. More precisely, we define the flow of characteristics F : Dom(F ) = (0, 1) × (-

1 3 d ds Θ L ∞ , 1 3 d ds Θ L ∞ ) → R 2 as F (s, t) = (s, 0) + tm(s, 0) ⊥ = s + ite iΘ(s) ∈ C for every (s, t) ∈ Dom(F ).
The choice of "time" range is done in order that F is a bi-Lipschitz homeomorphism onto its open range, denoted Ω (which implies that characteristics of m do not intersect in the domain Ω). Indeed, one computes that

∇F (s, t) = ( ∂ ∂s F ∂ ∂t F ) = 1 -t cos Θ(s) d ds Θ(s) -sin Θ(s) -t sin Θ(s) d ds Θ(s) cos Θ(s) with det ∇F (s, t) = cos Θ(s) -t d ds Θ(s) ≥ 1 2 -t d ds Θ(s) ≥ 1 6 in Dom(F ).
Here, we used that

|t| ≤ 1 3 d ds Θ L ∞ and cos Θ(s) ≥ 1/2. ( 14
)
Therefore, F is a local bi-Lipschitz homeomorphism. In order to be a global homeomorphism, we show that F is injective on Dom(F ). Assume by contradiction that there exist two points (s, t) = (s, t) in Dom(F ) so that F (s, t) = F (s, t). Then t, t = 0 and t t = cos Θ(s) cos Θ(s) . It follows that

|s -s| = |t(sin Θ(s) - t t sin Θ(s))| = t cos Θ(s) sin(Θ(s) -Θ(s)) ≤ 2 3 d ds Θ L ∞ |Θ(s) -Θ(s)| ≤ 2 3
|s -s| which would mean that s = s and then, t = t which is a contradiction. (In the above inequalities, we used again [START_REF] Giaquinta | Cartesian currents in the calculus of variations. II, volume 38 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF].) Denoting by G = (G 1 , G 2 ) : Ω → Dom(F ) the inverse of F , we have that

∇G(F (s, t)) = 1 cos Θ(s) -t d ds Θ(s) cos Θ(s) sin Θ(s) t sin Θ(s) d ds Θ(s) 1 -t cos Θ(s) d ds Θ(s) . (15)
We now define m : Ω → S 1 by m(F (s, t)) = m(s, 0) for every (s, t) ∈ Dom(F ).

Obviously, m is a Lipschitz function in Ω with values in S 1 . Since the open segment (0, 1)×{0} ⊂ Ω and Θ is not C 1 in (0, 1), then m is not a C 1 map in Ω. Finally, we check that

∇ • m = 0 in Ω. Indeed, if (x 1 , x 2 ) = F (s, t) ∈ Ω, then m(x) = m(G 1 (x), 0) = e iΘ(G1(x)) and ∇ • m(x) = ∂ ∂x 1 [cos Θ(G 1 (x))] + ∂ ∂x 2 [sin Θ(G 1 (x))] = -sin Θ(s) ∂ ∂x 1 G 1 (F (s, t)) + cos Θ(s) ∂ ∂x 2 G 1 (F (s, t)) d ds Θ(s) (15)
= 0.

H 1 vector fields (1) that are not globally Lipschitz

The case of non-convex domains Ω. We consider the kink domain Ω = {re iθ : r ∈ (0, 1), |θ| < r} and the boundary vortex configuration in the origin: m(x) = x ⊥ |x| for x ∈ Ω. Then m ∈ H 1 (Ω, S 1 ) and m is not globally Lipschitz in Ω (but only locally Lipschitz).

The case of convex domains Ω. There exist a convex domain Ω and a vector field m ∈ H 1 div (Ω, S 1 ), such that m is not globally Lipschitz in Ω. Indeed, we use the construction in the proof of Proposition 1. Let Θ : (0, 1/10) → (-∞, 0) be given by Θ(s) = -√ s for s ∈ (0, 1/10). On the "space" axis (0, 1) × {0}, we define m by m(s, 0) = (cos Θ(s), sin Θ(s)) = e iΘ(s) ∈ S 1 for every s ∈ (0, 1/10).

The flow of characteristics is defined by

F : Dom(F ) = {(s, t) : s ∈ (0, 1/10), 0 < t < √ s/10} → R 2 as F (s, t) = (s, 0) + tm(s, 0) ⊥ = s + ite iΘ(s) ∈ C for every (s, t) ∈ Dom(F ).
Then F is locally a bi-Lipschitz homeomorphism since

det ∇F (s, t) = cos Θ(s) -t d ds Θ(s) ≥ cos Θ(1/10) > 0 in Dom(F ).
We show that no characteristics intersect inside Dom(F ) (i.e., F is injective in Dom(F )): Assume by contradiction that there exist two points (s, t) = (s, t) in Dom(F ) so that F (s, t) = F (s, t). Then t, t = 0 and t t = cos Θ(s) cos Θ(s) which would imply

|s -s| = |t(sin Θ(s) - t t sin Θ(s))| = t cos √ s sin( √ s - √ s) ≤ √ s 5 |s -s| √ s + √ s ≤ 1 5
|s -s| and would mean that s = s and then t = t which is a contradiction. Here, we used that 0 < t < √ s 10 and cos Θ(s) ≥ 1/2 if s ∈ (0, 1/10). Let Ω be any convex domain included in F (Dom(F )) such that Ω contains the segment [0, 1/10] × {0} and G : Ω → F -1 (Ω) be the inverse of F . Then [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF] holds and we set m(F (s, t)) = m(s, 0) for every (s, t) ∈ G(Ω).

We compute that

∇m(F (s, t)) = dm ds (s, 0) 0 R 2 • ∇G(F (s, t))
so that [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF] implies

|∇m|(F (s, t)) = 1 det ∇F (s, t) dm ds (s) = 1 cos( √ s) + t 2 √ s dΘ ds (s) .
Then

Ω |∇m(x)| 2 dx = G(Ω) |∇m| 2 (F (s, t)) det ∇F (s, t) dsdt ≤ 1/10 0 dΘ ds (s) ds √ s/10 0 1 cos( √ s) -t dΘ ds (s) dΘ ds (s) dt ≤ 1/10 0 ds 2 √ s ln cos( √ s) + 1/20 -ln(cos( √ s)) < ∞.
Therefore, m ∈ H 1 div (Ω, S 1 ) and m is not globally Lipschitz in Ω since dΘ ds blows up at s = 0.

Vector fields (1) with arbitrary many vortices

The geometry of the domain Ω determines the number of vortices of a vector field [START_REF] Ambrosio | Line energies for gradient vector fields in the plane[END_REF].

Remark 6 (i) If Ω is a convex domain and m ∈ W 1,1 div (Ω, S 1 ) (or m ∈ H 1/2 div (Ω, S 1
)), then m has either no (interior) vortices, or one interior vortex in Ω. It is a consequence of the final step in the proof of Theorem 1 (see [START_REF] Jabin | Line-energy Ginzburg-Landau models: zero-energy states[END_REF] for more details). (ii) If Ω = R 2 is a smooth simply connected domain and additionally we impose the boundary condition m • n = 0 on ∂Ω, then either Ω is a disk and m has a vortex placed in the center of the disk, or Ω is a strip and m is constant (see [START_REF] Jabin | Line-energy Ginzburg-Landau models: zero-energy states[END_REF]).

Proof of Proposition 2. First of all, we construct domains Ω n and W 1,p vector fields (1) defined on Ω n with n vortices for every n ≥ 1. For n = 1, we choose Ω 1 := B(0, 2) to be the ball centered at the origin of radius 2 and

m 1 (x) = x ⊥ |x| in Ω 1 . Then m 1 ∈ W 1,p div (Ω 1 , S 1 ) and ∇m 1 p L p (Ω1) = 2π 2 2-p
2-p , for any p ∈ [1, 2). For n = 2, the construction is the following: Set f, g : R → R with f (t) = |t| and g(t) = 2(t + 1) for t ≤ -1 -1 2 (t + 1) for t ≥ -1 and define the curves

γ + = {(x 1 , f (x 1 )) : x 1 ∈ [-2, 1]} and γ -= {(x 1 , g(x 1 )) : x 1 ∈ [-2, 1]}.
Fixing the vortex points P 1 := (-2, 0) and P 2 = (1, 0), we define the domain

Ω 2 := B(P 1 , 2) ∩ {x 1 ≤ -2} ∪ (x 1 , x 2 ) : x 1 ∈ [-2, 1], g(x 1 ) < x 2 < f (x 1 )
∪ B(P 2 , 1) ∩ {x 1 ≥ 1} (see Figure 3). We define m 2 : Ω 2 → S 1 as follows:

2 (x) =        (x-P1) ⊥ |x-P1| in x ∈ Ω 2 : x 1 ≤ -1 or x 1 ∈ (-1, 0) and x 2 > 0 , -(x-P2) ⊥ |x-P2| in x ∈ Ω 2 : x 1 ≥ 0 or x 1 ∈ (-1, 0) and x 2 < 0 .
Then m 2 ∈ W 1,p div (Ω, S 1 ) has two vortices in P 1 and P 2 of degree 1, but different orientation. One easily checks that ∇m 2

p L p (Ω1) 1+2 2-p 2-p , p ∈ [1, 2).
For arbitrary n, the above construction is to be repeated by an inductive argument: for every positive integer n, we construct a non-convex Lipschitz simply-connected domain Ω n and a vector field m n ∈ W 1,p div (Ω n , S 1 ) having n vortex singularities and such that there exists a vortex point

P n ∈ Ω n where (P n + R + × R) ∩ Ω n is a half disk of radius 2 -n+2 and m n (x) = (-1) n-1 (x-Pn) ⊥ |x-Pn| for every x ∈ (P n + R + × R) ∩ Ω n and ∇m n p L p (Ω1) 2 2-p + n-2 k=0 2 -k(2-p) 2 -p ≤ C(p), p ∈ [1, 2),
where the constant C(p) > 0 is independent of n.

Letting n → ∞, one gets

Ω = x ∈ R 2 : ∃n(x) ≥ 1, ∀n ≥ n(x), x ∈ Ω n (16)
which is a non-convex piecewise Lipschitz simply-connected domain. We define m : Ω → S 1 as follows: for every x ∈ Ω, set m(x) = m n (x), for n ≥ n(x) (given in ( 16)). Then m ∈ W 1,p div (Ω, S 1 ) for every p ∈ [1, 2) and has infinitely many vortices {P 1 , P 2 , . . . }. (Here, the sequence of vortices {P k } accumulates on some point P ∈ ∂Ω, therefore ∂Ω \ {P } is a Lipschitz curve, but not ∂Ω).

Remark 7

In the above construction, the vortex singularities have alternative orientations. However, one can construct domains where the vortex singularities have the same orientation. Here is the example of two vortex configuration in P 1 = (-3, 0) and P 2 = (1, 0): ω = ω 1 ∪ ω 2 ∪ ω 3 with ω 1 be a union of a square and a rectangle ω 1 := (-5, -1) × (-2, 2) ∪ (-5, -3) × (-4, -2) and ω 2 := (0, 2) × (-1, 1) ∪ (1, 2) × (-4, -1) and ω 3 = (-3, 1) × (-4, -3). Then choose

m(x) =        (x-P1) ⊥ |x-P1| in ω 1 , (x-P2) ⊥ |x-P2| in ω 2 , (1, 0) in ω 3 .
Then m ∈ W 1,p div (ω, S 1 ) for every p ∈ [1, 2) (see Figure 4).

Let us explain why in general a domain Ω satisfying the properties in Proposition 2 (i.e. admitting configurations with infinitely many vortices) is not Lipschitz.

Proposition 7 If Ω is a Lipschitz domain and m ∈ W 1,1 div (Ω, S 1 ) (or m ∈ H 1/2 div (Ω, S 1
)), then m has only a finite number of interior vortex singularities.

Proof. Assume by contradiction that m has infinitely many interior vortex singularities {P 1 , P 2 , . . . }. Obviously, discarding a subsequence, we can assume that the points P k converge to a point P ∈ Ω. Claim: There exist two cones C k1 ⊂ Ω and C k2 ⊂ Ω centered in P k1 and P k2 (of some positive height and positive angle) such that C k1 ∩ C k2 has nonempty interior. Proof of Claim: The case of a limit point P belonging to the interior of Ω is obvious. Let us suppose that P ∈ ∂Ω. Since ∂Ω is Lipschitz, we may assume that there exists a ball B centered at P such that ∂Ω ∩ B is the Lipschitz graph {(x 1 , γ(x 1 )) : x 1 ∈ (-δ, δ)} with γ(0) = δ, P = (0, γ(0)) and

U = (x 1 , x 2 ) : 0 ≤ x 2 < γ(x 1 ), x 1 ∈ (-δ, δ) ⊂ Ω ∩ B.
Therefore, for every P k = (x 1,k , x 2,k ) ∈ U , the vertical segment S k between (x 1,k , 0) and P k belongs to U . Since γ is a Lipschitz function, there exists an angle

α = α(γ) := π/2 -arctan( dγ dx 1 L ∞ (-δ,δ) ) > 0
such that the cone C k centered at P k of angle α and having S k as height is included in Ω. Due to the fact that P k converges to P , it follows that there exists k 0 ≥ 1 such that two cones C k1 and C k2 have nonempty interior intersection for any k 1 , k 2 ≥ k 0 . This finishes the proof of Claim.

The contradiction comes from the structure proved in Theorem 1 since m(x) = ± (x-P k j ) ⊥

|x-P k j | in C k1 ∩ C k2 for both j = 1, 2 which is absurd.

Remark 8 For a Lipschitz domain Ω, we say that a point P ∈ ∂Ω is a boundary vortex of m if there exists a cone C ⊂ Ω (of some height > 0 and angle β > 0) centered at P such that m(x) = ± (x-P ) ⊥ |x-P | for x ∈ C \ {P }. By Jabin-Otto-Perthame strategy, a boundary vortex P of m is always assigned to a maximal cone C, in the sense that for every point x ∈ Ω \ C outside the maximal cone, the characteristic of m passing through x stays outside the cone C. For p < 2, one can construct W 1,p vector fields (1) in a Lipschitz simply connected domain with infinitely many boundary vortices (that accumulate on the boundary).

Non-density results. Proof of Propositions 4 and 5

Proof of Proposition 5. Assume by contradiction that such a sequence exists. By dominated convergence theorem, it would mean in particular that m n → m in L 1 (B 2 ), i.e., Up to a subsequence, there exists 0 < r < 1 such that

∂Br |m n -m| dH 1 → 0 as n → ∞. ( 17
)
The key point of the proof consists in a dynamical system argument related to the topology of the flow of m ⊥ n : Using the technique in [START_REF] Desimone | 2-d stability of the Néel wall[END_REF][START_REF] Ignat | A compactness result in thin-film micromagnetics and the optimality of the Néel wall[END_REF][START_REF] Ignat | Vortex energy and 360 • Néel walls in thin-film micromagnetics[END_REF], we will consider the autonomous system

Ẋ = m ⊥ n (X). ( 18
)
First of all, (18) has no critical point and no cycle (i.e., no closed loop): Since

|m ⊥ n | = 1 in B 2 and m ⊥
n is smooth, the degree of m ⊥ n on a closed curve in B 2 is zero and therefore, an orbit of ( 18) cannot be closed in B 2 . Now set X n be the orbit of ( 18) passing by 0 (see Figure 5), i.e.,

Ẋn (t) = m ⊥ n (X n (t)), X n (0) = 0.
Then either the orbit X n reaches the boundary ∂B 2 in finite time, or the limit points of X n (see [8], Chapter 16) belong to the boundary ∂B 2 : Suppose that this is not the case, i.e., there is a limit point inside the ball B 2 . Since [START_REF] Ignat | Vortex energy and 360 • Néel walls in thin-film micromagnetics[END_REF] has no critical point, Poincaré-Bendixson's theorem (see [START_REF] Coddington | Theory of ordinary differential equations[END_REF], Theorem 2.1) implies that the limit set of X n should contain a periodic orbit which is a contradiction with the nonexistence of cycles for [START_REF] Ignat | Vortex energy and 360 • Néel walls in thin-film micromagnetics[END_REF]. Hence, the orbit X n separates the ball B r into a right side G n (where m n is the inner normal vector to ∂G n ) and a left side B r \ G n (see Figure 5). We define

χ n = χ n =- G n m n
χ n =        1 2 in G n , -1 2 in B r \ G n , 0 in B 2 \ B r . Then χ n ∈ BV (B 2 ) with Dχ n = m n H 1 ({X n } ∩ B r ) + χ - n m ⊥ H 1 ∂B r = m n |Dχ n | B r + χ - n m ⊥ H 1 ∂B r in D ′ (B 2 ), (19)
where χ - n is the interior trace of χ n on ∂B r with respect to the normal outer vector x |x| = -m ⊥ (x). Notice that |χ - n | = 1 2 on ∂B r . Moreover, we have that

Br |Dχ n | = H 1 ({X n ∈ B r }) ≥ 2r and B 2 |Dχ n | = Br |Dχ n | + πr. ( 20
)
Integration by parts leads to 2r

(20) ≤ Br |Dχ n | (19) = Br m n • Dχ n = - ∂Br m n • m ⊥ χ - n dH 1 - Br ∇ • m n χ n dx (17)
= o(1) -

Br ∇ • m n χ n dx as n → ∞. (21)
If a) holds, then the contradiction follows immediately from ( 21): since

|χ n | = 1/2 in B r , then Br ∇ • m n χ n dx = o(1) which would mean that o(1) ≥ r > 0 that is absurd. If b) holds, we prove first that {χ n } is uniformly bounded in BV (B 2 ). Indeed, since {∇ • m n } is bounded in H -s (B 2 ) (here s ∈ [0, 1/2)), by Gagliardo-Nirenberg's inequality (see Proposition 8 in Section 7) we have Br ∇ • m n χ n dx = B 2 ∇ • m n χ n dx ≤ ∇ • m n H -s (B 2 ) χ n H s (B 2 ) P roposition 8 ≤ C ∇ • m n H -s (B 2 ) χ n 1/2 L ∞ (B 2 ) χ n 1/2 BV (B 2 ) + χ n L ∞ (B 2 ) (20)
≤ C 1 +

Br |Dχ n | 1/2
.

By [START_REF] Ignat | A compactness result in thin-film micromagnetics and the optimality of the Néel wall[END_REF], we deduce that {χ n } is uniformly bounded in BV (B r ) and also in BV (B 2 ) due to [START_REF] Ignat | Lower bound for the energy of Bloch walls in micromagnetics[END_REF]. Therefore, {χ n } is relatively compact in H s (B 2 ) (since s ∈ [0, 1/2), see Proposition 8 in Section 7) and we conclude by assumption b) that

Br ∇ • m n χ n dx = B 2 ∇ • m n χ n dx = o(1)
which is a contradiction with [START_REF] Ignat | A compactness result in thin-film micromagnetics and the optimality of the Néel wall[END_REF].

Proof of Proposition 4. Obviously, the statement of Proposition 4 is a direct consequence of Proposition 5 a). However, there is an easier proof in this particular case. More precisely, we assume by contradiction that there exists a sequence of divergence-free vector fields m n ∈ C ∞ (Ω, S 1 ) satisfying m n → m a.e. in B 2 . In particular, dominated convergence theorem leads to m n → m in L 3 (B 2 ). First of all, Poincaré's lemma yields the existence of smooth stream functions

ϕ n ∈ C ∞ (Ω) such that m n = ∇ ⊥ ϕ n in B 2 . Since |m n | = 1, ϕ n is a 1-Lipschitz function. Observe that m = ∇ ⊥ ϕ with ϕ(x) = |x| in B 2 .
Subtracting eventually a constant, we can assume that B 2 ϕ n dx = B 2 ϕ dx so that ϕ n → ϕ uniformly in B 2 : indeed, for every x ∈ B 2 , one has

|ϕ n (x) -ϕ(x)| = - B 2 [(ϕ n -ϕ)(x) -(ϕ n -ϕ)(y)] dy ≤ - B 2 1 0 |x -y| ∇(ϕ n -ϕ) (x + t(y -x)) dydt ≤ C 1 0 B 2 |∇(ϕ n -ϕ)| 3 (x + t(y -x)) dy 1/3 dt ≤ C m n -m L 3 (B 2 ) 1 0 dt t 2/3 → 0, as n → ∞.
Since 0 is a strict minimum of ϕ, it would imply that ϕ n has a minimum inside B 2 , i.e., ∇ϕ n vanishes somewhere inside B 2 which is a contradiction with the fact that

|∇ϕ n | = 1 in B 2 .
6 Dense subsets: Proof of Theorems 2 and 3

We start by proving Theorem 2:

Proof of Theorem 2. By Proposition 7, we know that m has only a finite number of (interior) vortex singularities in Ω (call this set A).

Case 1. There are no interior vortices, i.e., A = ∅. By Theorem 1 we know that m is locally Lipschitz in Ω. For d > 0 small, we choose a smooth simply connected subdomain Ω d of Ω that is close to Ω in the sense that dist (x, ∂Ω) ∈ (d, 3d/2) for every x ∈ ∂Ω d . Then, by Theorem 1, we have that m is globally Lipschitz in Ω d . For each characteristic of m passing through a point x ∈ Ω d we call extremal points in Ωd the two intersection points P (x) and Q(x) of the characteristic with ∂Ω d . By the Jabin-Otto-Perthame procedure (as recalled in the proof of Theorem 1), any two characteristics of m passing through two points x, y ∈ Ω d intersect outside Ω d at a distance larger than d with respect to their extremal points. As a consequence, |m(x)m(y)| ≤ 1 d |x -y| for every x, y ∈ Ω d , i.e., m is globally Lipschitz in Ω d with the Lipschitz constant ≤ 1/d (see [START_REF] Jabin | Line-energy Ginzburg-Landau models: zero-energy states[END_REF]).

A standard geometry argument shows the existence of a countable set of segments γ j : [0, 1] → Ωd , j ∈ J ⊂ N such that 1) every two (open) segments γ j ((0, 1)) and γ k ((0, 1)) are disjoint, j = k; 2) the characteristics of m passing through γ j (t) for every t ∈ [0, 1] and j ∈ J cover the whole domain Ωd , i.e., Ωd = ∪ j∈J,t∈[0,1] [P (t)Q(t)] where P (t) ∈ ∂Ω d and Q(t) ∈ ∂Ω d are the extremal points in Ωd of the characteristic passing through the point γ j (t).

3) for every t ∈ (0, 1) and j ∈ J, the characteristic [P (t)Q(t)] passing through γ j (t) intersects the set of segments ∪ k∈J {γ k ([0, 1])} only at {γ j (t)}.

One could have two end points P j and P k of segments γ j and γ k that coincide, or the characteristic passing through P j may intersect γ k at the end point P k . (There exist smooth vector fields m with (1) on C 2 simply connected domain such that any set of curves {γ j } satisfying 1), 2), 3) are necessary infinitely countable, i.e., J is not finite; one could think of a local boundary given by the graph {(x 1 , γ(x 1 ))} with γ : x 1 → x 4 1 sin 1 x1 around 0 and m be a small perturbation of the constant vector field e 2 .)

We consider the lifting Θ (unique up to a constant) of m in Ω d , so Θ ∈ W 1,∞ (Ω d ). The smoothing procedure is the following: for each segment γ j ([0, 1]), we approximate the lifting Θ γj in H 1 ({γ j }) (or any W 1,q with q < ∞) by a sequence of C ∞ liftings Θ n γj such that Θ n ∂γj = Θ ∂γj and the Lipschitz constant of Θ n is less than 1/(d-1/n). Then we set m n := e iΘn γj on the segment γ j ; after that, m n is uniquely smoothly extended along the characteristics starting from any point γ j (t) due to the initial value m n (γ j (t)) since these characteristics (passing through points γ j (t)) could intersect only outside Ω d at a distance ≥ d -1

n with respect to their extremal points in Ω d . The new vector fields m n satisfy [START_REF] Ambrosio | Line energies for gradient vector fields in the plane[END_REF] in Ω d and approximate m in H 1 (Ω d ); they are smooth on the domain covered by the characteristics passing through γ j ((0, 1)). However, globally in Ω d , they could be only Lipschitz (and not smooth) at the end points γ j ({0, 1}). Let us call these corresponding characteristics as "bad" characteristics of m n . In order to smooth everywhere m n , we will proceed as follows: we will restrict to a subdomain Ωd of Ω d such that 0 < dist (x, ∂Ω d ) < d/2 for every x ∈ ∂ Ωd . Then any "bad" characteristic S has the following property: S ∩ Ωd splits Ωd into two open subdomains ω n and ωn so that m n is locally smooth around S in ω n respectively in ωn . Then one considers a small segment S = [A Ã] orthogonal to S such that the middle point M := (A + Ã)/2 of S belongs to the "bad" characteristic S and m n is smooth on (AM ) ⊂ ω n respectively ( ÃM ) ⊂ ωn . Considering the lifting Θ n of m n on S, one repeats the same smoothing argument as above, but asking that the new vector field mn coincides with m n in a neighborhood of ∂ S. Therefore, we conclude that m can be regularized in H 1 ( Ωd ) by smooth vector fields with (1) defined in Ωd . Letting d → 0, the conclusion follows in Case 1 (the approximation holding in H 1 loc (Ω)).

Case 2. There are interior vortices, i.e., A = ∅. We recall that A is finite. As before, for d > 0 small, we choose a smooth simply connected subdomain Ω d of Ω that is close to Ω in the sense that dist (x, ∂Ω) ∈ (d, 3d/2) for every x ∈ ∂Ω Then Ω 1 (m) is an open subset of Ω d , Ω1 (m) contains all the (interior) vortex singularities of m (i.e. A ⊂ Ω 1 (m)) and m ∈ C ∞ (Ω 1 (m) \ A).

To construct the desired m n in Ω d , the idea is the following: Set first m n := m on Ω 1 (m) so that m n has the same (interior) vortices A as m and m n ∈ C ∞ (Ω 1 (m) \ A). It remains to smooth m on Ω 2 (m); since A is not included in Ω 2 (m) (in fact, A is placed at distance larger than d outside ∂Ω 2 (m)), it means that any two characteristics of m passing through Ω 2 (m) intersect at a distance larger than d outside Ω 2 (m), i.e., m is globally Lipschitz on Ω 2 (m) with a Lipschitz constant less than 1/d. We will construct m n on Ω 2 (m) as at Case 1: we find a countable set of segments γ j : [0, 1] → Ω2 (m), j ∈ J ⊂ N such that 1) every two (open) lines γ j ((0, 1)) and γ k ((0, 1)) are disjoint, j = k;

2) the characteristics of m passing through γ j (t) for every t ∈ [0, 1] and j ∈ J cover the whole domain Ω2 (m), i.e., Ω2 (m) = ∪ j∈J,t∈[0,1] P (t)Q(t) where P (t) ∈ ∂Ω 2 (m) and Q(t) ∈ ∂Ω 2 (m) are the extremal points of the characteristic passing through γ(t) in Ω 2 (m).

3) for every t ∈ (0, 1) and j ∈ J, the characteristic P (t)Q(t) passing through γ j (t) intersects the set of curves ∪ k∈J {γ k } only at {γ j (t)}.

It could happen that some lines γ j have their end points in ∂Ω 1 (m) (i.e., γ j ({0, 1}) ⊂ ∂Ω 1 (m)). Since Ω 1 (m) is open, it means that one can extend a little bit γ j by the line γ j ∪ γj in the interior of Ω 1 (m) (i.e., γj ⊂ Ω 1 (m)); for such lines, the smoothing process of Case 1 is to be repeated by adding the constraint that the approximation m n coincides with m on γj . On the other lines γ j (not intersecting ∂Ω 1 (m)), the smoothing process in Case 1 is to be repeated. As before, m n is not smooth around "bad" characteristics. Eventually by considering a subdomain Ωd ⊂ Ω d , one can also smooth m n around the "bad" characteristics. Therefore, by letting d → 0, the conclusion follows also in Case 2 (the approximation holding in W 1,p loc (Ω) for any p < 2) and the approximating smooth vector fields m n have the same vortices A as m. Moreover, since m n = m locally around A, one has that for any compact K ⊂ Ω and q < ∞, m nm W 1,q (K) → 0 as n → ∞.

Proof of Theorem 3. Let A = {P k : k ∈ K} be the set of interior vortices of m with K be a finite set. For each interior vortex P k of m, we can find a cone C k ⊂ Ω (of center A k small angle) such that P k ∈ int(C k ) and Ω \ C k is a Lipschitz simply-connected domain. Set d k = dist (P k , ∂C k ) > 0 and Ω := Ω \ ∪ k∈K C k . The smoothing process is the following: in a first step, we smooth m inside each cone C k by nonvanishing divergence vector fields, and in a second step we smooth m by divergence-free vector fields inside Ω.

Step 1. Smoothing inside the cones. We will use the strategy in [START_REF] Ignat | Vortex energy and 360 • Néel walls in thin-film micromagnetics[END_REF]. For simplicity of the writing, we suppose that P k = O(0, 0) is the origin, m(x) = x ⊥ |x| in C k and for every small ε, denoting

λ = λ(ε) = 1 | ln ε| 2 ,
we will assume that the cone C k contains the sets

ω 1 = ω 1 (ε) = {x ∈ R 2 : λ < |x| := x 2 1 + x 2 2 < 1, x 2 ≥ 0, |x 1 | ≤ λ} and ω 2 = ω 2 (ε) = {x ∈ R 2 : |x| < λ}
and ∂Ω ∩ {x 2 > 0, |x 1 | ≤ λ} is contained in ω1 (see Figure 6). We construct a family or equivalently, the rescaled phase is given by

{m ε ∈ H 1 (C k , S 1 )} such that m ε → m a.e. in C k and (∇ • m ε )1 C k → 0 in Ḣ-1/2 (R 2 ) as ε → 0.
ϕ ε (θ) = φδ θ -π 2 arcsin λ . (25)
The profile m ε is defined in terms of its components in radial direction r and angular direction θ for rotation around arcs with fixed radius,

m ε (r, θ) = e iΘε(r,θ) = u ε (θ) θ -v ε (θ) r in ω 1 .
Notice that in ω 1 (as well as in C k \ (ω 1 ∪ ω 2 )), the profile m ε (together with its phase Θ ε ) are invariant in r. Also, we have the following relation between ϕ ε and Θ ε :

Θ ε (r, θ) = θ + ϕ ε (θ) + π 2 in ω 1 (26)
and the phase Θ ε is continuous in C k \ ω 2 . Finally, in the core region ω 2 , we define the profile m ε (r, θ) = e iΘε(r,θ) in polar coordinates by

Θ ε (r, θ) = r λ Θ ε (λ, θ) in ω 2 , (27)
where we recall by (26

) that Θ ε (λ, θ) = θ + π 2 + φδ ( θ-π 2
arcsin λ ) for every θ ∈ (0, 2π). Energy estimates. We refer to [START_REF] Ignat | Vortex energy and 360 • Néel walls in thin-film micromagnetics[END_REF] for the following estimates:

C k |∇m ε | 2 dx = o( 1 ε| ln ε| ) and (∇ • m ε )1 C k 2 Ḣ1/2 (R 2 ) = O( 1 | ln ε| ). (28)
Notice that m ε is continuous, but not smooth in C k . Obviously, one can approximate m ε by mε,η ∈ C ∞ (C k , S 1 ) in H 1 (C k ) as η → 0, so that by a diagonal argument, m is approximated by smooth vector fields

m n ∈ C ∞ (C k , S 1 ) in L 1 such that (∇ • m n )1 C k → 0 in H -1/2 (R 2 ).
Step 2. Smoothing outside the cones. Observe that m doesn't have any interior vortices in Ω that is Lipschitz simply connected. By the proof of Theorem 2 our vector field m can be approximated in W 1,q loc ( Ω) for q < ∞ (in particular, a.e. in Ω) by smooth vector fields m n ∈ C ∞ ( Ω, S 1 ) of vanishing divergence. Moreover, m n = m in C k around ∂C k ∩∂ Ω and the characteristics of m passing through Ω around C k intersect at a distance larger than d k outside Ω so that the proof of Theorem 2 enables m n to be chosen C ∞ around the boundary of ∂ Ω ∩ ∂C k . Therefore, we have constructed smooth vector fields m n ∈ C ∞ (Ω, S 1 ) converging to m a.e. that are divergence-free in Ω. It remains to show that (∇

• m n )1 Ω → 0 in Ḣ-1/2 (Ω). Observe that (∇ • m n )1 Ω = (∇ • m n )1 ∪ k∈K C k ∈ L 2 (R 2 ) (since m n ∈ H 1 (C k ) for each k ∈ K). Therefore, by (28), it follows (∇ • m n )1 Ω Ḣ1/2 (R 2 ) → 0.
which concludes our statement.

Appendix

For s ∈ (0, 1 2 ] and Ω ⊂ R d a Lipschitz domain, the Sobolev space H s (Ω) is a Hilbert space defined as

H s (Ω) = {f ∈ L 2 (Ω) : f 2 Ḣs (Ω) := Ω Ω |f (x) -f (y)| 2 |x -y| d+2s dxdy < ∞} and endowed by the norm f 2 H s (Ω) := f 2 L 2 (Ω) + f 2 Ḣs (Ω) for every f ∈ H s (Ω). (Convention: H 0 (Ω) := L 2 (Ω).) It is known that H s (Ω) = H s 0 (Ω) := D(Ω) H s (Ω)
where

D(Ω) = C ∞ c (
ζu Ḣ1/2 ≤ ζ L ∞ u Ḣ1/2 + C(Ω) ∇ζ L ∞ u L 2 .
We denote by H -1/2 (Ω) the dual of (H 1/2 00 (Ω), |•| 00 ) with respect to < •, • > D ′ (Ω),D(Ω) . So, H -1/2 (Ω) is a closed subspace of H -1/2 (Ω). Recall that u → ∇u is continuous in H 1/2 (Ω) to H -1/2 (Ω) (see details in [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]).

The space BV (Ω) is the set of functions f ∈ L 1 (Ω) such that the derivative of f (in the sense of the distributions) is a finite Radon measure, i.e., One has that W 1,1 (Ω) = C ∞ (Ω) BV (Ω) . It is known that the zero extension operator T : BV (Ω) → BV (R d ) defined as T f (x) = f (x) for x ∈ Ω and T f (x) = 0 for x ∈ R d \ Ω is a linear continuous operator and we denote T BV (Ω) the norm of operator T (more details in [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]).

Proposition 8 (Gagliardo-Nirenberg's inequality) Let Ω ⊂ R d be a bounded Lipschitz domain and s ∈ [0, 1 2 ). Then the embedding BV ∩ L ∞ (Ω) ⊂ H s (Ω) is compact and one has

f 2 Ḣs (Ω) ≤ C T BV (Ω) diam (Ω) 1-2s 1 -2s f L ∞ (Ω) Ω |∇f | for every f ∈ BV ∩ L ∞ (Ω),
where C > 0 is universal positive constant. . Therefore, one obtains the embedding BV ∩ L ∞ (Ω) ⊂ H s (Ω). Moreover, this embedding is compact since for some s ′ ∈ (s, 1 2 ), one has BV ∩ L ∞ (Ω) ⊂ H s ′ (Ω) and H s ′ (Ω) ⊂ H s (Ω) is compact (see Proposition 4.4. in [START_REF] Taylor | Partial differential equations. I[END_REF]).

Proposition 9

For every open set Ω ⊂ R d , d ≥ 1, one has W 1,1 (Ω, S 1 ) \ H 1/2 (Ω, S 1 ) = ∅.

Proof. The idea is the following (as explained in [START_REF] Desimone | 2-d stability of the Néel wall[END_REF]): it is known that BV ∩ L ∞ ⊂ H 1/2 in any dimension and the counter-example is given by any jump function, e.g., ϕ 0 = 1 (0,1) in Ω = (-1, 1). Moreover, if one regularizes ϕ 0 by ϕ n = ϕ 0 in Ω \ (0, 1/n) and ϕ n (x) = nx for x ∈ (0, 1/n), then we have

Ω |ϕ ′ n | = 1, ϕ n L ∞ = 1 and ϕ n H 1/2 → ∞.
Based on this idea, one can construct a function ϕ ∈ W 1,1 (Ω) \ H 1/2 (Ω) with Ω = (-1, 1) and set m : Ω → R with m = e iϕ . Then m satisfies the desired properties. In higher dimensions, the same example (depending on a single variable) holds. ) and m n → m in H 1/2 then h m n → h m in H 1/2 (the last terms are to be interpreted as product of complex numbers).

1 / 2 loc

 12 (Ω, R 2 ) : m satisfies (1)}.

Figure 2 :

 2 Figure 2: Characteristics of m.

  m(y)| ≤ 1 d |x -y|, for every x, y ∈ ω (in this case, every two characteristics passing through ω may intersect only at distances ≥ d outside ω). Notice that m may have infinitely many vortex points P k and any vortex point has degree one, but the orientation α k of the vortex P k could change or not in Ω (see Section 4). 4 Several examples 4.1 Lipschitz vector fields (1) that are not C 1

Figure 3 :

 3 Figure 3: Two vortices of different orientation.

  m

Figure 4 :

 4 Figure 4: Two vortices with the same orientation.

1 0

 1 ∂Br |m n -m| dH 1 dr → 0 as n → ∞.

Figure 5 :

 5 Figure 5: The orbit X n of the vector field m ⊥ n passing by 0 in the ball B 2 .

  d and A ⊂ Ω d and dist (A, ∂Ω d ) ≥ d. Therefore, all the characteristics passing through Ω d intersect either in a vortex point of A, or at a distance larger than d outside Ω d . We decompose the domain in a partition Ω d := Ω 1 (m) ∪ Ω 2 (m) where Ω 1 (m) = {x ∈ Ω d : there exist r > 0, P ∈ A, m(z) = ± (z -P ) ⊥ |z -P | for all z ∈ B(x, r) ⊂ Ω d }.

Figure 6 :

 6 Figure 6: Smooth vortex approximation inside a cone using a 360 • Néel wall transition (described in the right picture).

H 1 / 2 00

 12 Ω). The dual space H -s (Ω) of H s (Ω) with respect <•, • > D ′ (Ω),D(Ω) is defined as H -s (Ω) = {u ∈ D ′ (Ω) : u H -s (Ω) := sup f ∈D(Ω) f H s (Ω) ≤1 < u, f > D ′ (Ω),D(Ω) < ∞}(for more details, see[START_REF] Taylor | Partial differential equations. I[END_REF], Chapter 4)We introduce a closed subspace ofH 1/2 (Ω): (Ω) = {f ∈ H 1/2 (Ω) : |f | 2 00 = f 2 Ḣ1/2 (Ω) + Ω |f (x)| 2 ρ(x) dx < ∞}where ρ(x) = dist(x, ∂Ω). In fact, H1/2 00 (Ω) can be seen as the closure ofC ∞ c (Ω) in H 1/2 (R d ). Recall that the map u → ζu is continuous from H 1/2 (Ω) to H 1/2 00 (Ω) for any ζ ∈ C ∞ c (Ω); indeed, this is due to the inequality

Ω

  |∇f | = sup Ω f ∇ • ζ dx : ζ ∈ C 1 c (Ω, R d ), |ζ(x)| ≤ 1, ∀x ∈ Ω < ∞ ,where | • | is the Euclidean norm in R d . We denote the BV -norm as follows:f BV (Ω) := f L 1 (Ω) + Ω |∇f |.

Proof.

  If f ∈ C ∞ (Ω), one has Ω Ω |f (x)f (y)| 2 |x -y| d+2s dxdy = Ω Ω |T f (x) -T f (y)| 2 |x -y| d+2s dxdy ≤ 2 f L ∞ (Ω) |h|≤diam (Ω) R n |T f (x + h) -T f (x)| |h| d+2s dhdx ≤ 2 f L ∞ (Ω) f )(x + th)| dtdx ≤ C T BV (Ω) diam (Ω) 1-2s 1 -2s f L ∞ (Ω) Ω |∇f |.The inequality holds true for general f ∈ BV ∩ L ∞ (Ω) due to the density of smooth functions in BV endowed with the topology induced by the strict convergence, i.e., there existsf n ∈ C ∞ ∩ BV (Ω) such that |f n | ≤ f L ∞ and f n → f a.e. in Ω and f n BV (Ω) → f BV (Ω) as n → ∞ (see Remark 3.22 in AFP). Obviously, f L 2 (Ω) ≤ C(Ω) f L ∞ (Ω)

  Proof. a) LetR(h, x, y):= h(x)h(y) |x -y| (d+1)/2 , x = y ∈ Ω. Observe that h n → h in H 1/2 yields h n → h in L 2 (Ω) and R(h n , •, •) → R(h, •, •) in L 2 (Ω × Ω).Discarding eventually a subsequence, we may assume that h n → h a.e. in Ω and|R(h n , •, •)| ≤ T for a.e. x, y ∈ Ω for some T ∈ L 2 (Ω × Ω, R + ). Therefore, |R(Ψ(h n ), •, •)| ≤ Ψ Lip T and R(Ψ(h n ), •, •) → R(Ψ(h), •, •) a.e. in Ω × Ω. By dominated convergence theorem, we get that R(Ψ(h n ), •, •) → R(Ψ(h), •, •) in L 2 (Ω × Ω). Since |Ψ(h n ) -Ψ(h)| ≤ Ψ Lip |h n -h| a.e. in Ω × Ω,

Acknowledgement: The author thanks David Chiron and Robert Jerrard for useful comments. He also acknowledges partial support by the ANR projects ANR-08-BLAN-0199-01 and ANR-10-JCJC 0106.

Definition of m ε . We will denote the phase of m ε by Θ ε , i.e. m ε = e iΘε . In the region C k \(ω 1 ∪ω 2 ), m ε will coincide with the vortex m (in particular, m ε is a smooth vector field [START_REF] Ambrosio | Line energies for gradient vector fields in the plane[END_REF] in C k \ (ω 1 ∪ ω 2 )). More precisely, in polar coordinates, the phase is given by

In the region ω 1 , m ε will turn clockwise as a transition layer of degree -1 (known in micromagnetic jargon as 360 • -Néel wall of initial angle 0, see [START_REF] Ignat | Vortex energy and 360 • Néel walls in thin-film micromagnetics[END_REF]) and in the region ω 2 (standing for the core of the vortex), we apply some linear cut-off in the radius for the phase of m ε .

Transition layer of degree

In ω 1 , we first denote by (ũ δ , ṽδ ) = e i φδ : R → S 1 , the following approximation of the 360 • -Néel wall of initial angle 0 (magnetization turning in clockwise direction, i.e., of topological degree -1):

and

where θδ :

(see Figure 7). In view of ( 22), we may assume that φδ (0) = -π. We then have φδ (-∞) = 0 and φδ (+∞) = -2π and since φδ + π is antisymmetric in t, we also get φδ (-t) + φδ (t) = -2π for every t ∈ R.

We rescale the transition layer (ũ δ , ṽδ ) so that it is contained in ω 1 : In polar coordinates, for each arc of radius r ∈ (λ, 1) fixed in ω 1 and where the angle θ varies inside the interval θ ∈ ( π 2 ±arcsin λ r ), we define the rescaled transition layer (u ε , v ε ) with phase ϕ ε by

we conclude that Ψ(h n ) → Ψ(h) in H 1/2 . (Since the limit is unique, the above argument holds for the whole sequence n in the metric space H 1/2 .) b) Discarding eventually a subsequence, we may assume as before that m n → m a.e. in Ω. Let v n := m n m where m is the complex conjugate of m. Then v n → 1 in H 1/2 . Indeed, we first have that |v n | = 1 and v n → 1 a.e. in Ω. Then R(v n , •, •) → 0 a.e. in Ω × Ω and


Loffhagen

M M Becker

G K Grubert

D Loffhagen

Boundary conditions for the electron kinetic equation using expansion techniques

Keywords: 52, 25, Dg Plasma kinetic equations -51, 10, +y Kinetic and transport theory of gases -52, 80, -s Electric discharges

Boundary conditions for the electron kinetic equation using expansion techniques

Introduction

The electron component determines decisively the behaviour of gas discharge plasmas. For a theoretical description, their kinetic equation has to be treated and different methods for its solution have been developed. On the one hand, there are expansion techniques which have been applied. Depending on the discharge conditions, expansions Correspondence to: markus.becker@inp-greifswald.de in spherical harmonics or in Legendre polynomials have generally been performed [START_REF] Shkarofsky | The Particle Kinetics of Plasmas[END_REF][START_REF] Winkler | [END_REF][3][START_REF] Loffhagen | Low Temperature Plasmas: Fundamentals, Technologies and Techniques[END_REF][START_REF] Matveenko | [END_REF], where the latter expansion represents a special case of the spherical harmonics expansion requiring e.g. the simplifying assumption that the electric field action and the inhomogeneity of the plasma occur in the same direction [START_REF] Winkler | [END_REF]. On the other hand, simulation techniques like Monte-Carlo methods [6][7][8][9][10][11] have been used to determine the kinetic properties of the electrons.

The treatment of boundary conditions and their representation in terms of measurable properties is essential for the solution of the electron Boltzmann equation and other kinetic approaches to model gas discharge plasmas. When solving the space-dependent kinetic equation of the electrons, different approximations for the electron momentum distribution function (EMDF) have been employed.

For instance, relations of the form [12][13][14][15] f 0 (r, U ) ∼ f r (r, U ) [START_REF] Shkarofsky | The Particle Kinetics of Plasmas[END_REF] have been adapted for the analysis of the radial inhomogeneous column plasma of glow discharges, in order to describe absorption and reflection of electrons at the wall of the discharge tube within the framework of the two-term approximation of the expansion of the EMDF in spherical harmonics. Here, f 0 and f r are the isotropic part and the radial contribution to the distribution anisotropy, which depend on the radial position r and the kinetic energy U of the electrons. An alternative condition fixing the radial anisotropyf r at the tube wall by an exponentially decreasing positive function g(U ) has been used in [16,17]. This condition has been taken to describe a positive current of low-energy electrons to the wall, where the shape of g(U )

has been chosen to correspond sufficiently to the shape of the positive part of f r in the inner part of the dc column plasma. However, its physical justification remained open.

In order to analyse the axial behaviour of the electrons in discharge between plane electrodes, relations similar to equation ( 1) have been employed in [18][19][20][21] to characterize the absorption and partial reflection of electrons at the anode. A generalization of this boundary condition at the an-ode for the multiterm approximation of the Legendre polynomial expansion of the EMDF has been reported in [22].

The assumption of complete absorption of electrons at the anode has been made in the kinetic studies of [23,24] using a two-term Legendre polynomial expansion of the EMDF.

Boundary conditions at infinity of the solution region for the multiterm analysis of non-hydrodynamic swarms of electrons in situations where both conservative and nonconservative collisions take place have been discussed e.g.

in [25]. But it is stated in that paper that these conditions do not represent conditions at any physical boundary.

Conditions characterizing the inflow of electrons from

an electron source or from the cathode of a discharge have been represented e.g. in [25][26][27][28][29]. However, a physical motivation for the condition of the emission of electrons from the cathode of the discharge has been given for the first contribution to the distribution anisotropy only.

The present work focuses on discharge conditions at which an expansion of the EMDF in Legendre polynomials can be applied. For the numerical determination of the corresponding expansion coefficients of the EMDF, the choice of suitable boundary conditions plays an important role. Based on an ansatz similar to that proposed by Marshak [30][31][32][33] and a fundamental integral relation for Legendre polynomials, explicit expressions for the expansion coefficients have been derived that can be used as boundary conditions and relate macroscopic properties of the electrons with the expansion coefficients at the spatial margins of the discharge volume. In particular, the boundary conditions presented make it possible to accu- rately treat physical effects like secondary electron emission, reflection and back-diffusion on the kinetic level for an arbitrary order of the expansion.

In order to demonstrate the superiority of the new boundary conditions, a comparison with commonly used boundary conditions is presented using the example of an abnormal oxygen glow discharge.

Electron kinetic equation

To describe the electron component in gas discharge plasmas at steady state, the Boltzmann equation

p m e • ∇ r + F ext • ∇ p f (r, p) = a I el a ( f ) + i I in i,a ( f ) (2)
for the EMDF f (r, p) has to be solved. Here, F ext = -e 0 E/m e is the electric field force acting on the plasma and I el a and I in i,a are the individual collision integrals for elastic scattering and the ith inelastic collision processes of electrons having the mass m e with gas particles of kind 'a'.

Considering the spatially one-dimensional geometry of discharges between plane electrodes illustrated in Fig. 1, where E and the plasma inhomogeneity are parallel to the z direction, the EMDF gets the reduced dependence f (z, p, p z /p) on the space coordinate z, the magnitude of the momentum p, and the direction cosine p z /p = cos ϑ. Thus, it can be expanded in Legendre polynomials

P n (p z /p) [4], that is f z, p, p z p = ∞ n=0 fn (z, p) P n p z p . (3)
The substitution of this expansion up to an arbitrary number l of expansion coefficients fn (z, p) into the kinetic equation ( 2) and the transformation to the electron kinetic energy U = p 2 /(2m e ) finally lead to the hierarchy of partial differential equations [34]

n 2n -1 U ∂ z f n-1 (z, U ) + n + 1 2n + 3 U ∂ z f n+1 (z, U ) - n 2n -1 e 0 m e E z U ∂ U f n-1 (z, U ) - n -1 2 f n-1 (z, U ) - n + 1 2n + 3 e 0 m e E z U ∂ U f n+1 (z, U ) + n + 2 2 f n+1 (z, U ) = a 2N a m e M a ∂ U U 2 Q mt a (U )f n (z, U ) δ 0n - a U N a Q mt a (U )θ(n -1) + i Q in i,a (U ) f n (z, U ) + a N a i (U + U in i,a )Q in i,a (U + U in i,a ) × f n (z, U + U in i,a ) δ 0n (4)
for the expansion coefficients

f n (z, U ) = 2π(2m e ) 3/2 fn (z, p(U )) (5)
with n = 0, . . . , l -1. Here, N a is the density of the heavy particle of kind 'a' with the mass M a . Q mt a and Q in i,a denote the momentum transfer cross section for elastic collisions and the total cross section for the ith inelastic collision processes with the particle component 'a', respectively. U in i,a is the corresponding energy loss of electrons due to the inelastic collision process. In particular, isotropic scattering of the electrons in inelastic collisions processes has been assumed. The system of partial differential equations ( 4) can be solved numerically as an initial-boundary value problem provided that appropriate initial values and boundary conditions are given at the spatial margins z = 0 and z = z max of the solution

U = 0 U max z = 0 (cathode)
region (0 ≤ z ≤ z max , 0 ≤ U ≤ U max ) represented in Fig. 2.
Here, U max is an appropriate upper energy limit of the kinetic energy above which all components of the EMDF can be assumed to be already negligibly small, i.e., f n (z, U ≥ U max ) = 0. Such a value U max can be found approximately, for example, from the solution of the corresponding spatially homogeneous Boltzmann equation.

For the numerical solution of system (4) it is convenient to replace the kinetic energy U by the total energy

ε = U + W (z) with W (z) = e 0 z 0 E z (z ′ )dz ′
being the potential energy of the electrons [35]. After the transformation of Eq. (4) to the (z, ε) coordinates and the discretization with respect to z and ε on the centred grid points (z i+1/2 , ε k+1/2 ) with 1 ≤ i ≤ i max and 1 ≤ k ≤ k max , the numerical solution as initial-boundary value problem takes place by advancing from higher to lower total energies [34] within the transformed solution region rep- resented in Fig. 3. The space-dependence of the electric field and of the heavy particle densities are assumed to be given.

0 z max z ε min 0 ε max ε ε = W(z) solution area ε = U max + W(z)
The numerical solution starts at ε max where all transformed expansion coefficients fn (z, ε) = f n (z, U (z, ε)) are negligibly small [34]. At the boundary ε = W (z), i.e. at vanishing kinetic energy, the condition f n (z, U = 0) = 0 for all n ≥ 1 can immediately be derived when analysing system (4) in the limit of zero kinetic energy [35]. For the numerical solution this boundary condition is only required for all odd components.

In the following appropriate conditions at the spatial margins z = 0 and z = z max are discussed. Although system (4) is numerically solved after transformation to the total energy space, the boundary conditions are derived for the natural solution region shown in Fig. 2. The corresponding relations derived can easily be transferred to the total energy region.

Derivation of the boundary conditions

The system of hierarchy equations (4) represents a set of partial differential equations of first order. That is, the number of necessary boundary conditions equals the the number l of expansion coefficients. Here, ⌊l/2⌋ conditions are applied at z = 0 to the coefficients f 2n+1 (z, U ) with an odd index and ⌊(l + 1)/2⌋ conditions are applied at z = z max to the coefficients f 2n (z, U ) with an even index.

According to the physical situation considered, suitable boundary conditions have to characterize (i) an influx of electrons leaving the cathode at z = 0 due to the impact of, e.g., ions, fast neutrals, metastable atoms, or photons impinging onto this electrode and (ii) the absorption or reflection of electrons at the anode.

Following the approach of Marshak [30], the boundary conditions are derived from macroscopic properties by considering the moments of the electron kinetic equation (2) of odd order. In general, the z component of the kth order moments of Eq. ( 2) is given by [

2] G z,k (z) = 1 2 2 m e k/2 ∞ 0 dU U (k+1)/2 × 1 -1 dx x k f (z, U, x) (6)
with x = p z /p. Starting from this expression for k = 2̺+1, the l boundary conditions needed are found from requiring appropriate conditions at both the spatial margins.

Influx condition at the cathode

At the cathode side, a specification of the influx of electrons provides a meaningful boundary condition. The in-flux of electrons leaving the cathode and moving towards the anode can be characterized by the moments

G CA z,2̺+1 (0) = 1 2 2 m e ̺+1/2 ∞ 0 dU U ̺+1 × 1 0 dx x 2̺+1 f (0, U, x) (7)
of the order 2̺ + 1 with ̺ ∈ N 0 , where the integration of the angular variable x of the momentum space occurs over the half-range [31] as shown in Fig 1 . The substitution of f (0, U, x) in Eq. ( 7) by its l-term approximation according to Eq. ( 3) and the application of the integral relation (29) given in the appendix yield

G CA z,2̺+1 (0)
= (2̺ + 1)! 2 2 m e ̺+1/2 ∞ 0 dU U ̺+1 × l-1 n=0 f n (0, U ) (2̺ + 1 -n)!!(2̺ + n + 2)!! . (8)
Note that the expansion in Legendre polynomials is not affected by the discontinuity of the distribution function separating ingoing and outgoing particles because the halfrange of the angular variable of the momentum space is taken into account only.

Applying an appropriate normalized distribution function f S (U, x) at the cathode, e.g., a distribution with a Gaussian-like energy profile and a forward directed or an isotropic inflow of electrons into the half-range, the rela-

tion ∞ 0 dU U ̺+1 1 0 dx x 2̺+1 f S (U, x) = ∞ 0 dU U ̺+1 l-1 n=0 (2̺ + 1)!f n (0, U ) (2̺ + 1 -n)!!(2̺ + n + 2)!! (9)
is obtained by equating expressions ( 7) and (8) using

f (0, U, x) = f S (U, x).
Clearly, this equation is fulfilled if the sum at the right-hand side of Eq. ( 9) equals the halfrange integral at the left-hand side of Eq. ( 9) for all values of U , i.e., if the relation

l-1 n=0 (2̺ + 1)! f n (0, U ) (2̺ + 1 -n)!!(2̺ + n + 2)!! = C 2̺+1 (U ) (10)
with

C 2̺+1 (U ) = 1 0 dx x 2̺+1 f S (U, x) (11)
holds. The required boundary conditions at the cathode are finally obtained by splitting Eq. ( 10) into sums of expansion coefficients with exclusively odd and even indices, respectively. Using definition (28) given in the appendix, the energy-dependent boundary conditions for the expan-

sion coefficients f n (z, U ) with odd n reads ̺ i=0 (2̺ + 1)! f 2i+1 (0, U ) (2̺ -2i)!!(2̺ + 2i + 3)!! = C 2̺+1 (U ) - ⌊(l-1)/2⌋ i=0 (2̺ + 1)! f 2i (0, U ) (2̺ -2i + 1)!!(2̺ + 2i + 2)!! ( 12
)
for ̺ = 0, . . . , ⌊l/2⌋ -1. Eq. ( 12) represents relations that allow the determination of f n (0, U ) with odd n in dependence on f n (0, U ) with even n and an additional term whose energy dependence is explicitly known. For instance, the first contribution f 1 (0, U ) to the distribution anisotropy of the EMDF is given by the relation

f 1 (0, U ) = 3C 1 (U ) -3 ⌊(l-1)/2⌋ i=0 f 2i (0, U ) (-2i + 1)!!(2i + 2)!! ( 13
)
describing the microscopic particle flux of the electrons.

The corresponding macroscopic particle flux of electrons is obtained by means of an appropriate energy space averaging of Eq. ( 13).

In contrast to previous studies assuming vanishing odd expansion coefficients f n (z, U ) with n = 3, 5, . . . at the cathode [26,27,34], Eq. ( 12) provides a much better approximation of the boundary condition

f l (0, U, x) = f S (U, x) (14)
where f l (0, U, x) denotes the representation of f (0, U, x) in l-term approximation. Notice, that the derivation of boundary conditions that provide a mathematically exact fulfillment of Eq. ( 14) seems to be impossible when a truncated Legendre polynomial expansion for solving Eq. ( 2)

is used [32,[START_REF] Robson | Introductory transport theory for charged particles in gases[END_REF].

Reflection condition at the anode

At the anode side, relations that describe the reflection or absorption of electrons provide suitable boundary conditions. To characterize partial reflection of electrons at the anode, the flux of electrons can be decomposed into a flux towards the wall and away from it. The latter can be described by the moments

G AC z,2̺+1 (z max ) = 1 2 2 m e ̺+1/2 ∞ 0 dU U ̺+1 × 0 -1 dx x 2̺+1 f (z max , U, x) . ( 15
)
It is related to the flux G CA z,2̺+1 (z max ) towards the anode according to

G AC z,2̺+1 (z max ) = -rG CA z,2̺+1 (z max ) ( 16
)
for arbitrary moments of the order 2̺ + 1 with ̺ ∈ N 0 .

Here, r ∈ [0, 1[ is the reflection coefficient of electrons at the anode. Starting from Eq. ( 16), boundary conditions for the expansion coefficients with an even index can be derived by means of a procedure similar to that described in section 3.1 using the relation (32) given in the appendix.

The energy-dependent conditions at z = z max read

1 -r 1 + r ⌊(l-1)/2⌋ i=0 (2̺ + 1)! f 2i (z max , U ) (2̺ -2i + 1)!!(2̺ + 2i + 2)!! = ̺ i=0 (2̺ + 1)! f 2i+1 (z max , U ) (2̺ -2i)!!(2̺ + 2i + 3)!! ( 17
)
for ̺ = 0, . . . , ⌊(l -1)/2⌋. These equations relate a sum of expansion coefficients f n (z max , U ) with only even indices n to a sum of expansion coefficients with only odd indices n. They provide an approximation of the boundary condition [32] f l (z max , U, x)

x<0 = rf l (z max , U, -x) , (18)
where f l (z max , U, x) denotes the representation of f (z max , U, x) in l-term approximation. Eq. ( 17) is in agreement with the relation (21) of Ref. [22], which was also justified by comparison with Monte Carlo simulations in that paper.

In the limit of pure diffusion, i.e. when E z = 0, the relation

f 1 (z, U ) = -λ m (z, U ) ∂ z f 0 (z, U ) + 2 5 ∂ z f 2 (z, U ) (19)
with the mean free path of the electrons

λ m (z, U ) = 1 a N a (z) Q mt a (U ) + i Q in i,a (U ) (20)
results from (4) for n = 1. The combination of (17) for ρ = 0 and (19) at z = z max and the subsequent integration over the energy space finally yield in the limit of the twoterm approximation the equation

∇ z n e (z) n e (z) z=zmax = - 3 2λ m 1 -r 1 + r (21)
for the electron density n e (z) = ∞ 0 √ U f 0 (z, U )dU , where an average electron collisional mean free path λ m has been used. Equation ( 21) is in agreement with the formula for diffusion calculations involving wall reflection and low density derived in [START_REF] Chantry | [END_REF].

Comparison of boundary conditions at the cathode

In this section, cathode-sided boundary conditions are evaluated by comparison with the distribution function f S (U, x) assumed for the electrons emitted from the cathode. In order to illustrate the effect of boundary conditions, an abnormal oxygen glow discharge was investigated by numerically solving the set of equations ( 4) for given spatial profiles of the electric field E z and the heavy particle densities N a . Details of the discharge parameters and the underlying cross sections can be found in Ref. [34]. In particular, the set of electron-heavy particle collisions in the plasma includes both conservative processes, namely elastic collisions, vibrational and electronic excitation as well as dissociation, and non-conservative electron processes, i.e. ionization and dissociative electron attachment.

The numerical solution of the equation system (4) was obtained by an approximation with 20 terms in the Legendre polynomial expansion (3). To solve system (4), the boundary conditions (12) were used for each odd expansion coefficient at the cathode including the angular dependence of the electron influx. The required source function f S (U, x) at the cathode was chosen as

f S (U, x) = U exp - U -U w U c 2 (22)
with the energetic width and centre of the distribution U w = 5 eV and U c = 2 eV, respectively. This source function corresponds to an isotropic influx of electrons into the half-range with a Gaussian-like energy profile. The consideration of an isotropic inscattering is motivated by its usage in a lot of theoretical plasma investigations, see, e.g., Ref. [7]. The corresponding relation for the electron flux determined by the first anisotropic component of the EMDF reads

f 1 (0, U ) = 3 2 U exp - U -U w U c 2 -3 ⌊(l-1)/2⌋ i=0 f 2i (0, U ) (-2i + 1)!!(2i + 2)!! . ( 23
)
Furthermore, the conditions ( 17) were employed at the anode side using r = 0.36.

In the literature [26,27,34], the inscattering profile (22) is commonly applied to f 1 (0, U ) only, i.e.,

f 1 (0, U ) = U exp - U -U w U c 2 (24)
and all further expansion coefficients f n (0, U ) with odd n ≥ 3 are set to zero. Higher order corrections by f n (0, U ) with even n as present in Eq. ( 23) are completely neglected.

In Fig. 4, numerical results for the distribution function f (z = 0, U, x) at the cathode, normalized to the maximum of the source function f S (U, x), are compared with f S (U, x) at x = 1/2. For the numerical computations, the boundary conditions ( 12) and ( 24), respectively, have been used. The comparison shows that the results based on the new boundary conditions ( 12) match the distribution function f S (U, x = 1/2). In contrast to that, appreciable obtained by a 20-term approximation using the boundary conditions (BC) according to ( 12) and ( 24) with the inscattering profile (22) of the electrons at the angular variable x = 1/2.

differences between the results based on the commonly used boundary condition (24) neglecting the contributions of higher odd expansion coefficients and the analytically given function f S (U, x = 1/2) are found. That is, expansion coefficients of higher order have to be included if the electron kinetic equation ( 2) is solved.

Notice, that the results presented here are based on an isotropic angular dependence of the inscattering profile (12) corresponding to P 0 (x) = 1. In the special case that the angular distribution is assumed to be P 1 (x) = x, see, e.g., Ref. [38], the commonly used boundary condition (24) yields exact expressions for the odd expansion coefficients.

The influence of the approximation order of the Legendre expansion (3) is presented in Fig. 5. It can be seen, that the commonly used two-term approximation is not suitable to describe the prescribed distribution function f S (U, x) at the cathode. Hence, approximately 20 terms are required to obtain good numerical results at the cathode-sided boundary and the effect of the higher order terms has to be included at the boundaries. At the same time, less than about ten terms have been found to be sufficient in general to yield convergent results in the plasma volume [22,34,39].

The results presented have been obtained by prescribing an isotropic influx of electrons into the half-range x = p z /p with 0 ≤ x ≤ 1 using the function (22), which results from the impact of e.g. ions, fast neutrals or photons impinging onto the cathode. Electrons, which diffuse back to the cathode e.g. after collisions and strike the electrode, are assumed to be totally reflected and to produce no further electrons at the cathode. An extension of the current approach to this more general case is immediately possible by modifying the source function (22) for ̺ = 0, . . . , ⌊(l -1)/2⌋. Note that these relations differ from equations ( 17) by the minus sign on the left-hand side of ( 25) only.

Summary

The kinetic equation of the electrons is frequently treated by using an expansion of the electron momentum distribution function (EMDF) in Legendre polynomials. This approach leads to a system of partial differential equations whose numerical solution requires suitable boundary conditions. Based on relations for the macroscopic properties of the electrons and a fundamental integral relation of Legendre polynomials using a half-range integration technique in the angular space, relations between the expansion coefficients of the EMDF with an odd index and the expansion coefficients with an even index have been derived. These energy-dependent relations can be used as boundary conditions for spatially one-dimensional discharge configurations with plane-parallel electrodes. In particular, the new boundary conditions at the cathode yield a numerical solution of the electron kinetic equation which almost matches the prescribed profile. It has be shown that an improvement of numerical solutions can be achieved at a slightly higher computational cost when using Eq. ( 12). The approach presented makes it possible to treat correctly the

Fig. 1 .

 1 Fig. 1. Axially inhomogeneous discharge geometry.

Fig. 2 .

 2 Fig. 2. Natural solution region in (z, U ) coordinates.

Fig. 3 .

 3 Fig. 3. Solution region in the transformed (z, ε) space. The dotted line represents the total energy ε = Umax + W (z).

Fig. 4 .

 4 Fig. 4. Comparison of the distribution functions f (0, U, x)

Fig. 5 .

 5 Fig. 5. Distribution functions f (0, U, x) obtained by different orders of the Legendre expansion (3) using the boundary conditions (12) with the inscattering profile (22) of the electrons at the angular variable x = 1/2.

-

  to include the secondary electron emission coefficients and different energy profiles for the different species as well as by combining the resulting influx conditions with the corresponding energy-dependent reflection conditions at z = 0 1)! f 2i (0, U ) (2̺ -2i + 1)!!(2̺ + 2i + 2)!! = ̺ i=0(2̺ + 1)! f 2i+1 (0, U ) (2̺ -2i)!!(2̺ + 2i + 3)!!(25)

influx of electrons due to, e.g., ion impact onto the cathode or thermionic emission as well as the partial reflection of electrons at the wall on the microscopic level. Its extension to obtain boundary conditions for arbitrary moments of the kinetic equation is immediately possible.

This work was supported by the Deutsche Forschungsgemeinschaft within the SFB TR 24.

Appendix

The well known double factorial notation

can be extended to negative integers using the relations [START_REF] Weisstein | CRC Concise Encyclopedia of Mathematics[END_REF] (-

for n ∈ N. Eq. ( 28) is a direct consequence of the general

With the definitions ( 26)-( 28) and n/∞ = 0 for every -∞ < n < +∞, the fundamental integral relation

for the Legendre polynomials holds for arbitrary k, n ∈ N 0 . Obviously, Eq. ( 29) is fulfilled for n ∈ {0, 1} and any k ∈ N 0 . Its general validity is proven by induction. Using the recurrence formula [START_REF] Weisstein | CRC Concise Encyclopedia of Mathematics[END_REF]

for the Legendre polynomials with n ≥ 1, it follows that

In the final step, the equation n!! = n • (n -2)!! has been used. It even holds for the double factorial of negative integer arguments.

Note that Eq. ( 29) provides a much simpler expression for the integral at the left-hand side than the well known solution of this integral in terms of the Gamma function [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF]. Furthermore, notice that the relation 0 -1 dx x k P n (x) = (-1) k+n 1 0 dx x k P n (x) (32) holds for k, n ∈ N.