Marc Briane
email: mbriane@insa-rennes.fr

Graeme W Milton
email: milton@math.utah.edu

Isotropic realizability of current fields in R 3

Keywords: current field, isotropic conductivity, Frobenius' condition, dynamical flow Mathematics Subject Classification: 35B27, 78A30, 37C10

come

Introduction

In the theory of composite conductors (see, e.g., [START_REF] Milton | The Theory of Composites[END_REF]), we are naturally led to study periodic composites. The effective properties of a periodic composite are obtained by passing from a local Ohm's law j = σ e,

between a periodic divergence free current field j and a periodic electric field e, to an effective Ohm's law j = σ * e , with • the average over the period cell, (

where σ(y) is the local conductivity which is isotropic, and σ * is the (constant) effective conductivity of the composite which is in general anisotropic. In this context, it is natural to characterize the periodic current fields arising in the solution of these equations among all the divergence free fields. More precisely, the paper deals with the following question: Given a periodic regular divergence free field j from R 3 into R 3 , under which conditions j is an isotropically realizable current field, namely there exists an isotropic conductivity σ > 0 and a gradient field such that j = σ∇u? An additional motivation comes from the success of transformation optics (see, e.g., [START_REF] Pendry | Controlling Electromagnetic Fields[END_REF][START_REF] Pendry | Calculation of material properties and ray tracing in transformation media[END_REF]) where the objective is to choose moduli (in our case the conductivity σ) to achieve desired fields (in our case the prescribed current field j).

In [START_REF] Briane | Which electric fields are realizable in conducting materials?[END_REF] we have studied the isotropic realizability of a given regular electric field e = ∇u in R d , for any d ≥ 2. The key ingredient of our approach was the associated gradient system X ′ (t, x) = ∇u X(t, x) , for t ∈ R, X(0, x) = x, (1.3) which allowed us to prove the following isotropic realizability result for a gradient field in the whole space and in the torus: satisfies the conductivity equation div (σ∇u) = 0 in R d . On the other hand, when ∇u is periodic, the conductivity σ can be chosen periodic if and only if there exists a constant C > 0 such that ˆτ(x) 0 ∆u X(t, x) dt ≤ C, ∀ x ∈ R d .

(1.7)

In the case of a gradient field, the local isotropic realizability which follows from the nonvanishing condition (1.4) thanks to the rectification theorem (see [START_REF] Briane | Which electric fields are realizable in conducting materials?[END_REF], Theorem 2.2 i)), is thus equivalent to the global realizability given by the previous theorem. The case of a regular divergence free field j in R 3 is much more intricate. First of all, a necessary condition for the isotropic realizability is the orthogonality of j and curl j in R 3 . Conversely, if j is non-zero and orthogonal to curl j in R 3 , then Frobenius' theorem implies that j is isotropically realizable locally in R 3 (see Proposition 2.2). However, contrary to the case of a gradient field, these two conditions are not sufficient to ensure the global realizability (see Section 3.2 for a counter-example). This strictly local nature of Frobenius' theorem is strongly connected to cohomology which is outside the scope of this paper. On the other hand, we cannot use for a current field the properties of a gradient system which permits us in particular to define the time τ (x) satisfying (1.5).

Our approach concerning the isotropic realizability of a current field is still based on dynamical systems. But now, the procedure to construct an admissible conductivity associated with a given regular divergence free field j, uses a combination of three dynamical flows which are not of gradient type. To this end, we need that the three fields j, curl j and j × curl j make an orthogonal basis of R 3 , including in this way Frobenius' condition. Then, the method consists in flowing from a fixed point x 0 ∈ R 3 , first with the flow X 1 along the direction j/|j| during a time t 1 , next with the flow X 2 along the direction curl j/|curl j| during a time t 2 , finally with the flow X 3 along the direction j/|j| 2 × curl j during a time t 3 . So, we obtain the triple time dynamical flow

(t 1 , t 2 , t 3 ) → X 32 (t 3 , t 2 , t 1 ) = X 3 t 3 , X 2 (t 2 , X 1 (t 1 )) , with X 32 (0, 0, 0) = x 0 , (1.8)
which is assumed to be a C 1 -diffeomorphism onto R 3 . Under these assumptions we prove (see Theorem 2.4) that the field j is isotropically realizable with the conductivity σ defined by

σ X 32 (t 1 , t 2 , t 3 ) := exp ˆt3 0 |curl j| 2 |j| 2 X 32 (s, t 2 , t 1 ) ds , for (t 1 , t 2 , t 3 ) ∈ R 3 , (1.9)
This result can be regarded as a global Frobenius' theorem, and is illustrated by the very simple current field j of Section 3.1, which yields an infinite set of (not obvious) admissible conductivities. Unfortunately, Section 3.3 shows that the approach with the triple time dynamical flow fails for a periodic regular field j of a particular form which is everywhere perpendicular to a constant vector, since curl j does vanish in R 3 (see Remark 3.3). However, in this case the divergence free field can be written as an orthogonal gradient, which allows us to apply Theorem 1.1 for a two-dimensional electric field.

When the field j is periodic, the isotropic realizability in the torus needs an extra assumption as in the case of a periodic electric field [START_REF] Briane | Which electric fields are realizable in conducting materials?[END_REF] (Theorem 2.17). Under the former conditions which ensure the isotropic realizability of j in the whole space R 3 , we prove (see Corollary 2.9) that the field j is isotropically realizable in the torus, namely reads as σ∇u with both σ and ∇u periodic, if and only if

sup (t 1 ,t 2 )∈R 2 ˆ∞ -∞ |curl j| 2 |j| 2 X 32 (s, t 2 , t 1 ) ds < ∞, (1.10)
which is equivalent to the boundedness from below and above of the conductivity (1.9) in R 3 . The sharpness of condition (1.10) is illustrated by Proposition 3.5 and Example 3.6 below. The paper is divided in two parts. In Section 2 we study the validity of the isotropic realizability of a regular divergence free field first in the whole space R 3 , then in the torus when the current field is assumed to be periodic. Section 3 is devoted to examples and counterexamples which illustrate the theoretical results of Section 2.

Notations

• Y := [0, 1] 3 and Y ′ := [0, 1] 2 . • • denotes the average over Y . • C k ♯ (Y ) denotes the space of k-continuously differentiable Y -periodic functions on R d . • L 2 ♯ (Y ) denotes the space of Y -periodic functions in L 2 (R d ), and H 1 ♯ (Y ) denotes the space of functions ϕ ∈ L 2 ♯ (Y ) such that ∇ϕ ∈ L 2 ♯ (Y ) d .
• For any open set Ω of R d , C ∞ c (Ω) denotes the space of smooth functions with compact support in Ω, and D ′ (Ω) the space of distributions on Ω.

2 Results of isotropic realizability

Realizability in the whole space

Let us start by the following definition: Definition 2.1. Let j be a divergence free field in L ∞ (R 3 ) 3 -j will be taken regular in the sequel -The field j is said to be isotropically realizable in R 3 as a current field, if there exist an isotropic conductivity σ > 0 with σ, σ -1 ∈ L ∞ (R 3 ), and a potential u ∈ W 1,∞ (R 3 ) 3 , such that j = σ∇u. Moreover, when j is Y -periodic, j is said to be isotropically realizable in the torus if σ and ∇u can be chosen Y -periodic.

First of all we have the following result which provides a criterion for the local isotropic realizability of a regular current field: Proposition 2.2. Let j be a vector-valued function such that j ∈ C 2 (R 3 ) 3 , j = 0 in R 3 , and div j = 0 in R 3 .

(2.1)

Then, a necessary and sufficient condition for the current field j to be locally isotropically realizable in R 3 with some positive C 1 conductivity σ is that

j • curl j = 0 in R 3 . (2.2)
Proof. If j is isotropically realizable with some conductivity σ ∈ C 1 (R 3 ), then j = σ∇u with σ ∈ C 1 (R 3 ), and

curl j = curl (σ∇u) = ∇σ × ∇u + σ curl (∇u) = ∇σ σ × j in R 3 , (2.3)
which yields immediately (2.2). Conversely, if (2.1) and (2.2) are both satisfied, then by Frobenius' theorem (see, e.g., [START_REF] Cartan | Calcul Différentiel[END_REF], Theorem 6.6.2 and example p. 279) there exists locally a non-zero C 1 function σ and a C 1 function u, such that j = σ∇u. The function σ can be chosen positive by a continuity argument, which shows the isotropic realizability of j locally in R 3 . Actually, the divergence free condition is not necessary to obtain the local realizability.

Frobenius' condition (2.2) implies the local isotropic realizability for a current field j satisfying condition (2.1). However, contrary to the case [START_REF] Briane | Which electric fields are realizable in conducting materials?[END_REF] of an electric field for which the local realizability and the global realizability turn out to be equivalent, these two conditions are not sufficient to ensure the global isotropic realizability of j, as shown by the counter-example of Section 3.2. To overcome this difficulty we will use an alternative approach based on the flows along the three orthogonal directions j, curl j and j × curl j under suitable assumptions which are detailed below:

Let j be a current field satisfying conditions (2.1). Beyond condition (2.2) we assume that j, curl j, j × curl j is an orthogonal basis of R 3 .

(2.4)

Then, for a fixed x 0 ∈ R 3 and for any (t 1 , t 2 , t 3 ) ∈ R 3 , consider the flows X 1 (t, x), X 2 (t, x), X 3 (t, x) along the orthogonal directions j/|j|, curl j/|curl j|, j/|j| 2 × curl j respectively, that is

                 ∂X 1 ∂t (t, x) = j |j| X 1 (t, x) , X 1 (0) = x, ∂X 2 ∂t (t, x) = curl j |curl j| X 2 (t, x) , X 2 (0) = x, ∂X 3 ∂t (t, x) = j × curl j |j| 2 X 3 (t, x) , X 3 (0, x) = x. for (t, x) ∈ R × R 3 . (2.5)
Note that the flows X 1 and X 2 are well defined in the whole set R × R 3 , since by (2.1) j/|j| and curl j/|curl j| belong to C 1 (R 3 ) and are bounded in R 3 (see, e.g., [START_REF] Arnold | Ordinary differential equations[END_REF], Chap. 2.6). In the sequel, we will assume that the flow X 3 is also defined in the whole set R × R 3 . That is the case if for example j ∈ C 2 ♯ (Y ) 3 .

Remark 2.3. In view of the normalization of the flows X 1 , X 2 , and to avoid the latter assumption, it seems a priori more logical to renormalize the flow X 3 with |j||curl j| rather than |j| 2 . The derivation of the isotropic realizability is quite similar in both cases (see Theorem 2.4 and Remark 2.5 just below). However, the normalization by |j| 2 arises naturally in the orthogonal decomposition (2.18) which is a key-ingredient for the construction of an admissible conductivity associated with the isotropic realizability of j. Moreover, it gives a necessary condition for the isotropic realizability in the torus without the need to assume that curl j does not vanish in R 3 (see the first part of Corollary 2.9 below). Actually, there are lots of examples where curl j vanishes somewhere (see Section 3.3 below), but the normalization of the flow X 3 by |j| 2 may be relevant in some cases (see Proposition 3.2 and Remark 3.3).

Next, denote for a fixed point

x 0 ∈ R 3 ,        X 1 (t 1 ) := X 1 (t 1 , x 0 ) X 23 (s 2 , s 3 , t 1 ) := X 2 s 2 , X 3 (s 3 , X 1 (t 1 , x 0 )) X 32 (t 3 , t 2 , t 1 ) := X 3 t 3 , X 2 (t 2 , X 1 (t 1 , x 0 )) , for (s 1 , s 2 , s 3 , t 1 , t 2 , t 3 ) ∈ R 6 . (2.6)
So, the dynamical flow X 32 is obtained by flowing from the point x 0 along the direction j/|j| during the time t 1 , then from the point X 1 (t 1 ) along the direction curl j/|curl j| during the time t 2 , finally from the point X 2 (t 2 , X 1 (t 1 )) along the direction (j/|j| 2 ) × curl j during the time t 3 . The end point is thus X 32 (t 3 , t 2 , t 1 ). A similar construction holds for X 23 (t 2 , t 3 , t 1 ) by commuting the flows X 2 and X 3 . Now, the main assumption is that any point x in R 3 can be attained by the composition of the three flows, so that x can be represented in a unique way by the system of coordinates (t 1 , t 2 , t 3 ), that is

(t 1 , t 2 , t 3 ) → X 32 (t 3 , t 2 , t 1 ) is a C 1 -diffeomorphism onto R 3 . (2.7)
Then, we have the following sufficient condition for the global isotropic realizability in R 3 :

Theorem 2.4. Let j be a field in R 3 satisfying (2.1) and (2.4). Also assume that condition (2.7) holds true. Then, the field j is isotropically realizable in R 3 with the conductivity e w ∈ C 1 (R 3 ), where the function w is defined by

w(x) = w X 32 (t 3 , t 2 , t 1 ) := ˆt3 0 |curl j| 2 |j| 2 X 32 (s, t 2 , t 1 ) ds, for (t 1 , t 2 , t 3 ) ∈ R 3 . (2.8) Remark 2.5.
In view of Remark 2.3, if we renormalize the flow X 3 by |j||curl j|, it is replaced by the flow X3 defined by

∂ X3 ∂t (t, x) = j × curl j |j||curl j| X3 (t, x) , X3 (0, x) = x, (2.9)
which by condition (2.4) is defined in R × R 3 . Then, similarly to (2.7), assuming that the triple flow X32 :

(t 1 , t 2 , t 3 ) → X3 t 3 , X 2 (t 2 , X 1 (t 1 , x 0 ) (2.10) is a C 1 -diffeomorphism onto R 3
, we obtain that the field j is isotropically realizable in R 3 with the conductivity e w ∈ C 1 (R 3 ), where

w(x) = w X32 (t 3 , t 2 , t 1 ) := ˆt3 0 |curl j| |j| X32 (s, t 2 , t 1 ) ds, for (t 1 , t 2 , t 3 ) ∈ R 3 . (2.11)
The proof is quite similar to the proof of Theorem 2.4 replacing formula (2.12) by

∇ w • j × curl j |j||curl j| = |curl j| |j| = j × curl j |j| 2 • j × curl j |j||curl j| in R 3 .
(2.12)

Remark 2.6. Alternatively, we can replace the diffeomorphism condition (2.7) by

(t 1 , t 2 , t 3 ) → X 32 (t 3 , t 2 , t 1 ) is a homeomorphism onto R 3 of class C 1 , (2.13)
so that the Jacobian of X 32 may vanish somewhere. In compensation we have to assume that the function w of (2.8) belongs to C 1 (R 3 ). See the application to the example of Section 3.1.

Remark 2.7. Condition (2.7) is not sharp to ensure the isotropic realizability of the current field. Indeed, the planar example of Proposition 3.2 below shows that the isotropic realizability can be satisfied while condition (2.4) is violated. See also Example 3.6 below.

Proof of Theorem 2.4.

First step: Construction of an admissible conductivity.

Let j be a field satisfying (2.1). Assume that j is isotropically realizable in R 3 , namely there exists u, w ∈ C 1 (R 3 ) such that j = e w ∇u in R 3 .

(2.14)

It seems that we can choose the potential u arbitrarily along the trajectory X 1 (t), provided ∇u does not vanish along this trajectory. So, define u X 1 (t) := ˆt 0 j X 1 (s) ds, for t ∈ R.

(2.15)

Taking the derivative with respect to t of (2.15) and using (2.5), (2.14), we get that

∂ ∂t u X 1 (t) = j X 1 (t) = ∇u X 1 (t) • ∂ ∂t X 1 (t) = e -w(X 1 (t)) j • j |j| X 1 (t) , (2.16)
which implies that w X 1 (t) = 0, for t ∈ R.

(2.17)

Next, taking the curl of (2.14) we get that curl j = ∇w × j, hence

j × curl j |j| 2 = ∇w -Π j (∇w) in R 3 , (2.18)
where Π j is the orthogonal projection on the subspace Rj. Hence, integrating (2.18) along the trajectory X 3 , we have

w X 32 (t 3 , t 2 , t 1 ) -w X 32 (0, t 2 , t 1 ) = w X 32 (t 3 , t 2 , t 1 ) -w X 2 (t 2 , X 1 (t 1 )) = ˆt3 0 ∇w X 32 (s, t 2 , t 1 ) • ∂X 32 ∂s (s, t 2 , t 1 ) ds = ˆt3 0 j × curl j |j| 2 2 X 32 (s, t 2 , t 1 ) ds = ˆt3 0 |curl j| 2 |j| 2 X 32 (s, t 2 , t 1 ) ds.
Then, integrating (2.18) along the trajectory X 2 and using (2.2), we get that

w X 2 (t 2 , X 1 (t 1 )) -w X 2 (0, X 1 (t 1 )) = w X 2 (t 2 , X 1 (t 1 )) -w X 1 (t 1 ) = ˆt2 0 ∇w X 2 (s, t 1 ) • ∂X 2 ∂s (s, t 1 ) ds = ˆt2 0 ∇w X 2 (s, t 1 ) • curl j |curl j| X 2 (s, t 1 ) ds = 0.
The two previous equalities combined with (2.17) yield the desired expression (2.8).

Second step: Construction of a grid on the surface {t 1 = c 1 }, generated by the flows X 2 , X 3 . Let us prove that for any (c 1 , t ′ 2 , t 2 , t 3 ) ∈ R 4 , the flows X 2 and X 3 generate on the regular surface {t 1 = c 1 }, a thin grid whose:

• step is of small enough size ν > 0,

• horizontal lines are trajectories along the flow X 2 ,

• vertical lines are trajectories along the flow X 3 ,

• two opposite vertices are the points

x ′ := X 2 (t ′ 2 , X 1 (c 1 )) and x := X 3 (t 3 , X 2 (t 2 , x ′ )) = X 32 (t 3 , t ′ 2 + t 2 , c 1 ). (2.19)
First, we divide the flows in small time steps, as shown the diagram

x  •  •  •  x ′ -→ • -→ • -→ • -→ • -→ • (2.20)
where the horizontal arrows represent the flow X 2 , and the vertical ones the flow X 3 . Then, we commute step by step the flows X 2 and X 3 , while remaining on the surface {t 1 = c 1 }, as shown the commutation diagram

• • • q • X 2 -→ q . . .  X 3 =⇒ X 3   X 3 p -→ X 2 • p -→ X 2 • (2.21)
to finally obtain the desired grid

y ′ -→ • -→ • -→ • -→ • -→ x       • -→ • -→ • -→ • -→ • -→ •       • -→ • -→ • -→ • -→ • -→ •       • -→ • -→ • -→ • -→ • -→ •       x ′ -→ • -→ • -→ • -→ • -→ y (2.22)
where by (2.19) there exists (s 2 , s 3 ) ∈ R 2 such that

x ′ := X 2 (t ′ 2 , X 1 (c 1 )), y ′ = X 3 (s 3 , x ′ ), y := X 2 (t 2 , x ′ ), x = X 3 (t 3 , y) = X 2 (s 2 , y ′ ). (2.
23) The vertices x ′ , y ′ , y, x of the grid (2.22) satisfy the commutative diagram

y ′ = X 3 (s 3 , x ′ ) X 2 -→ X 2 (s 2 , y ′ ) = X 3 (t 3 , y) = x X 3   X 3 x ′ = X 2 (t ′ 2 , X 1 (c 1 )) -→ X 2 X 2 (t 2 , x ′ ) = y.
(2.24)

Note that the grid (2.22) is schematic. A more realistic grid is represented in figure 1 below.

Frobenius' theorem will allow us to make the local switching (2.21) thanks to a potential u satisfying j = σ∇u. To this end, we proceed by induction on the number n of switchings, for an appropriate time step ν > 0 which will be chosen later. The induction hypothesis, for a given n ∈ N, consists in the existence of a partial grid G n (ν), with n switchings, represented by the diagram x

 •  • • • q -→ • -→ • . . .    • -→ • -→ p -→ • -→ • -→ •       x ′ -→ • -→ • -→ • -→ • -→ • (2.25)
which lies on the surface {t 1 = c 1 }.

First, the result holds for n = 0. Indeed, the points x ′ and x defined by (2.19) clearly belong to the surface {t 1 = c 1 }, so does the initial diagram (2.20) for any time step τ .

Next, assume that after a number n of switchings, we are led to the grid G n (ν) (2.25) which, by the induction hypothesis, lies on the surface {t 1 = c 1 }. By virtue of Frobenius' theorem there exist an open neighborhood V of p, and a potential u ∈ C 1 (V ) such that j = σ∇u in V . Then, we may chose τ > 0 small enough so that

q ∈ S 32 (p, τ ) := X 3 (δ 3 , X 2 (δ 2 , p)) : |δ 2 | + |δ 3 | < τ ⊂ V S 23 (p, τ ) := X 2 (δ 2 , X 3 (δ 3 , p)) : |δ 2 | + |δ 3 | < τ ⊂ V, (2.26)
independently of the point p in a given compact set of R 3 . Since the potential u is constant along the flows X 2 and X 3 , we have for any point r ∈ S 32 (p, τ ),

u(r) = u X 3 (δ 3 , X 2 (δ 2 , p)) = u X 3 (0, X 2 (δ 2 , p)) = u X 2 (δ 2 , p) = u X 2 (0, p) = u(p).
The same equality holds for any point r ∈ S 23 (p, τ ), hence

q ∈ S 32 (p, τ ) ⊂ u = u(p) ∩ V and S 23 (p, τ ) ⊂ u = u(p) ∩ V. (2.27)
Moreover, since the trajectory along the flow X 2 , passing through the point p in diagram (2.25) lies on the surface {t 1 = c 1 } (by the induction hypothesis), thanks to the semi-group property satisfied by the flow X 3 , we have that for any

|δ 2 | + |δ 3 | < t, there exists (t 2 , t 3 ) ∈ R 2 such that X 3 δ 3 , X 2 (δ 2 , p) = X 3 δ 3 , X 32 (t 3 , t 2 , c 1 ) = X 32 (δ 3 + t 3 , t 2 , c 1 ) ∈ t 1 = c 1 . (2.28)
Hence, by the definition (2.26) of S 32 (p, τ ), we get that

∀ t > 0, S 32 (p, t) ⊂ t 1 = c 1 .
(2.29) However, thanks to the condition (2.4) which yields ∇u = 0 in V , the sets S 32 (p, τ ), S 23 (p, τ ) and {u = u(p)} ∩V are regular open surfaces in V . Hence, from the inclusions (2.27) we deduce that the surfaces S 32 (p, τ ) and S 23 (p, τ ) actually agree with the equipotential {u = u(p)} in some neighborhood of p containing q, but independent of p in a given compact set K of R 3 . This combined with the condition (2.29) (which does not depend on time), implies that the time step ν < τ may be chosen small enough, but independently of the point p in K, so that

q ∈ S 23 (p, ν) ⊂ t 1 = c 1 .
(2.30)

Therefore, the following grid G n+1 (ν) which completes (2.25)

x  •  • -→ q -→ • -→ •     • -→ • -→ p -→ • -→ • -→ •       x ′ -→ • -→ • -→ • -→ • -→ • (2.31)
also lies on the surface {t 1 = c 1 }. The induction proof is thus done, which establishes the existence of the grid (2.22).

Third step: Proof of the isotropic realizability with the conductivity e w . Let us prove that the function w defined by (2.8) combined with condition (2.7) satisfies the equality (2.18). This yields the global isotropic realizability of j, since by (2.1), (2.2) and (2.18) we have curl (e -w j) = e -w (curl j -∇w × j) = e -w curl j -

(j × curl j) × j |j| 2 = 0. (2.32)
On the one hand, taking the derivative of (2.8) with respect to t 3 , we have

∇w • j × curl j |j| 2 X 32 (t 3 , t 2 , t 1 ) = |curl j| 2 |j| 2 X 32 (t 3 , t 2 , t 1 ) , ∀ (t 1 , t 2 , t 3 ) ∈ R 3 , (2.33)
which together with condition (2.7) implies that

∇w • j × curl j |j| 2 = |curl j| 2 |j| 2 = j × curl j |j| 2 2 in R 3 . (2.34)
Moreover, taking the derivative of (2.8) with respect to t 2 for t 3 = 0, we get that

0 = ∇w X 32 (0, t 2 , t 1 ) • ∂X 32 ∂t 2 (0, t 2 , t 1 ) = ∇w X 2 (t 2 , X 1 (t 1 )) • ∂X 2 ∂s (t 2 , X 1 (t 1 )) = ∇w X 2 (t 2 , X 1 (t 1 )) • curl j |curl j| X 2 (t 2 , X 1 (t 1 )) , Figure 1: The square Q split in small squares Q k on the surface {t 1 = c 1 } which yields ∇w • curl j = 0 on X 2 (t 2 , X 1 (t 1 )) : (t 1 , t 2 ) ∈ R 2 . (2.35)
On the other hand, for a constant c 1 ∈ R, consider on the surface {t 1 = c 1 } the curvilinear integral over ∂Q defined by

ˆ ∂Q ω := ˆt3 0 |curl j| 2 |j| 2 (X 3 (s, y) ds - ˆs3 0 |curl j| 2 |j| 2 X 3 (s, x ′ ) ds, (2.36)
where Q = Q x ′ ,y ′ ,y,x , with x ′ , y ′ , y, x defined by (2.23), is the "square" lying on the surface {t 1 = c 1 }, whose:

• horizontal sides are trajectories associated with the flow X 2 along curl j/|curl j|,

• vertical sides are trajectories associated with the flow X 3 along j/|j| 2 × curl j,

• vertices x ′ , y ′ , y, x make the loop (2.24). Now, consider the thin grid on Q, associated with the grid (2.22), whose lines are parallel to the trajectories along curl j/|curl j| and (j/|j| 2 ) × curl j, and which splits Q in small squares Q k , as shown in figure 1. According to the second step of the proof, we may choose the time step of the grid so that Frobenius' theorem applies in some open neighborhood V k of R 3 , containing the closed square Q k . Namely, there exists

w k ∈ C 1 (V k ) such that curl (e -w k j) = 0 in V k ,
which implies formulas (2.18) and (2.34) for w k . Using the loops induced by the boundaries ∂Q k , the contributions of the interior vertical lines (along j/|j| 2 × curl j) two by two cancel (see figure 1), which leads to

ˆ ∂Q ω = k ˆ ∂Q k ω.
(2.37) However, since

∇w k • curl j = 0 in V k ⊃ Q k ,
the curvilinear integrals of dw k on the two horizontal lines of ∂Q k (along curl j/|curl j|) are equal to 0, while by putting (2.34) in (2.36) the curvilinear integrals of dw k on the two vertical lines of ∂Q k (along j/|j| 2 × curl j) agree with the curvilinear integral of ω over ∂Q k , which yields

ˆ ∂Q k ω = ˆ ∂Q k dw k = 0. (2.38)
The last equality is due to the fact that dw k is an exact differential on the closed loop ∂Q k . Hence, from (2.37) and (2.38) we deduce that

ˆ ∂Q ω = 0. (2.39)
Next, consider the function w defined by (2.8). Taking the loop (2.24) in the anti-clockwise direction, the contributions of the vertical lines (along (j/|j| 2 )×curl j) of the curvilinear integral of dw over ∂Q in the anti-clockwise direction, read as, in view of (2.34) and (2.36),

ˆt3 0 ∇w • j × curl j |j| 2 X 3 (s, y) ds - ˆs3 0 ∇w • j × curl j |j| 2 X 3 (s, x ′ ) ds = ˆ ∂Q ω. (2.40)
Moreover, again by (2.24) the contributions of the horizontal lines (along curl j/|curl j|) of the curvilinear integral of dw over ∂Q in the anti-clockwise direction, read as

ˆt2 0 ∇w • curl j |curl j| X 2 (t 2 , x ′ ) ds - ˆs2 0 ∇w • curl j |curl j| X 2 (s, y ′ ) ds. ( 2
ˆs2 0 ∇w • curl j |curl j| X 2 (s, X 3 (s 3 , x ′ )) ds = 0, where x ′ = X 2 (t ′ 2 , X 1 (c 1 )). (2.43)
Then, making t 2 → 0 -which, by the continuity of the flows in the diagram (2.24), implies that s 2 → 0 and X 3 (s 3 , x ′ ) → X 3 (t 3 , x ′ ) -and using the continuity of the integrand in the left-hand side of (2.43), we get that

∇w • curl j X 32 (t 3 , t ′ 2 , c 1 ) = ∇w • curl j X 3 (t 3 , x ′ ) = 0, ∀ (c 1 , t ′ 2 , t 3 ) ∈ R 3 . (2.44)
This combined with the diffeomorphism condition (2.7) leads to

∇w • curl j = 0 in R 3 . (2.45)
Finally, combining the equalities (2.45) and (2.34) with (2.4), we derive the formula (2.18). Therefore, thanks to (2.32) we may conclude that the current field j is isotropically realizable with the conductivity e w defined by (2.8), which concludes the proof of Theorem 2.4.

Isotropic realizability in the torus

As in the case of an electric field [START_REF] Briane | Which electric fields are realizable in conducting materials?[END_REF] the isotropic realizability of a periodic current field in R 3 does not imply in general the isotropic realizability in the torus, as shown in Example 3.6 below.

First of all, we have the following criterion of isotropic realizability in the torus:

Proposition 2.8. Let j be a Y -periodic divergence free field in L ∞ ♯ (Y ) 3 . Then, the field j is isotropically realizable in the torus with a conductivity σ > 0 satisfying σ, σ -1 ∈ L ∞ ♯ (Y ), if and only if there exists w ∈ L ∞ (R) such that curl (e -w j) = 0 in R 3 .

(2.46)

Proof. Assume that j is isotropically realizable in the torus with a conductivity σ > 0 satisfying σ, σ -1 ∈ L ∞ ♯ (Y ). Then, defining w := ln σ ∈ L ∞ (R 3 ), the function e -w j is a gradient, which implies (2.46).

Conversely, assume that (2.46) holds with w ∈ L ∞ (R). Then, there exists u ∈ W 1,∞ (R 3 ) such that e -w j = ∇u, or equivalently j = e w ∇u in R 3 . It is not clear that e w is periodic. However, we can construct a suitable periodic conductivity by adapting the average argument of [START_REF] Briane | Which electric fields are realizable in conducting materials?[END_REF] (proof of Theorem 2.17). To this end, define the sequence (w n ) n∈N\{0} , by

w n (x) := 1 (2n + 1) 3 k∈Z 3 , |k|∞≤n w(x + k), for x ∈ R 3 , where |k| ∞ := max (|k 1 |, |k 2 |, |k 3 |) .
Since w is in L ∞ (R 3 ), the sequence w n is bounded in L ∞ (R), and thus converges weakly- * to some function w ♯ in L ∞ (R 3 ). It is easy to check that w ♯ is Y -periodic. Moreover, by the periodicity of j we have curl (e -wn j) = 1 (2n + 1) 3

k∈Z 3 , |k|∞≤n curl e -w(•+k) j(• + k) = 1 (2n + 1) 3 k∈Z 3 , |k|∞≤n curl ∇u(• + k) = 0.
As e -wn j converges weakly- * to e -w ♯ j in L ∞ (R 3 ) 3 , the previous equality leads to curl (e -w ♯ j) = 0 in D ′ (R 3 ) 3 .

Therefore, e -w ♯ j is a periodic gradient and j is isotropically realizable in the torus with the conductivity e w ♯ .

As a consequence of Proposition 2.8 and Theorem 2.4, we have the following result: Corollary 2.9. Let j be a Y -periodic current field satisfying conditions (2.1) and (2.2). i) Assume that the field j is isotropically realizable in the torus with a positive conductivity σ ∈ C 1 ♯ (Y ). Then, there exists a constant C > 0 such that

ˆ∞ -∞ |curl j| 2 |j| 2 X 3 (s, x) ds ≤ C, ∀ x ∈ R 3 , (2.47)
where the flow X 3 is defined by (2.5).

ii) Alternatively, assume that the conditions (2.4) and (2.7) are satisfied. Then, the field j is isotropically realizable in the torus if and only if estimate (2.47) holds true.

Remark 2.10. Corollary 2.9 with the boundedness condition (2.47) is illustrated in Proposition 3.5 below.

Proof of Corollary 2.9. Proof of i). Denoting σ = e w ∈ C 1 (R 3 ), we have j = e w ∇u, with ∇u ∈ C 1 (R 3 ) 3 . Then, from (2.18) we deduce that

∂ ∂t w X 3 (t, x) = |curl j| 2 |j| 2 X 3 (t, x) , ∀ (t, x) ∈ R × R 3 , which yields w X 3 (t, x) -w(x) = ˆt 0 |curl j| 2 |j| 2 X 3 (s, x) ds, ∀ (t, x) ∈ R × R 3 . (2.48)
Therefore, due to the boundedness of w ∈ C 1 ♯ (Y ), the estimate (2.47) holds. Proof of ii). By Theorem 2.4 we already know that j is isotropically realizable with the conductivity σ = e w ∈ C 1 (R 3 ) defined by (2.8). If the field j is isotropically realizable in the torus, then by i) the estimate (2.47) holds. Conversely, in view of the estimate (2.47) combined with the definition (2.6) of X 32 , the function w defined by (2.8) clearly belongs to L ∞ (R 3 ). Therefore, applying Proposition 2.8 with the conductivity e w , we get that j is isotropically realizable in the torus.

Examples and counter-examples

In this section a point of R 3 is denoted by the coordinates (x, y, z), and (e x , e y , e z ) denotes the canonical basis of R 3 .

Example of global realizability in the space

We will illustrate the construction of Theorem 2.4 with the current field j(x, y, z)

:=   1 sinh x 0   , for (x, y, z) ∈ R 3 , (3.1)
which is clearly non-zero and divergence free in R 3 . We have curl j(x, y, z) :=

  0 0 cosh x   and j × curl j(x, y, z) =   sinh x cosh x -cosh x 0   , (3.2)
so that the field j satisfies (2.1) and Frobenius' condition (2.2).

Noting that |j| = |curl j| = | cosh x|, and using the solutions to one-dimensional first-order odes, the flows X 1 , X 2 , X 3 of (2.5) associated with the field j are given by

X 1 (t) =    argsinh t + sinh (x 1 (0)) 1 + t + sinh (x 1 (0)) 2 + y 1 (0) -cosh(x 1 (0)) z 1 (0)    , X 2 (t) =   x 2 (0) y 2 (0) t + z 2 (0)   , X 3 (t) =      argsinh e t sinh (x 3 (0)) - ˆt 0 ds 1 + e 2s sinh 2 (x 3 (0)) + y 3 (0) z 3 (0)      . (3.3)
Hence, starting from the point x 0 = (0, 1, 0), the composed flows X 32 , X 23 of (2.6) are given by

X 32 (t 3 , t 2 , t 1 ) =     argsinh t 1 e t 3 - ˆt3 0 ds 1 + e 2s t 2 1 + 1 + t 2 1 t 2     , for (t 1 , t 2 , t 3 ) ∈ R 3 . (3.4)
Making classical changes of variables in the integral term, formula (3.4) can be written

X 32 (t 3 , t 2 , t 1 ) =                            0 1 -t 3 t 2   if t 1 = 0      argsinh t 1 e t 3 argtanh 1 1 + t 2 1 e 2t 3 + f (t 1 ) t 2      if t 1 = 0, (3.5)
where

f (t) := -argtanh 1 √ 1 + t 2 + √ 1 + t 2 , for t = 0. (3.6)
The function f satisfies

f (t) = ln |t| + 1 -ln 2 + o(1) and f ′ (t) = √ 1 + t 2 t = 1 t + o(1), (3.7)
and is a C 1 -diffeomorphism from (0, ∞) onto R, and from (-∞, 0) onto R. Hence, we have

∀ (x, y) ∈ R \ {0} × R, ∃ ! t 1 ∈ R, x t 1 > 0 and y -argtanh 1 cosh x = f (t 1 ), (3.8)
which together with (3.5) implies that for this t 1 ,

(x, y, z) = X 32 (t 3 , t 2 , t 1 ) ⇔                         x = 0 y = 1 -t 3 z = t 2 if t 1 = 0        sinh x = t 1 e t 3 y -argtanh 1 cosh x = f (t 1 ) z = t 2 .
if t 1 = 0.

(3.9)

As a consequence of (3.5), (3.6), the first equality of (3.7), (3.8) and (3.9), the mapping X 32 define a homeomorphism onto R 3 , which is of class C 1 by (3.4). Unhappily, we can check that the Jacobian of X 23 vanishes (exclusively) on the line {t 1 = 0}. However, taking into account the equality sinh x = t 1 e t 3 , X 23 does establish a C 1 -diffeomorphism from the half-spaces {± t 1 > 0} onto the half-spaces {± x > 0}. Therefore, condition (2.7) holds true restricting ourselves on these half-spaces. On the other hand, since |j| = |curl j|, the function w defined by (2.8) reads as

w(x, y, z) = t 3 =      1 -y if x = 0 ln sinh x t if x = 0, where x t > 0, y -argtanh 1 cosh x = f (t).
(3.10)

It is easy to check that the asymptotic expansions (3.7) satisfied by f imply that w ∈ C 1 (R 3 ). Hence, the conditions of Theorem 2.4 are fulfilled in the two half-spaces {± x > 0}. Therefore, the field j defined by (3.1) is isotropically realizable in the half-spaces {± x > 0}, with the conductivity σ ∈ C 1 (R 3 ) given by σ(x, y, z) :=

     e 1-y if x = 0 sinh x t if x = 0, where x t > 0, y -argtanh 1 cosh x = f (t). (3.11)
Finally, the C 1 -regularity of j and σ ensure the isotropic realizability in the whole space R 3 . Since X 32 is a homeomorphism onto R 3 of class C 1 and the function w of (3.10) is in C 1 (R 3 ), we can also conclude thanks to Remark 2.6.

Remark 3.1. The previous example allows us to show that there exists an infinite set of admissible conductivities which cannot be derived from a multiple of the conductivity (3.11) defined with the function (3.6). To this end, consider any function f : R \ {0} → R which is a C 1 -diffeomorphism from (0, ∞) onto R and from (-∞, 0) onto R, which satisfies the following asymptotic expansions around 0:

f (t) = ln |t| + c + o(1) and f ′ (t) = 1 t + c ′ + o(1), for some c, c ′ ∈ R. (3.12)
Then, define the conductivity σ f by

σ f (x, y, z) :=      2 e c-y if x = 0 sinh x t if x = 0, where x t > 0, y -argtanh 1 cosh x = f (t). (3.13)
Thanks to (3.6), (3.7) and (3.12) the function σ f belongs to C 1 (R). Moreover, we have for any

(x, y) ∈ R \ {0} × R, curl σ -1 f j = curl    t sinh x t 0    =     0 0 ∂t ∂x - 1 sinh x ∂t ∂y     . (3.14) This combined with - ∂ ∂x argtanh 1 cosh x = 1 sinh x = ∂t ∂x f ′ (t) and 1 = ∂t ∂y f ′ (t), (3.15)
yields that curl (σ -1 f j) = 0 in R \ {0} × R. Since σ f is in C 1 (R)
, the equality holds in R 2 . Therefore, the field j defined by (3.1) is isotropically realizable in R 3 with any conductivity σ f defined by (3.13) and (3.12).

Example of non-global realizability under Frobenius' condition

The following example shows that condition (2.2) is not sufficient to derive a global realizability result in accordance with the local character of Frobenius' theorem. This example is an extension to a divergence free non-vanishing field in R 3 of the counterexample of [START_REF] Boyling | Carathéodory's principle and the global existence of an integrating factor[END_REF] obtained for a non-vanishing field in R 2 .

Define the function j in R 3 by j(x, y, z) :=

  f (x) f ′ (x) -z f ′ (x)   , where f (x) := 8 x 3 -6 x 4 -1, (3.16)
which satisfies condition (2.1). We have

curl j(x, y, z) =   0 z f ′′ (x) f ′′ (x)   ,
so that Frobenius' condition (2.2) holds, but not (2.4) since f ′′ ( 2/3) = 0. Now, assume that j can be written σ∇u for some positive continuous function σ in the closed strip {0 ≤ x ≤ 1}. We have for any x ∈ (0, 1), f ′ (x) = 0 and

       f (x) = σ ∂u ∂x f ′ (x) = σ ∂u ∂y , hence ∂u ∂x = f (x) f ′ (x) ∂u ∂y . (3.17)
Then, the method of characteristics implies that for a fixed z ∈ R, there exists a function

g z in C 1 (R) such that u(x, y, z) = g z y + F (x) , ∀ (x, y) ∈ (0, 1) × R, (3.18)
where F is the primitive of f /f ′ on (0, 1) defined by

F (x) := x 2 8 - x 12 - ln(1 -x) 24 - ln x 24 + 1 24x
, for x ∈ (0,

As x → 0, x > 0, we have by (3.19)

t := y + F (x) ∼ x→0 1 24 x .
This combined with (3.16), (3.17), (3.18) yields

g ′ z (t) = 1 σ(x, y, z) f ′ (x) ∼ t→∞ 1 24 σ(0, y, z) 1 t 2 . As x → 1, x > 1, we have by (3.19) t = y + F (x) = y + 1 12 - ln(1 -x) 24 + o(1), hence 1 -x ∼ t→∞ e 24y+2 e -24t ,
which implies that

g ′ z (t) = 1 σ(x, y, z) f ′ (x) ∼ t→∞ 24 e 24y+2 σ(1, y, z) e -24t .
Therefore, the two asymptotic expansions of g z (t) as t → ∞, lead to a contradiction.

3.3

The case where the current field lies in a fixed plane Consider a periodic field j in C 2 ♯ (Y ) 3 satisfying (2.1) and (2.2) which remains perpendicular to a fixed direction. By an orthogonal change of variables we may assume that j lies in the plane {z = 0}, namely j = (j x , j y , 0). From now on, any vector of R 2 will be identified to a vector of R 3 with zero z-coordinate. Hence, we deduce that

j • curl j = 0 ⇔ ∂ z j j in R 3 .
(3.20)

Then, using the representation of a divergence-free field by an orthogonal gradient in R 2 , the most general expression for a divergence free field j satisfying (3.20) is

j(x, y, z) = α(x, y, z) ∇ ⊥ v(x, y) = α(x, y, z)   -∂ y v(x, y) ∂ x v(x, y) 0   for (x, y, z) ∈ R 3 , with ∂ x α ∂ y v -∂ y α ∂ x v = 0 in R 3 , (3.21)
where α ∈ C 2 ♯ (Y ′ ), α > 0, and v ∈ C 3 (R 2 ) 2 , with ∇v Y ′ -periodic. For the sake of simplicity, we assume from now on that the function α only depends on the coordinate z. Therefore, we are led to j(x, y, z) = α(z) ∇ ⊥ v(x, y), for (x, y, z

) ∈ R 3 , (3.22)
where α ∈ C 2 ♯ ([0, 1]), α > 0, and v ∈ C 3 (R 2 ) 2 , with ∇v Y ′ -periodic and ∇v = 0 in R 2 by (2.1). Following the isotropic realizability procedure for a gradient field [START_REF] Briane | Which electric fields are realizable in conducting materials?[END_REF], consider the gradient system Z ′ (t, x, y) = ∇v Z(t, x, y) , for t ∈ R, satisfies the conductivity equation div (e -wv ∇v) = 0 in R 2 .

Z(0, x, y) = (x, y) ∈ R 2 . ( 3
We have the following isotropic realizability result:

Proposition 3.2. If the function w v of (3.25) is in L ∞ (R 2
), then the field j defined by (3.22) is isotropically realizable in the torus. Conversely, if j is isotropic realizable in the torus with a positive conductivity σ ∈ C 1 ♯ (Y ), then the function w v is in L ∞ (R 2 ). On the other hand, if w v belongs to L ∞ (R 2 ), then condition (2.47) holds.

Remark 3.3. The criterion for the isotropic realizability of j in the torus given in Proposition 3.2, that is the boundedness of w v , implies condition (2.47). The converse is not clear. The reason is that the field j defined by (3.22) does not satisfy condition (2.4). More precisely, the curl of j curl j(x, y, z) = -α ′ (z) ∇v(x, y) + α(z) ∆v(x, y) e z , for (x, y, z) ∈ R 3 , (3.26) does vanish in R 3 . Indeed, due to the periodicity of α and ∇v we have

ˆ1 0 α ′ (z) dz = ˆY ′ ∆v(x, y) dxdy = 0.
This combined with the continuity of α ′ and ∆v implies the existence of a point (x, y, z) ∈ R 3 such that α ′ (z) = ∆v(x, y) = 0. Therefore, we cannot use Theorem 2.4.

Proof of Proposition 3.2. Assume that the function w v of (3.25) is in L ∞ (R 2 ). Then, by Theorem 1.1 the gradient field ∇v is isotropically realizable in the torus, namely there exists a periodic positive conductivity σ, with σ, σ -1 ∈ L ∞ ♯ (Y ′ ), such that div (σ∇v) = 0 in R 2 . Hence, there exists a stream function u, with ∇u ∈ L 2 ♯ (Y ′ ) 2 , such that σ∇v = -∇ ⊥ u in R 2 . Therefore, by (3.22) j = α σ -1 ∇u is isotropically realizable in the torus.

Conversely, assume that j is isotropically realizable in the torus with a positive conductivity σ ∈ C 1 ♯ (Y ). Then, since j z = 0, we have j = σ∇u in R 3 , with ∇u ∈ C 1 ♯ (Y ′ ) 2 . Equating this equality with (3.22) at z = 0, we get that

α(0) σ(x, y, 0) ∇v(x, y) = -∇ ⊥ u(x, y), for (x, y) ∈ R 2 .
Thus, ∇v is isotropically realizable in the torus with the conductivity α(0) σ -1 (•, 0) ∈ C 1 ♯ (Y ′ ), which by virtue of Theorem 1.1 implies that w v belongs to L ∞ (R 2 ). Now, assume that w v is in L ∞ (R 2 ). By (2.5), (3.22) and (3.26) we have

∂X 3 ∂t = j × curl j |j| 2 (X 3 ) = ∆v(X 3 ) |∇v(X 3 )| 2 ∇v(X 3 ) + α ′ (X 3 • e z ) α(X 3 • e z ) e z .
Moreover, by (3.25) we have div (e wv ∇v) = 0 in R 2 , or equivalently ∆v = -∇w w • ∇v in R 2 , hence

∂X 3 ∂t = -Π ∇v (∇w v )(X 3 ) + α ′ (X 3 • e z ) α(X 3 • e z ) e z , (3.27)
where Π ∇v denotes the orthogonal projection on R∇v. However, since ∇w v -Π ∇v (∇w v ) is parallel to j by (3.22) and thus orthogonal to ∂ t X 3 by (2.5), we get that

∂X 3 ∂t 2 = -Π ∇v (∇w v )(X 3 ) • ∂X 3 ∂t + α ′ (X 3 • e z ) α(X 3 • e z ) ∂X 3 ∂t • e z = -∇w v (X 3 ) • ∂X 3 ∂t + α ′ (X 3 • e z ) α(X 3 • e z ) ∂X 3 ∂t • e z = ∂ ∂t -w(X 3 ) + ln α(X 3 • e z )
It follows that for any (t, x, y, z) ∈ R 4 ,

ˆt 0 ∂X 3 ∂s (s, x, y, z) 2 ds = w v (x, y, z) -w v X 3 (t, x, y, z) + ln α X 3 (t, x, y, z) • e z α(z) . (3.28)
The function ln α is periodic and continuous in R, hence it is bounded. Therefore, equality (3.28) shows that the boundedness of w v implies condition (2.47).

Remark 3.4. Note that the field current j defined by (3.1) has a zero z-coordinate, and

j = ∇ ⊥ v with v(x, y) := cosh x -y, (3.29)
so that we could a priori use the above method. However, in this case the solution of the gradient system (3.23) is given by Z(t, x, y) = 2 argtanh e t tanh (x/2) -t + y , (3.30) which is clearly not a global solution. Therefore, the present two-dimensional approach does not work for the very simple current field (3.1).

A particular class of current fields

Let f, g, h ∈ C 2 (R) be three 1-periodic functions such that f has only isolated roots in R which are not roots of f ′ , while g, h do not vanish in R. Then, the following isotropic realizability result holds:

Proposition 3.5. The Y -periodic field j defined by

j(x, y, z) :=   g(y) h(z) f (x) h(z) f (x) h(z)   , for (x, y, z) ∈ R 3 , (3.31)
satisfies the conditions (2.1) and (2.2). On the other hand, consider the following assertions:

(i) the function f does not vanish in R 3 , (ii) the field j is isotropically realizable in the torus with a positive conductivity σ ∈ C 1 ♯ (Y ), (iii) condition (2.47) holds.

Then, (i) and (ii) are equivalent conditions and they imply assertion (iii). Moreover, when g = h = 1 all three assertions (i), (ii) and (iii) are equivalent.

Proof. Condition (2.1) clearly holds. We have

curl j =   f (x) g ′ (y) -h ′ (z) g(y) h ′ (z) -f ′ (x) h(z) f ′ (x) -g ′ (y)   , for (x, y, z) ∈ R 3 , (3.32)
hence condition (2.2) is also satisfied. However, note that condition (2.7) does not hold.

(i) ⇒ (ii). If f does not vanish in R, we can write

j(x, y, z) = f (x) g(y) h(z) ∇u(x, y, z), with u(x, y, z) := ˆx 0 dt f (t) + ˆy 0 dt g(t) + ˆz 0 dt h(t) . (3.33)
Therefore, j is isotropically realizable in the torus with the conductivity σ(x, y, z) := f (x) g(y) h(z) > 0, which belongs to C 1 ♯ (Y ). (ii) ⇒ (i). More precisely, we will prove that if f vanishes in R, then j is not isotropically realizable with any positive continuous conductivity σ(x, y, z) which is Y ′ -periodic with respect to (y, z).

To this end, assume by contradiction that both f vanishes in R and j is isotropically realizable with a positive continuous conductivity σ(x, y, z) which is Y ′ -periodic with respect to (y, z), Y ′ = [0, 1] 2 . Let a < b be such that f (a) = 0 and f (b) = 0. Starting from the equality curl (σ -1 j) = 0, integrating by parts over the cube [a, b] × Y ′ , and denoting by n the outside normal, we have 0 (ii) ⇒ (iii). This is a straightforward consequence of Corollary 2.9 i).

= ˆ[a,b]×Y ′ curl (σ -1 j) • e y dx dy dz = ˆ∂([a,b]×Y ′ ) (n × σ -1 j) • e y ds = ˆ{a,b}×Y ′ (n × σ -1 j) • e y ds + ˆ[a,b]×∂Y ′ (n × σ -1 j) • e y ds.
(iii) ⇒ (i), when g = h = 1. Assume by contradiction that f vanishes at some point a ∈ R. Since f ′ (a) = 0, we may assume that, for instance, there exists a real number b > a such that f > 0 in (a, b] and f ′ > 0 in [a, b]. By (3.31) and (3.32) we have

j × curl j =   2f (x)f ′ (x) -f ′ (x) -f ′ (x)   , for x ∈ R,
hence the flow X 3 of (2.5) reads as Therefore, the L 2 (R)-norm of t → ∂ t X 3 (t, x, y, z) is infinite for any x ∈ (a, b]. This proves the implication (iii) ⇒ (i), when g = h = 1.

                 x ′ (t) = 2f (x(t))f ′ (x(t)) 2f 2 (x(t)) + 1 , x(0) = x y ′ (t) = - f ′ (x(t)) 2f 2 (x(t)) + 1 , y(0) = y z ′ (t) = - f ′ (x(t))
Example 3.6. Consider the particular case of (3.31) where g = h = 1. When the function f vanishes in R, the field j ∈ C 2 ♯ (Y ) 3 still satisfies conditions (2.1) and (2.2). Since condition (2.4) does not hold, Theorem 2.4 does not apply. However, the field j is actually isotropically realizable in the whole space R 3 , but not in the torus due to Proposition 3.5.

It is not obvious how to derive an explicit conductivity associated with j, but we now proceed to do so. To this end, we may assume that f ∈ C 2 (R) is a 1-periodic function satisfying f (0) = 0, f ′ (0) = 0, and f > 0 in (0, 1). Then, j lies in the plane {j y = j z }, so that we can apply the procedure of Section 3.3 based on the representation of two-dimensional divergence-free functions by orthogonal gradients. This combined with the approach of [START_REF] Briane | Which electric fields are realizable in conducting materials?[END_REF] (Proposition 2.11) allows us to construct a conductivity σ ∈ C 1 (R 3 ) such that curl (σ -1 j) = 0 in R 3 , as follows:

Let F be the function defined in (0, 1) by F (x) := ˆx 1 2 ds f (s) , for x ∈ (0, 1).

(3.38)

The function F is a C 1 -diffeomorphism from (0, 1) onto R. Then, denoting by F -1 its reciprocal, an admissible conductivity σ is given by σ(x, y, z) :=

     f (x) f F -1 (y + z + F (x -n)) if x ∈ (n, n + 1) e -f ′ (0)(y+z) if x = n, for n ∈ Z, (3.39)
which is 1-periodic with respect to x. Let us prove that σ ∈ C 1 (R 3 ), and curl (σ -1 j) = 0 in R 3 . For x ∈ (0, 1) and (y, z) ∈ R 2 , set t := F -1 (y + z + F (x)) ∈ (0, 1). Since f ′ (0) = 0 and f ∈ C 2 (R), we have for n = 0, 1, or equivalently, curl (e -w j) = 0 in R 3 . Therefore, the current field j = 1, f (x), f (x) is isotropically realizable with the conductivity σ ∈ C 1 (R 3 ) defined by (3.38) and (3.39). Note that the function σ is 1-periodic with respect to x, but is not periodic with respect to (y, z) in accordance with Proposition 3.5. Finally, the isotropic realizability of j in R 3 can be written

y + z - 1 f ′ (0) ln t -n x -n = F (t) -F (x) - 1 f ′ (0) ln t -n x -n = ˆt x 1 f (s) - 1 f ′ (0)(s -n) ds ≤ C |t -x|,
  1 f (x) f (x)   = σ(x, y, z) ∇ x + ˆy+z 0 f F -1 (s + F (x -n)) ds , for x ∈ (n, n + 1), n ∈ Z,
where the conductivity σ is defined by (3.39).

Theorem 1 . 1 ( 4 )

 114 [START_REF] Briane | Which electric fields are realizable in conducting materials?[END_REF], Theorems 2.15 & 2.17). Let u be a function in C 3 (R d ) satisfying the non-Then, there exists a unique function τ ∈ C 1 (R d ) such that for any x ∈ R d , the trajectory t → X(t, x) meets the equipotential {u = 0} at the times τ (x), namely u X(τ (x), x) = 0.(1.5)Moreover, the positive function σ defined in R d by σ(x) := exp ˆτ(x) 0 ∆u X(s, x) ds , for x ∈ R d , (1.6)

  .23) Then, by virtue of Theorem 1.1 there exists a unique function τ v ∈ C 1 (R 2 ) such that v Z(τ v (x, y), x, y) = 0, for any (x, y) ∈ R 2 , (3.24) and the function w v defined by w v (x, y) := -ˆτv(x,y) 0 ∆v Z(s, x, y) ds, for (x, y) ∈ R 2 , (3.25)

( 3 .

 3 34)The integral over [a, b] × ∂Y ′ in (3.34) is equal to zero due to the Y ′ -periodicity of σ -1 j with respect to (y, z). Hence, using that f (a) = 0 and (e x × j) • e y = -j z = -f (x) g(y), it follows that the integral over {a, b} × Y ′ in (3.34) satisfies0 = -ˆY ′ f (b) g(y) σ -1 (b, y, z) dy dz.(3.35)This leads to a contradiction, since the function (y, z) → f (b) g(y) σ -1 (b, y, z) is continuous and has a constant sign in Y ′ .

  2f 2 (x(t)) + 1 , z(0) = z, for (x, y, z) ∈ R 3 . (3.36) Define the function F in (a, b] byF (x) := ˆx b 2f 2 (s) + 1 2f (s)f ′ (s) ds, for x ∈ (a, b].The function F is an increasing bijection from (a, b] onto (-∞, 0], and the solution of the first equation of (3.36) is given byx(t) = F -1 t + F (x) , for t ≤ 0,where F -1 denotes the reciprocal of F . Making the change of variables r = x(s), we have for any t ≤ 0, x(t) tends to a as t → -∞ and f (r) ∼ a f ′ (a) (r -a), we get thatˆ0 -∞ ∂X 3 ∂s (s, x, y, z) 2 ds ≥ -lim t→-∞ ˆt 0 y ′ (s) 2 ds = -ˆa x f ′ (r) 2f (r) 2f2 (r) + 1 dr = ∞. (3.37)

  x -n = e f ′ (0)(y+z) , for n = 0, 1.(3.41)This shows the continuity of the function σ. Moreover, we have for any x ∈ (0, 1),∂σ ∂x = f ′ (x) -f ′ (t)(3.41) implies that ∇σ has finite limits as x → 0 or 1. Therefore, the function σ belongs to C 1 (R 3 ).Set w := ln σ. By (3.39) and (3.42) we have for any x ∈ (0, 1), 31) and (3.32) with g = h ≡ 1, equations (3.43) lead to the equation curl j -∇w × j = 0 in R 3 ,

Acknowledgments. GWM thanks the National Science Foundation for support through grant DMS-1211359. Also the authors thank Andrejs Treibergs for his comments.


Maggi ⋆ Lorenzo

Francesco De Pellegrini

Alexandre Reiffers

Jean-Jacques Herings

Eitan Altman

Coordination of Epidemic Control Policies: A Game Theoretic Perspective

Keywords: Epidemic games, Nash bargaining, Mechanism Design, Truth revealing

We consider two neighbouring countries in which a pandemic disease spreads. Countries face a trade-off between the social costs of the epidemic diffusion and the monetary costs in order to avoid the insurgence of pandemics. However, due to migration of people across countries, the treatment efforts by one country generate a positive externality for the neighbouring country. Both countries can negotiate on the healthcare cost that each has to sustain. But, they do so subject to a central authority (CA) who can impose penalties to both countries whenever they cannot reach an agreement. We analyse the outcome of such situation via the Nash bargaining concept. Next we show how the CA should design penalties to i) ensure that revealing the true migration flow data is a self-enforcing behaviour, and to ii) enforce that the NB solution adheres to certain fairness properties.

I. INTRODUCTION

The dynamics of contagious diseases over a population of individuals under health treatment is a traditional topic in mathematical biology [1].

On the one hand, when a certain disease spreads across its population, a country incurs healthcare monetary costs stemming from hospitalization and treatment of infected individuals. On the other hand, countries also need to face societal costs which increase with the number of infected individuals living on their soil. One of them is clearly of ethical nature: countries should guarantee the permanence of a healthy environment within their borders. Moreover, there exist also several indirect costs related to the temporary reduction of workforce. Such costs altogether amount to the social cost incurred by every country subject to an epidemic outbreak.

In this paper, we explore an epidemic model where we also account for the effect of mobility across countries. Due to individuals moving across borders, epidemic processes developing in each country are subject to coupling effects. Hence, healthcare treatments performed in one country also have an impact on the pandemic spread in the neighbouring country. In economic terminology, we can say that healthcare efforts in one country generate positive externalities for the other country. Hence, this gives rise to a game situation. We consider the situation where two countries bargain on the healthcare costs that each one has to sustain. If negotiations succeed, then the countries come up with an allocation which is Pareto optimal. If instead the two countries are not able to come up with an agreement, then a Central Authority (CA) has the power to inflict monetary penalties to both countries. We resort to the concept of Nash bargaining solution (NBS) to predict the outcome of such negotiations.

The penalty imposition has a three-fold objective. The CA wants to ensure that the two countries make efficient usage of resources for the control of epidemics, i.e., by ensuring Pareto optimality. Secondly, by choosing appropriate penalties the CA can induce a malicious country possessing private migration flow data to reveal them, so as to guarantee a correct healthcare cost sharing. Finally, the penalties imposed (or threatened) by the CA can enforce the resulting NBS to adhere to some socially desirable fairness properties.

We derive our results under the assumption that the infection process evolves at a time scale which is much slower than the one underlying the migration process. This allows us to exploit a technique that aggregates in a simple manner all the parameters at stake in a single differential equation, i.e., migration rates, healthcare effort costs, and epidemic spread speeds, in the single countries.

The paper is organized as follows. In Section I-A we resume the related works on the optimal control and games models developed in literature for epidemics. Section II introduces our epidemic and game-theoretic model. In Section III we compute the Nash bargaining solution (NBS) to the healthcare cost sharing negotiations, under both convex and linear cost assumptions. Section IV provides insights on the application of our solution concept to the design of penalties inflicted by the Central Authority (CA). Specifically, in Section IV-A we show the expression of self-enforcing truth-revealing penalties, that minimize the probability that a malicious country has an incentive not to disclose private migration flow statistics. Finally, in Section IV-B we show how to enforce a fairness criterion. The reader may find all the proofs in the extended version of this paper at [START_REF] Maggi | Coordination of epidemic control policies: A game theoretic perspective[END_REF].

A. Related works

Epidemic diffusion is a mainstream research topic in mathematical biology [1]. Also, the presence of critical recovery rates is well known [START_REF] Pastor-Satorras | Epidemic spreading in scale-free networks[END_REF]: the so called epidemic threshold poses a recovery rate below which a pandemic affects the whole population, and above which it dies out. Results on epidemic thresholds have been derived for networked systems using the NIMFA mean-field approach [START_REF] Van Mieghem | Virus spread in networks[END_REF]- [START_REF] Bonaccorsi | Epidemic outbreaks in two-scales community networks[END_REF]. The notion of epidemic threshold is core in the model we propose. Optimal treatment campaigns for the susceptibleinfected-susceptible (SIS) model was solved in the seminal work [START_REF] Sethi | Optimal Control Theory: Applications to Management Science and Economics[END_REF]: the cost model we adopt resembles the standard cost proposed there. Recent developments appeared in [START_REF] Khouzani | Optimal control of epidemic evolution[END_REF] for the SIR model. Similarly, [START_REF] Verriest | Control of epidemics by vaccination[END_REF] provides optimal pulse control policies for vaccination. Recent studies have investigated the key role of migration in order to describe the epidemic process which may develop across communities [START_REF] Colizza | Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations[END_REF], [START_REF] Poletto | Host mobility drives pathogen competition in spatially structured populations[END_REF]. Our research is partially motivated by this observation. The paper [START_REF] Rowthorn | Optimal control of epidemics in metapopulations[END_REF] provides optimal control strategies for coupled populations under budget constraints. The authors also demonstrate that a strategy giving preferential treatment in the region with the higher prevalence of infected proves the worst possible strategy. Because of immigration flows, healthcare efforts in one country have positive externalities for the other country. Game theory is the standard tool to deal with situations involving externalities. Very few papers address a game-theoretical analysis of epidemic diffusion. The paper [START_REF] Omic | Protecting against network infections: A game theoretic perspective[END_REF] discusses malware prevention for computer networks: the key observation is that the health of a player depends on the recovery rate of the other peers. The authors of [START_REF] Smith | Strategic interactions in multi-institutional epidemics of antibiotic resistance[END_REF] study a game for the so called spillover effect due to the coupling of infection levels and drug resistance among hospitals. [START_REF] Wang | Decentralized resource allocation to control an epidemic: A game theoretic approach[END_REF] studies a two-countries game with no migration. Similar to our case, objective for each country is to minimize the number of infective individuals over the entire time horizon given a finite budget for vaccination. Each country allocates such finite budget so as to minimize the number of infected individuals within their own population.

Main contribution:

We provide a framework for studying the orchestration of healthcare efforts among neighbouring countries with cross-migration. Countries face a trade-off between the social costs for the epidemic diffusion and the monetary costs for healthcare efforts. We further provide tools for an optimal and fair design of penalties that a CA should inflict to the countries when an agreement is not reached, in such a way that disclosing the real migration statistics is a self-enforcing behaviour.

II. SYSTEM AND GAME MODEL

We consider hereafter two neighbouring countries, in which a pandemic disease spreads. People migrate at a constant rate Kp ij from country i to country j. Our model is developed under a fast mixing assumption: we assume that the migration process occurs at a much faster scale than the infection process. The parameter K determines the relative speed of the two processes. We denote N i (t) as the number of people in country i at time t. Then, we can write

Ṅ1 (t) = K p 21 N 2 (t) -p 12 N 1 (t) (1) Ṅ2 (t) = K p 12 N 1 (t) -p 21 N 2 (t) = -Ṅ1 (t).
(2) We first note that, under the flow conservation argument in (2), the total number of people in the system is constant, and we denote it as N :

N := N 1 (t) + N 2 (t), ∀ t ≥ 0.
Let us now compute the stationary solutions N 1 , N 2 by setting Ṅ1 (t) = Ṅ2 (t) = 0:

N i = α i N, i = 1, 2 (3)
α 1 = p21 p12+p21 = 1 -α 2 .
The evolution dynamics of the number X 1 (t), X 2 (t) of infected individuals at time t in country 1,2, respectively, are coupled via the mobility of individuals between the two countries. In order to make the model more realistic, we should assume that, during the incubation period 1 people migrate with the same rate as the healthy individuals, hence they are potentially able to spread the disease in the neighbouring country. On the other hand, individuals who already show symptoms should have a reduced mobility. Nevertheless, in order to keep the model analytically tractable, we assume that the incubation period is negligible, and the same mobility pattern holds for both infected and non-infected individuals. Hence, we are allowed to write the same equations (1) and (2) for X 1 (t), X 2 (t), i.e.,

Ẋ1 (t) = K p 21 X 2 (t) -p 12 X 1 (t) (4) Ẋ2 (t) = K p 12 X 1 (t) -p 21 X 2 (t) = -Ẋ1 (t), (5)
We model the epidemic spread evolution in each country as a classic SIS compartmental model [1]. We call λ i the rate of infection in country i and we denote γ i the control variable representing the recovery rate of infected individuals in country i. It is induced by the treatment efforts operated over the population. When K is big enough to have Kp ij ≫ λ k , γ k for all i, j, k, then our fast mixing assumption holds, and we can exploit the aggregation technique utilized in [START_REF] Auger | Methods of aggregation of variables in population dynamics[END_REF] to approximate i) the number of people in country i as constant in t and equal to its asymptotic value N i , i = 1, 2. Moreover, from (4,5) we can consider ii) the number of infected individuals in the two countries as proportional to their total population, i.e., X 1 (t)/X 2 (t) = N 1 /N 2 , for all t ≥ 0. Therefore, we can write

Ẋ1 (t) = λ 1 X 1 (t) N 1 -X 1 (t) -γ 1 X 1 (t) (6) Ẋ2 (t) = λ 2 X 2 (t) N 2 -X 2 (t) -γ 2 X 2 (t) (7) X 1 (t) X 2 (t) = N 1 N 2 = p 21 p 12 . (8)
We remark that the recovery rate is static, i.e., γ i does not depend on t. The disease spread and its treatment are considered over the finite time interval [0, T ].

Finally, if we define X(t) := X 1 (t) + X 2 (t), and sum equations (6) [START_REF] Sethi | Optimal Control Theory: Applications to Management Science and Economics[END_REF], we obtain

Ẋ(t) = λ X(t) N -X(t) -γX(t) X(0) = X 1 (0) + X 2 (0) := x 0 (9)
where

λ = λ 1 α 2 1 + λ 2 α 2 2 (10) γ = α 1 γ 1 + α 2 γ 2 . ( 11
)
Hence, the aggregation technique in [START_REF] Auger | Methods of aggregation of variables in population dynamics[END_REF] allows us to express with the single equation in ( 9) the epidemic spread dynamics in both countries, since clearly X i (t) = α i X(t) for all t ∈ [0; T ]. The solution of the aggregated ODE dynamics ( 9) is given by the following standard logistic dynamics:

X(t) = V 1 + V x0 -1 e -λV t
where

V := V (γ 1 , γ 2 ) = N -γ λ .
The threshold effect on the epidemic control of the SIS system is resumed in the following.

Lemma 1 (Epidemic Threshold). X(t) is monotone and converges asymptotically to max(V, 0), i.e.,

lim t→∞ X(t) = max(V, 0) (12)
Ẋ(t) > 0 ∀ t ≥ 0, if X 0 < max(V, 0) Ẋ(t) < 0 ∀ t ≥ 0, if X 0 > max(V, 0).

A. Game model

In our framework, we assume that the healthcare systems of each country perform treatments against a certain pandemic disease. As mentioned before, each country incurs a social cost as well as a healthcare cost. The cost function that country i seeks to minimize can be expressed as a linear combination of the total number of infected individuals over time and of the healthcare costs [START_REF] Sethi | Optimal Control Theory: Applications to Management Science and Economics[END_REF], i.e.,

J i (γ i , γ -i ) = T 0 X i (v)dv + f i (γ i ) (13)
where f i is a convex and strictly increasing continuous function defined as

f i = f ′′ i • f ′ i
where f ′′ i and f ′ i are also strictly increasing and continuous, with

f ′ i (0) = f ′′ i (0) = 0. The function f ′ i : R + 0 → R + 0
maps the recovery rate γ i into the total healthcare expenditure f ′ i (γ i ) for country i over the whole time window [0; T ]. On the other hand, f ′′ i : R + 0 → R + 0 accounts for the sensitivity of country i to the healthcare expenditure versus the social cost of epidemics. For instance, we can expect that a country with efficient healthcare system is characterized by a slowly increasing function f ′ i , while an advanced social welfare program is associated with a slowly decreasing function f ′′ i . We notice that the cost function J i depends on the level of treatment effort of both countries through X i . We point out that f ′ i can be thought as an increasing function of the time window length T ; nevertheless, for simplicity of notation, we prefer keep this dependence implicit. We also remark that, given the monotonicity of (f ′ 1 , f ′ 2 ), we can identify the recovery rate γ i with the cost f ′ i (γ i ) of healthcare efforts for country i = 1, 2. Hence, we will make use of the two terms interchangeably.

After some calculations, it is possible to derive the closed form of (13) as follows:

J i (γ i , γ -i ) = α i Ω V (γ i , γ -i ) + f i (γ i ) (14)
where

Ω(V ) = 1 λ log 1 - x 0 V 1 -e λV T . (15)
It is apparent from the expression of the cost function in ( 14) that the healthcare investments γ -i of country -i generates a positive externality for country i via the total number of infected individuals α i Ω(V (γ i , γ -i )).

Cost function approximation:

We now provide an approximation for the cost function J i that will prove itself useful later in the paper. Let γ -i be fixed, and consider J i as a function of γ i only. Let γi the minimum recovery rate that guarantees the epidemics to die out, i.e., V (γ i , γ -i ) = 0. Then J i can be approximated as

J i (γ i , γ -i ) ≈ α i Ω + f i (γ i ) ∀ γ i > γi (V < 0), α i Ω V (γ i , γ -i ) + f i (γ i ) ∀ γ i ≤ γi (V ≥ 0). ( 16
)
where Ω := Ω(0). This approximation, that preserves the continuity of J i in V = 0, follows the following observations:

• γ i ≥ γi (healthcare cost dominates): this is the socially desirable condition V ≤ 0 where Ω(V ) is bounded by Ω(0) and varies very slowly in V . Hence Ω(V ) can be approximated as Ω(0). • γ i < γi (social cost dominates): this case corresponds to the outbreak of the epidemic disease, i.e., V > 0. The number of infected shows a fast increase with V , since Ω(V ) ≈ V T for V > 0, and this term dominates the healthcare cost. Hence, under the approximation (16), J(., γ -i ) reaches its minimum at γi , i.e., V = 0.

B. Pareto optimality

In the previous section we have derived the expression of the per country cost function for a given healthcare cost pair (γ 1 , γ 2 ). Now, we shall characterize the set of Pareto optimal cost allocations. We will then study the case when the two countries come up with one cost allocation among the Pareto ones via bargaining. Definition 1. The set of Pareto optimal cost allocations is the set of costs (γ 1 , γ 2 ) such that there does not exist

(γ ′ 1 , γ ′ 2 ) with J i (γ ′ 1 , γ ′ 2 ) ≤ J i (γ 1 , γ 2 ) for i = 1, 2
, where the strict inequality holds for at least one country.

In order to compute set of Pareto optimal allocations we first need to introduce further concepts and notation. Let γ * be the aggregated healthcare cost associated to V = 0, i.e., from [START_REF] Poletto | Host mobility drives pathogen competition in spatially structured populations[END_REF]:

γ * = λN = (λ 1 α 2 1 + λ 2 α 2 2 )N. (17)
According to Lemma 1, the healthcare cost γ * can be also interpreted as the minimum aggregated cost that ensures the total number of infected individuals across countries X(t) to asymptotically vanish, i.e.,

γ * = inf γ : lim t→∞ X(t) = 0 .
We notice that the recovery rate γ * is actually a linear combination between the recovery rates in the single countries, as it is clear from [START_REF] Poletto | Host mobility drives pathogen competition in spatially structured populations[END_REF]. Therefore, there are infinite ways for sharing γ * between the two countries. We call Γ the set of all feasible pairs of recovery rates, and hence of all cost feasible healthcare cost divisions, giving rise to γ * :

Γ = (γ 1 , γ 2 ) ≥ 0 α 1 γ 1 + α 2 γ 2 = γ * .
We observe that the cost for both countries within Γ simplifies, and writes

J i (γ 1 , γ 2 ) = α i Ω + f i (γ i ) ∀ (γ 1 , γ 2 ) ∈ Γ, (18)
where the integral cost Ω is given by the expression (15) evaluated at V = 0, i.e.,

Ω := Ω(0) = T 0 X(v)dv V =0 = 1 λ log 1 + x 0 λT . (19)
Finally, let J Γ be the set of cost allocations associated to Γ, i.e.,

J Γ := (J 1 , J 2 ) : J i = J i (γ 1 , γ 2 ), i = 1, 2, (γ 1 , γ 2 ) ∈ Γ . Fact 1.
The set of cost allocations J Γ associated to Γ can be expressed as

J Γ = (J 1 , J 2 ) : J 2 = α 2 Ω + f 2 γ * α2 -α1 α2 f -1 1 (J 1 -α 1 Ω) , α 1 Ω ≤ J 1 ≤ α 1 Ω + f 1 γ * α1 . ( 20
)
Now we are ready to characterize the set of Pareto healthcare costs (or, equivalently, recovery rates).

Theorem 1. Under the approximation [START_REF] Auger | Methods of aggregation of variables in population dynamics[END_REF], the set of Pareto optimal recovery rate allocations coincides with Γ.

Evidently, if the two countries start a negotiation on how to share the healthcare costs, they can never agree on any allocation that does not lie on the Pareto frontier, since by jointly deviating they can both be better off. On the other hand, it is still unclear which allocation inside Γ should be chosen by the two countries. Predicting the outcome of such negotiation is not a trivial task. In fact, each country contributes to the overall disease spread through different migration rates Kp ij . Also, each country perceives differently the trade-off between social and healthcare costs, in our framework, via the shape of functions f ′′ i . We suggest to study the outcome of the bargaining problem by resorting to the well known concept of Nash bargaining solution (NBS) [START_REF] Nash | The bargaining problem[END_REF].

D 2 α 2 Ω + f 2 γ * α 2 J 2 α 2 Ω J 1 α 1 Ω D 1 J(0) D J(γ * T ) J Γ 0 , linear costs J Γ 0 , convex costs α 1 Ω + f 1 γ * α 1 J ′

III. NASH BARGAINING

The Nash bargaining solution (NBS) [START_REF] Nash | The bargaining problem[END_REF] is arguably one of the prominent bargaining solutions in the game theoretical literature. It is the unique cost allocation that satisfies the four axioms of Pareto efficiency, symmetry, invariance to equivalent pay-off representation, and independence of irrelevant alternatives. Moreover, the NBS is the outcome of a non-cooperative bargaining protocol where the players take turns in making proposals and decide at each bargaining round whether to accept or to reject the proposal currently on the table [START_REF] Binmore | The Nash bargaining solution in economic modelling[END_REF], [START_REF] Britz | Non-cooperative support for the asymmetric Nash bargaining solution[END_REF].

The NBS heavily relies on the situation that arises when an agreement is not reached by the two bargaining parties, that is commonly called disagreement point. Hence, before delving into the computation of NBS we first need to describe the scenario at the disagreement.

A. Disagreement scenario

When no agreement is reached by the two countries on how to share the recovery rate (or equivalently the healthcare costs) within Γ, the CA is assumed to have the power to inflict monetary sanctions P i ≥ 0 to each countries i = 1, 2. Here below we provide three motivations for this model choice.

i) By threatening to impose penalties, the CA wishes to enforce a socially desirable healthcare cost allocation. In fact, any allocation which is not Pareto optimal can be improved upon by both countries by jointly deviating and sharing the costs differently.

ii) By tilting the value of the penalties P 1 , P 2 , it turns out that the CA can ensure that ii.a) disclosing private migration data is a self-enforcing behaviour for a malicious country, that otherwise may be enticed not to reveal them to its own advantage (see Section IV-A); ii.b) the outcome of the bargaining adheres to some fairness principles (see Section IV-B). We call D = (D 1 , D 2 ) the pair of values of the cost function at the disagreement, i.e., when negotiations between the two countries on the healthcare cost sharing do not succeed. At the disagreement, we assume that each country i = {1, 2} has no visibility on the healthcare strategy of the neighbour. Therefore, each country i = 1, 2 needs to face the possible insurgence of the worst case scenario, in which the neighbour country chooses not to treat its citizens at all, i.e., γ -i = 0. Therefore, the country i needs to implement the minimax strategy γ d i that, under the approximation in ( 16), can be computed as follows:

γ d i = γ * α i ≈ argmin γi≥0 max γ-i≥0 J i (γ i , γ -i ) = argmin γi≥0 J i (γ i , γ -i = 0). (21)
Thus, it turns out that the conservative minimax healthcare plan at the disagreement point induces each country to actually invest on the healthcare system for the two countries together. Finally, the cost D i for country i at the disagreement point writes

D i = T 0 X (γ d i ,γ-i=0) (v)dv + f ′′ i f ′ i (γ d i ) + P i = α i Ω + f ′′ i f ′ i γ * αi + P i .
We are now in the position of characterizing the set of achievable cost allocations. Via side-payments, both countries can always achieve any cost J which is higher than the one obtained at the disagreement (see Eq. 18). We call J the set of achievable cost allocations:

J = J = (J 1 , J 2 ) : J ≥ J ′ J ′ ∈ J Γ .
On the other hand, each country cannot possibly accept as an outcome of the bargain a cost function value which is higher than the one it can achieve in the absence of an agreement. Therefore, we can characterize the feasible cost set J as

J ′ = J ∩ J = (J 1 , J 2 ) : J ≤ D
By combining equations in [START_REF] Binmore | The Nash bargaining solution in economic modelling[END_REF], we come up with an explicit expression for J ′ , which is the set of utilities (J 1 , J 2 ) fulfilling the following system of nonlinear equations:

J ′ = (J 1 , J 2 ) :      α 1 Ω ≤ J 1 ≤ D 1 α 2 Ω ≤ J 2 ≤ D 2 J 2 ≥ α 2 Ω + f 2 γ * α2 -α1 α2 f -1 1 (J 1 -α 1 Ω) (22)

B. Nash bargaining computation

We can now formulate the NBS as the cost pair

(J N B 1 , J N B 2
) that solves the following optimization problem [START_REF] Nash | The bargaining problem[END_REF]:

(J N B 1 , J N B 2 ) = argmax (J1,J2)∈J ′ (D 1 -J 1 )(D 2 -J 2 ). ( 23
)
In practice, the NBS maximizes the product of the individual gains with respect to the disagreement point.

We remark that the feasibility region J ′ is always nonempty, since D ∈ J , hence the NBS exists. We also observe that the third inequality in (22) generate a convex set, since both f 1 and f 2 are convex and strictly increasing. The following result easily follows.

Proposition 1. The feasibility region J ′ is convex.

Let us now compute the expression of the NBS. Since the region of feasible allocations J ′ is convex then the NBS is well defined and lies on the Pareto boundary of the feasibility region ( 23), that we depicted in Figure 1. Then we first compute the expression of the Pareto boundary J Γ in parametric form:

J 1 (θ) = α 1 Ω + f 1 θ α1 J 2 (θ) = α 2 Ω + f 2 γ * -θ α2 , θ ∈ [0; γ * ].
We notice that, when θ assumes values at its boundaries then just one country takes charge of the whole healthcare costs.

In fact, if θ = 0 then γ 1 = 0 and γ 2 = γ * /α 1 . On the other hand, θ = γ * corresponds to the case γ 1 = γ * and γ 2 = 0. Since there exists a bijective map between the Pareto boundary J Γ and the set of optimal healthcare costs Γ, we are then allowed to define the healthcare costs at the NBS as γ N B = γ(J N B ), as described below.

Theorem 2. The cost allocation at the NBS, γ N B , can be expressed as

γ N B 1 = θ * α1 γ N B 2 = γ * α2 -θ * α2
where θ * is the solution of the uni-dimensional optimization problem:

θ * = argmax 0≤θ≤γ * D 1 -J 1 (θ) D 2 -J 2 (θ) . ( 24
)
C. Linear costs From this section onwards we will consider the special case in which both countries are characterized by increasing linear cost functions

f ′ i , f ′′ i : f ′ i (x) = ǫ ′ i x f ′′ i (x) = ǫ ′′ i x where x ≥ 0, ǫ ′ i , ǫ ′′ i > 0, i = 1, 2. ( 25
)
By analogy with the nonlinear case, let us call ǫ i = ǫ ′ i ǫ ′′ i . Under the linear assumption (25), we are able to find the explicit expression of the NBS. We first observe that in the linear case the cost function J i for both countries can be rewritten as

J i (γ i , γ -i ) = α i Ω + ǫ i γ i , ∀ (γ i , γ -i ) ∈ Γ, i = 1, 2. (26)
The parametric form of J, where θ ∈ [0; γ * ], writes

J(θ) = J 1 (θ) J 2 (θ) = α 1 Ω α 2 Ω + ǫ2 α2 γ * + ǫ1 α1 -ǫ2 α2 θ. (27)
Moreover, the disagreement point D = (D 1 , D 2 ) can be expressed as

D i = α i Ω + ǫ i γ * α i + P i ǫ ′ i , i = 1, 2.
Now we are finally ready to compute explicitly the NBS in the form of the following Theorem. Theorem 3. Let us define θ 0 as

θ 0 = 1 2 α 1 P 1 ǫ ′ 1 -α 2 P 2 ǫ ′ 2 + γ * .
Then, under the linear costs assumption in (25), i. if θ 0 ≤ 0, then J N B = J(θ = 0), and country 2 covers all the healthcare costs, i.e.,

γ N B 1 = 0, γ N B 2 = γ * α 2 ; ii. if 0 < θ 0 < γ * , then for i = 1, 2 γ N B i = 1 2α i α i P i ǫ ′ i -α -i P -i ǫ ′ -i + γ * ;
iii. if θ 0 ≥ γ * , then J N B = J(θ = γ * ) and country 1 covers all the healthcare costs, i.e.,

γ N B 1 = γ * α 1 , γ N B 2 = 0.
Remark 1. The expression of the cost allocation at the NBS in Theorem 3 does not depend on the sensitivity of both countries to the social-monetary cost trade-off, i.e., the value of ǫ ′′ 1 , ǫ ′′ 2 , which can reasonably assumed to be unknown at the CA side. On the other hand, if we assume that the CA has at its disposal good estimate of λ 1 , λ 2 , and α 1 (and hence of α 2 ), then the outcome of the bargaining in the linear case can be well predicted and anticipated by the CA itself.

IV. PENALTY DESIGN

In this section we explore the role of penalties that are inflicted at the disagreement by the CA to both countries in order to induce a desired behaviour from both countries regarding their healthcare program. We first deal with the scenario in which the CA aims at eliciting the truthful disclosure of migration flow data from a malicious country that may want to utilize the private information to its own advantage. In the following we will still assume that the cost functions f ′ i and f ′′ i are linear, i.e., conditions (25) hold.

A. Self-enforcing private migration data disclosure

So far, in this paper we have assumed that the true values of α 1 , α 2 are public and available to the two countries and to the CA. In reality, the statistical information on ingoing and outgoing migration flow available to the countries may differ. For example, country m may be in possession of private, unofficial migration data that it may not be willing to disclose, since they may give it an edge over the neighbour country during the negotiation phase. Hence, in this case, the malicious country m would be able to convince country -m that some αm = α m = 1 -α -m is really the migration level from country -m to country m. In our model, the CA suspects that country m has private migration flow data, and we compute the penalties that the CA should impose to both countries such that country m, in case it really possesses private information, finds (with highest probability) no benefit in cheating and declaring fake migration figures. In other words, we wish to find the penalties ensuring that disclosing the private migration data is a selfenforcing behaviour for the malicious country m. The main goal of the CA is acting so as to guarantee a correct healthcare cost sharing.

In order to restrict our search of optimal penalties from a 2to a 1-dimensional space while still adhering to some desirable fairness property, we consider that the CA penalizes at the disagreement the country with the least efficient healthcare system. More specifically, we force the penalties to each country i to be proportional to ǫ ′ i .

Fairness property 1 :

P 1 ǫ ′ 1 = P 2 ǫ ′ 2 := P. (28)
We can reasonably assume that the CA has at its disposal some a priori information on the real value α m , described by an a priori probability distribution p αm (.), i.e., We notice that this model choice incorporates the extreme cases in which a) the CA has no information at all on α m (i.e., p αm is uniform) and b) the CA knows the exact value of α m (i.e., p αm is the Dirac's delta centred in α m ).

For simplicity, we assume that the epidemic disease spreads in both countries with the same speed:

λ 0 := λ 1 = λ 2 .
Let us delve into the details of this new scenario. The negotiation between the two countries on the healthcare cost sharing is carried out on the basis of the officially declared migration values (α m , 1 -αm ). We then call γNB m (α m ) the Nash solution of the bargaining problem when αm is the officially declared migration flow level, i.e.,

γNB m (α m ) = γ N B m (α m = αm ).
Remarkably, the malicious country m also modifies the Pareto global healthcare investment level γ * (αm) (α m ) which is dif-ferent from the optimal γ * , i.e., γ * (αm) (α m ) = γ * (α m = αm ) = N λ 0 2 α2 m -2 αm + 1 . Hence, under the unfortunate case γ * (αm) (α m ) < γ * , the resulting recovery rate is not sufficient to extinguish the pandemics. We call Ω (αm) (α m ) the total number of infected individuals over the time interval [0; T ] when αm has been declared, i.e.,

Ω(αm) (α m ) = Ω( Ṽ (αm) (α m ))
where max( Ṽ (αm) (α m ), 0) is the asymptotic value of X(t), i.e.,

Ṽ (αm) (α m ) = N - γ * (αm) ( αm) λ = N 1 - 2 α2 m -2 αm+1 2α 2 m -2αm+1 .
We point out that, when the true value α m is declared, Ṽ (αm) (α m ) = V = 0. We also remark that Ω (αm) (α m ) depends on the true value α m through the epidemic spread speed λ, while it is a function of the fictitious value αm via the Pareto healthcare cost γ * . The cost function that country m perceives when declaring αm instead of the real value α m is called J(αm) m (α m ), and equals

J(αm) m (α m ) = α m Ω(αm) (α m ) + ǫ m γNB m (α m ).
Depending on the value of the penalties P m , malicious country m may find profitable not to disclose its private information on migration flows -so that the officially declared value αm is different from the real one, α m -in order to incur a smaller healthcare cost share. We hence define below the set of selfenforcing truth-revealing penalties.

Definition 2. The self-enforcing truth-revealing set-valued function P : [0; 1] → R + 0 defines, for each input value of α m , the set of penalties satisfying (28) such that the best strategy for malicious country m is declaring the real value of α m :

P(α m ) = P : α m = argmin αm∈[0;1] J(αm) m αm , P1 ǫ ′ 1 = P2 ǫ ′ 2 = P .
Our main objective is figuring out how the CA can exploit its a priori information on α m in order to properly design the penalty that minimizes the probability that malicious country m has an incentive to declare a fictitious migration parameter αm = α m , i.e., disclosing private data is not a self-enforcing behaviour for country m. This idea is inspired to the very popular concept of mechanism design (see e.g. [START_REF] Fudenberg | Game theory[END_REF], Chap.7). Definition 3. Assume that the CA has an a priori probability distribution p αm on the real value of α m . The penalty P * is a mechanism for the malicious country m ∈ {1, 2} whenever its probability of being truth-revealing is maximal, i.e.,

P * = argmax P ∈R Pr P ∈ P(α m ) , ( 29
)
where the probability is computed w.r.t. p αm .

Let α m = min(α m , 1 -α m ) and α m = max(α m , 1 -α m ). In order to find the mechanism P * we need to rely on the

   α m Ω + ǫ m γNB m (α m ), ∀ αm ∈ [0; α m ] ∪ [α m ; 1] α m Ω Ṽ (αm) (α m ) + ǫ m γNB m (α m ), ∀ αm ∈ [α m ; 1 2 ] α m Ω Ṽ (αm) (α m ) + ǫ m γNB m (α m ), ∀ αm ∈ [ 1 2 ; α m ] (30)
In fact, the expression αm ∈ [0; α m ] ∪ [1 -α m ; 1] corresponds to Ṽ (αm) (α m ) ≤ 0, while the condition αm ∈ (α m ; α m ) corresponds to the outbreak of the epidemic disease, since in this case Ṽ (αm) (α m ) > 0.

Therefore, we can already claim that malicious country m never has an incentive in declaring αm ∈ (α m ; 1 -α m ), since the epidemic outbreak would cause its cost function to rapidly increase. This tells us that, even in the case that the CA does not manage to design truth-revealing penalties (P 1 , P 2 ) ∈ P, the epidemic spread outbreak is forestalled, i.e., for any value of α m and P , then lim t↑∞ X(t) = 0 even if country m acts maliciously.

Before characterizing the set P(α m ) of truth-revealing penalties, let us first show the following auxiliary result. Lemma 2. Assume that 0 ≤ P ≤ N λ 0 . Then, the NBS equals

γNB m (α m ) = 1 2 αm P (2 αm -1) + N λ 0 (2 α2 m -2 αm + 1) .
Now we are finally ready to characterize the set of truthrevealing penalties. Theorem 4. Under the approximation in (30), the set P(α m ) of truth-revealing penalties can be expressed, when 0 < α m ≤ 1/2, as

P(α m ) = P : 2α 2 m -2α m + 1 ≤ P N λ0 ≤ 1 -2α 2 m ,
and, when 1/2 < α m < 1, as

P(α m ) = P : max(1 -2α 2 m , 0) ≤ P N λ0 ≤ 2α 2 m -2α m + 1 .
and, when α m = {0, 1}, as P(α m ) = {P ≥ 0}.

After characterizing in Theorem 4 the set-valued function P(α m ) of truth-revealing penalties, we can finally compute the expression of the mechanism P * as the outcome of an optimization problem. The proof is direct consequence of Definition 3 and Theorem 4. ; 1 .

We remark that if the CA has full information on the migration flow, then the CA knows P(α m ). Hence, by choosing any P * ∈ P(α m ), P * is truth-revealing with probability 1, and the malicious country m is never enticed to declare a fictitious value αm = α m . Hence, under the penalty P * , disclosing all the private migration information is always a self-enforcing behaviour.

Corollary 2. If p αm = δ αm , where δ αm is the Dirac's delta centred in α m , then Pr (P * ∈ P(α m )) = 1.

Let us now consider the other extreme case, where the CA has no prior information about the true value of α m , i.e., the a priori distribution function p αm is uniform. Remark 2. We highlight that the information needed at the CA's side to compute the truth-revealing penalty P * is very limited, and restricted to the total population size N and the epidemic spread rate λ 0 (along with the prior p αm ). Therefore, the CA does not need to estimate the cost function J i for each country i, which would clearly represent a daunting task.

B. Fairness

In this final section we show another example of penalty design. Unlike the previous section, we assume that no country possesses private mobility information. We now aim at designing the penalties that ensure the NBS to adhere to a certain fairness property. Specifically, we suggest that the recovery rates ensured by the two countries should be proportioned to the expected number of infected individuals per unit of time in each country in the absence of migration, that is formalized in the following.

Fairness property 2 :

γ

N B 1 γ N B 2 = λ 1 N 1 (0) λ 2 N 2 (0) (31)
In order to understand the rationale behind the fairness condition (31), consider the case when λ 2 N 2 (0) ≫ λ 1 N 1 (0), i.e., country 2 contributes to the pandemic spread much more consistently than country 1. Hence, according to property (31), γ N B 2 ≫ γ N B 1 should hold. But, if there is a large immigration rate from country 2 to 1 (α 1 ≫ α 2 ) and both countries have the same sensitivity to healthcare costs (ǫ ′ 1 = ǫ ′ 2 , ǫ ′′ 1 = ǫ ′′ 2 ), for penalties equal and sufficiently large, then the whole cost for the healthcare program is sustained by country 1, i.e. γ N B 1 = γ * /α 1 and γ N B 2 = 0. This solution would clearly be perceived as unfair by the country 1.

Proposition 2. In order to enforce the fairness condition (31), the penalties (P 1 , P 2 ) need to satisfy the following relation:

α 2 ǫ ′ 2 P 2 - α 1 ǫ ′ 1 P 1 = γ * α 2 λ 2 N 2 (0) -α 1 λ 1 N 1 (0) α 2 λ 2 N 2 (0) + α 1 λ 1 N 1 (0) .
If we also impose the fairness condition (28) under which Pi ǫ ′ i = P for i = 1, 2, we conclude that

P = γ * α 2 -α 1 α 2 λ 2 N 2 (0) -α 1 λ 1 N 1 (0) α 2 λ 2 N 2 (0) + α 1 λ 1 N 1 (0) .

Figure 1 :

 1 Figure 1: The feasibility region with penalties P1, P2 > 0. The Nash bargaining solution lies on the Pareto frontier.

x 0 p

 0 αm (v)dv = Pr(α m ≤ x), x ∈ [0; 1].

Figure 2 :

 2 Figure 2: The set-valued function P(αm) of self-enforcing truth-revealing penalties is shown in light blue. same kind of approximation on the cost function J(αm) m (α m ) that we utilized in (16), i.e., J(αm) m (α m ) approximately equals

Corollary 1 .

 1 Let p αm be a bounded probability distribution function. The mechanism P * for malicious country m equalsP * = argmax 0≤P ≤N λ0 P -1 (P ) p αm (v) dv,where, if 0 ≤ x ≤ 1 2 , P -1 (N λ 0 x) =

Corollary 3 . 2 N λ 0 ,

 320 If p αm (v) = 1 for all v ∈ [0; 1], then P * = 1 Pr (P * ∈ P(α m )) = 1 2 .

Table I :

 I Main notation used throughout the paper

	Symbol	Meaning
	X i	number of infected individuals in country i
	N i	population of country i at the stationary regime
	N	total (and constant) number of citizens
	λ i	infection rate in country i
	γ i	recovery rate in country i
	γ *	Pareto optimal aggregated recovery rate
	γ NB i	recovery rate at Nash bargaining solution (NBS) for country i
	α i	migration level from country -i to i
	P i	monetary penalty to country i
	f ′ i (γ i )	healthcare costs for country i to ensure a recovery rate γ i
	J i	cost function for country i
	αm	migration level declared by malicious country m
	P(αm) set of self-enforcing truth-revealing penalties at αm

During the incubation period, an individual already got exposed to a pathogenic organism but does not show any symptoms yet

This work has been partially supported by the European Commission within the framework of the CONGAS project FP7-ICT-2011-8-317672, see www.congas-project.eu.