Victor P Éron

Clair Poignard

ON A MAGNETIC SKIN EFFECT IN EDDY CURRENT PROBLEMS : THE MAGNETIC POTENTIAL IN MAGNETICALLY SOFT MATERIALS

Keywords: Asymptotic Expansions, Impedance Conditions, Eddy Current Problems, Magnetic Potential, Ferromagnetic Material

This work is concerned with the time-harmonic eddy current problem in a bidimensional setting with a high contrast of magnetic permeabilities between a conducting medium and a dielectric medium. We describe a magnetic skin effect by deriving rigorously a multiscale expansion for the magnetic potential in power series of a small parameter ε which represents the inverse of the square root of a relative permeability. We make explicit the first asymptotics up to the order ε 3 . As an application we obtain impedance conditions up to the fourth order of approximation for the magnetic potential. Finally we measure this skin effect with a characteristic length that depends on the scalar curvature of the boundary of the conductor.

 [START_REF] Chabassier | Equivalent boundary conditions for acoustic media with exponential densities. Application to the atmosphere in helioseismology[END_REF]. A measure of the magnetic skin effect 3. Proofs of uniform estimates and stability results for asymptotic models 3.1. Proof of uniform estimates 3.2. Proof of stability results for asymptotic models 4. First terms of the multiscale expansion 4.1. Notations 4.2. First terms 5. Existence and regularity of the asymptotics 5.1. Regularity results for the first terms 5.2. Synthesis of the multiscale expansion 6. Elements of derivations for the multiscale expansion and impedance conditions 6.1. Expansion of the operators in power series of ε 6.2. Equations for the coefficients of the magnetic potential 6.3. First terms of the asymptotics for the magnetic potential 6.4. Construction of impedance conditions 1. INTRODUCTION Numerical simulation of eddy current problems is crucial for the design of industrial systems. We intend to work in the context of this application to devise numerical models for the magnetic potential in (linear) magnetically soft materials. These materials are used for instance for the cores of transformers in order to reduce the energy losses associated with the reversing fields of the alternating currents. However such magnetic materials may have high magnetic permeability and a small skin depth. This raises the difficulty of applying a finite element method on a mesh that combines thin cells inside a skin depth and much larger cells outside this zone. To overcome this difficulty, it is possible to develop an asymptotic method to derive reduced models that can be solve numerically by using a coarse mesh.

To this end, we consider the derivation of an asymptotic expansion for the magnetic potential in high relative permeability (µ r = µ -/µ + 1). We are interested in this issue for the magnetic potential in a soft magnetic conducting body with constant permeability µ = µ - µ 0 (µ 0 is the vacuum permeability) embedded in a magnetic dielectric medium with constant permeability µ = µ + , in a bi-dimensional setting. We consider the conductivity σ, as well as the angular frequency ω, as given parameters.

We emphasize that the above framework will be helpful to devise ε-parameterizations (ε = 1 √ µ r 1) for the magnetic potential by combining the new asymptotic models with finite element solutions. We previously developed δ-parameterizations (δ 1 is the skin depth) for the magnetic potential in non-magnetic materials in high frequency or high conductivity (i.e. when ωµ 0 σ tends to infinity), see e.g. [START_REF] Krähenbühl | On the asymptotic expansion of the magnetic potential in eddy current problem: a practical use of asymptotics for numerical purposes[END_REF].

Roughly speaking, the aim of this paper is to provide the asymptotic expansion of the magnetic potential as the parameter ε tends to zero when the surface of the conducting body is smooth. We exhibit the first terms of a singular expansion in power series of ε to describe a boundary layer inside the magnetic conductor in a vicinity of its surface. In the dielectric medium Ω + , the magnetic potential has a regular expansion in powers of ε. This asymptotic expansion is validated by proving error estimates. As a by-product of this expansion, we infer high order impedance boundary conditions on the surface of the conductor.

Impedance Conditions (ICs) are usually introduced to reduce the computational domain, see for instance [START_REF] Lafitte | Équations de Maxwell et opérateur d'impédance sur le bord d'un obstacle convexe absorbant[END_REF][START_REF] Senior | Institution of Electrical Engineers. Approximate Boundary Conditions in Electromagnetics[END_REF][START_REF] Ida | Impedance boundary conditions for transient scattering problems[END_REF][START_REF] Antoine | High-frequency asymptotic analysis of a dissipative transmission problem resulting in generalized impedance boundary conditions[END_REF][START_REF] Haddar | Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case[END_REF] for scattering problems from highly absorbing obstacles. The main idea consists in replacing the "exact" model inside a part of the domain by an approximate boundary condition. This idea is pertinent when the IC can be readily handled for numerical computations, for instance when this condition is local, see for instance [START_REF] Engquist | Effective boundary condition for acoustic and electromagnetic scattering in thin layers[END_REF][START_REF] Bendali | The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation[END_REF].

We refer the reader to [START_REF] Maccamy | A skin effect approximation for eddy current problems[END_REF][START_REF] Haddar | Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case[END_REF][START_REF] Péron | Asymptotic expansion for the magnetic potential in the eddy current problem : the ferromagnetic case[END_REF] for similar issues in the context of a non-magnetic conductor when the product ωσ tends to infinity (large frequency or high conductivity limit). In this case, a boundary layer phenomenon occurs near the surface of a conductor body; this is the so-called skin effect [START_REF] Rytov | Calcul du skin effect par la méthode des perturbations[END_REF][START_REF] Leontovich | Approximate boundary conditions for the electromagnetic field on the surface of a good conductor[END_REF]. In [START_REF] Maccamy | A skin effect approximation for eddy current problems[END_REF] the authors validate rigorously a multiscale expansion at any order for the solution of an eddy current problem which is addressed in a bi-dimensional setting where the domain is unbounded and the solution grows logarithmically at infinity. In the work in Ref. [START_REF] Haddar | Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case[END_REF] the authors address the issue of impedance conditions for the skin effect problem. They derive rigorously ICs up to the fourth order of approximation for a scalar transmission problem (Helmholtz equation) in 3D.

In this paper, we obtain a family of "Dirichlet-to-Neumann" IBCs (Section 2.3). We observe that there is a main difference between the nature of the limit problem when the product ωµ 0 σ tends to infinity (large frequency or high conductivity limit) [START_REF] Péron | Asymptotic expansion for the magnetic potential in the eddy current problem : the ferromagnetic case[END_REF] and the nature of the limit when the relative permeability µ r tends to infinity. Indeed, the limit in the first case is simply the homogeneous Dirichlet problem for the Laplace operator set in the dielectric medium Ω + (corresponding to a Perfect conductor impedance boundary condition (IBC) on the surface of the conductor) whereas the limit in the second case is the homogeneous mixed Dirichlet-Neumann problem for the Laplace operator set in Ω + (Problem (4.1)), corresponding to a Perfect magnetic conductor IBC on the surface of the conductor. Nonetheless, the second order IBC in high relative permeability (Eq. (2.13)) is nothing but the so-called Leontovich BC as in large frequency / conductivity [14, Eq. (A.2)] (Section 2.3.2). The third order IBC in high relative permeability is a modified Leontovich BC (Eq. (2.14)). The fourth order IBC in high relative permeability is a Ventcell condition (Eq. (2.15)). We recover the higher order IBCs in large frequency / high conductivity [START_REF] Péron | Asymptotic expansion for the magnetic potential in the eddy current problem : the ferromagnetic case[END_REF], which are of "Neumann-to-Dirichlet" nature, from the higher order IBCs in high relative permeability by truncating Taylor expansions of the formal inverses of the impedance operators (Section 2.3.2).

With this work, we observe that the asymptotic expansion for the magnetic potential in high relative permeability (Section 4) is different from the asymptotic expansion in the large frequency / high conductivity case [START_REF] Péron | Asymptotic expansion for the magnetic potential in the eddy current problem : the ferromagnetic case[END_REF]. There is also a main difference between the regularity in Sobolev spaces of the first asymptotics in high relative permeability (Section 5) and in the second case [START_REF] Péron | Asymptotic expansion for the magnetic potential in the eddy current problem : the ferromagnetic case[END_REF] since the derivate consuming processes are different. Contrary to the high conductivity case, the regular part of the expansion in high relative permeability does not consume any derivative up to the order ε 2 . In particular, for given L 2 data and in smooth domains, the first four terms of the regular expansion are well-defined in Sobolev spaces based on H 1 in the first case (Section 5) whereas only the first two terms are well-defined in Sobolev spaces in the second case.

The outline of the paper proceeds as follows. In Section 2, we introduce the studied problem and we present the main results. We state uniform estimates with respect to the small parameter ε for solutions of the problem in Lipschitz domains. We apply this result to the convergence study of an asymptotic expansion in power series of ε as ε tends to zero. As a by-product of this expansion, we infer a hierarchy of impedance boundary conditions (IBCs) up to the fourth order of approximation. In Section 2.4, we measure the magnetic skin effect by introducing a characteristic length that turns out to depend on the scalar curvature of the interface. In Section 3, we prove uniform estimates, stability and convergence results for reduced models with IBCs. In Section 4, we present the first terms of the asymptotic expansion. Then we prove existence and regularity results for the asymptotics and we present a synthesis of the multiscale expansion, Section 5. In Section 6, we give elements of derivations for the multiscale expansion and we present a formal derivation of impedance conditions.

STATEMENT OF THE PROBLEM AND MAIN RESULTS

Let Ω be a smooth bounded domain of the plane R 2 with boundary ∂Ω, and Ω -⊂⊂ Ω be a smooth connected subdomain of Ω, with boundary Σ. We denote by Ω + the complementary of Ω -in Ω, see Fig. 1.

The magnetic vector potential A reduced to a single scalar component satisfies the following problem (2.1)

-div 1 µ ∇A + iωσ1 Ω -A = J in Ω . Ω - Ω + Σ Γ Figure 1 -
The domain Ω and its subdomains Ω -, Ω +

Here the magnetic permeability µ is piecewise constant on Ω ± with positive real values : µ = µ ± > 0 in Ω ± . In this paper we shall use the following notation for the relative magnetic permeability, which is assumed to be a large parameter

(2.2) µ r = µ -/µ + 1 .
In (2.1), the frequency ω > 0 and the conductivity σ > 0 (in Ω -) are given data. For the sake of simplicity, we assume that the right-hand side J (which represents a current source term) is a smooth function which has a support in Ω + .

Notation 2.1. We shall denote by h + (resp. h -) the restriction of any function h to Ω + (resp. Ω -).

We introduce the small parameter ε = 1 √ µ r > 0 inside the problem (2.1).

Then the magnetic vector potential A ε = (A + ε , A - ε ) satisfies the following boundary value problem

(2.3)                  -∆A + ε = µ + J in Ω + , -ε 2 ∆A - ε + iωσµ + A - ε = 0 in Ω -, A + ε = A - ε on Σ , ∂ n A + ε = ε 2 ∂ n A - ε on Σ , A + ε = 0 on Γ ,
where the differential operator ∆ is the Laplace operator in cartesian coordinates x = (x, y) ∈ R 2 . As a convention, the unit normal vector n on the interface Σ is inwardly oriented to Ω -.

Notation 2.2. We shall denote by H 1 0,Γ (Ω + ) the Sobolev space {u ∈ H 1 (Ω + ) | u = 0 on Γ}. 2.1. Uniform estimates. In this section we assume that the domains Ω -and Ω + are Lipschitz. We present in this part uniform a priori estimates with respect to the parameter ε for the solution of the following boundary value problem

           -∆ϕ + = µ + f + in Ω + , -ε 2 ∆ϕ -+ iωσµ + ϕ -= f - in Ω -, ϕ + = ϕ - on Σ , ∂ n ϕ + = ε 2 ∂ n ϕ -+ g on Σ ,
associated with Dirichlet external boundary conditions on Γ. The functional space suitable for its variational formulation is the Sobolev space V = H 1 0 (Ω). Then the variational problem writes: Find ϕ ∈ V such that (2.4) ∀ψ ∈ V,

Ω + ∇ϕ + • ∇ψ + dx + ε 2 Ω - ∇ϕ -• ∇ψ -dx + iωσµ + Ω - ϕ -ψ -dx = µ + Ω f ψ dx + g, ψ H -1 2 (Σ), H 1 2 (Σ)
.

Here the right-hand sides f and g satisfy the regularity assumption

(2.5) f ∈ L 2 (Ω) and g ∈ H -1 2 (Σ)
. In this framework, there exists a unique solution ϕ in V to Problem (2.4) for any parameter ε > 0.

Statement of uniform estimates. In the framework above it is possible to prove H 1 a priori estimate, uniform with respect to the parameter ε.

Theorem 2.3. Let us assume that ω, σ, µ + = 0. Then there exists ε 0 > 0 such that for all ε ∈ (0, ε 0 ), the problem (2.4) with data f and g satisfying (2.5) has a unique solution ϕ ∈ V which satisfies the uniform estimate

(2.6) ϕ + 1,Ω + + ε ∇ϕ - 0,Ω -+ ϕ - 0,Ω - C(|µ + | f 0,Ω + g -1 2
,Σ ) with a constant C > 0, independent of ε, f , and g.

Note that these estimates are different from uniform estimates in large frequency or high conductivity [START_REF] Péron | Asymptotic expansion for the magnetic potential in the eddy current problem : the ferromagnetic case[END_REF]. We prove Theorem 2.3 in section 3.1.

Application: Convergence of multiscale expansions.

In this section we assume that Σ is a smooth curve. Then it is possible to derive a multiscale expansion for the solution

A ε = (A + ε , A - ε ) of the model problem (2.
3). The solution A ε possesses an asymptotic expansion in power series of the small parameter ε

A + ε (x) = A + 0 (x) + εA + 1 (x) + ε 2 A + 2 (x) + • • • in Ω + , (2.7) A - ε (x) = A - 0 (x; ε) + εA - 1 (x; ε) + ε 2 A - 2 (x; ε) + • • • in Ω -, (2.8) with A - j (x; ε) = χ(ν)A j (ξ, ν ε ) . (2.9)
Here x ∈ R 2 are the cartesian coordinates and (ξ, ν) is a "curvilinear coordinate system" to the curve Σ on the domain Ω -: ξ is a curvilinear abscissa on Σ and ν ∈ (0, ν 0 ) (where ν 0 is sufficiently small, see Section 6) is the distance to the curve Σ. In Eq. (2.9), χ is a smooth cutoff function, its support is included in a tubular neighborhood of Σ, and equal to 1 in a smaller neighborhood. The first-order terms A j and A + j are explicited in Sec. 4.2. Each term A j is a "profile" defined on Σ × I where I = (0, +∞). Furthermore, for any j ∈ N

A + j ∈ H 1 (Ω + ) and A j ∈ H 1 (Σ × R + ) , see Prop.
r m;ε = A ε - m n=0 ε n A n in Ω .
It is possible to prove estimates at any order m ∈ N * for the remainder r m;ε . We can prove the following convergence result as a consequence of Th. 2.3.

Theorem 2.4. We assume that J is a smooth data and the interface Σ is a smooth curve. There exists ε 0 > 0 such that for all m ∈ N there exists a constant C m > 0 independent of ε such that for all ε ∈ (0, ε 0 ) the remainder r m;ε satisfies the optimal estimate

(2.10) r + m;ε 1,Ω + + ε ∇r - m;ε 0,Ω -+ r - m;ε 0,Ω - C m ε m+1 .
Proof. This proof is done by an evaluation of the right-hand side when the operator -div( 1 µ ∇)+ iωσ1 Ω -is applied to the remainder r m;ε . By construction of the expansion (2.7)-(2.8) we obtain

(setting δ 0 = 1 √ ωσµ + ) (2.11)                  -∆r + m;ε = 0 in Ω + , -ε 2 ∆r - m;ε + iδ -2 0 r - m;ε = f - m;ε in Ω -, r + m;ε = r - m;ε on Σ , ∂ n r + m;ε = ε 2 ∂ n r - m;ε + ε m+1 ∂ n A + m+1 on Σ , r + m;ε = 0 on Γ .
Since J is smooth, the right-hand side f - m;ε belongs to L 2 (Ω -) and it is roughly of the order ε m+1 ; the right-hand side ∂ n A + m+1 belongs to H -1 2 (Σ) and it is independent of ε. Then using estimates (2.6) we infer the error estimate (2.10).

Impedance boundary conditions.

As a by-product of the multiscale expansion it is possible to derive high-order impedance boundary conditions (ICs) set on the interface Σ for the solution of the problem (2.3). Elements of formal derivations are given in Section 6.4.

2.3.1.

Statement of impedance boundary conditions. This section presents a hierarchy of ICs up to the fourth order of approximation with respect to the parameter ε, See Thm. 2.7 ( §2.3.4) for precise estimates. (2.13)

∂ n A ε 1 + ε λ 0 A ε 1 = 0 on Σ where λ 0 = √ ωσµ + e iπ 4 .
Order 3: Modified Leontovich b.c.

(2.14)

∂ n A ε 2 + ε λ 0 - k(ξ) 2 ε A ε 2 = 0 on Σ
Here k denotes the scalar curvature of the curve Σ (see Not. 4.1).

Order 4: Ventcell b.c.

(2.15)

∂ n A ε 3 + ε λ 0 - k(ξ) 2 ε - ε 2 2λ 0 ∂ 2 ξ + k(ξ) 2 4 I A ε 3 = 0 on Σ 2.3.2.
Comparison with the large frequency / high conductivity case [START_REF] Péron | Asymptotic expansion for the magnetic potential in the eddy current problem : the ferromagnetic case[END_REF]. The Leontovich b.c.

(2.13) writes

∂ n A ε 1 + ωσ µ - µ + e iπ 4 A ε 1 = 0 on Σ .
This IC is nothing but the second order IC for the large frequency / high conductivity case [14, Eq. (A.

∂ n A ε 2 + ωσ µ - µ + e iπ 4 1 - 1 √ ωσµ - k(ξ) 2 e -iπ 4 A ε 2 = 0 on Σ .
From this IC, we recover the third order IC in the large frequency / high conductivity case [14, Eq. (A.

3)] by performing a first order truncated Taylor expansion of the formal operator

I - 1 √ ωσµ - k(ξ) 2 e -iπ 4 I -1
as the parameter 1 √ ωσ tends to zero :

A + µ - ωσ 1 µ + e -iπ 4 1 + 1 √ ωσµ - k(ξ) 2 e -iπ 4 ∂ n A = 0 on Σ .
This Neumann-to-Dirichlet IC can also be obtained from the modified Leontovich b.c. (2.14) by truncating a first order Taylor expansion of the formal operator

λ 0 I -k(ξ) 2 εI -1
as ε tends to zero.

The Ventcell b.c. writes

∂ n A ε 3 + ωσ µ - µ + e iπ 4 1 - 1 √ ωσµ - k(ξ) 2 e -iπ 4 + 1 ωσµ - i 2 ∂ 2 ξ + k(ξ) 2 4 I A ε 3 = 0 . on Σ
From this IC, we recover the fourth order IC in the large frequency / high conductivity case [14, Eq. (A.4)] by performing a second order truncated Taylor expansion of the formal operator

I - 1 √ ωσµ - k(ξ) 2 e -iπ 4 I + 1 ωσµ - i 2 ∂ 2 ξ + k(ξ) 2 4 I -1
as the parameter 1 √ ωσ tends to zero :

A + µ - ωσ 1 µ + e -iπ 4 1 + 1 √ ωσµ - k(ξ) 2 e -iπ 4 - 1 ωσµ - i 2 ∂ 2 ξ + k(ξ) 2 4 I ∂ n A = 0 on Σ .
This Neumann-to-Dirichlet IC can also be obtained from the Ventcell b.c. (2.15) by truncating a second order Taylor expansion of the formal operator

λ 0 I -k(ξ) 2 εI - ε 2 2λ 0 ∂ 2 ξ + k(ξ) 2 4 I -1
as ε tends to zero.

       -∆A ε k = f in Ω + , N k,ε (A ε k ) + ∂ n A ε k = 0 on Σ , A ε k = 0 on Γ , (with f = µ + J) where N k,ε is a Dirichlet-to-
N 0 = 0 , (2.17a) N 1,ε = λ 0 εI , (2.17b) N 2,ε = ε λ 0 - k(ξ) 2 ε I , (2.17c) N 3,ε = ε λ 0 - k(ξ) 2 ε I - ε 2 2λ 0 ∂ 2 ξ + k(ξ) 2 4 I . (2.17d)
We can prove a stability result for this problem in the following functional framework which is defined as

V k = H 1 0,Γ (Ω + ) (Not. 2.
2) when k = 0, 1, 2, and

V k = {ϕ ∈ H 1 0,Γ (Ω + ) | ∇ Σ ϕ ∈ L 2 (Σ)} when k = 3.
Theorem 2.5. For all k ∈ {0, 1, 2, 3} there are constants ε k , C k > 0 such that for all ε ∈ (0, ε k ), the problem (2.16) with a data f ∈ L 2 (Ω + ) has a unique solution A ε k ∈ V k which satisfies the uniform estimates:

A ε k 1,Ω + C k f 0,Ω + for all k ∈ {0, 1, 2} , (2.18a) A ε 3 1,Ω + + ε 3 2 ∇ Σ A ε 3 0,Σ C k f 0,Ω + . (2.18b) Remark 2.1. Estimate (2.18a) holds for all ε > 0 when k = 0, i.e. ε 0 = +∞. Estimate (2.18a) holds for all ε ∈ 0, √ 2 1 k(ξ) ∞,Σ when k = 1.
The key for the proof of this result is the following Lemma, that is uniform L 2 -estimates for any solution of problem (2.16) Lemma 2.6. For all k ∈ {0, 1, 2, 3} there are constants ε k , C k > 0 such that for all ε ∈ (0, ε k ),

any solution A ε k ∈ V k of problem (2.16) with a data f ∈ L 2 (Ω + ) satisfies the uniform estimate (2.19) A ε k 0,Ω + C k f 0,Ω + .
This Lemma is proved in Section 3.2. As a consequence of this Lemma, any solution of the problem (2.16) satisfies uniform H 1 -estimates (2.18a), (2.18b), respectively when k = 0, 1, 2, 3. Then, the proof of the Thm. 2.5 is obtained as a consequence of the Fredholm alternative since problem (2.16) is Fredholm of index zero.

2.3.4. Convergence result. We remind that A ε is the solution of the problem (2.3). As a consequence of Thm. 2.4 and Thm. 2.5, we can deduce the following convergence result for the solution A ε k of the asymptotic model (2.16) Theorem 2.7. We assume that J is a smooth data and the interface Σ is a smooth curve. For all k ∈ {0, 1, 2, 3} there are constants ε k , C k > 0 such that for all ε ∈ (0, ε k ), A ε k ∈ V k satisfies the uniform error estimates

(2.20) A ε -A ε k 1,Ω + C k ε k+1 .

2.4.

A measure of the magnetic skin effect. As another application of the multiscale expansion, we make the link in this section between the parameter ε and the "pointwise energy" of the solution of the problem (2.3) across the boundary layer. We measure this boundary layer phenomenon by introducing a characteristic length that turns out to depend on the scalar curvature (see Not. 4.1) of the boundary of the domain Ω -. An asymptotic expansion when ε → 0 for this function shows the influence of the geometry of the interface : the characteristic length is larger for small values of ε when the scalar curvature of the interface Σ is larger. We refer also to the works [START_REF] Dauge | Comportement asymptotique à haute conductivité de l'épaisseur de peau en électromagnétisme[END_REF][START_REF] Caloz | On the influence of the geometry on skin effect in electromagnetism[END_REF][START_REF] Chabassier | Equivalent boundary conditions for acoustic media with exponential densities. Application to the atmosphere in helioseismology[END_REF] where the authors measure similarly boundary layer phenomena (in 3D) by introducing suitable "skin depth" functions.

Introduction of a characteristic length. For a data J of the problem (2.3), let us define A ε (ξ, ν) = A - ε (x), ξ ∈ Σ, 0 ν < ν 0 for ν 0 small enough. Definition 2.8. Let J be a smooth data of problem (2.3) such that for all ξ ∈ Σ, A ε (ξ; 0) = 0. The characteristic length is the length L(ε; ξ) defined on Σ and taking the smallest value such that

(2.21) |A ε (ξ; L(ε; ξ))| = |A ε (ξ; 0)|e -1 .
Thus this length L(ε; ξ) is the distance from the interface where |A ε | has decreased of a fixed rate e. It depends on ε, of each point ξ ∈ Σ and a priori on the data J. However it is possible to exhibit the asymptotic behavior of the characteristic length L for small values of ε independently of both ξ and the data J.

Asymptotic behavior of the characteristic length function when ε → 0. Theorem 2.9. Let J be a smooth data of problem (2.3). We assume that A - 0 X(ξ) = 0 on Σ. Then characteristic length has the following behavior for small values of ε

(2.22) L(ε; ξ) ≈ 2 ωσµ - 1 + k(ξ) 2 2 ωσµ - + O µ + µ - , ε = µ + µ - → 0 .
Remark 2.2. This result is comparable to the skin effect in 2D. The truncated expansion formed by the two first terms in (2.22) coincides with the asymptotic behavior in high conductivity of the "skin depth" function introduced in [START_REF] Dauge | Comportement asymptotique à haute conductivité de l'épaisseur de peau en électromagnétisme[END_REF][START_REF] Caloz | On the influence of the geometry on skin effect in electromagnetism[END_REF] when the permeability µ -is the vacuum permeability.

Proof. The proof of this theorem is based on the formula

(2.23) |A ε (ξ; ν)| = |A ε (ξ; 0)|m(ξ; ν) e -ν λ 0 ε with m(ξ; ν) = 1 + k(ξ) 2 λ 0 ν + O (ε + ν) 2 .
This formula (2.23) is a consequence of equations (4.2) and (4.4). According to Definition 2.8, the characteristic length L(ε; .) satisfies

(2.24) |A ε (ξ; L(ε; ξ))| = |A ε (ξ; 0)| e -1 .
Hence, m(ξ; L(ε; ξ)) e -L(ε; ξ) λ 0 /ε = e -1 .

Performing an asymptotic expansion when ε tends to zero, we infer the asymptotic behavior of the characteristic length (2.22)

L(ε; ξ) ≈ ε( λ 0 ) -1 1 + k(ξ) 2 λ 0 ε + O(ε 2 ) , ε → 0 .

PROOFS OF UNIFORM ESTIMATES AND STABILITY RESULTS FOR ASYMPTOTIC MODELS

3.1. Proof of uniform estimates. The proof of Theorem 2.3 is based on the following Lemma, that is uniform L 2 estimates.

Lemma 3.1. There exists ε 0 > 0 such that for all ε ∈ (0, ε 0 ), any solution ϕ ∈ V of problem (2.4), with data f and g satisfying (2.5), satisfies the uniform estimate

(3.1) ϕ 0,Ω C( f 0,Ω + g -1 2
,Σ ) with a constant C > 0, independent of ε, f , and g.

The proof of this result involves a compactness argument. Using estimates (3.1) we can deduce estimates (2.6) and Thm. 2.3 is obtained as a consequence of the Fredholm alternative since the problem (2.4) is Fredholm of index zero.

3.1.1. Proof of Lemma 3.1 : Uniform L 2 estimates for ϕ ε . Without loss of generality we can restrict our consideration by proving this lemma when g = 0 (since it is still possible to absorb the H 1 2 norm of ϕ on Σ with the H 1 semi-norm of ϕ + on Ω + from the right-hand side to the left-hand side when g = 0). We adopt a proof by contradiction : We assume that there exists a sequence (ϕ m ) ∈ V , m ∈ N, of solutions of problem (2.4) associated with a parameter ε m and a right-hand side f m ∈ L 2 (Ω):

(3.2) ∀ψ ∈ V, Ω + ∇ϕ + m • ∇ψ + dx + ε 2 m Ω - ∇ϕ - m • ∇ψ -dx + iωσµ + Ω - ϕ - m ψ -dx = µ + Ω f m ψ dx satisfying the following conditions ε m → 0 as m → ∞, (3.3a) ϕ m 0,Ω = 1 for all m ∈ N, (3.3b) f m 0,Ω → 0 as m → ∞. (3.3c)
Choosing tests functions ψ = ϕ m in (3.2) and taking the real part, we obtain the following convergence results on the gradients :

(i) the sequence {∇ϕ + m } converges to 0 in L 2 (Ω + ). Then taking the imaginary part, we obtain with the help of conditions (3.3b)-(3.3c) the following convergence result :

(ii) the sequence {ϕ - m } converges to 0 in L 2 (Ω -). As a consequence of (3.3b) and (i), the sequence {ϕ + m } is bounded in H 1 (Ω + ).

Limit of the sequence and conclusion. The domain Ω + being bounded, the embedding of H 1 (Ω + ) in L 2 (Ω + ) is compact (Rellich Lemma). Hence we can extract a subsequence of {ϕ + m } (still denoted by {ϕ + m }) which is strongly converging in L 2 (Ω + ) and weakly converging in H 1 (Ω + )

(3.4) ∇ϕ + m ∇ϕ + in L 2 (Ω + ) ϕ + m → ϕ + in L 2 (Ω + )
and (i) implies that this sequence is strongly converging in H 1 (Ω + ) and ∇ϕ + = 0 in Ω + .

A consequence of the strong convergence in L 2 (Ω) and (3.3b) is that ϕ + 0,Ω + = 1. Since ∇ϕ + = 0 in Ω + and ϕ + = 0 on ∂Ω, we infer that ϕ + = 0 in Ω + which contradicts ϕ + 0,Ω + = 1 and finally proves estimate (3.1).

3.2.

Proof of stability results for asymptotic models. We focus on the proof of Lemma 2.6 for k = 3 since the proof is simpler when k = 0, 1, 2. We present a proof which is based on a compactness argument. To prepare for this proof, we introduce the variational formulation for u = A ε 3 . If u ∈ V 3 is a solution of (2.16), then it satisfies for all v ∈ V 3

(3.5)

Ω + ∇u•∇v dx+ε Σ λ 0 - k(ξ) 2 ε - k(ξ) 2 8λ 0 ε 2 u v + ε 2 2λ 0 ∂ ξ u ∂ ξ v dξ = Ω + f v dx .
Reductio ad absurdum: We assume that there is a sequence {u m } ∈ V 3 , m ∈ N, of solutions of the problem (2.16) (when k = 3) associated with a parameter ε m and a right-hand side

f m ∈ L 2 (Ω + ) -∆u m = f m in Ω + , (3.6a) ∂ n u m + J m u m - ε 3 m 2λ 0 ∆ Σ u m = 0 on Σ , (3.6b) u m = 0 on ∂Ω (3.6c)
satisfying the following conditions

ε m → 0 as m → ∞ , (3.7a) u m 0,Ω + = 1 for all m ∈ N , (3.7b) f m 0,Ω + → 0 as m → ∞ . (3.7c) In (3.6b), J m = ε m λ 0 -k(ξ) 2 ε m - k(ξ) 2 8λ 0 ε 2 m is a scalar function defined on Σ.
Estimates of the sequence {u m }. We first prove that the sequence {u m } is bounded in H 1 (Ω + ).

We particularize the weak formulation (3.5) for the sequence {u m } and choosing v = u m we obtain

(3.8) ∇u m 2 0,Ω + + Σ J m |u m | 2 + ε 3 m 2λ 0 |∂ ξ u m | 2 dξ = Ω + f u m dx .
Then taking the real part, we obtain with the help of condition (3.7a)-(3.7b)-(3.7c) the convergence result

(3.9) ∇u m -→ 0 in L 2 (Ω + )
since J m → 0. We infer that the sequence {u m } is bounded in

H 1 (Ω + ) (3.10) u m 1,Ω + C .
Limit of the sequence and conclusion. The domain Ω + being bounded, the embedding of

H 1 (Ω + ) in L 2 (Ω + ) is compact (Rellich Lemma).
As a consequence of (3.10), using the Rellich Lemma we can extract a subsequence of {u m } (still denoted by {u m }) which is weakly converging in H 1 (Ω + ) and strongly converging in L 2 (Ω + ). We deduce that there is u ∈ L 2 (Ω -) such that

u m → u in L 2 (Ω + ) (3.11a) ∇u m ∇u in L 2 (Ω + ) . (3.11b) A consequence of the strong convergence result in L 2 (Ω + ) (3.11a) and (3.7b) is that u 0,Ω + = 1.
As a consequence of (3.11b)-(3.9) we deduce that ∇u = 0 in Ω + . Since u = 0 on Γ we infer u = 0 in Ω + which contradicts u 0,Ω + = 1 and ends the proof.

FIRST TERMS OF THE MULTISCALE EXPANSION

In this section we assume that Σ is a smooth curve. Hereafter, we present the construction of the first profiles A j and of the first terms A + j for the multiscale expansion (2.7)-(2.8). We remind that n(ξ) denotes the unit normal at X(ξ) ∈ Σ which is inwardly oriented to Ω -.

First terms.

In this section, we present the construction of the asymptotics (A + j , A j ) when j = 0, 1, 2, 3. Elements of formal derivations are given in Section 6. The first term A + 0 solves the problem (4.1)

     -∆A + 0 = µ + J in Ω + , ∂ n A + 0 = 0 on Σ , A + 0 = 0 on Γ . Then the first profile A 0 is (4.2) A 0 (ξ, Υ) = A + 0 X(ξ) e -λ 0 Υ , where λ 0 = ωσµ + 2 (1 + i).
The next term which is determined in the asymptotics is A + 1 which solves

(4.3)      ∆A + 1 = 0 in Ω + , ∂ n A + 1 = -λ 0 A + 0 on Σ , A + 1 = 0 on Γ .
Then the profile A 1 satisfies (4.4)

A 1 (ξ, Υ) = A + 1 X(ξ) + k(ξ) 2 A + 0 X(ξ) Υ e -λ 0 Υ .
The next term A + 2 solves the problem (4.5)

       ∆A + 2 = 0 in Ω + , ∂ n A + 2 = k(ξ) 2 A + 0 -λ 0 A + 1 on Σ , A + 2 = 0 on Γ .
Then the profile A 2 satisfies (4.6)

A 2 (ξ, Υ) = A + 2 + k(ξ) 2 A + 1 + 1 2λ 0 ∂ 2 ξ + k(ξ) 2 4 I A + 0 Υ + 3k(ξ) 2 8 A + 0 Υ 2 X(ξ) e -λ 0 Υ .
The next term which is determined is

A + 3 (4.7)        ∆A + 3 = 0 in Ω + , ∂ n A + 3 = -λ 0 A + 2 + k(ξ) 2 A + 1 + 1 2λ 0 ∂ 2 ξ + k(ξ) 2 4 I A + 0 on Σ , A + 3 = 0 on Γ .

EXISTENCE AND REGULARITY OF THE ASYMPTOTICS

5.1. Regularity results for the first terms. It is possible to prove that Problem (4.1) (resp. (4.3), (4.5), (4.7)) has a unique weak solution A + 0 (resp. A + l , l = 1, 2, 3) in the space H 1 0,Γ (Ω + ) (Not. 2.2) when the right-hand side J belongs to L 2 and the domain Ω + is sufficiently smooth. We first describe briefly the derivate consuming process for the first terms of the expansion (2.7).

The derivate consuming process of the expansion. We assume that J ∈ L 2 (Ω + ). Then, the problem (4.1) is solvable in the space H 1 0,Γ (Ω + ). Since this problem is elliptic, the solution A + 0 has an optimal regularity which depends on the regularity of the data. Hence, A + 0 ∈ H 2 (Ω + ) ∩ H 1 0,Γ (Ω + ) and this solution satisfies an a priori estimate: (5.1)

A + 0 2,Ω + C J 0,Ω +
with a constant C > 0 independent of J. Then, the trace A + 0 Σ has to be inserted as a Neumann data into problem (4.3) defining A + 1 . Since A + 0 Σ belongs to H 3/2 (Σ), the problem (4.3) is solvable in the space H 1 0,Γ (Ω + ) and the solution A + 1 belongs to the Sobolev space H 3 (Ω + ). Furthermore, the solution satisfies the estimate:

A + 1 3,Ω + C A + 0 3/2,Σ
and, from (5.1), we infer the estimate:

(5.2)

A + 1 3,Ω + C 1 J 0,Ω + .
Hence we win one rank of regularity to derive A + 1 from the first asymptotic A + 0 . Since A + 0 Σ ∈ H 3/2 (Σ) and A + 1 Σ ∈ H 5/2 (Σ), the right-hand side on Σ in Problem (4.5) belongs to H 3/2 (Σ).

Hence, the problem (4.5) is also solvable in H 1 0,Γ (Ω + ) and its solution A + 2 belongs also to the Sobolev space H 3 (Ω + ). Furthermore, the solution satisfies the estimate:

A + 2 3,Ω + C A + 0 3/2,Σ + A + 1 5/2,Σ C 2 J 0,Ω + .
Finally, the problem (4.7) has a unique solution A + 3 in the space H 1 0,Γ (Ω + ) since the righthand side on Σ belongs to the Sobolev space H -1/2 (Σ). Furthermore, the solution satisfies the estimate:

A + 3 1,Ω + C 3 J 0,Ω + .
Using general elliptic regularity result in Sobolev spaces for the Laplace operator, we prove regularity results in Sobolev spaces H s for the first terms A + l and A l , l = 0, 1, 2, 3 Proposition 5.1. Let k be an integer such that k 1. We assume that the domain

Ω + is smooth (of class C ∞ ). If J belongs to H k (Ω + ) then A + 0 ∈ H k+2 (Ω + ) and A 0 ∈ H k+ 3 2 (Σ × R + ) , A + 1 ∈ H k+3 (Ω + ) and A 1 ∈ H k+ 3 2 (Σ × R + ) , A + 2 ∈ H k+3 (Ω + ) and A 2 ∈ H k-1 2 (Σ × R + ) , A + 3 ∈ H k+1 (Ω + )
. It is also possible to prove regularity results in analytic classes. We assume that Ω + is analytic. If J is an analytic function on Ω + then each term A + l is an analytic function on Ω + , see for instance [START_REF] Costabel | Corner Singularities and Analytic Regularity for Linear Elliptic Systems[END_REF]Th 3.4.2]. 5.2. Synthesis of the multiscale expansion. Following the process of construction described above it is possible to derive the asymptotics A ± j at any order when the right-hand side J is of class C ∞ . In this section we specify this result in Prop. 5.2.

The following definition for the resolvent operator R makes sense and define a bounded operator R : H

1 2 (Σ) → {ϕ ∈ H 2 (Ω + ) | ϕ = 0 on Γ} h → ϕ
where ϕ is the unique solution (in distributional sense) of the boundary value problem

     ∆ϕ = 0 in Ω + , ∂ n ϕ = h on Σ , ϕ = 0 on Γ .
For all t 2, the resolvent operator R is still a bounded operator in the following spaces

H t-3 2 (Σ) → {ϕ ∈ H t (Ω + ) | ϕ = 0 on Γ} Proposition 5.2.
Let s 2 and assume that J ∈ H s-2 (Ω) has a support in Ω + . Assume that the interface Σ is a curve of class C ∞ . For all j = 0, • • • , s +1, it is possible to derive successively the following terms which are independent of ε A + j ∈ H s-j (Ω + ) and A j ∈ H s-j-3 2 Σ, C ∞ (I) .

For j = 0, A + 0 ∈ H s (Ω + ) ∩ H 1 0,Γ (Ω + ) is the solution of problem (4.1), and A 0 (ξ, Υ) = A + 0 X(ξ) e -λ 0 Υ . For all j = 1, • • • , s + 1, there exists a tangential operator T j which is defined as follow

T j : H s (Ω + ) → H s-j-3 2 (Σ) A + 0 → ∂ n A + j | Σ such that (5.3) A + j = R • T j (A + 0 )
. In particular for j = 1, 2, 3, the operator T j is defined as

(5.4) T 1 (A) = -λ 0 A| Σ T 2 = T 1 • R - k(ξ) 2λ 0 I • T 1 T 3 = T 1 • R • T 2 - k(ξ) 2λ 0 T 1 - 1 2λ 2 0 ∂ 2 ξ + k(ξ) 2 4 I • T 1
Furthermore A 1 (resp. A 2 ) is defined by (4.4) (resp. (4.6)) and for all n 3, the unique solution of the ODE (6.10) (see Sec. 6.2) such that A n (•, Υ) -→ 0 as Υ → ∞ writes

(5.5) A n (ξ, Υ) = (a 0 (ξ) + Υa 1 (ξ) + Υ 2 a 2 (ξ) + • • • + Υ n a n (ξ))e -λ 0 Υ .
The solution A ε of Problem (2.3) possesses the following asymptotic expansion

A + ε (x) = A + 0 (x) + εA + 1 (x) + ε 2 A + 2 (x) + ε 3 A + 3 (x) + • • • in Ω + , A - ε (x) = χ(ν)A 0 (ξ, ν ε ) + εχ(ν)A 1 (ξ, ν ε ) + ε 2 χ(ν)A 2 (ξ, ν ε ) + + • • • in Ω -, (2.7
)-(2.8), see Theorem 2.4 for precise estimates.

ELEMENTS OF DERIVATIONS FOR THE MULTISCALE EXPANSION AND IMPEDANCE

CONDITIONS

We assume that Σ is a smooth (C ∞ ) curve of length L. For υ 0 > 0 small enough, the tubular neighborhood V(Σ) of Σ of diameter υ 0 in the inner domain Ω -is parameterized by V(Σ) = {x(ξ, υ) = X(ξ) + υn(ξ), (ξ, υ) ∈ Σ × (0, υ 0 )} , See Notation 4.1. The system of coordinates (ξ, υ) is orthogonal, and the first fundamental form writes

Q 2 (ξ, υ) (dξ) 2 + (dυ) 2 , with Q(ξ, υ) = (1 -υk(ξ)),
where k(ξ) denotes the curvature at the point X(ξ) (Notation 4.1). For υ 0 < 1/ k ∞ the change of coordinates (6.1)

Ψ : x = (x, y) -→ (ξ, υ) is a C ∞ -diffeomorphism from V(Σ) into the cylinder T L × (0, υ 0 ): V(Σ) = Ψ -1 T L × (0, υ 0 ) .
Hereafter, in Sec. 6.2 we set the ansatz (2.7)-(2.8)-(2.9) for the solution of problem (2.3). The function (ξ, υ) → χ(υ) which appear in (2.9) is a smooth cut-off function whose support is included in V(Σ), and equal to 1 in a smaller tubular neighborhood of the interface.

6.1. Expansion of the operators in power series of ε. In order to exhibit the formal expansion of A - ε we first use the change of variables Ψ in order to write the equations in the cylinder T L × (0, υ 0 ). We then perform the rescaling (6.2) Υ = υ ε in the equations set in Ω -and Σ in order to make appear the small parameter ε in the equations. The Laplace operator writes in coordinates (ξ, ν) as

∆ = Q -1 ε -2 ∂ Υ (Q∂ Υ ) + ∂ ξ (Q -1 ∂ ξ ) , (6.3)
where Q = (1 -εΥk(ξ)). This operator ∆ (6.3) expands in power series of ε with coefficients intrinsic operators with respect to Σ (6.4)

∆ = 1 ε 2 ∂ 2 Υ + n -1 ε n D n .
Note that this expansion (6.4) is a convergent power series expansion when Υ ∈ I ε = (0, ν 0 ε ). Here for all n ≥ -1, D n = D n (ξ, Υ; ∂ ξ , ∂ Υ ) is an operator with at most one derivative in the Υ-variable, which has smooth coefficients in ξ, and polynomial coefficients of degree n -1 in Υ. Note that in particular D -1 = -k(ξ)∂ Υ . In the following we shall use the operator

A n := D n-2 ε 2 ∆ = ∂ 2 Υ + n 1 ε n A n .
Observe that

A 1 = -k(ξ)∂ Υ , (6.5a) A 2 = ∂ 2 ξ -k(ξ) 2 Υ∂ Υ , (6.5b) A 3 = -k(ξ) 3 Υ 2 ∂ Υ + Υk (ξ)∂ ξ + 2Υk(ξ)∂ 2 ξ . (6.5c) Similarly we write ∂ n (ξ; ∂ Υ ) = 1 ε ∂ Υ
on the interface Σ.

6.2.

Equations for the coefficients of the magnetic potential. In this section, define v ε by v

ε (ξ, Υ) = A - ε (x) in V(Σ). After the scaling υ → Υ = υ ε in V(Σ), the problem (2.3) writes (6.6)      -∆A + ε = µ + J in Ω + , ∂ n A + ε = ε∂ Υ v ε on Σ , A + ε = 0 on Γ , and
(6.7)    (-∂ 2 Υ + i δ 2 0 )v ε - n 1 ε n A n v ε = 0 in T L × (0, +∞), v ε = A + ε on T L × {0} .
We insert the Ansatz

A + ε ∼ n 0 ε n A + n (x) in Ω + , and v ε ∼ n 0 ε n A n (ξ, Υ) in V(Σ)
with A n (•, Υ) -→ 0 as Υ → ∞, in equations (6.6) and (6.7), and we perform the identification of terms with the same power in ε. Setting (6.8)

λ 0 = 1 δ 0 √ 2 2 (1 + i)
in equation (6.7), then the terms A n and A + n satisfy the following family of problems coupled by their conditions on the interface Σ (corresponding to Υ = 0):

(6.9)      -∆A + n = µ + Jδ 0 n in Ω + , ∂ n A + n = ∂ Υ A n-1 on Σ , A + n = 0 on Γ , and
(6.10)      -∂ 2 Υ A n + λ 2 0 A n = n p=1 A p A n-p in Σ × (0, +∞) , A n = A + n on Σ .
In (6.9) δ 0 n denotes the Kronecker symbol and we use the convention A -1 = 0. Hereafter, we make explicit the first asymptotics A n and A + n for n = 0, 1, 2 by induction.

6.3. First terms of the asymptotics for the magnetic potential. The first terms which are determined are A + 0 and A 0 . We obtain from (6.9) that A + 0 solves the problem (6.11)

     -∆A + 0 = µ + J in Ω + , ∂ n A + 0 = 0 on Σ , A + 0 = 0 on Γ .
Then according to (6.10) in the case n = 0, A 0 solves the ODE (6.12)

-∂ 2 Υ A 0 + λ 2 0 A 0 = 0 in Σ × (0, +∞) , A 0 = A + 0 on Σ .
The unique solution of (6.12) such that A 0 -→ 0 when Υ → ∞ is (6.13)

A 0 (ξ, Υ) = A + 0 X(ξ) e -λ 0 Υ .
The next term which is determined is A + 1 . According to (6.9) in the case n = 1 and according to (6.13), A + 1 solves the following problem (6.14)

     ∆A + 1 = 0 in Ω + , ∂ n A + 1 = ∂ Υ A 0 = -λ 0 A + 0 on Σ , A + 1 = 0 on Γ .
Then according to (6.10) for n = 1, A 1 solves the ODE (6.15) -∂ 2 Υ A 1 (•, Υ) + λ 2 0 A 1 (•, Υ) = λ 0 k(ξ)A + 0 X(ξ) e -λ 0 Υ for Υ ∈ (0, +∞) , A 1 (•, 0) = A + 1 X(ξ) .

The unique solution of (6.15) such that A 1 → 0 when Υ → ∞ is (6.16) A 1 (ξ, Υ) = A + 1 X(ξ) + k(ξ) 2 A + 0 X(ξ) Υ e -λ 0 Υ .

The next term which is determined is A + 2 . According to (6.9) in the case n = 2 and according to (6.16), A + 2 solves the following problem (6.17)

       ∆A + 2 = 0 in Ω + , ∂ n A + 2 = ∂ Υ A 1 = k(ξ) 2 A + 0 -λ 0 A + 1 on Σ , A + 2 = 0 on Γ .
According to (6.10) for n = 2, A 2 solves the ODE (6.18)

-∂ 2 Υ A 2 (•, Υ) + λ 2 0 A 2 (•, Υ) = -k(ξ)∂ Υ A 1 + A 2 A 0 for Υ ∈ (0, +∞) , A 2 (•, 0) = A + 2 X(ξ) ,
We can explicit the right-hand side of this ODE. According to (6.16)

∂ Υ A 1 = k(ξ) 2 A + 0 -λ 0 A + 1 + k(ξ) 2 A + 0 Υ e -λ 0 Υ
and according to (6.5b) and (6.13) we infer successively Then the unique solution of the ODE (6.18) such that A 2 → 0 when Υ → ∞ is (6.19)

A 2 (ξ, Υ) = A + 2 + k(ξ) 2 A + 1 + 1 2λ 0 ∂ 2 ξ + k(ξ) 2 4 I A + 0 Υ + 3k(ξ) 2 8 A + 0 Υ 2 X(ξ) e -λ 0 Υ .
The next term which is determined is A + 3 . According to (6.9) in the case n = 3 and according to (6.19), A + 3 solves the following problem (6.20)

       ∆A + 3 = 0 in Ω + , ∂ n A + 3 = ∂ Υ A 2 = -λ 0 A + 2 + k(ξ) 2 A + 1 + 1 2λ 0 ∂ 2 ξ + k(ξ) 2 4 I A + 0 on Σ , A + 2 = 0 on Γ .
6.4. Construction of impedance conditions. In this section, we derive high-order impedance boundary conditions (see Sec. 2.3.1) set on the interface Σ for the solution of the problem (2.3).

Order 1. Since A + 0 solves the limit problem (4.1), the condition of order 1 is the perfect magnetic conductor boundary condition, see (2.12)

∂ n A 0 = 0 on Σ .
∂ n A + 2,ε + ε λ 0 - k(ξ) 2 ε A + 2,ε = ε 3 - k(ξ) 2 A + 1 + λ 0 - k(ξ) 2 ε A + 2 on Σ
We neglect the terms in O(ε 3 ) in the previous right-hand side. Then we infer the impedance boundary condition of order 3, see (2.14)

∂ n A ε 2 + ε λ 0 - k(ξ) 2 ε A ε 2 = 0 on Σ .

  5.2 for precise results. The validation of the asymptotic expansion (2.7)-(2.8) consists in proving estimates for the remainder r m;ε defined at any order m ∈ N * as

Order 1 :

 1 Perfect magnetic conductor b.c.

(2. 12 )Order 2 :

 122 ∂ n A 0 = 0 on Σ Leontovich b.c.

  2)]. The modified Leontovich b.c. (2.14) writes

4. 1 .Notation 4 . 1 .

 141 Notations. We denote by X(ξ) ∈ Σ an arc-length parametrization of Σ. Here X is a smooth function from T L := R/LZ into R 2 and L is the length of the curve Σ. The scalar curvature k(ξ) at the point X(ξ) ∈ Σ is well-defined by dn dξ = -k(ξ) dX dξ .

A 2 A

 2 0 (ξ, Υ) = ∂ 2 ξ -k(ξ) 2 Υ∂ Υ A 0 (ξ, Υ) = ∂ 2 ξ A + 0 + λ 0 k(ξ) 2 ΥA + 0 X(ξ) e -λ 0 Υ .

Order 2 .

 2 According to (4.1)-(4.3), the truncated expansion A + 1,ε := A + 0 +εA + 1 solves the Laplace equation -∆A + 1,ε = µ + J in Ω + together with the boundary condition set on Σ∂ n A + 1,ε + λ 0 εA + 1,ε = λ 0 ε 2 A + 1 on Σ .Neglecting the term of order 2 in the previous right-hand side, we infer the Leontovich boundary condition, see (2.13)∂ n A ε 1 + λ 0 εA ε 1 = 0 on Σ .Order 3. According to (4.1)-(4.3)-(4.5), the truncated expansion A + 2,ε := A + 0 + εA + 1 + ε 2 A + 2 solves the Laplace equation -∆A + 2,ε = µ + J in Ω + together with the boundary condition set on Σ

Order 4 . 1 + ε 2 A + 2 + ε 3 A + 3 ( 6 = 4 I

 412364 We identify a simpler problem satisfied by the truncated expansionA + 3,ε := A + 0 + εA + µ + J in Ω + , ∂ n A + 3,ε + N 3,ε (A + 3,ε ) = O(ε 4) on Σ , . We neglect the terms in O(ε 4 ) in the right-hand sides. Then we infer the impedance boundary condition of order 4, see (2.15).

  Neumann operator which is given explicitly by (2.12)-(2.13)-(2.14)-(2.15) when k = 0, 1, 2, 3, respectively

Acknowledgements References

ACKNOWLEDGEMENTS

The author V. P. thanks very warmly Monique Dauge for her well-considered suggestions.


Yevgeniy Goryakin
email: ygoryakin@yahoo.com

Sophie P Thie ´baut

M Alie ´nor Lerouge

Michele Cecchini

Andrea B Feigl

Bruno Ventelou

Assessing the future medical cost burden for the European health systems under alternative exposure-to-risks scenarios

Background

Ageing populations and rising prevalence of non-communicable diseases (NCDs) increasingly contribute to the growing cost burden facing European healthcare systems. Few studies have attempted to quantify the future magnitude of this burden at the European level, and none of them consider the impact of potential changes in risk factor trajectories on future health expenditures.

Methods

The new microsimulation model forecasts the impact of behavioural and metabolic risk factors on NCDs, longevity and direct healthcare costs, and shows how changes in epidemiological trends can modify those impacts. Economic burden of NCDs is modelled under three scenarios based on assumed future risk factors trends: business as usual (BAU); best case and worst case predictions (BCP and WCP).

Findings

The direct costs of NCDs in the EU 27 countries and the UK (in constant 2014 prices) will grow under all scenarios. Between 2014 and 2050, the overall healthcare spending is expected to increase by 0.8% annually under BAU. In the all the countries, 605 billion Euros can be saved by 2050 if BCP is realized compared to the BAU, while excess spending under the WCP is forecast to be around 350 billion. Interpretation: Although the savings realised under the BCP can be substantial, population ageing is a stronger driver of rising total healthcare expenditures in Europe compared to scenario-based changes in risk factor prevalence.

Introduction

In Western Europe, NCDs account for about 91% of all deaths and 88% of all disabilityadjusted life years (DALYs) [START_REF] Vos | Global, regional, and national incidence, prevalence, and years lived with disability for 328 diseases and injuries for 195 countries, 1990-2016: A systematic analysis for the Global Burden of Disease Study[END_REF]. They also strain healthcare budgets, with five major NCDscardiovascular and endocrine conditions, cancers, respiratory and mental disorders-contributing to over a third of total health spending in high income countries [START_REF] Garg | What is the impact of non-communicable diseases on National Health Expenditures: A synthesis of available data[END_REF]. The societal impacts of NCDs will become more severe in decades to come, as population ageing and the rising prevalence of NCDs are expected to put mounting financial pressure on European healthcare systems and on the labor markets [START_REF] Ghebreyesus | Acting on NCDs: counting the cost[END_REF][START_REF] Sassi | Obesity and the economics of prevention: Fit not fat[END_REF][START_REF] Devaux | The Labour Market Impacts of Obesity, Smoking, Alcohol Use and Related Chronic Diseases[END_REF]. However, forecasting the future magnitude of this burden is difficult, due to demanding data requirements and the need to make strong assumptions about future trajectories of health expenditures and underlying risk factors. For example, since a large part of the NCD burden is preventable [START_REF] Forouzanfar | Global, regional, and national comparative risk assessment of 79 behavioural, environmental and occupational, and metabolic risks or clusters of risks, 1990-2015: a systematic analysis for the Global Burden of Disease Study[END_REF], future medical costs are likely to be driven by the evolution of underlying risk factors such as obesity, smoking and harmful alcohol consumption. In addition, future medical costs will depend on the evolving size and structure of the population and its aging process; pattern and volume of healthcare use, as well as changes in the medical technology and the associated costs.

Although various efforts to forecast medical costs have been made, both on country [START_REF] Wang | Health and economic burden of the projected obesity trends in the USA and the UK[END_REF] and regional level [START_REF] Abegunde | The burden and costs of chronic diseases in low-income and middle-income countries[END_REF], a general criticism of the existing forecasting methods is that they do not take into account major epidemiological changes [START_REF] Goldman | Consequences of health trends and medical innovation for the future elderly[END_REF], thus treating future risk factor exposure as fixed. In addition, a number of macro-level socioeconomic determinants such as regulatory role of government, income inequality, urbanization and globalization may have a strong influence on the future evolution of risk factors [START_REF] Popkin | Urbanization, lifestyle changes and the nutrition transition[END_REF][START_REF] Wilkinson | Income inequality and population health: A review and explanation of the evidence[END_REF][START_REF] Goryakin | The impact of economic, political and social globalization on overweight and obesity in the 56 low and middle income countries[END_REF].

A distinguishing feature of our study is its consideration of the evolution of medical costs under various plausible scenarios on the distribution of risk-factors across the whole population of the 27 EU countries and the UK, while taking into account both demographic and socioeconomic future trends. Having said that, although we will provide results according to the central predictions of our model, generating precise health expenditure forecasts until 2050 is not our goal, as such efforts can become notoriously unreliable even under much shorter time horizons. Rather, we are more interested in potential monetary savings and losses under alternative scenario assumptions, where we consider how direct medical costs may evolve in the future given what we call best case prediction (BCP) and worst case prediction (WCP), compared to the BAU trajectory based on current socioeconomic trends.

This analysis formed part of the FRESHER H2020 consortium, funded by the European Union H2020 program, linking several institutions in Europe and gathering the evidence from relevant disciplines in the domain of healthcare need projections, including public health and epidemiology, foresight sciences, economics and sociology. As a result, a dynamic microsimulation model designed to estimate the impacts of behavioral risk factors on chronic diseases and their direct economic costs in the EU 27 countries and the UK as well as in the subregions, was developed. This model also fills the gap in the existing tools in that it combines the benefits of microeconomic analyses with their focus on individual behaviors, with macroeconomic analyses more concerned about future projection of aggregate trends.

Methods

Microsimulation model design

The microsimulation model was developed to forecast future chronic disease burden, longevity and direct economic costs in 27 EU countries and the UK until the year 2050, as well as the extent to which future changes in epidemiological trends and specific policies can modify these outcomes. This modelling platform, initially based on previous OECD work [START_REF] Cecchini | Assessing the impacts of alcohol policies: a microsimulation approach[END_REF], was further improved as part of the FRESHER project [START_REF] Cecchini | Scientific paper on the methodology, results and recommendation for future research[END_REF]. The model uses case-based microsimulation to create representative synthetic life histories from birth to death, and relies on detailed epidemiological and demographic information from various sources. The model also uses prevalence-based direct cost estimates as an input into the model to forecast incidence-based health expenditures associated with various scenarios/policy interventions, from the health system perspective. More detailed information on data and the modelling assumptions can be found at a technical report posted online [START_REF] Cecchini | Scientific paper on the methodology, results and recommendation for future research[END_REF]15], the accompanying paper [START_REF] Devaux | How will the main risk factors contribute to the burden of non-communicable diseases under different scenarios by 2050? A modelling study[END_REF], as well as in the S6 Appendix in S1 File.

Diseases modelled

In general, attempt has always been made to estimate the direct medical costs for the following NCDs, although in some countries we evaluated costs only for a subset of these diseases due to the limited data (for more information, see S1 Appendix in S1 File):

• Ischemic stroke

• Haemorrhagic stroke

• Myocardial infarction and chronic heart disease

• Cancers

• Diabetes

• Chronic kidney disease (CKD)

• Chronic Obstructive Pulmonary Disease (COPD)

• Cirrhosis

• Depression

• Neurologic disorders (including Alzheimer disease and dementia)

• Injuries (including intentional and unintentional injuries and self-harm)

We selected these diseases due to their large contribution to the total NCD burden, as well as because many of them have common aetiologies and potentially modifiable risk factors such as obesity and harmful alcohol consumption [START_REF] Murray | Disability-adjusted life years (DALYs) for 291 diseases and injuries in 21 regions, 1990-2010: A systematic analysis for the Global Burden of Disease Study[END_REF][START_REF] Horton | The neglected epidemic of chronic disease[END_REF], and therefore are likely to be amenable to the scenario assumptions. In addition, since injuries (including intentional, unintentional and self-harm) are significantly linked with alcohol use disorders [START_REF] Schwarzinger | Alcohol use disorders and associated chronic disease-A national retrospective cohort study from France[END_REF], representing about 10% of the total disease burden caused by the alcohol use [START_REF] Evaluation | Global Burden of Disease[END_REF], the cost of injuries was modelled as well. Additional information on how the injury costs were estimated is provided in S2 Appendix in S1 File.

Estimation strategy

The microsimulation model requires individual-level total medical costs, conditional on age, gender and disease status as an input. In a standard bottom-up approach to cost estimation, patient-level units of health care attributable to a specific disease are multiplied by a price for this unit, and then summed up. However, in practice administrative coding is often not very informative about the underlying reason for incurring health expenditures. For example, falls can occur excessively in people suffering from Alzheimer's disease, but health expenditures may be recorded under alternative coding, thus giving an incomplete picture of the health expenditure burden facing this group. Alternatively (and also incorrectly), all the recorded costs for a person with a disease can be attributed to this disease. To overcome this limitation, we estimated costs using regressions, as the mean marginal difference of the predicted medical costs with a disease variable switched on or off (i.e., using between subjects comparison). This approach is commonly used to estimate incremental costs for select diseases and risk factors [START_REF] Manning | Health insurance and the demand for medical care: evidence from a randomized experiment[END_REF][START_REF] Finkelstein | Annual medical spending attributable to obesity: Payer-and service-specific estimates[END_REF][START_REF] Finkelstein | The costs of obesity in the workplace[END_REF]. In our study, the outcome variable is the total cost of hospital and ambulatory care (including costs of drugs) at the patient level.

We also account for the possibility that having two or more diseases may lead to costs that are greater than a simple sum of costs for separate diseases [START_REF] Cortaredona | The extra cost of comorbidity: Multiple illnesses and the economic burden of non-communicable diseases[END_REF]. For more information about how this super-additivity was modelled, see the S1 Appendix in S1 File.

Finally, we estimated age-and gender-specific residual medical costs for people who had no model-defined diagnosed diseases (but could have other diagnosed conditions, or were otherwise healthy). Taking into account such costs is important because reducing risk factor prevalence may still lead to accruing costs for unrelated medical conditions, either because people will live longer, or because they can be affected by unrelated competing disease risks. For more details about the estimation approach, see the S1 Appendix in S1 File.

Data sources and cost extrapolation

Although average healthcare spending per capita was accessible for all European countries, precise cost-of-illness estimates were not available, or even calculable, for a great majority of countries. A cost extrapolation model was therefore used to generalize the incremental costs of our 10 chronic conditions from the three "anchor countries" where the data was available -France, Estonia and the Netherlands-to the remaining EU-28 countries.

Our extrapolation methodology assumes that the annual treatment cost differentials between countries are time-invariant, and mostly driven by the differences in two components: costs per unit of treatment received, as well as population-level intensity of the treatments provided. To estimate these differentials, we used the OECD data [25] on the inpatient curative and rehabilitative care spending per capita, outpatient curative and rehabilitative care spending per capita as well as medical goods spending per capita. To ensure comparison and to smooth over temporary data variability, we used this data averaged over 3 years: 2012, 2013 and 2014, expressed in current prices.

Given a significant amount of cost-based extrapolations and uncertainties associated with future modelled scenarios, it was decided to present results not separately for each country, but aggregated over three regions: Southern Europe, Central/Eastern Europe and Northern Europe. There is no consensus on how such a grouping should be done, therefore members of the Fresher consortium decided to follow simple geographical criteria at the beginning of the project. First, Southern Europe is composed of EU27 countries bordering the Mediterranean Sea, as well as Portugal. Central/Eastern Europe counties include those which joined the EU since 2004, except Slovenia and Croatia (which belong in the Southern Europe group). The remaining countries belong in Northern Europe group. The final groupings were as follows:

1. Southern Europe (France, Italy, Spain, Portugal, Cyprus, Greece, Croatia, Malta, Slovenia); 2. Central/Eastern Europe (Estonia, Bulgaria, Poland, Romania, Slovakia, Hungary, Latvia and Lithuania);

3. Northern Europe (the Netherlands, Austria, Belgium, Czech Republic, Denmark, Finland, Germany, Ireland, Luxembourg, Sweden and United Kingdom).

This setup is not supposed to indicate that countries belonging to one group are always more similar to each other than to countries in another group. For example, healthcare system in France can have much more in common with Belgium than with Croatia. Nevertheless, deciding on group composition is likely to involve at least some degree of arbitrariness, and we tried to minimize it by using simple geographical proximity criteria. In any case, grouping decisions are not likely to affect the quality of the extrapolation so long as the differentials in costs between countries are taken into account, as we describe in the S1 Appendix in S1 File.

Disease identification

In source databases, we had to identify the occurrence of NCDs through a medical diagnosis and/or an associated treatment. Disease definitions were standardized across countries, with disease identification based on criteria such as International Classification of Diseases (ICD) diagnostic codes [26] and/or codes from the drug prescription databases. Specifically, in France, disease identification was done using ICD-version 10 codes; hospital discharge data as well as drug discharge Anatomical Therapeutic Chemical (ATC) codes, using Huber's classification method [START_REF] Huber | Identifying patients with chronic conditions using pharmacy data in Switzerland: An updated mapping approach to the classification of medications[END_REF]. In Estonia, identification was done using ICD-10 codes provided in the Estonian Health Insurance Fund (EHIF) dataset. In the Netherlands, identification of NCDs was only possible using ATC codes of medications prescribed to patients outside of hospitals recorded, which precluded reliable identification of diseases, such as CKD and cirrhosis. Therefore, for the Netherlands, we borrowed these later costs from France (only those missing). For more details about country-specific diseases identification, see the S1 Appendix in S1 File.

Model scenarios

We compare the outcomes under three scenarios for the evolution of the following key risk factors: smoking; obesity rates; alcohol use; physical inactivity and high blood pressure. Our BAU scenario is identical with the "the rich get healthier" scenario described in more detail in the accompanying paper [START_REF] Devaux | How will the main risk factors contribute to the burden of non-communicable diseases under different scenarios by 2050? A modelling study[END_REF]. Briefly, a group of 90 health experts were asked to predict the future evolution of five key risk factors based on the narrative description of four different scenarios of the future, previously identified after consulting a large number of experts on societal trends likely to impact the future evolution of NCDs. The "the rich get healthier" projection, in which it was assumed that the societal inequalities will continue to widen, and mostly marketbased solutions to societal and environmental problems will be offered, was chosen as the BAU, as it was viewed as the most likely to represent the future based on the continuation of the current trends.

To implement the best (worst) case scenarios, we first calculated the spread between the Europe-wide average rate for a given risk factor in 2015, and the lowest (highest) rate for that risk factor. For example, if the average obesity rate in Europe in 2015 was 20%, and the highest observed obesity rate that year was 30%, then we assume that every country will see its obesity rate linearly increase so that in 2050 it's greater by 10 percentage points than its current rate. This approach was implemented to ensure comparability with the other scenarios described in the accompanying paper [START_REF] Devaux | How will the main risk factors contribute to the burden of non-communicable diseases under different scenarios by 2050? A modelling study[END_REF].

Table 1 presents the average risk factor prevalence in 2015, as well as their evolution under the three scenarios. See S3 Appendix in S1 File for a graphical depiction of the changes in risk factor prevalence over time. Fig 1 depicts an example of a model-implemented link between an increase in one risk factor-the BMI-and an increased risk of developing related conditions such as heart disease or diabetes. Each additional disease arising out of an increased BMI is then associated with the disease-specific cost, or in the case of multiple conditions, with the cost for multiple diseases, which have been estimated for all 28 countries. A total cost saving attributable to the BCP can likewise be estimated.

Results

All the costs are expressed in constant, 2014 prices. Between 2014 and 2050, the overall healthcare spending is forecast to grow at an annual rate of about 0�8% in the BAU case, from 966 billion Euros in 2014 to 1258 billion Euros in 2050 (Table 2).Total healthcare spending under the BCP is predicted to be lower than under BAU by 2.73% in 2050, while spending under the WCP is predicted to be higher by about 1.62%. Another way to look at this is that under BAU scenario, medical spending in the EU-27 countries and the UK will be greater in 2050 by 291 billion Euros than in 2014. On the other hand, WCP will lead to 312 billion greater spending in 2050 compared to 2014, which means that it only contributes about 7% more spending on top of the BAU scenario over this period. Likewise, BCP will only lead to 12% lower spending compared to the BAU scenario over this period.

This narrow range of variation between BCP and WCP, each compared to BAU, was a first striking result. It indicates that population ageing is the major driver of the overall healthcare spending in Europe, with a relatively weak margin attributable to healthy behaviour improvements. By region, in 2050, the largest reductions in the healthcare budget under the BCP compared to BAU are predicted to occur Northern Europe (3.76%); followed by Southern Europe (1.65%) and Eastern Europe (0.98%). The same regional ranking will hold when considering excess spending under WCP scenario.

During the studied period (2014-2050), the population demographic will change, with countries experiencing various trends in their population composition (e.g. in age, gender and size). It is therefore important to look at the impact of the scenarios on predicted medical expenditures per capita. As only the injury costs were available for people younger than 18, we estimated per capita costs for people 18 years of age and older. In all the modelled countries, in the BAU case, these costs are expected to increase by about 30%, from 2,331 Euros in 2014 to 3,040 in 2050 (Fig 2). This increase will be mostly driven by an increasingly ageing population (S5 Appendix in S1 File). At the same time, by 2050, changes in risk factors consistent with the scenarios will lead to a significant deviation from the BAU trend: in the BCP, average cost will be about 4�6% lower than under BAU, and in the WCP-about 2�9% higher. The different trajectories in average predicted costs by region are shown in S4 Appendix in S1 File.

Cumulatively, 605 billion Euros can be saved by 2050 (or 16.8 billion annually) if the BCP is realized across all the modelled countries compared to the BAU, while excess spending under the WCP is forecast to be about 350 billion Euros ( Fig 3,left panel). By looking at specific regions, the largest cumulative savings by 2050 (when comparing BCP to BAU scenarios) can be obtained in Northern Europe (about 453 billion) and Southern Europe (153 billion). The lowest cumulative savings are expected to be obtained in Central and Eastern Europe (about 10.5 billion). Excess spending under the WCP is predicted to be in Northern Europe (242 bn), followed by Southern (92 bn) and Central and Eastern Europe (7�2 bn). The rate of annual savings will accelerate with time, with as much as 34 billion potentially saved in all the modelled countries in 2050, compared to only about 14 billion in 2030 (Fig 3, right panel). Therefore, the economic effect from the improved risk factor profile in the population will take some time materialize, but the savings can be substantial if the long-term view is taken.

On Fig 3, the scenario effect on health spending is masked by the differences in the total health spending between regions, which is also a function of the differences in the size of the population and the economies between regions. On Fig 4, one can see that there is much less variability between regions when scenario-specific cumulative health savings/losses are expressed as a proportion of total cumulative regional health expenditures. Nevertheless, it does appear that the savings are somewhat larger in Northern Europe, followed by Southern and then Eastern Europe. Table 3 suggests that one important reason for this is that the 10 modelled NCDs account for the largest proportion of total health costs in Northern Europe, so that any scenario-specific effect is also likely to be translated into larger savings in that region. In addition, it seems that for one risk factor in particular-hazardous drinking-there is going to be much less improvement under the BCP in Eastern Europe, compared to the other regions (S3 Appendix in S1 File).

Figs 5 and 6 further illustrate scenario-specific cost trends. Thus, older age is generally associated with larger savings and losses (Fig 5). This is not surprising, as diseases associated with the modelled risk factors mostly manifest themselves in the older ages. The only exception is for the oldest age groups, i.e. older than 80, as with time the BCP is expected to lead to cost losses, and the WCP-to cost savings. This can be explained by the fact in this age group, cost savings resulting from better risk factor profiles are outweighed by losses resulting from longer life expectancy.

Finally, Fig 6 suggests that both cost savings and losses depending on the scenarios are expected to be larger for men than for women. This is consistent with the hypothesis that baseline risk factor profile (and health in general) is generally worse among men, so the modelled scenarios are likely to lead to larger epidemiological and cost changes in this group.

Discussion

As the data availability and modelling capacity improve, there is a growing interest in forecasting future trends in health outcomes and projecting health expenditures. For example, Devaux et al [START_REF] Devaux | How will the main risk factors contribute to the burden of non-communicable diseases under different scenarios by 2050? A modelling study[END_REF] used qualitative and quantitative foresight techniques to project how future societal trends will affect the future burden of NCDs in Europe between 2015 to 2050. Relying on the same microsimulation modelling approach as used in this paper, they concluded that population ageing will be a dominant driver of population health, although the future risk factor trends will also play an important role.

Although a number of studies projecting future health expenditures exist, they tend to be restricted to a single country [START_REF] Harris | Estimating the future health and aged care expenditure in Australia with changes in morbidity[END_REF][START_REF] Guemmegne | Modeling the Dynamics of the U.S. Healthcare Expenditure Using a Hyperbolastic Function[END_REF][START_REF] Colombier | Projecting health-care expenditure for Switzerland: Further evidence against the "red-herring" hypothesis[END_REF]. In addition, they often assume that future risk factor prevalence will remain fixed [START_REF] Guemmegne | Modeling the Dynamics of the U.S. Healthcare Expenditure Using a Hyperbolastic Function[END_REF], and rarely take competing risks in individual health histories, as well as patient-level cost data into account, mostly due to the challenges with obtaining the data needed to populate microsimulation models [START_REF] Dieleman | Trends in future health financing and coverage: future health spending and universal health coverage in 188 countries[END_REF][START_REF] Przywara | Projecting future health care expenditure at European level: drivers, methodology and main results[END_REF]. As such, our study has numerous novel features: first, it adopts a life-long perspective on the evolution of the population health, allowing for the fact that people who do not get diseases as a result of an improved risk factor profile may nevertheless accumulate healthcare costs later in life for unrelated reasons. Second, it relaxes the assumption that the risk factor exposure will stay fixed in the future, making it conditional on three plausible scenarios. Third, unlike in most other cost of illness studies, direct disease costs are estimated from medical reimbursement data in a consistent methodological framework, rather than borrowed from a range of disparate sources. Finally, extra cost of comorbidities are explicitly taken into account.

Our micro-epidemiologic simulation exercise shows that the modelled scenarios can nontrivially affect direct medical cost burden in the EU. Depending on a series of alternative assumptions on the distribution and evolution of risk factors across the population, we estimate that by 2050, about 600 billion Euros can be saved cumulatively in the EU-27 and the UK under the BCP compared to BAU, with the amount of savings accelerating over time. In Northern Europe, these savings will be relatively more pronounced: by 2050, total healthcare budgets will be lower there by 3.8% of under BCP compared to BAU, revealing a marked sensitivity of Northern Europe to its current distribution of risk factors.

Nevertheless, total health expenditures (both per capita and aggregate) in the EU-27 and the UK will continue to increase even under the optimistic BCP. The main driver of this will be population ageing, although in Eastern Europe, the effect will be somewhat countered by the emigration-and low birth-rate fuelled population decline.

One of the main strengths of our microsimulation tool is that it allows modelling real-life counterfactuals. For example, in the absence of exposure to smoking, the same individual might still be afflicted with heart disease due to other risk factors, or live longer and continue accumulating healthcare costs for diseases that we are not modelling explicitly. To some extent, this mechanism explains why the BCP does not yield particularly large reduction in spending: people whose morbidity and mortality is reduced however continue to get older and to consume healthcare [START_REF] Cortaredona | The extra cost of comorbidity: Multiple illnesses and the economic burden of non-communicable diseases[END_REF][START_REF] Russell | Perspective: Preventing chronic disease: An important investment, but don't count on cost savings[END_REF]. This lowers the impact of risk-factor specific policy scenarios compared to more traditional cost-of-illness approaches, while at the same time representing a much more realistic cost savings estimates.

The model further contributes to the field of modelling the cost burden of chronic diseases by incorporating the cost of co-morbidities, answering the pertinent question whether there is additionality when two or more disease co-exist in one person 21 . Only a limited amount of high quality results exist in this area, mainly from the US and Nordic countries, and for only a sub-set of conditions.

Limitations

This analysis only considers the impact of scenarios on direct medical costs. However, it is well known that improvements in health are likely to lead to improvements in other economic outcomes [START_REF] Popkin | Measuring the full economic costs of diet, physical activity and obesity-related chronic diseases[END_REF]. For example, alcohol-related economic burden may also include direct non-medical costs, e.g. relating to property damage, or "indirect" costs arising from the lost or underutilized labour market resources. Simply put, indirect costs arise when no money physically changes hands, but some resources are still lost/underutilized. We do not estimate such costs in this paper, but they can contribute substantially to the economic impact of the modelled scenarios, often surpassing the direct burden [START_REF] Rehm | Global burden of disease and injury and economic cost attributable to alcohol use and alcohol-use disorders[END_REF]. In addition, although direct costs do contribute to the GDP, in their absence, the money could be invested in other areas, with potentially higher returns, and we do not take this potential "opportunity cost" aspect into account. In the same vein, we also do not take into account the potential cost of policies needed to obtain the variations in risk factors for the two alternate scenarios, which are likely to be very country-dependent. We also do not take into account the potential impact of other variables on the evolution of costs in the future, e.g. the impact of medical technology, which may well differ depending on the scenarios, or possible changes in prices of drugs (e.g. when patents expire). Another limitation is that most available data on the relationships between the modelled risk factors and diseases are from observational studies and therefore causality cannot usually be taken for granted, although the most recent literature we used generally tries to establish reliable causal impacts rather than simple associations. Finally, there were some limitations related to the disease identification in some countries or assumptions behind the cost extrapolation approach.

Policy implications

All in all, our modelling exercise is generating three important messages. Firstly, unless there is an exceptional effort to reduce risk factor prevalence in Europe which goes beyond the BCP assumption taken in this paper, future health expenditures in this region will be dominated by the population ageing trend, and therefore will continue to grow rapidly. Nevertheless, various population-level preventative interventions may still be cost-effective of even cost-saving, and better value for money than treating a chronic disease once it has developed [START_REF] Cecchini | Tackling of unhealthy diets, physical inactivity, and obesity: Health effects and cost-effectiveness[END_REF]. Indeed, with the current health system focus in many countries on curative interventions, there is significant room for shifting the focus towards prevention [START_REF] Hm Government | Healthy Lives, Healthy People: Our strategy for public health in England[END_REF], and various innovative intervention packages may be developed to reduce the prevalence of NCDs yet further [START_REF] The | The heavy burden of obesity[END_REF]. In addition, there should be a search for more sustainable modes of care in the future, including greater attention paid to improving rehabilitative and supportive care to keep the elderly independent and well-functioning for as long as possible. Secondly, by taking into account the cost of unrelated competing NCDs as well as co-morbidities, and showing that additionality and superadditionality is significant for a subset of the modelled conditions, the model provides results that not only improve on existing models available to date, but also constitute immediate policy relevance. Super-additionality in disease costs highlights the importance of targeting not only single diseases, but also overall comorbidities, with appropriate policy interventions. The model can provide further guidance about which co-morbidity reduction might result in the highest cost savings. Thirdly, as shown in the accompanying paper [START_REF] Devaux | How will the main risk factors contribute to the burden of non-communicable diseases under different scenarios by 2050? A modelling study[END_REF], although the expected positive effect of an improved risk factor profile is relatively modest for the financial sustainability of health systems, the impact on epidemiological burden as measured by changes in disease incidence, life expectancy and premature mortality is potentially very significant. Likewise, evidence suggests that economic returns to higher labour market productivity may significantly exceed those from healthcare system savings [START_REF] The | The heavy burden of obesity[END_REF]. As the methodological approach we use in this paper does not value this impact in monetary terms (which may include both nonmarket value of improved health, as well as potential improvements in productivity and/or labour force participation), the full economic and welfare benefit from the more favourable risk factor profile is likely to be much greater than the estimated health expenditure saving may suggest.

Fig 2 .Fig 3 .

 23 Fig 2. Change in average direct costs in the EU-27 countries and the UK for people 18 years and older, 2014-2050. Base refers to BAU scenario; Best-to BCP and Worst-to WCP. https://doi.org/10.1371/journal.pone.0238565.g002

Fig 4 .

 4 Fig 4. Total savings and losses as a % of total health expenditures, 2014-2050. EE means Central and Eastern Europe; EU: European Union; NE is Northern Europe and SE is Southern Europe. Best refers to BCP and worst-to WCP. Blue bars reprsent savings, red bars-losses. https://doi.org/10.1371/journal.pone.0238565.g004

Fig 5 .Fig 6 .

 56 Fig 5. Annual cost savings by age cohorts in the EU, 2014-2050. Best refers to BCP and Worst to WCP. https://doi.org/10.1371/journal.pone.0238565.g005

  PLOS ONEFuture medical cost burden for the European health systems under alternative scenarios Epi Visualisation Tool at https://vizhub.healthdata. org/epi/. The Institute for Health Metrics and Evaluation GBD Results Tool can be found at http:// ghdx.healthdata.org/gbd-results-tool. The International Agency for Research on Cancer at https://gco.iarc.fr/today/home. Click on the 'table' button and download the data (option on the right).

	The WHO Global Information System on Alcohol
	and Health at http://apps.who.int/gho/data/node.
	gisah.GISAHhome?showonly=GISAH. The NCD
	Risk Factor Collaboration data at http://ncdrisc.org/
	index.html. Click on the 'data downloads' button
	and select the risk factor of choice. The 'individual
	country data' option should be chosen. Risk factors
	are available from the literature as listed in A6
	Appendix. The datasets to calculate costs are not
	publicly available, but researches can submit
	requests for access to the relevant institutions.
	fr)
	from Aix-Marseille University, CNRS, EHESS,
	Centrale Marseille, IRD, AMSE, Marseille, France,
	had access to the data through his institutional
	affiliation. For Estonia, the main data file was
	available from the Estonian Health Insurance Fund
	(EHIF) https://www.haigekassa.ee/. We obtained
	access to EHIF thanks to Taavi Lai (taavi.lai@gmail.
	com) and Natalja Eigo (natalja.eigo@tai.ee) from
	the National Institute of Health Development in
	Estonia. For Netherlands, the data was provided
	online by Statistics Netherlands https://www.cbs.
	nl/en-gb. The main contact email address is
	Microdata@cbs.nl.

PLOS ONE | https://doi.org/10.1371/journal.pone.0238565 September 11, 2020 2 / 14 Information on the datasets can be found at: https://www.amse-aixmarseille.fr/sites/default/files/ research/cost_calculations_france.pdf (for France); https://www.amse-aixmarseille.fr/sites/default/files/ research/cost_calculations_estonia.pdf (for Estonia) and https://www.cbs.nl/-/media/_pdf/ 2016/39/zvwkostentab.pdf (for the Netherlands). All cost data are national health systems data sets which means it is systematically available to researchers completing usual data confidentiality and other requirements. For France, the bureau of statistics has to give agreement on specific projects and/or researchers. In this respect, researchers would have to complete the data access requirement. In our case, Sebastien Cortaredona (sebastien.cortaredona@inserm.

Table 1 . Risk factor prevalence (%) in 2015 (actual) and in 2050 (predicted).

 1

	Region	Risk factor	BAU, 2015	BCP, 2050	BAU, 2050	WCP, 2050
		Inactive	28.4	12	35.1	54.5
		Overweight	62	59.9	65.9	64.2
	EU	HBP	27.9	24.5	35	44.6
		Smoker	22	10	17.8	31.2
		Hazard drinking	12.1	9	12.3	12.6
		Inactive	20.6	6.7	27.8	47.1
		Overweight	59.6	58.5	64.4	62.7
	EE	HBP	35.1	33.3	43.7	53.1
		Smoker	24.4	12.3	20	33.1
		Hazard drinking	9.8	8.7	9.5	9.5
		Inactive	27	10.1	33.2	52.5
		Overweight	61.7	59.6	65.6	63.9
	NE	HBP	24.9	20.6	31.2	40.8
		Smoker	20.6	9	17.1	30.4
		Hazard drinking	13.2	9.3	13.5	14
		Inactive	33.5	17	40.2	59.6
		Overweight	63.1	60.8	66.7	65
	SE	HBP	28.3	25.2	35.8	45.4
		Smoker	22.4	10.2	17.8	31.1
		Hazard drinking	11.5	7.4	11.8	12.3

EE means Central and Eastern Europe; EU: European Union; NE is Northern Europe and SE is Southern Europe https://doi.org/10.1371/journal.pone.0238565.t001 Fig 1. Conceptual framework for high BMI attributable direct costs. https://doi.org/10.1371/journal.pone.0238565.g001

Table 2 . Evolution of total annual predicted costs (bn Euros) by region, year and scenario.

 2

	Region	Year	WCP	BAU	BCP	Excess spending �	Cost Savings ��
		2014	966	966	966	0.00%	0.00%
	EU	2030	1097	1089	1075	0.72%	1.29%
		2050	1278	1258	1223	1.62%	2.73%
		2014	36	36	36	0.00%	0.00%
	EE	2030	37	37	37	0.53%	0.90%
		2050	38	37	37	0.80%	0.98%
		2014	547	547	547	0.00%	0.00%
	NE	2030	627	622	612	0.81%	1.56%
		2050	743	728	701	1.98%	3.76%
		2014	366	366	366	0.00%	0.00%
	SE	2030	423	421	417	0.57%	0.95%
		2050	498	493	485	1.02%	1.65%

� WCP vs BAU �� BCP vs BAU. Excess spending is presented as % of cost difference under WCP and BAU scenarios, relative to BAU; cost savings are difference between costs under BCP and BAU scenarios, relative to baseline. EE means Central and Eastern Europe; EU: European Union; NE is Northern Europe and SE is Southern Europe. https://doi.org/10.1371/journal.pone.0238565.t002

Table 3 . Contribution of the 10 NCDs as % of total healthcare spending, by region and scenario (over 2014- 2050).

 3

	Region	WCP	BAU	BCP
	EU	46.67%	46.00%	44.83%
	North	52.81%	52.22%	51.09%
	South	36.83%	36.15%	35.01%
	East	36.44%	35.83%	34.84%

https://doi.org/10.1371/journal.pone.0238565.t003

PLOS ONE | https://doi.org/10.1371/journal.pone.0238565September 11, 2020

Acknowledgments

The opinions expressed herein are solely those of the authors and do not necessarily reflect the official views of the OECD or its member countries.

FRESHER ("FoResight and Modelling for European Health policy and Regulation") is an interdisciplinary research project funded by the European Union's Horizon 2020 research and innovation programme under grant agreement No 643576 for a duration of 36 months (2015-2018).

The results presented in the study are the outputs of a micro-simulation model that is informed by multiple data sources. Specifically, the demographic data are available from the United Nations at https://population.un. org/wpp. Click on the 'download data files' button and choose the standard population projections. Mortality data are available from the Human Mortality Database at http://www.mortality.org/. Epidemiology data are available from the following sources: Institute for Health Metrics and

Supporting information

S1 File. (DOCX)

Author Contributions

Conceptualization: Yevgeniy Goryakin.

Formal analysis: