Maryna Kachanovska
email: maryna.kachanovska@ensta.fr

STABLE PERFECTLY MATCHED LAYERS FOR A CLASS OF ANISOTROPIC DISPERSIVE MODELS. PART II: ENERGY ESTIMATES *

Keywords: 1991 Mathematics Subject Classification. 65M12, 35Q60. perfectly matched layers, Maxwell equations, Laplace transform, energy, Fourier analysis

Stable perfectly matched layers for a class of anisotropic dispersive models. Part II: Energy estimates

Introduction

The method of the perfectly matched layers (PML), introduced by Bérenger [START_REF] Bérenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF][START_REF] Bérenger | Three-dimensional perfectly matched layer for the absorption of electromagnetic waves[END_REF], is used ubiquitously in engineering and physics communities to compute a solution to a problem posed in an unbounded domain. However, it is well-known that for some classes of problems (e.g. the wave propagation in anisotropic and/or dispersive media) the classical PML method may result in instabilities [START_REF] Bécache | Stability of perfectly matched layers, group velocities and anisotropic waves[END_REF][START_REF] Bécache | On the analysis of perfectly matched layers for a class of dispersive media[END_REF]. For a class of anisotropic dispersive models PMLs were stabilized [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF].

In this work we continue the analysis of the stable PMLs constructed in [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF]. This class of systems describes the wave propagation in metamaterials and plasmas in 2D. In the frequency domain, they can be written in the form of an anisotropic dispersive wave equation:

ε 1 (ω) -1 ∂ 2 x u + ε 2 (ω) -1 ∂ 2 y u + ω 2 µ(ω)u = 0, ω ∈ R,
where ε 1 (ω), ε 2 (ω) have a meaning of a dielectric permittivity and µ(ω) is a magnetic permeability. They depend on frequency non-trivially and satisfy Im(ωε j (ω)) > 0, j = 1, 2, Im(ωµ(ω)) > 0 for Im ω > 0, an assumption connected to the stability of the model, cf. [START_REF] Welters | Speed-of-light limitations in passive linear media[END_REF].

In the time domain, such systems correspond to the Maxwell's equations with currents, which are coupled to the electromagnetic field through the ordinary differential equations.

In [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF] a method of construction of stable perfectly matched layers for this class of models is suggested. It is based on the ideas of Bécache et al. [START_REF] Bécache | On the analysis of perfectly matched layers for a class of dispersive media[END_REF] for isotropic metamaterials, which, in turn, generalizes the work of Cummer [START_REF] Cummer | Perfectly matched layer behavior in negative refractive index materials[END_REF] for the Drude model. However, the analysis in the aforementioned works does not cover the question of the decay of an energy associated with the newly constructed PML systems.

This questions was answered for the classical PML applied to isotropic, non-dispersive Maxwell equations by Bécache and Joly [START_REF] Bécache | On the analysis of Bérenger's perfectly matched layers for Maxwell's equations[END_REF]. In a more general case, the derivation of energy estimates for PML systems does not seem trivial. For example, in [START_REF] Bécache | On the analysis of Bérenger's perfectly matched layers for Maxwell's equations[END_REF] the authors have obtained energy estimates for the PML for isotropic Maxwell equations written in the Zhao-Cangellaris formulation [START_REF] Zhao | Gt-pml: generalized theory of perfectly matched layers and its application to the reflectionless truncation of finite-difference time-domain grids[END_REF]. The derivation of the energy estimates for this formulation is simpler compared to e.g. Bérenger's split formulation. Our approach is different from that of [START_REF] Bécache | On the analysis of Bérenger's perfectly matched layers for Maxwell's equations[END_REF], where the PML system was given a priori. We suggest to look for a formulation that would be equivalent to the original system and would allow to obtain the energy in a simpler way. We make use of the stability analysis in the Laplace domain, as it was done in the work [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF], which provides intuition on derivation of explicit (without appeal to the Plancherel's theorem) energy estimates.

An alternative way to derive energy estimates is offered by the use of the method of Hagstrom and Appelö [START_REF] Hagstrom | Automatic symmetrization and energy estimates using local operators for partial differential equations[END_REF]. It can be applied to a very general class of systems, however, it results in energy estimates on the spatial derivatives of unknowns [START_REF] Appelö | Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability[END_REF], while most of the estimates we obtain involve the original unknowns. Moreover, the application of this approach to dispersive models does not seem trivial, due to the need to rewrite the original problem as a single equation, and then apply the method of [START_REF] Hagstrom | Automatic symmetrization and energy estimates using local operators for partial differential equations[END_REF] to the resulting problem.

This work is organized as follows. In Section 1 we state the problem under consideration. In Section 2 we show how to construct stable PMLs for so-called generalized Lorentz models and derive the energy estimates for the corresponding time-domain system. In Section 3 we extend these results to general passive materials. Finally, in Section 4 we demonstrate how the ideas of the previous sections can be used to prove the stability of the PMLs of [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF] for 1D dispersive systems, for a non-constant absorption PML parameter.

Problem Setting

The wave propagation in 2D dispersive media is described by the Maxwell system:

∂ t D x = ∂ y H z , ∂ t D y = -∂ x H z , ∂ t B z = -∂ x E y + ∂ y E x . (1)
In the above we use the scaling c = ε 0 = µ 0 = 1. The relations between the electric field and the electric displacement field, as well as between the magnetic field and the magnetizing field are given in the Laplace domain. Denoting by s the Laplace variable, we set û(s) = (Lu) (s) the Laplace transform of u(t) and introduce C + = {s ∈ C : Re s > 0}. With this notation, the constitutive relations read D = ε(s) Ê, Ĥz = µ -1 (s) Bz . We will assume that the dielectric permittivity is a diagonal matrix

ε(s) = ε x (s) 0 0 ε y (s) .
The multiplication in the Laplace domain becomes a convolution in the time domain, which we formally denote

D = ε(∂ t )E, B z = µ(∂ t )H z .
The system [START_REF] Bérenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF] in the Laplace domain can be rewritten as an anisotropic dispersive wave equation:

s 2 µ(s) Ĥz -ε y (s) -1 ∂ 2 x Ĥz -ε x (s) -1 ∂ 2 y Ĥz = 0. ( 2
)
To explain the assumption on the coefficients of the above equation, let us introduce the following definition. In this work we will assume that ε x , ε y , µ are passive. One of the examples of passive models is offered by so-called generalized Lorentz materials [START_REF] Bécache | Stable perfectly matched layers for a cold plasma in a strong background magnetic field[END_REF]:

ε α (s) = 1 + nα =0 ε α s 2 + ω 2 α , ε α > 0, ω α ∈ R, = 0, . . . , n α , α ∈ {x, y}, µ(s) = 1 + nµ =0 µ s 2 + ω 2 µ , µ > 0, ω µ ∈ R, = 0, . . . , n µ . (3)
The generalized Lorentz materials include Drude materials [START_REF] Bécache | On the analysis of perfectly matched layers for a class of dispersive media[END_REF] and a 2D uniaxial cold plasma [START_REF] Bécache | Stable perfectly matched layers for a cold plasma in a strong background magnetic field[END_REF][START_REF] Bécache | Perfectly matched layers in negative index metamaterials and plasmas[END_REF].

Remark 1.2. The passivity assumption covers nonhyperbolic models as well (e.g. the diffusion equation, with

x (s) = y (s) = 1, µ(s) = s -1 ).
In [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF] it was demonstrated that the well-posedness and stability of (1) follows from the passivity of coefficients of the sesquilinear form associated with [START_REF] Bérenger | Three-dimensional perfectly matched layer for the absorption of electromagnetic waves[END_REF]. Introducing the scalar product

(u, v) = R 2 uvdx, u, v ∈ L 2 ,
we can define the following class of sesquilinear forms.

Definition 1.3. We will call a sesquilinear form

A(u, v) = a(s)(∂ x u, ∂ x v) + b(s)(∂ y u, ∂ y v) + s 2 c(s)(u, v), u, v ∈ H 1 (R 2 ), passive if a(s) -1 , b(s) -1 , c (
Instead of resorting to the Laplace-domain analysis of [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF], one can prove stability results directly in the time domain, using energy techniques, cf. [START_REF] Cassier | On properties of passive systems In preparation[END_REF]. Extending these results to the PMLs is the subject of present work.

Stability of PMLs for Lorentz Materials

The analysis for the generalized Lorentz materials (3) will play a crucial role in this work. This is due to the fact that any passive material can be approximated by generalized Lorentz materials, see Section 3. Moreover, Lorentz materials are the only representatives of a fairly general class of passive materials. To demonstrate this, let us assume that ε x , ε y , µ satisfy the following assumptions, justified in [START_REF] Bécache | On the analysis of perfectly matched layers for a class of dispersive media[END_REF][START_REF] Bécache | Stable perfectly matched layers for a cold plasma in a strong background magnetic field[END_REF][START_REF] Cassier | On properties of passive systems In preparation[END_REF].

Assumption 2.1. A function r(s) is a rational function r(s) = 1 + pr(s 2 )
qr(s 2 ) , where the polynomials p r and q r have real coefficients, no common roots, and also deg p r < deg q r . All its zeros and poles lie in iR.

A class of passive functions satisfying the above condition is quite narrow, see [START_REF] Cassier | On properties of passive systems In preparation[END_REF] and references therein. Thus, passive functions that satisfy Assumption 2.1 correspond to Lorentz materials (3). This section is organized as follows. In Section 2.1 we formulate the time-domain system corresponding to Lorentz materials without the PML. Next, in Section 2.2 we discuss the construction of stable PMLs for dispersive systems, concentrating in particular on Lorentz models. In Section 2.3 we show the main idea of the derivation of the PML system that would allow to obtain simplified energy estimates. Finally, in Sections 2.4 and 2.5 we derive the energy estimates for systems corresponding to different PMLs.

Time-Domain System without PMLs

We consider the Maxwell system (1) with Lorentz parameters (3). In the time domain, we obtain the Maxwell equations with currents, coupled to E, H z via ODEs, cf. [START_REF] Bécache | On the analysis of perfectly matched layers for a class of dispersive media[END_REF]:

∂ t D x = ∂ y H z , (5a)
∂ t D y = -∂ x H z , (5b)
∂ t B z = ∂ y E x -∂ x E y , (5c)
∂ t D α = ∂ t E α + nα =0 ε α λ α , (5d)
∂ t λ α + ω 2 α p α = E α , ∂ t p α = λ α , = 0, . . . n α , α ∈ {x, y}, (5e)
∂ t B z = ∂ t H z + nµ =0 µ λ µ , (5f)
∂ t λ µ + ω 2 µ p µ = H z , ∂ t p µ = λ µ , = 0, . . . n µ . ( 5g
)
Let us clarify the time-domain realization of Bz = µ(s) Ĥz (the time-domain equivalent of D = ε(s)E can be obtained similarly). The latter equivalently reads s Bz = sµ(s)

Ĥz = s + nµ =0 µ s+ω 2 µ s -1
Ĥz . One defines

λµ = 1 s + ω 2 µ s -1 Ĥz , or, equivalently, s λµ + ω 2 µ pµ = Ĥz , sp µ = λµ .
Provided that λ µ | t=0 = 0 = p µ | t=0 , the above coincides with (5g) in the time domain. Similarly, for B z | t=0 = H z | t=0 , we verify that (5f) coincides in the Laplace domain with Bz = µ(s) Ĥz . Let us formulate the following known result for the system (5a-5g), see also [START_REF] Bécache | On the analysis of perfectly matched layers for a class of dispersive media[END_REF]. We provide its proof since it will be of use later.

Theorem 2.3. An energy E of (5a-5g), defined below, satisfies d dt E = 0. Here

E = 1 2 E x 2 + E y 2 + H z 2 + E x + E y + E z , E α = 1 2 nα =0 ε α λ α 2 + ω 2 α p α 2 , α ∈ {x, y}, E z = 1 2 nµ =0 µ λ µ 2 + ω 2 µ p µ 2 .
Proof. Test the equation (5c) with H z , to obtain

(∂ t B z , H z ) -(∂ y E x , H z ) + (∂ x E y , H z ) = (∂ t B z , H z ) + (E x , ∂ y H z ) -(E y , ∂ x H z ) = 0. ( 6
)
We consider the first term of the above expression; using (5f),

(∂ t B z , H z ) = 1 2 d dt H z 2 + nµ =0 µ (λ µ , H z ) (5g) = 1 2
d dt H z 2 + nµ =0 µ λ µ , ∂ t λ µ + ω 2 µ p µ (5g) = 1 2
d dt H z 2 + 1 2 d dt nµ =0 µ λ µ 2 + nµ =0 µ ω 2 µ (∂ t p µ , p µ ) = 1 2 d dt H z 2 + d dt E z . (7)
The term (E x , ∂ y H z ) in ( 6) can be rewritten exactly like in the previous case:

(E x , ∂ y H z ) (5a) = (E x , ∂ t D x ) (5d) = 1 2
d dt E x 2 + 1 2 d dt nµ =0 ε λ x 2 + ω 2 x p x 2 = 1 2 d dt E x 2 + d dt E x . (8)
Similarly one shows that

-(E y , ∂ x H z ) = 1 2 d dt E y 2 + d dt E y .

Construction of Stable PMLs for Dispersive Models

Consider the system (1) written in the Laplace domain [START_REF] Bérenger | Three-dimensional perfectly matched layer for the absorption of electromagnetic waves[END_REF]. To construct a PML in the region x ≥ 0, one assumes that (2) is valid for x = x ∈ C. The corresponding analytic continuation of Ĥz , denoted by Ĥz , solves (2), with x = x. In [START_REF] Bécache | On the analysis of perfectly matched layers for a class of dispersive media[END_REF] it is proposed to choose x as follows, for some analytic function ψ(s),

x =        x + s -1 ψ x (s) x 0 σ x (x )dx , x ≥ 0, x,
x < 0,

σ x (x) =    σ(x) ≥ 0, x ≥ 0, 0, x < 0. (9)
The classical PML can be obtained taking ψ x (s) = 1. For dispersive materials this choice may be unstable. We discuss how to choose ψ x (s) in further sections. With this change of variables, (2) transforms into

ε -1 y 1 + σ x s -1 ψ x -1 ∂ x 1 + σ x s -1 ψ x -1 ∂ x Ĥz + ε -1 x ∂ yy Ĥz -s 2 µ Ĥz = 0, x ∈ R. (10)
By analytic continuation, Ĥz (x, y) = Ĥz (x, y) for x < 0. The resulting system should be rewritten in the time domain. We postpone this question to future sections. The PML in other directions (e.g. for y ≥ 0) can be constructed similarly. In corners the changes of both of the variables x and y is used. Under (mild) assumptions on the coefficients of the sesquilinear form, one can show that the time-domain system corresponding to [START_REF] Hagstrom | Automatic symmetrization and energy estimates using local operators for partial differential equations[END_REF] is well-posed, for any σ x (x) ∈ L ∞ (R), see [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I. Necessary and Sufficient Conditions of Stability[END_REF]. It seems significantly more difficult to show the stability of the corresponding problem for variable σ x (x), cf. [START_REF] Diaz | A time domain analysis of PML models in acoustics[END_REF]. Finer results can obtained by studying the sesquilinear form corresponding to [START_REF] Hagstrom | Automatic symmetrization and energy estimates using local operators for partial differential equations[END_REF] with σ x (x) = const, x ∈ R, when (9) becomes, cf. [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF],

x → x 1 + s -1 ψ x σ x , σ x ≡ const ≥ 0, x ∈ R. (11)
In this case the sesquilinear form corresponding to [START_REF] Hagstrom | Automatic symmetrization and energy estimates using local operators for partial differential equations[END_REF] becomes

A(u, v) = ε y (s) -1 (1 + s -1 ψ x (s)σ x ) 2 (∂ x u, ∂ x v) + ε x (s) -1 (∂ y u, ∂ y v) + s 2 µ(s)(u, v), u, v ∈ H 1 (R 2 ). ( 12
)
As most of the results of this work are derived for [START_REF] Appelö | Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability[END_REF], let us discuss the reasons for this choice and its place in the analysis of the stability of PMLs. To our knowledge, so far only the energy estimates for the classical PMLs for non-dispersive systems are available. These results are of either of two types:

(1) the energy estimates are derived for a PML system obtained from the change of variables [START_REF] Appelö | Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability[END_REF], which, strictly speaking, is not what is used in practice. Moreover, the analysis is available only in 2D, for ψ(s) = 1 and non-dispersive models. Typically, such energy estimates result in the bound of the form V(t) 1 ≤ C(t) V(0) 2 , with C(t) depending polynomially on t. They constitute one of the steps in the analysis of the stability of the PML models with σ = const. See [START_REF] Bécache | On the analysis of Bérenger's perfectly matched layers for Maxwell's equations[END_REF][START_REF] Appelö | Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability[END_REF][START_REF] Bécache | On the long-time behavior of unsplit perfectly matched layers[END_REF] for this kind of analysis.

(2) the energy estimates are obtained for a PML system where the change of variables ( 9) is performed (again with ψ(s) = 1 and in non-dispersive case). This is indeed closer to the case of practical interest. For 1D problems these estimates are optimal and imply stability, see [START_REF] Ervedoza | Perfectly matched layers in 1-d: energy decay for continuous and semi-discrete waves[END_REF]. However, in higher dimensions they result in the bounds of the form V(t) 1 ≤ C(t) V(0) 2 , with C(t) depending on t exponentially. Thus, such estimates are non-optimal, since they do not reflect the stability of the underlying PMLs (the latter fact suggested by numerous experiments). This kind of results was obtained e.g. in [START_REF] Bécache | On the analysis of Bérenger's perfectly matched layers for Maxwell's equations[END_REF][START_REF] Huang | Mathematical analysis of a PML model obtained with stretched coordinates and its application to backward wave propagation in metamaterials[END_REF]. Our goal is to extend the results of the first type to 2D dispersive models, and show the stability of the PMLs for 1D dispersive problems for σ x = const (see Section 4). In this section we concentrate on the case σ x = const.

Construction of Stable PMLs for Lorentz Materials in One Direction

The difficulty in the construction of the stable PML for dispersive models is the choice of the function ψ x (s) in [START_REF] Zhao | Gt-pml: generalized theory of perfectly matched layers and its application to the reflectionless truncation of finite-difference time-domain grids[END_REF]. This question had been studied in [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF], and for corners in particular in [START_REF] Kachanovska | A note on the stability of PMLs for a class of anisotropic dispersive models in corners[END_REF]. Let us provide several main results on the stability of PMLs, which are formulated as passivity of the corresponding sesquilinear forms, see [START_REF] Bécache | On the analysis of perfectly matched layers for a class of dispersive media[END_REF]. One of the choices of ψ x (s) resulting in stable PMLs is offered by the following statement.

Theorem 2.4 ( [5]

). Let ε x , ε y , µ be passive and satisfy Assumption 2.1. Then the sesquilinear form [START_REF] Bécache | Stable perfectly matched layers for a cold plasma in a strong background magnetic field[END_REF] with ψ x (s) = ε y (s) -1 is passive for any σ x ≥ 0.

Remark 2.5. The passivity of the sesquilinear form [START_REF] Bécache | Stable perfectly matched layers for a cold plasma in a strong background magnetic field[END_REF], with ψ(s) = ε y (s) -1 , implies its coercivity for all s ∈ C + . To see this, one tests [START_REF] Bécache | Stable perfectly matched layers for a cold plasma in a strong background magnetic field[END_REF] with su, to obtain

Re (A(u, su)) = Re sε y (s) -1 (1 + s -1 ε y (s) -1 σ x ) 2 ∂ x u 2 + sε x (s) -1 ∂ y u 2 + sµ(s)|s| 2 u 2 = Re sε y (s) -1 (1 + s-1 εy (s) -1 σ x ) 2 |1 + s -1 ε y (s) -1 σ x | 4 ∂ x u 2 + sε x (s) -1 ∂ y u 2 + sµ(s)|s| 2 u 2 = |1 + s -1 ε y (s) -1 σ x | -4 Re(sε y (s) -1 ) + 2σ x |ε y (s)| -2 + σ 2 x |ε y (s)| -2 Re(s -1 εy (s) -1 ) ∂ x u 2 + Re(sε x (s) -1 ) ∂ y u 2 + Re(sµ(s)) u 2 .
Notice that Re(sε -1 y ) = Re sε y -1 = Re sε y |ε y | -2 > 0, for all s ∈ C + , thanks to passivity. For the same reason Re((sε y ) -1 ) > 0, Re(sε -1 x ) and Re(sµ(s)) > 0, s ∈ C + . From this the coercivity of Re A(u, sv) follows. The coercivity alone is not sufficient for the well-posedness or stability of the corresponding time-domain system. In particular, one should show the analyticity of the Laplace-domain solution in C + , see [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF]. This is indeed not the only stable PML change of variables. While the theorem below does not show how to construct ψ x (s), in [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF] it is shown that its conditions can be ensured by choosing ψ x (s) that verifies certain sign conditions on the imaginary axis. We omit these details, since they are not important for the energy estimates.

Theorem 2.6 ( [5]

). Let ε x , ε y , µ be passive and satisfy Assumption 2.1. Let additionally (1) ψ x (s) -1 satisfy Assumption 2.1 and be passive;

(2) ε x (s)ε y (s) -1 ψ x (s) -1 be passive;

(3) µ(s)ε y (s)ψ x (s) be passive. Then the sesquilinear form

Ã(u, v) = ψ x (s)ε y (s)A(u, v), u, v ∈ H 1 (R 2 ),
with A(u, v) given by [START_REF] Bécache | Stable perfectly matched layers for a cold plasma in a strong background magnetic field[END_REF], is passive for any σ x ≥ 0.

Construction of Stable PMLs for Lorentz Materials in Corners

Similar results can be formulated for the corner PMLs, which are obtained by the following change of variables:

x → x 1 + σ x ψ x (s)s -1 , y → y 1 + σ y ψ y (s)s -1 , σ x , σ y ≥ 0. ( 13
)
The associated sesquilinear form reads

A c (u, v) = ε y (s) -1 (1 + s -1 ψ x (s)σ x ) 2 (∂ x u, ∂ x v) + ε x (s) -1 (1 + s -1 ψ y (s)σ y ) 2 (∂ y u, ∂ y v) + s 2 µ(s)(u, v), u, v ∈ H 1 (R 2 ). (14)
The corner PML analogue of Theorem 2.4 can be obtained by a change of variables with ψ x = ε -1 y and ψ y = ε -1

x . Theorem 2.7 ( [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF]). Let ε x , ε y , µ be passive and satisfy Assumption 2.1. Then the sesquilinear form A c (u, v) of ( 14) with ψ x (s) = ε y (s) -1 and ψ y (s) = ε x (s) -1 is passive for any σ x ≥ 0 and σ y ≥ 0.

Notice that the above result covers the stability of the PML in one direction (by setting σ x = 0 or σ y = 0) and in corners. A trivial analogue of Theorem 2.6 can be formulated as follows.

Theorem 2.8 ( [20]

). Let ε x , ε y , µ be passive and satisfy Assumption 2.1. Let additionally (1) ψ x (s) -1 , ψ y (s) -1 be passive and satisfy Assumption 2.1;

(2)

ψ x ε y = ψ y ε x ; (3) µ(s)ε y (s)ψ x (s) be passive. Then the sesquilinear form Ãc (u, v) = ψ x (s)ε y (s)A c (u, v) = ψ y (s)ε x (s)A c (u, v), u, v ∈ H 1 (R 2 ),
with A c (u, v) given by ( 14), is passive for any σ x ≥ 0 and σ y ≥ 0.

The Main Idea of the Energy Derivation for a PML System

There exists many ways to write a PML system for a given problem, see e.g. [START_REF] Bécache | On the analysis of Bérenger's perfectly matched layers for Maxwell's equations[END_REF] and references therein, or [START_REF] Bécache | On the analysis of perfectly matched layers for a class of dispersive media[END_REF]. To derive the energy conservation in an easy manner, it is crucial to introduce a very special set of auxiliary unknowns. One of such sets can be obtained by studying the coercivity of the corresponding sesquilinear form.

A Toy Example: a 2D Isotropic Non-dispersive Wave Equation (Second-Order Formulation)

Let us explain our idea based on the example of the isotropic non-dispersive 2D wave equation, to which the classical PMLs are applied in the x-direction. In the Laplace domain it reads (with f being its source term):

-1 + s -1 σ x -1 ∂ x 1 + s -1 σ x -1 ∂ x Ĥz -∂ 2 y Ĥz + s 2 Ĥz = f , s ∈ C + , (15)
where we assumed

H z | t=0 = 0, ∂ t H z | t=0 = 0. The corresponding variational formulation reads A σx ( Ĥz , v) = 1 + s -1 σ x -2 (∂ x Ĥz , ∂ x v) + (∂ y Ĥz , ∂ y v) + s 2 ( Ĥz , v) = ( f , v), Ĥz , v ∈ H 1 (R 2 ).
The above sesquilinear form coincides with (12) for a particular case ε x = ε y = µ = 1, where the change of variables of Theorem 2.4 is performed. To show the coercivity of the sesquilinear form for all s ∈ C + , one tests the above with v = s Ĥz (and then takes the real part), see Remark 2.5,

A σx ( Ĥz , s Ĥz ) = Re 1 + s -1 σ x -2 (∂ x Ĥz , s∂ x Ĥz ) + (∂ y Ĥz , s∂ y Ĥz ) + s(s Ĥz , s Ĥz ) = Re f , s Ĥz . ( 16
)
Testing with v = s Ĥz corresponds in the time domain to testing the corresponding equation with

∂ t H z (if H z | t=0 = 0),
which is exactly what one does in order to obtain the energy estimates for the problem without the PML. On the other hand, it is possible to obtain some energy identities directly from ( 16), using the Plancherel's equality. To proceed with this idea, let us rewrite the first term of ( 16) in a more convenient form:

1 + s -1 σ x -2 ∂ x Ĥz , s∂ x Ĥz = ∂ x Ĥz (1 + s -1 σ x ) 2 , s 1 + s -1 σ x 2 (1 + s -1 σ x ) 2 ∂ x Ĥz (17) = ∂ x Ĥz (1 + s -1 σ x ) 2 , s + 2σ x + σ 2 x s -1 (1 + s -1 σ x ) 2 ∂ x Ĥz = (s + 2σ x ) ∂ x Ĥz (1 + s -1 σ x ) 2 2 + σ 2 x s ∂ x Ĥz s (1 + s -1 σ x ) 2 2 . ( 18
)
Thus, ( 16) can be rewritten as follows:

Re (s + 2σ x ) ∂ x Ĥz (1 + s -1 σ x ) 2 2 + σ 2 x Re s ∂ x Ĥz s (1 + s -1 σ x ) 2 2 + Re s ∂ y Ĥz 2 + Re s s Ĥz 2 = Re f , s Ĥz . ( 19
)
Recall the Plancherel's identity (here η > 0) and its implication:

+∞ 0 e -2ηt (H z (t), v(t))dt = 1 2πi η+iR ( Ĥz (s), v(s))ds =⇒ +∞ 0 e -2ηt Re(H z (t), v(t))dt = 1 2πi η+iR Re( Ĥz (s), v(s))ds.
Before applying the last identity of the above to [START_REF] Huang | Mathematical analysis of a PML model obtained with stretched coordinates and its application to backward wave propagation in metamaterials[END_REF], let us introduce two auxiliary unknowns:

 = ∂ x Ĥz (1 + s -1 σ x ) 2 , Ĵ = ∂ x Ĥz s (1 + s -1 σ x ) 2 . ( 20
)
This allows us to obtain a formal energy identity (assuming zero initial conditions for j, J):

+∞ 0 e -2ηt (η + 2σ x ) j 2 + σ 2 x η J 2 + η ∂ y H z 2 + η ∂ t H z 2 dt = +∞ 0 e -2ηt (f, ∂ t H z )dt.
Importantly, this time-domain equality involves newly introduced unknowns (20). This suggests that when writing the PML system corresponding to [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I. Necessary and Sufficient Conditions of Stability[END_REF], it may be advantageous to introduce the unknowns [START_REF] Kachanovska | A note on the stability of PMLs for a class of anisotropic dispersive models in corners[END_REF]. Let us rewrite [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I. Necessary and Sufficient Conditions of Stability[END_REF] in time using this idea (now taking f = 0 but assuming non-zero initial conditions):

∂ 2 t H z -∂ x j -∂ 2 y H z = 0, ( 21a
)
∂ t j + 2σ x j + σ 2 x J = ∂ x ∂ t H z , (21b)
∂ t J = j. (21c)
Testing (21a) with ∂ t H z , we obtain 1 2

d dt ∂ t H z 2 + ∂ y H z 2 + (j, ∂ x ∂ t H z ) = 0, (j, ∂ x ∂ t H z ) (21b) = (j, ∂ t j + 2σ x j + σ 2 x J) (21c) = 1 2 d dt j 2 + 2σ x j 2 + σ 2 x (∂ t J, J) = 1 2 d dt j 2 + σ x J 2 + 2σ x j 2 .
Combining the above, we get the following energy decay law

d dt E = -2σ x j 2 , E = 1 2 ∂ t H z 2 + ∂ y H z 2 + j 2 + σ 2 x J 2 . ( 22
)
Indeed, when σ x = const, the formulation ( 21) is not perfectly matched. However, for σ x = const, it is equivalent to another, perfectly matched formulation.

Remaking the Toy Example: A First-Order Formulation

Our goal is to derive the PML system in the first order formulation, like the original system (5a-5g). On one hand, we would like to introduce unknowns that would correspond to s -1 , s -1 Ĵ (the energy of the first-order formulation for the wave equation coincides with the energy of the second-order formulation with u replaced by ∂ -1 t u). On the other hand, we would like to simplify the corresponding PML system of equations. First, the system of equations in the Laplace domain with the PML change of variables, and then comment on the introduction of the time-domain unknowns:

s Ĥz = - ∂ x Êy 1 + s -1 σ x + ∂ y Êx , (23a)
s Êy = - ∂ x Ĥz 1 + s -1 σ x , (23b)
s Êx = ∂ y Ĥz . (23c)
In this first-order formulation we suggest to introduce an auxiliary unknown E * y (it corresponds to -s -1 Ĵ), s.t.

E * y = - ∂ x Ĥz (s + σ x ) 2 (23b) = Êy s + σ x . ( 24
)
This choice is advantageous for two reasons:

• there is no need to introduce an unknown corresponding to s -1 , since it can be represented as a linear combination of the existing unknowns

s -1  = ∂ x Ĥz (s + σ x )(1 + s -1 σ x ) (23b) = - Êy 1 + s -1 σ x = -1 - σ x s + σ x Êy (24) = -Êy + σ x Ê * y .
• the above linear combination can replace the first term in the right-hand side of (23a), namely

1 + s -1 σ x -1 ∂ x Êy by ∂ x ( Êy -σ x Ê * y ).
Here we make use of the assumption σ x = const. Therefore, (23a-23c) can be rewritten in the time domain as follows:

∂ t H z = -∂ x E y + σ x ∂ x E * y + ∂ y E x , (25a)
∂ t E y + σ x E y = -∂ x H z , (25b)
∂ t E x = ∂ y H z , (25c)
∂ t E * y + σ x E * y = E y . (25d)
Let us show that the energy is consistent with [START_REF] Cockburn | Maxwell equations in polarizable media[END_REF]. We test the first equation above with H z , to obtain

(∂ t H z , H z ) + (∂ x E y -σ x ∂ x E * y , H z ) -(∂ y E x , H z ) = 0. ( 26
)
The second term in the above

(∂ x E y -σ x ∂ x E * y , H z ) = -(E y -σ x E * y , ∂ x H z ) (25b) = (E y -σ x E * y , ∂ t E y + σ x E y ) = (E y -σ x E * y , ∂ t (E y -σ x E * y ) + σ x ∂ t E * y + σ x E y ) (25d) = 1 2 d dt E y -σ x E * y 2 + (E y -σ x E * y , σ x ∂ t E * y + σ x (∂ t E * y + σ x E * y )) = (E y -σ x E * y , 2σ x ∂ t E * y ) + (E y -σ x E * y , σ 2 x E * y ).
Applying (25d) to each of the two terms above yields

(E y -σ x E * y , 2σ x ∂ t E * y ) = 2σ x E y -σ x E * y 2 , (E y -σ x E * y , σ 2 x E * y ) = (∂ t E * y , σ 2 x E * y ) = σ 2 x 2 d dt E * y 2 .
Finally, combining (26) and the latter expressions, we obtain:

d dt E = -2σ x E y -σ x E * y 2 , E = H z 2 + E x 2 + E y -σ x E * y 2 + σ x E * y 2 .
As expected, the above corresponds to [START_REF] Cockburn | Maxwell equations in polarizable media[END_REF], with all the unknowns substituted by their primitives in time.

Remarks on the Connection Between the Formulation (25a-25d) and the Bérenger's Split Formulation

Recall the Bérenger's split formulation for the 2D Maxwell's equations:

H z = H zx + H zy , (27a)
∂ t E y + σ x E y = -∂ x H z , (27b)
∂ t E x = ∂ y E x , (27c)
∂ t H zx + σ x H zx = -∂ x E y , (27d)
∂ t H zy = ∂ y E x . (27e)
To see how from (25a-25d) one can obtain the above system, we apply ∂ x to (25d), and use σ x = const,

∂ t ∂ x E * y + σ x ∂ x E * y = ∂ x E y .
Then we rewrite (25a) as

∂ t H z = -∂ x E y + σ x ∂ x E * y + ∂ y E x = -∂ t ∂ x E * y + ∂ y E x .
Finally, we can introduce the additional unknown H zy as in (27e). Thus, (25a-25d) can be rewritten as

∂ t H z = -∂ t ∂ x E * y + ∂ t H zy , ∂ t E y + σ x E y = -∂ x H z , ∂ t E x = ∂ y H z , ∂ t ∂ x E * y + σ x ∂ x E * y = ∂ x E y , ∂ t H zy = ∂ y E x .
Then, choosing the initial conditions ∂ x E * y t=0 = -H zx | t=0 , and

H z | t=0 = H zx | t=0 + H zy | t=0
, we can ensure that the solution of the above system coincides with the solution of (27a-27e), and, in particular, ∂ x E * y = -H zx .

2.4. Stability of the PML of Theorem 2.7, with ψ x = ε -1 y , ψ y = ε -1

x

In this section we extend the ideas of the construction of the PML systems of Section 2.3 to the PMLs for problems with the Lorentz dispersion (3). Let us write a PML system corresponding to (5a-5g) with the PML change of variables of Theorem 2.7. We perform the PML change of variables in both directions, however, assume that σ x , σ y ≥ 0. Setting in the resulting system σ y = 0 and discarding the auxiliary unknowns associated with the PML in the direction y, we will get the PML system in the direction x (similarly we can obtain the PML system in the direction y). Let us start with the first equation (5a), which in the Laplace domain with the PML change of variables of Theorem 2.7 and with the use of the constitutive relation s Dx = sε x Êx reads

s Dx = sε x Êx = 1 + σ y sε x -1 ∂ y Ĥz .
In the time-domain this is equivalent to

∂ t D x + σ y E x = ∂ y H z .
Remark 2.9. When deriving the PML system, one ignores all the source terms and initial conditions, since they are necessarily supported outside of the domain where the PML is used.

The equation (5b) can be treated similarly. To deal with (5c), we suggest to employ the splitting idea of Bérenger [START_REF] Bérenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF]. This is motivated by results of Section 2.3.3. Let us introduce split fields Ĥzx and Ĥzy so that they satisfy in the Laplace domain the following equations: Finally, the time-domain expression of the operators sε x (s), sε y (s) is done similarly to the system (5a-5g), via the introduction of the auxiliary unknowns. After the PML change of variables, (5a-5g) becomes:

∂ t D x + σ y E x = ∂ y H z , (29a)
∂ t D y + σ x E y = -∂ x H z , (29b)
∂ t B z + σ y H zy + σ x H zx = ∂ y E x -∂ x E y , (29c)
∂ t H zx + ny =0 ε y h x + σ x H zx = -∂ x E y , (29d)
∂ t H zy + nx =0 ε x h y + σ y H zy = ∂ y E x , (29e)
∂ t h α + ω 2 β b α = H zα , ∂ t b α = h α , = 0, . . . , n β , (α, β) ∈ {(x, y), (y, x)}, (29f)
which we equip with the equations (5d -5g). In the derivation of the above system we did not use the assumption σ x , σ y = const, hence it is perfectly matched. However, even in the case σ x , σ y = const, obtaining the energy estimates for it directly requires to solve additional difficulties, besides the dispersive behaviour. We will report how to resolve them elsewhere.

sε x Ê * x + σ y Ê * x = Êx , sε y Ê * y + σ x Ê * y = Êy , s Bz = ∂ y Êx -∂ x Êy -σ y ∂ y Ê * x + σ x ∂ x Ê * y . ( 30
)
The realization of the operators sε x , sε y is done similarly to (5a-5g). We thus obtain the following PML system:

∂ t D x + σ y E x = ∂ y H z , (31a)
∂ t D y + σ x E y = -∂ x H z , (31b)
∂ t B z + σ y ∂ y E * x -σ x ∂ x E * y = ∂ y E x -∂ x E y , (31c)
∂ t E * x + nx =0 ε x λ * x + σ y E * x = E x , (31d)
∂ t E * y + ny =0 ε y λ * y + σ x E * y = E y , (31e)
∂ t λ * α + ω 2 α p * α = E * α , ∂ t p * α = λ * α , = 0, . . . , n α , α ∈ {x, y}, (31f)
∂ t D α = ∂ t E α + nα =0 ε α λ α , (31g)
∂ t λ α + ω 2 α p α = E α , ∂ t p α = λ α , = 0, . . . , n α , α ∈ {x, y}, (31h)
∂ t B z = ∂ t H z + nµ =0 µ λ µ , (31i)
∂ t λ µ + ω 2 µ p µ = H z , ∂ t p µ = λ µ , = 0, . . . , n µ . (31j)
In the above we introduced new PML unknowns E * x , (λ * x , p * x ), = 0, . . . , n x and E * y , (λ * y , p * y ), = 0, . . . , n y . Remark 2.10. The equivalence of (29a-29f) and (31a-31j) for constant σ x , σ y , i.e. that with properly chosen initial data, the solution E, D, B z , H z of (29a-29f) solves (31a-31j) and vice versa, can be shown as in [START_REF] Bécache | On the analysis of Bérenger's perfectly matched layers for Maxwell's equations[END_REF].

Remark 2.11. In the system (31a-31j) the PML in the direction y is realized via the introduction of the auxiliary unknowns E * y , λ * y , p * y , = 0, . . . , n y . To obtain the PML in the direction x only, one sets σ y = 0. In this case (31e) and (31f) for α = y are decoupled from the rest of the equations and can be discarded.

Stability of the PML System (31a-31j)

The energy non-growth result is summarized in the following theorem. Theorem 2.12. An energy associated with (31a-31j) defined as

E = 1 2 E x -σ y E * x 2 + σ y E * x 2 + E y -σ x E * y 2 + σ x E * y 2 + H z 2 + E * x + E * y + E z , (32)
E * x = 1 2 nx =0 ε x λ x -σ y λ * x 2 + ω 2 x p x -σ y p * x 2 + σ y λ * x 2 + ω 2 x σ y p * x 2 , E * y = 1 2 ny =0 ε y λ y -σ x λ * y 2 + ω 2 y p y -σ x p * y 2 + σ x λ * y 2 + ω 2 y σ x p * y 2 , E z = 1 2 nµ =0 µ λ µ 2 + ω 2 µ p µ 2 ,
does not grow, provided σ x , σ y ≥ 0:

d dt E = -2σ y E x -σ y E * x 2 -2σ x E y -σ x E * y 2 .
Proof. First, we test (31c) with H z to obtain

(∂ t B z , H z ) -(∂ y E x -σ y ∂ y E * x , H z ) + (∂ x E y -σ x ∂ x E * y , H z ) = 0. ( 33
)
Let us consider each term of the above expression separately. The first term is exactly as in ( 7):

(∂ t B z , H z ) (7)
= d dt

1 2 H z 2 + E z .
The second term, after the integration by parts, becomes

-(∂ y E x -σ y ∂ y E * x , H z ) = (E x -σ y E * x , ∂ y H z ) (31a) = (E x -σ y E * x , ∂ t D x + σ y E x ) = (E x -σ y E * x , ∂ t D x ) + σ y (E x -σ y E * x , E x -σ y E * x + σ y E * x ) = (E x -σ y E * x , ∂ t D x ) + σ y E x -σ y E * x 2 + σ 2 y (E x -σ y E * x , E * x ) = T 1 + σ y E x -σ y E * x 2 + T 2 . (34)
Let us first consider

T 1 = (E x -σ y E * x , ∂ t D x ) (31g) = E x -σ y E * x , ∂ t E x + nx =0 ε x λ x = (E x -σ y E * x , ∂ t (E x -σ y E * x ) + σ y ∂ t E * x ) + E x -σ y E * x , nx =0 ε x λ x (31d) = 1 2 d dt E x -σ y E * x 2 + σ y E x -σ y E * x , E x -σ y E * x - nx =0 ε x λ * x + E x -σ y E * x , nx =0 ε x λ x = 1 2 d dt E x -σ y E * x 2 + σ y E x -σ y E * x 2 + nx =0 ε x (E x -σ y E * x , λ x -σ y λ * x ) .
The last term in the above corresponds to a weighted sum of norms of a linear combination of auxiliary unknowns:

nx =0 ε x (E x -σ y E * x , λ x -σ y λ * x ) (31h,31f ) = nx =0 ε x ∂ t (λ x -σ y λ * x ) + ω 2 x (p x -σ y p * x ), λ x -σ y λ * x (31h,31f ) = 1 2 d dt nx =0 ε x λ x -σ y λ * x 2 + nx =0 ε x ω 2 x (p x -σ y p * x , ∂ t (p x -σ y p * x )) = 1 2 d dt nx =0 ε x λ x -σ y λ * x 2 + ω 2 x p x -σ y p * x 2 . ( 35
)
Summarizing the above,

T 1 = 1 2 d dt E x -σ y E * x 2 + σ y E x -σ y E * x 2 + 1 2 d dt nx =0 ε x λ x -σ y λ * x 2 + ω 2 x p x -σ y p * x 2 .
It remains to obtain an explicit energy-like expression for the term T 2 in (34):

T 2 = σ 2 y (E x -σ y E * x , E * x ) (31d) = σ 2 y ∂ t E * x + nx =0 ε x λ * x , E * x (31f ) = σ 2 y 2 d dt E * x 2 + nx =0 ε x λ * x 2 + ω 2 x p * x 2 ,
where the derivation of the latter identity is done similarly as e.g. in [START_REF] Cummer | Perfectly matched layer behavior in negative refractive index materials[END_REF] or (35). Summarizing the above, we obtain the following expression for the second term in (33):

-(∂ y E x -σ y ∂ y E * x , H z ) = 1 2 d dt E x -σ y E * x 2 + σ y E * x 2 + 2σ y E x -σ y E * x 2 + d dt E * x .
Repeating almost verbatim the above computations, one can show that the third term in (33) reads:

(∂ x E y -σ x ∂ x E * y , H z ) = 1 2 d dt E y -σ x E * y 2 + σ x E * y 2 + 2σ x E y -σ x E * y 2 + d dt E * y .
Summing up all the terms of (33) we obtain the desired statement.

Remark 2.13. The above result can be generalized to the case of Lorentz materials with dissipation, i.e.

ε α (s) = 1 + nα =0 ε α s 2 + 2ν α s + ω 2 α , ε α > 0, ν α ≥ 0, ω α ≥ 0, = 0, . . . , n α , α ∈ {x, y}, µ(s) = 1 + nµ =0 µ s 2 + 2ν µ s + ω 2 µ , µ > 0, ν µ ≥ 0, ω µ ≥ 0, = 0, . . . , n µ .
In this case the choice of ψ x (s) = ε y (s) -1 , ψ y = ε x (s) -1 would lead to a stable system.

Remark 2.14. With the help of the Young's inequality, one can show that the norm of the field E is controlled with the help of the stability result of Theorem 2.12. E.g. consider the first term in (32):

E x -σ y E * x 2 + σ y E * x 2 ≥ E x 2 + 2 σ y E * x 2 -2 E x σ y E * x ≥ E x 2 + 2 σ y E * x 2 - 2 3 E x - 3 2 σ y E * x = 1 3 E x 2 + 1 2 σ y E * x 2 .
2.5. Stability of the PML of Theorem 2.8, for General ψ x (s), ψ y (s)

An exact expression of the corresponding to the PML with the change of variables as in Theorem 2.8 is slightly more complicated. It is easier to express the energy of this formulation in terms of new, 'effective' electric and magnetic fields, rather than the original unknowns. However, this requires the reformulation of the corresponding system without the PML. Similar ideas are used in [START_REF] Cassier | On properties of passive systems In preparation[END_REF].

A Preliminary Reformulation of the System (5a-5g)

Let us rewrite the Maxwell's equations in a more convenient form, equivalent to the original Lorentz system (5a-5g), which, however, would allow to handle easier the energy of the corresponding PML system. In particular, notice that the PML change of variables of Theorem 2.8 results in a passivity of a special scaled sesquilinear form. This form can be viewed as a sesquilinear form resulting from the PML of Theorem 2.7 applied to the problem with (passive) coefficients ε e y = ψ -1 x , ε e x = ε x ψ -1 x ε -1 y = ψ -1 y and µ e = ε y ψ x µ. Let us consider the solution (E, H z ) of the system (5a-5g) with the initial conditions (E, H z )| t=0 = (E 0 , H z0 ). As discussed before the derivation of (5a-5g), we choose D 0 = E 0 , B z0 = H z0 and zero initial conditions for the rest of unknowns. Then, in the Laplace domain, the system (5a-5g) reads

sε x (s) Êx -E x0 = ∂ y Ĥz , sε y (s) Êy -E y0 = -∂ x Ĥz , sµ(s) Ĥz -H z0 = ∂ y Êx -∂ x Êy .
The above can be split into two systems:

sε x (s) Ê(1) x = ∂ y Ĥ(1) z , sε y (s) Ê(1) y = -∂ x Ĥ(1) z , sµ(s) Ĥ(1) z -H z0 = ∂ y Ê(1) x -∂ x Ê(1) y , (36)
and

sε x (s) Ê(2) x -E x0 = ∂ y Ĥ(2) z , sε y (s) Ê(2) y -E y0 = -∂ x Ĥ(2) z , sµ(s) Ĥ(2) z = ∂ y Ê(2) x -∂ x Ê(2) y . ( 37
)
Due to linearity, E = E (1) + E (2) ,

H z = H (1) z + H (2)
z . Let us consider the first equation of ( 36), which we multiply by ψ x (s) -1 ε y (s) -1 , to obtain

sε x (s)ψ x (s) -1 ε y (s) -1 Ê(1) x = ∂ y ψ -1 x (s)ε y (s) -1 Ĥ(1) z = ∂ y Ĥe z , Ĥe z = ψ -1 x (s)ε y (s) -1 Ĥ(1) z .
Here the index 'e' stands for 'effective'. Setting ε e x (s) = ε x (s)ψ x (s) -1 ε y (s) -1 , we rewrite the above as

sε e x (s) Ê(1) x = ∂ y Ĥe z . (38)
Next, we repeat the procedure with the second equation of (36), defining ε e y = ψ x (s) -1 :

sε e y (s) Ê(1) y = -∂ x Ĥe z . ( 39
)
Finally, for the third equation of (36), it suffices to introduce µ e (s) = ψ x (s)ε y (s)µ(s) and rewrite

sµ(s)ψ x (s)ε y (s)ψ x (s) -1 ε y (s) -1 Ĥ(1) z -H z0 = sµ e (s) Ĥe z -H z0 = ∂ y Ê(1) x -∂ x Ê(1) y . (40)
Therefore, the system (36) is equivalent to the system of equations (38, 39, 40), provided that the initial conditions are chosen as H e z (0) = H z0 . Similarly the system (37) can be rewritten:

sε e x (s) Êe x -E x0 = ∂ y Ĥ(2) z , sε e y Êe y -E y0 = -∂ x Ĥ(2) z , sµ e (s) Ĥ(2) z = ∂ y Êe x -∂ x Êe y , with Êe x = ε y (s)ψ x (s) Ê(2) x , Êe y = ε y (s)ψ x (s) Ê(2)
y . Since the above two systems have similar structures, from now on we will concentrated only on one of them, namely (38,39,40). Thanks to Theorem 2.8, the quantities ε e

x , ε e y , µ e are passive. Therefore, they are of the following form, see Theorem 2.2:

ε e α (s) = 1 + n e α =0 ε e α s 2 + (ω e α ) 2 , ε e α > 0, ω e α ∈ R, = 0, . . . , n e α , α ∈ {x, y}, µ(s) = 1 + n e µ =0 µ s 2 + (ω e µ ) 2
, µ e > 0, ω e µ ∈ R, = 0, . . . , n e µ .

The PML System and Its Stability

The application of the PML of Theorem 2.8 to (41) results in the system that has the same structure as (31a-31j). Indeed, ψ x = (ε e y ) -1 and ψ y = (ε e x ) -1 . We can immediately write the corresponding PML system:

∂ t D e x + σ y E (1) x = ∂ y H e z , ∂ t D e y + σ x E (1) y = -∂ x H e z , ∂ t B e z = ∂ y E (1) x -σ y ∂ y E * x -∂ x E (1) y + σ x ∂ x E * y , ∂ t E * x + n e x =0 ε e x λ * x + σ y E * x = E (1) x , ∂ t E * y + n e y =0 ε e y λ * y + σ x E * y = E (1) y , ∂ t λ * α + (ω e α ) 2 p e α = E (1) α , ∂ t p * α = λ * α , = 0, . . . , n α , α ∈ {x, y}, ∂ t D e α = ∂ t E α + n e α =0 ε e α λ e α , ∂ t λ e α + (ω e α ) 2 p e α = E (1) α , ∂ t p e α = λ e α , = 0, . . . n e α , α ∈ {x, y}, ∂ t B e z = ∂ t H e z + n e µ =0 µ e λ e µ , ∂ t λ e µ + (ω e µ ) 2 p e µ = H e z , ∂ t p e µ = λ e µ , = 0, . . . n e µ . (44)
As before, the physical field H

(1) z can be recovered using (42-43). An energy associated with the above system can be derived as in Theorem 2.12, and the bound on H

(1) z as in Proposition 2.17.

Theorem 2.19. An energy associated with (44) defined as

E = 1 2 E (1) x -σ y E * x 2 + σ y E * x 2 + E (1) y -σ x E * y 2 + σ x E * y 2 + H e z 2 + E e, * x + E e, * y + E e z , (45)
E e, * x = 1 2 nx =0 ε e x λ e x -σ y λ * x 2 + (ω e x ) 2 p e x -σ y p * x 2 + σ y λ * x 2 + (ω e x ) 2 σ y p * x 2 , E e, * y = 1 2 ny =0 ε e y λ e y -σ x λ * y 2 + (ω e y ) 2 p e y -σ x p * y 2 + σ x λ * y 2 + (ω e y ) 2 σ x p * y 2 , E e z = 1 2 nµ =0 µ e λ e µ 2 + (ω e µ ) 2 p e µ 2
does not grow, provided σ x , σ y ≥ 0:

d dt E = -2σ y E (1) x -σ y E * x 2 -2σ x E (1) y -σ x E * y 2 .
Moreover, for the solution of (42-43) coupled with (44), it holds

Ẽz (t) := 1 2 H (1) z (t) -H e z (t) 2 + nµ =0 µ λ (1)
µ (t) 2 + ω 2 µ p (1)
µ (t) 2 ≤ 2 Ẽz (0) + CE(0)t 2 , t ≥ 0,
where C > 0 is constant.

General Passive Materials: Stability with Energy Techniques

In this section we summarize some results on the materials whose dielectric permittivity and magnetic permeability do not satisfy the conditions of Section 2. One of the examples is given by Warburg's law, when the dielectric permittivity ε(s) = 1+ 1+a √ s s(1+b √ s) , a, b > 0, see e.g. [START_REF] Bamberger | Numerical solution of Maxwell's equations in a conductive and polarizable medium[END_REF][START_REF] Cockburn | Maxwell equations in polarizable media[END_REF] for the corresponding numerical method. The construction of the PML for this types of systems is done as before, via a special change of variables [START_REF] Zhao | Gt-pml: generalized theory of perfectly matched layers and its application to the reflectionless truncation of finite-difference time-domain grids[END_REF]. In this section we will show the stability of the following PML.

Theorem 3.1 ( [5]

). Let ε x , ε y , µ be passive. Then the sesquilinear form [START_REF] Cassier | On properties of passive systems In preparation[END_REF] with ψ x (s) = ε y (s) -1 and ψ y (s) = ε x (s) -1 is passive for any σ x ≥ 0 and σ y ≥ 0.

The stability of the corresponding system without the PML was shown in [START_REF] Cassier | On properties of passive systems In preparation[END_REF] with the help of a Herglotz-Nevanlinna representation formula. This integral representation allows to write any passive system (even dissipative) in the form (5a-5g), and thus, associate with any system an energy that is conserved (while at the same time, there may exist another energy that decays). A conservative extension for a general class of dissipative systems had been constructed, though in a slightly different form, in the seminal work by Figotin and Schenker in [START_REF] Figotin | Spectral theory of time dispersive and dissipative systems[END_REF]. Such an approximation of an arbitrary material with the help of Lorentz materials will be used to construct a PML system similar to (31a-31j), see Section 3.2.

Time-Domain System without PMLs

The results of this section are due to [START_REF] Cassier | On properties of passive systems In preparation[END_REF], where it is suggested that passive functions corresponding to the Laplace transform of real-valued in the time domain distributions (i.e., in the Laplace domain, f (s) = f (s), s ∈ C + ) satisfy the property formulated below. The following lemma is nothing more but a version of the Herglotz-Nevanlinna representation for such functions.

f (s) = a + ∞ -∞ dν f (ξ) s 2 + ξ 2 , a ≥ 0. ( 46
)
Here a = lim sr→+∞ f (s r ), and the measure

ν f satisfies +∞ -∞ dν f (ξ) 1+ξ 2 < ∞.
Proof. Notice that f (s) is passive if and only if Re (-iωf (-iω)) > 0 for Im ω > 0, or Im(ωf (-iω)) > 0 for Im ω > 0. Hence, ωf (-iω) is Herglotz, and one can apply the results of [14, Section 4.1] to f (-iω).

For instance, to obtain the Lorentz magnetic permeability (3), we can take a sum of δ-measures ν µ (ξ) = nµ =0 µ δ(ξ -ω µ ). Without loss of generality, let us assume that [START_REF] Welters | Speed-of-light limitations in passive linear media[END_REF][START_REF] Cassier | On properties of passive systems In preparation[END_REF]

ε x (s) → 1, ε y (s) → 1, µ(s) → 1, as Re s → +∞.
Then Lemma 3.2 yields (where we take into account ε x (s r ), ε y (s r ), µ(s r ) → 1 as s r → +∞)

ε α (s) = 1 + +∞ -∞ dν α s 2 + ξ 2 , α ∈ {x, y}, µ(s) = 1 + +∞ -∞ dν µ s 2 + ξ 2 , s ∈ C + .
These identities generalize the corresponding definitions for the Lorentz materials (3), with the exception that the weighted sums are substituted by integrals with Borel measures. Our goal is to write the system (1) in the time domain, using the above representation of ε, µ. Let us recall from [START_REF] Cassier | On properties of passive systems In preparation[END_REF] how the equation s Dx = sε x (s) Êx can be written in the time domain. Let us introduce λx (ξ) and px (ξ), which map ξ ∈ R into a set of analytic functions from s ∈ C + into L 2 (R 2 ), so that they satisfy the following identities:

s λx (ξ) + ξ 2 px (ξ) = Êx , sp x (ξ) = λx (ξ), ξ ∈ R.
Then, in the time domain, we can rewrite (formally) s Dx = sε Êx as:

∂ t D x = ∂ t E x + ∞ -∞ λ x (ξ)dν x (ξ), ∂ t λ x (ξ) + ξ 2 p x (ξ) = E x , ∂ t p x (ξ) = λ x (ξ), ξ ∈ R. (47)
If the measure ν x has a density f νx , we can define the quantities λ x (ξ), p x (ξ) only for ξ ∈ supp f νx (ξ) (this is the case for the Lorentz materials, where the density has a discrete support). Thus, extending the above idea to the rest of the constitutive relations, we write (1) in the form (5a-5g):

∂ t D x = ∂ y H z , ∂ t D y = -∂ x H z , ∂ t B z = ∂ y E x -∂ x E y , ∂ t D α = ∂ t E α + ∞ -∞ λ α (ξ)dν α (ξ), ∂ t λ α + ξ 2 p α = E α , ∂ t p α = λ α , ξ ∈ R, α ∈ {x, y}, ∂ t B z = ∂ t H z + ∞ -∞ λ µ (ξ)dν µ (ξ), ∂ t λ µ + ξ 2 p µ = E µ , ∂ t p µ = λ µ , ξ ∈ R. (48)
Here λ α (ξ), p α (ξ), α ∈ {x, y, µ}, are functions from R to C R; L 2 (R 2 ) . An associated energy is conserved [START_REF] Cassier | On properties of passive systems In preparation[END_REF].

Proposition 3.3. An energy E associated with (48), defined below, satisfies d dt E = 0. Here

E = 1 2 E x 2 + E y 2 + H z 2 + E x + E y + E z , E α = 1 2 ∞ -∞ λ α 2 + ξ 2 p α 2 dν α (ξ), α ∈ {x, y}, E z = 1 2 ∞ -∞ λ µ 2 + ξ 2 p µ 2 dν µ (ξ).

The PML System and Its Stability

Let us perform the PML change of variables for the system (48) as per Theorem 3.1. This results in the following analogue of the system (31a-31j) (where again the difference is that the sums are substituted by integrals). The derivation of this system follows the derivation of (31a-31j), combined with the use of (47).

∂ t D x + σ y E x = ∂ y H z , ∂ t D y + σ x E y = -∂ x H z , ∂ t B z + σ y ∂ y E * x -σ x ∂ x E * y = ∂ y E x -∂ x E y , ∂ t D α = ∂ t E α + ∞ -∞ λ α (ξ)dν α (ξ), ∂ t λ α + ξ 2 p α = E α , ∂ t p α = λ α , ξ ∈ R, α ∈ {x, y}, ∂ t E * x + ∞ -∞ λ * x (ξ)dν x (ξ) + σ y E * x = E x , ∂ t E * y + ∞ -∞ λ * y (ξ)dν y (ξ) + σ x E * y = E y , ∂ t λ * α + ξ 2 p * α = E * α , ∂ t p * α = λ * α , ξ ∈ R, α ∈ {x, y}, ∂ t B z = ∂ t H z + ∞ -∞ λ µ (ξ)dν µ (ξ), ∂ t λ µ + ξ 2 p µ = E µ , ∂ t p µ = λ µ , ξ ∈ R. (49)
We can formulate the following analogue of Theorem 2.12, with the proof almost verbatim the same.

Theorem 3.4. An energy E associated with (49), defined as

E = 1 2 E x -σ y E * x 2 + σ y E * x 2 + E y -σ x E * y 2 + σ x E * y 2 + H z 2 + E x + E y + E z , E x = 1 2   +∞ -∞ λ x -σ y λ * x 2 + ξ 2 p x -σ y p * x 2 + σ y λ * x 2 + ξ 2 σ y p * x 2 dν x   , E y = 1 2   +∞ -∞ λ y -σ x λ * y 2 + ξ 2 p y -σ x p * y 2 + σ x λ * y 2 + ξ 2 σ x p * y 2 dν y   , E z = 1 2 +∞ -∞ λ µ 2 + ξ 2 p µ 2 dν µ ,
does not grow, provided any σ x , σ y ≥ 0:

d dt E = -2σ x E y -σ x E * y 2 -2σ y E x -σ y E * x 2 .

Stability of PMLs for a Non-constant Absorption Parameter for the 1D Dispersive System

While we have proven the stability of the new PMLs under the assumption σ x = const, obtaining the energy estimates for σ x = const remains an open question. It seems that showing the stability for varying σ x is not too trivial even for 1D dispersive models (being almost obvious for classical PMLs for a 1D non-dispersive wave equation). The goal of this section is to prove the stability of the PMLs, that were proposed and analyzed in [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF] and in this work for σ x ≡ const, in the case when σ x = const for a 1D dispersive system of equations.

All over this section, without loss of generality, we will assume that the permittivity and the permeability are Lorentz [START_REF] Bécache | Stability of perfectly matched layers, group velocities and anisotropic waves[END_REF]. Moreover, we will consider only the PMLs suggested in Theorem 2.7. However, all the results can be extended to more general ε x , ε y , µ, and other PMLs (e.g. given in Theorem 3.1).

The 1D analogue of the dispersive system of equations (5a-5g) in the Laplace domain reads:

sε y (s) Êy = -∂ x Ĥz , sµ(s) Ĥz = -∂ x Êy . (50)
The time-domain equivalent of the above is given by the system of equations

∂ t D y = -∂ x H z , ∂ t B z = -∂ x E y , ∂ t D y = ∂ t E y + ny =0 ε y λ y , ∂ t λ y + ω 2 y p y = E y , ∂ t p y = λ y , = 0, . . . , n y , ∂ t B z = ∂ t H z + ny =0 µ λ µ , ∂ t λ µ + ω 2 µ p µ = H z , ∂ t p µ = λ µ , = 0, . . . , n µ . ( 51
)
This section is organized as follows. First, we consider a simplified case ε y = µ, for which the derivation of the energy estimates can be done as for the 1D non-dispersive wave equation. Next, we will concentrate on a general case ε y = µ, and show how to deal with it by rewriting the PML system of equations in a different form and deriving the energy estimates for this new system.

4.1. Stability of PMLs with ψ x = ε -1 y for a Special Case ε y = µ

Let us assume that ε y (s) = µ(s), and apply the PML change of variables ( 9) with ψ x (s) = ε y (s) -1 to (50):

sε y (s) Êy = -1 + σ x sε y (s) -1 ∂ x Ĥz , sε y (s) Ĥz = -1 + σ x sε y (s) -1 ∂ x Êy .
This results in the following system: sε y (s) Êy + σ x Êy = -∂ x Ĥz , sε y (s) Ĥz + σ x Ĥz = -∂ x Êy , x ∈ R.

In the time domain, the above reads:

∂ t D y + σ x E y = -∂ x H z , ∂ t B z + σ x H z = -∂ x E y , (52a)
∂ t D y = ∂ t E y + ny =0 ε y λ y , ∂ t λ y + ω 2 y p y = E y , ∂ t p y = λ y , (52b)
∂ t B z = ∂ t H z + ny =0 ε y λ z , ∂ t λ z + ω 2 y p z = H z , ∂ t p z = λ z , = 0, . . . , n y . (52c)
The energy nongrowth result follows trivially.

Proposition 4.1. An energy E associated with the system (52a-52c) defined as

E = E y 2 + H z 2 + E y + E z , E y = ny =0 ε y λ y 2 + ω 2 y p y 2 , E z = ny =0 ε y λ z 2 + ω 2 y p z 2 ,
does not grow, provided any non-negative σ x ∈ L ∞ (R):

d dt E = - +∞ 0 |E y | 2 + |H z | 2 σ x (x)dx.
Proof. The energy identity is deduced by testing the first equation of (52a) with E y , and the second equation of (52a) with H z , and summing up the obtained identities. The details are left to the reader.

4.2. Stability of PMLs with ψ x = ε -1 y for a General Case when ε y = µ

The case ε y = µ is less trivial. Applying the PML change of variables [START_REF] Zhao | Gt-pml: generalized theory of perfectly matched layers and its application to the reflectionless truncation of finite-difference time-domain grids[END_REF] with ψ x = ε -1 y to (50), we can rewrite the corresponding system in the second order formulation, cf. [START_REF] Hagstrom | Automatic symmetrization and energy estimates using local operators for partial differential equations[END_REF],

∂ x ε y (s) -1 1 + σ x (x) sε y (s) -1 ∂ x Ĥz -s 2 µ(s) 1 + σ x (x) sε y (s) Ĥz = 0, x ∈ R,
with σ(x) ≡ 0 for x < 0. Following the ideas of Section 2.3, we will show the coercivity of the corresponding sesquilinear form for all s ∈ C + , namely

A( Ĥz , v) = ε y (s) -1 1 + σ x (x)(sε y (s)) -1 ∂ x Ĥz , ∂ x Ĥz + s 2 µ(s) 1 + σ x (x)(sε y (s)) -1 Ĥz , v , Ĥz , v ∈ H 1 (R). ( 53
)
So far, we were able to show only a very special estimate, derived by testing (53) with v = s Ĥz and using a proper scaling. Defining the branch cut of the square root as R ≤0 , we multiply the above with sεy(s) sµ(s) and test the result with s Ĥz . This results in the following:

Re sε y sµ A Ĥz , s Ĥz = Re   s sεy sµ ε -1 y 1 + σ x (x)(sε y ) -1 |1 + σ(x)(sε y ) -1 | 2 ∂ x Ĥz , ∂ x Ĥz   + Re s sε y sµ s 2 µ 1 + σ x (x) sε y Ĥz , Ĥz .
We use the notation sεy sµ to underline that the real parts of the numerator and denominator are positive in C + , and hence sεy sµ does not cross the branch cut of the square root when s ∈ C + . In this case we can use the identity √ z 1 z 2 = √ z 1 √ z 2 for z 1 , z 2 ∈ C + , and rewrite the above as follows:

Re sε y sµ A Ĥz , s Ĥz = Re   |s| 2 ((sµ)(sε y )) -1 2 ∂ x Ĥz 1 + σ x (x)(sε y ) -1 2   + Re   |ε y | -2 ((sµ) -1 (sε y )) 1 2 R σ x (x) Ĥz 1 + σ x (x)(sε y ) -1 2 dx   + Re   |s| 2 ((sε y )(sµ)) 1 2 Ĥz 2 + |s| 2 ((sε y ) -1 (sµ)) 1 2 R σ x (x) Ĥz 2 dx   . ( 54
)
Thanks to Re (sµ)(sε y ) > 0 and Re (sµ) -1 (sε y ) > 0 for s ∈ C + we can see that the real part of every term in the above is positive. This implies the coercivity of the scaled sesquilinear form A( Ĥz , v). Moreover, the apparatus of the work [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF] can be used to show the well-posedness and the stability of the resulting PML system in the time domain. However, to perform the energy analysis, it is necessary to rewrite the original system of 1D equations in a very special form. This is the subject of the following section.

Similarly, the energy conservation is a corollary of Proposition 3.3, since (56) is a particular case of (48).

The PML System and Its Stability

Let us now apply the PMLs to the time-domain system (56), or, in the Laplace domain (55). Performing the change of variables [START_REF] Zhao | Gt-pml: generalized theory of perfectly matched layers and its application to the reflectionless truncation of finite-difference time-domain grids[END_REF] with ψ x = ε -1 y results in the following system (recall that H z0 is supported outside of the perfectly matched layer) inside the perfectly matched layer:

(sµ)(sε y ) Ê( 1

-H z0 = -∂ x Ê (1) y .
(58)

To write the above in the time domain, we again apply Lemma 3.2 to the passive function s -1 (sε y ) -1 (sµ), with the use of lim sr→+∞ s -1 (sε y ) -1 (sµ) = 0. This results in the following representation:

s -1 (sε y ) -1 (sµ) = +∞ -∞ dν * (ξ) s 2 + ξ 2 . ( 59
)
With the help of the above we can write the system (58) in the time domain. We make use of s(s 2 +ξ 2 ) -1 Ê(

In order to recover H

z , we use (57). The following stability result holds for the above system. First of all, in the above we can clearly see the role of the condition σ x (x) ≥ 0: it is required for the stability of the PML system. Unlike the results of previous sections for σ x ≡ const, the above result shows the conservation of a certain energy, rather than its decay. In practice this may appear non-optimal.

The non-optimality becomes clear when one considers the case µ(s) = ε y (s), for which one can obtain a finer energy decay result in Section 4.1, but for which the results of Theorem 4.4 still remain valid. For instance, in this case, (59) degenerates to a very special representation of the symbol s -1 , namely s

-1 = π -1 +∞ -∞ dξ s 2 +ξ 2 , s ∈ C + .
In this case, on one hand (sε y ) -1 (sµ) Êy = Êy in the time domain corresponds to E y , and on the other hand it can be represented as π -1 +∞ -∞ λ e y dξ, see (60). Nonetheless, the energy conservation result obtained in this section is not surprising, and is a consequence of the fact that the system (60) can be viewed as a conservative extension of an equivalent PML system, see [START_REF] Cassier | On properties of passive systems In preparation[END_REF].

As for the original unknown H

z , the corresponding stability result mimics Proposition 2.17. Proposition 4.5. For the solution of the system (60) coupled with (57), the following holds true: Proof. See the proof of Proposition 2.17 in Appendix A.

Ẽz
The above result indicates a possible linear growth of the field H

z . At the moment it is not clear whether this estimate on the behaviour of the field H

(1) z is optimal, cf. Remark 2.18.

Conclusions and Open Questions

In this work we have examined the stability of the PMLs of [START_REF] Bécache | Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability[END_REF] for dispersive models with the help of the energy techniques. We exploit the coercivity of the corresponding sesquilinear form in the Laplace domain, and based on its analysis, we show how a set of auxiliary PML unknowns can be introduced to simplify the derivation of the energy estimates. This technique allows to obtain the stability result for a constant absorption parameter in 2D, and can be extended to the analysis of the PML with a non-constant absorption parameter in 1D.

There are many open questions remaining, among them the derivation of the energy estimates for a nonconstant absorption parameter in two dimensions, even for classical PMLs and non-dispersive case. Another interesting question is whether the technique of this work can be extended to analyze the stability of classical PMLs in 3D. To our knowledge, no energy estimates are known in this case even for a constant absorption parameter. This constitutes a subject of a future research.

Definition 1 . 1 .

 11 A function c : C + → C is passive if it is analytic in C + and satisfies Re (sc(s)) > 0 there.

  s) are passive. Informally, Passivity of a Sesquilinear Form =⇒ Time-Domain Stability of the Corresponding System.

Theorem 2 . 2 .

 22 A function r(s) satisfying Assumption 2.1 is passive if and only if r(s) = 1 + n =0 r s 2 +ω 2 , where ω ∈ R and r > 0 for all = 0, . . . , n.

2. 4 . 1 .

 41 Construction of a PML System Corresponding to Theorem 2.7, with ψ x = ε -1 y , ψ y = ε -1x

  sε x (s) Ĥzy = ∂ y Êx , sε y (s) Ĥzx = -∂ x Êy , s Bz = ∂ y Êx -∂ x Êy = sε x (s) Ĥzy + sε y (s) Ĥzx . (28) Applying the PML change of variables of Theorem 2.7 to the above equations, we obtain the following: sε x Ĥzy + σ y Ĥzy = ∂ y Êx , sε y Ĥzx + σ x Ĥzx = -∂ x Êy , s Bz = sε x (s) Ĥzy + sε y (s) Ĥzx = ∂ y Êx -∂ x Êy -σ y Ĥzy -σ x Ĥzx .

Lemma 3 . 2 (

 32 Herglotz-Nevanlinna Representation). The function f (s), which satisfies f (s) = f (s), with s ∈ C + , is passive if and only if there exists a non-decreasing function of a bounded variation ν f (ξ), for which the Stiltjes integral ∞ -∞ dν f (ξ) 1+ξ 2 is finite and f (s) is represented as the following Stiltjes integral:

Proposition 4 . 3 . 2 + ξ 2 p e α 2

 4322 An energy E associated with (56), defined below, satisfies d dt E = 0. dν(ξ), α ∈ {x, y}.

Theorem 4 . 4 .2 + H e z 2 +| 2 +

 4422 Let σ x (x) ∈ L ∞ (R). An energy of the PML system (60) defined byE = E (1) y E y + E z + E * y + E * z , ξ 2 |p e α | 2 σ x (x)dxdν * (ξ), α ∈ {x, z},remains constant: d dt E = 0. Proof. The result is obtained by testing the first equation of (60) with E(1) y , the second equation with H e z and summing up the two obtained identities. The details are left to the reader.

  Itdiffers from (29a-29f) only by treatment of the equation for B z (5c). More precisely, we introduce Ê *

	Ĥzy = ∂ y	Ê * x , Ĥzx = -∂ x	Ê *

To simplify the discussion, we suggest to consider another formulation, equivalent to (29a-29f) for constant absorption parameters (see Remark 2.10 or Section 2.3.3). x , Ê * y , s.t. y , cf. (28). Then, formally, using σ x = const, σ y = const,

  )

	y	1 +	σ x (x) sε y	= -∂ x	Ĥe z ,	(sµ)(sε y ) Ĥe z 1 +	σ x (x) sε y	= -∂ x	Ê(1)

y , or, alternatively, in the whole domain:

(sµ)(sε y ) Ê(1) y + σ x (x) (sµ)(sε y ) -1 Ê(1) y = -∂ x Ĥe z ,

(sµ)

(sε y ) Ĥe z + σ x (x) (sµ)(sε y ) -1 Ĥe z

Acknowledgements

I am grateful to Patrick Joly for many useful suggestions and remarks on this work.

* This work was supported by a public grant as part of the Investissement d'avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, as well as a co-financing program PRESTIGE 1 Laboratoire Poems (UMR 7231

The system without the PML mimics the Lorentz system (5a-5g): (41)

Then the following result holds true.

Proposition 2.15. Let (E (1) , H e z , B e z ) solve the system (41) with the initial conditions chosen as H e z | t=0 = B e z | t=0 = H z0 and as zero for the rest of unknowns. Let E, H z be the solution to the system (5a-5g) with the initial conditions chosen as H z | t=0 = B z | t=0 = H z0 and as zero for the rest of unknowns. Then E (1)

z , we can use the identity B z = B e z , or sµ(s) Ĥ

= sµ e (s) Ĥe z , which gives in the time domain:

µ , = 0, . . . n µ .

(43)

With the system (41) we can associate a conservation of a certain energy.

Theorem 2.16. An energy E of (41), defined below, satisfies d dt E = 0. Here

x

As for the field H

z , we were not able to deduce the conservation of its norm from the equations (42-43). However, the following stability result which will be of use later.

Proposition 2.17. The following bound holds for the solution of the system (42-43) coupled with (41):

where E(t) is defined in Theorem 2.16 and C > 0 is constant.

Proof. See Appendix A.

Remark 2.18. The result of (at most) linear growth of the field H

(1) z is non-optimal, at least when the initial conditions are chosen as in Proposition 2.15. This follows from the fact that H

A Preliminary Reformulation of the System (51)

For constant σ x = const, the estimate (54) would imply the passivity of the scaled sesquilinear form sεy sµ A( Ĥz , v), cf. Theorem 2.8. This suggests that to write the time-domain PML system for which the energy estimates are derived in an easy manner, it may be first necessary to reformulate the original system without the PML (51) in an equivalent form, making use of the ideas of Section 2.5.1.

As in Section 2.5.1, we will make use of the linearity, and rewrite the corresponding system in the time domain, assuming the initial conditions (E y , H z )| t=0 = (0, H z0 ). The case (E y0 , 0) can be treated similarly. We thus start with (50) with the initial conditions (0, H z0 ), using the same notation as in Section 2.5.1:

and then define Ĥe

Since for µ = ε y , the function (sµ)(sε y ) is not rational, to rewrite the above system in the time domain, we will make use of the ideas of Section 3. In particular, notice that Re (sε y )(sµ) > 0 for s ∈ C + . Lemma 3.2 applied to s -1 (sε y )(sµ), with the use of lim sr→+∞ s -1 (sε y )(sµ) = 1, yields

Then the system (55) in the time domain reads

z , we can make use of the identities similar to (42-43), more precisely

The equivalence result of this system to the system (51) mimics the result of Proposition 2.15.

y , H e z ) solve the system (56) with the initial conditions chosen as H e z | t=0 = B z | t=0 = H z0 and as zero for the rest of unknowns. Let E y , H z be the solution to the system (51) with the initial conditions chosen as H z | t=0 = B z | t=0 = H z0 and as zero for the rest of unknowns. Then E Appendix A. Proof of Proposition 2.17

Proof. Testing (42) with H

z -H e z , we obtain ∂ t (H (1) z -H e z ), H (1) z -

µ , H

µ e λ e µ , H (1) z -H e z .

(61)

The left-hand side of the above is d dt Ẽz , see also [START_REF] Cummer | Perfectly matched layer behavior in negative refractive index materials[END_REF]. As for the right-hand side, let us consider each term separately. The first term, with the use of the Cauchy-Schwartz formula, can be bounded

where we used the energy identity of Theorem 2.19 and the definition of Ẽz . Similarly, for some C > 0, the second term of the right-hand side of (61) satisfies

µ e λ e µ , H (1) z -H e z ≤ C H (1) z -H e z n e µ =0 µ e λ e µ .

To bound the above, we notice that Theorem 2.19 implies that λ e µ (t) ≤ C * E(0), with t ≥ 0 and C * independent of t, see also Remark 2.14. Thus, for some constant C 1 > 0,

µ e λ e µ , H (1) z -H e z ≤ C 1 E(0) Ẽz .

Thus we get from (61) the following differential inequality:

From the above we obtain Ẽz (t) ≤ 2C 2 E(0)t + Ẽz (0) 2 ≤ 2 4C 2 2 E(0)t 2 + Ẽz (0) .

The above result shows that H

(1) z (t)-H e z (t) grows at most linearly, and since H e z (t) is uniformly bounded in t, the growth of H