UNIFORM STABILIZATION OF ANISOTROPIC MAXWELL’S EQUATIONS WITH BOUNDARY DISSIPATION

We consider the Maxwell system with variable anisotropic coefficients in a bounded domain Ω of R3. The boundary conditions are of Silver-Muller’s type. We proved that the total energy decays exponentially fast to zero as time approaches infinity. This result is well known in the case of isotropic coefficients. We make use of modified multipliers with the help of an elliptic problem and some technical assumptions on the permittivity and permeability matrices.


1.
Introduction.Maxwell's equations provide mathematical foundations for the analysis of a broad range of devices which include wave guides, energy conversion devices, electronic components, electro-optic devices, among many others.In many situations it is enough to consider the isotropic Maxwell's equations, that is, when the permittivity E(x) and the permeability µ(x) are scalar real-valued functions, strictly positive.In recent years, industrial applications of the so-called "smart materials" [1] and the research on electromagnetic waves in crystal optics [12] have stimulated the interest in the so-called anisotropic Maxwell's equations, which we describe next: Let Ω ⊆ R 3 be an open bounded set with smooth boundary and let E(x, t) and H(x, t) be vector-valued functions defined in Ω × (0, T ) → R 3 denoting the electric field intensity and magnetic field intensity, respectively.The electric permittivity and the magnetic permeability are denoted by E(x) and µ(x) and are 3 × 3 matrices, symmetric and uniformly positive definite.The entries of E(x) and µ(x) are real-valued functions.We assume that Ω is filled with a medium with zero conductivity and there are no electrical charges in Ω.

CLEVERSON R. DA LUZ AND GUSTAVO P. MENZALA
The evolution of the electromagnetic field is governed by Maxwell's equations with boundary conditions and initial conditions at t = 0 given by E(x, 0) = E 0 (x), H(x, 0) = H 0 (x) in Ω. ( Here η = η(x) is the exterior unit normal at x ∈ ∂Ω, × is the usual vector product, curl denotes the rotational operator and div the divergence operator.Observe that the boundary conditions 2 have a dissipative character for the above model.In fact, formally, by taking the inner product of the first equation 1 by E and the second by H, integrating over Ω and using the divergence theorem give us 2 is known as the classical Silver-Muller condition which arises as a first order approximation to the so-called "transparent" boundary condition.Silver-Muller condition allows for reflections back into the region Ω.
Our main result in this article is the uniform decay of the total energy associated I(t) with problem 1-3 where The behavior of I(t) as t → +∞ has been previously considered only in the isotropic case with internal or boundary dissipation ( [6], [9], [10], [11], [13] and the references therein).In this isotropic case, Maxwell's equations 1 can be reduced (by differentiating once more in time) to a vector wave equation which can be treated using well-known standard results in literature.This argument does not work in the anisotropic case.Consequently, in order to analyze I(t), we have to deal with a first order hyperbolic system with variable coefficients at all times.There are not so many articles in the mathematical literature giving rigorous results on the anisotropic case for Maxwell's equations.Let us mention some of them: V. Vogelsang [16] and T. Okaji [14] proved strong unique continuation for time-harmonic anisotropic Maxwell's equations, V. G. Yakhno [17] gave a construction of a Green's function for the time-dependent anisotropic Maxwell's system, M. Eller [4,5] proved a unique continuation of the system across non-characteristic surfaces and also a boundary observability inequality for the anisotropic case.This result implies the exact boundary controllability of an electromagnetic field in Ω, M. Eller and M. Yamamoto [8] obtained a Carleman estimate for the time-harmonic anisotropic Maxwell system.
A classical reference describing the importance of the anisotropic Maxwell's equations in crystal optics is the book [12] by M. Kline and D. Kay.
This paper is organized as follows: global solvability of problem 1-3 in the required class is briefly indicated in Section 2. Uniform exponential decay of the total energy I(t) is established in Section 3.

2.
Well-posedness of 1-3.Let Ω ⊆ R 3 be an open bounded set with smooth boundary ∂Ω (class C 2 is enough) and E(x), µ(x) be symmetric 3 × 3 matrices whose entries are real-valued functions belonging to W 1,∞ (Ω).Furthermore the condition (uniformly positive definite) is assumed: There exists positive constants A 1 , A 2 in such a way that Here Let us describe the function spaces where we will consider the solution of problem 1-3: If and there exists A 0 > 0 such that ξ t α(x)ξ ≥ A 0 |ξ| 2 , for any ξ ∈ R 3 and all x ∈ Ω a.e., then we consider the weighted space: and norm (Ω; µ) be the Hilbert space with the inner product We also consider the Hilbert spaces The following lemma is well-known (see [3] or [11]).
Observe that using the results of [11] it follows that whenever u = (u 1 , u 2 ) ∈ Y then u 2 ×η is well defined on ∂Ω.Also, since the relation u 1 −(η ×u 1 )×η = η(u 1 •η) always holds then (η × u 1 ) × η is well defined on ∂Ω.This observation makes it possible to introduce the subspace In • Y we define the unbounded operator A with domain Here E −1 (x) and µ −1 (x) denote the inverse matrices of E(x) and µ(x) respectively.
Matrices E(x) and µ(x) are invertible a.e. in Ω because they are positive definite, therefore the eigenvalues of E(x) and µ(x) are positive (see [15]).Consequently the determinant of those matrices are positive.Hence E(x) and µ(x) are invertible a.e. in Ω.We can also prove that the entries of E −1 (x) and µ −1 (x) belong to L ∞ (Ω).
Following the same ideas of [11] we can prove the following.
Lemma 2.2.a) The domain of the adjoint operator A * coincides with the subspace and Using Lumer-Phillips' Theorem, Lemma 2.2 implies that A is the infinitesimal generator of a C 0 semigroup.Remains to fulfill the requirement that EE and µH are divergent free.However, taking the divergent of each of the first two equations in 1 we obtain Thus, it is natural to consider the subspace Theorem 2.3.Under the assumptions at the beginning of this section on Ω, E and µ (here it is only necessary that the entries are in L ∞ (Ω)).If we take initial data (E 0 , H 0 ) ∈ D(A) ∩ W then, system 1-3 has a unique strong solution

Stabilization.
In this section we will prove our main result.Our strategy will be to use convenient multipliers with the help of an elliptic problem.The result, then, will follow by assuming a geometric condition on the domain ("substarlike") and an extra technical assumption on the matrices E(x) and µ(x).
Let us consider a solution Φ ∈ C 2 (Ω) ∩ C 1 (Ω) of the Neumann problem Area(∂Ω) on ∂Ω.Let δ > 0 and x 0 ∈ Ω (to be chosen later) and consider the auxiliary function Clearly h(x) has the following properties: 3 where all entries of E and µ belong to W 1,∞ (Ω).Suppose that we can find k > 0 in such a way that holds for some x 0 ∈ Ω and are valid for any v ∈ R 3 , x ∈ Ω a.e. .Then, there exist positive constants C and α such that, the total energy satisfies ≤ CI(0) exp(−αt) for any t ≥ 0.
Remark 1. Condition 8 is called "substarlike".It is a geometric condition on the region Ω.It was first introduced by B. Kapitonov in [10].If k = 0 then 8 is reduced to the well known starlike condition on Ω. Likely, the final result of Theorem 3.1 may not be optimal with respect to assumptions 8 and 9.
Remark 2. Suppose that instead of 7 we just choose h(x) = 1 2 |x − x 0 | 2 then condition 9 can be written as which are the assumptions introduced by M. Eller in [5] to prove a boundary observability inequality.See also [6] (Theorem 1.1) for the isotropic case.
Proof of Theorem 3.1.We present the proof in three steps.
Step 1: Let h(x) as in 7. We use the multipliers ∇h × µH and ∇h × EE to obtain an identity for Those kind of multipliers were previously used by M. Eller and J. Masters in [7] to obtain a result on exact boundary controllability for Maxwell equations and by M. Eller in [5] to deduce a continuous observability for the anisotropic Maxwell system.Although the stabilization problem was not considered in [5] it could have been studied in the presence of boundary dissipation.
We take the inner product of EE t −curl H = 0 with ∇h×µH and µH t +curl E = 0 with ∇h × EE.By adding and integrating in space/time Ω × (0, T ) we obtain We can rewrite 10 as Let us rewrite the integrand of the last term on the left hand side of 11 in a convenient way: direct calculation give us the identity where here [h xixj ] denotes the Hessian matrix of h(x).

The terms 2[grad(E • ∇h)] • EE and [grad(EE • E)]
• ∇h on the right hand side of 12 can be replaced by using the identities because div(EE) = 0. Thus, from 12 and the above identities we obtain Similar calculations give us the identity where By substituting identities 13 and 14 into 11 and using the divergence theorem give us here we used the vector identities and Step 2: We estimate all terms on the right hand side of identity 15 by I(0) (the initial energy) and |E × η|, |H × η| on L 2 (0, T ; L 2 (∂Ω)).We estimate Clearly because I(t) is decreasing.Let Φ defined as in the beginning of this Section and consider the number ρ given by Next, we estimate the term 2 By integrating identity 17 over Ω × (0, T ) gives us several terms.One is All others are of the form 2δ where i, j, k = 1, 2, 3. We can estimate each one of these terms as follows: Let In a similar procedure we can estimate where C 4 is a positive constant depending on µ.
Next, we estimate the term We can choose a positive constant C 0 in such a way that ∂h ∂η (x) ≥ C 0 |∇h(x)| for any x ∈ ∂Ω.In fact, by 8 we know that ∂h ∂η (x) ≥ 0 for any x ∈ ∂Ω, then if ∇h(x) = 0 the inequality always holds.If ∇h(x) = 0 then we define C 0 = min x∈∂Ω due to the choice of C 0 and our assumption E ij (x) belong to L ∞ (Ω).
From 21 we deduce the estimate where A 1 > 0 was defined in Section 2. We choose Similar discussion proves that where Step 3: The conclusion.
Using estimates 16, 18, 19, 23 and 24 together with 15 we obtain where  Hence, the last term on the right hand side of 27 reduces to 2γ for some positive constant C. A classical result due to R. Datko [2] says that 28 implies the conclusion of Theorem 3.1.
Remark 3. We observe that once we got the crucial estimate 25 and by choosing δ = δ 0 in the appropriate interval we could assume K • ∇h + K • ∇h ≥ 0 in Ω × (0, T ) to conclude the above proof.However, assumption 9 allows us the possibility that the quantity K • ∇h + K • ∇h may assume small negative values.