The time dependent Maxwell system with measurable coefficients in Lipschitz domains

J. Math. Anal. Appl. 452 (2017) 941–956

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

The time dependent Maxwell system with measurable coefficients in Lipschitz domains ✩ Eric Stachura Department of Mathematics and Statistics, Haverford College, Haverford, PA 19041, United States

a r t i c l e

i n f o

Article history: Received 20 October 2016 Available online 22 March 2017 Submitted by P. Sacks Keywords: Maxwell system Sobolev spaces Lipschitz domains Anisotropy

a b s t r a c t Semigroup theory is employed to study inhomogeneous boundary conditions for the time-dependent Maxwell equations with anisotropic material parameters in a non-smooth domain. Only boundedness and measurability of the coefficients are assumed, and the boundary data is assumed to be time independent. When the material parameters are positive definite, a higher order Sobolev theory is developed for the time-dependent Maxwell system. © 2017 Elsevier Inc. All rights reserved.

1. Introduction In the Maxwell system of electromagnetism, the material parameters ε and μ are characteristic quantities which determine the propagation of light within a specified material. In an anisotropic medium, ε and μ are 3 × 3 positive definite matrices, with entries depending on position [9]. This note is devoted to solving the full time-dependent Maxwell equations with variable ε and μ and nonzero boundary data in a bounded, Lipschitz domain in R3 . We also consider the case of constant, positive definite ε and μ, and develop a higher order Sobolev theory in this case. We give a rigorous analysis of function spaces related to the Maxwell system in this setting, and prove useful characterizations of these spaces. The guiding motivation for this problem is an electromagnetic inverse problem, originating in the work of A. P. Calderón in [3]. The electromagnetic Dirichlet-to-Neumann map is the boundary map which sends the tangential component of the electric field to the tangential component of the magnetic field on the boundary, Λ : ν × E|∂Ω → ν × H|∂Ω ✩

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.jmaa.2017.03.052 0022-247X/© 2017 Elsevier Inc. All rights reserved.

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The inverse problem is as follows: given two sets of material parameters (ε1 , μ1 ) and (ε2 , μ2 ), assume that the corresponding maps Λ1 = Λ2 everywhere on the boundary. Then one hopes to prove that this map uniquely determines the material parameters, i.e. ε1 ≡ ε2 and μ1 ≡ μ2 . To solve the inverse problem, one first needs to solve the forward (Cauchy) problem, which is what we establish in this paper. There is a vast literature on this subject in the time harmonic regime, but what we wish to mention is that the time dependent case in this general setting (i.e. a non-smooth domain) to our knowledge has not yet been studied. When the domain is of class C 2 , though, various boundary conditions for the time dependent problem (perfect conductor and Silver–Müller) are analyzed in [17], with results similar to our Theorem 4.1. The outline of this paper is as follows. In Section 2 we discuss the necessary function spaces for the Cauchy problem. Then in Section 3 we formulate the Maxwell equations as an abstract evolution equation in a Hilbert space. We then proceed in Section 4 to solve the homogeneous boundary value problem. Section 5 is our main focus, where we impose nonzero boundary data on a Lipschitz domain and solve the resulting initial boundary value problem; the main result is Theorem 5.3. A key tool in the proof is a surjectivity result of Tartar [19]. Finally, we discuss in Section 6 how to extend the previous results to higher order Sobolev spaces if the material parameters are positive definite matrices with constant entries.1 The main result in this section for the Maxwell system is Theorem 6.4. We would like to thank the referee for pointing out the reference [17]. 2. Function spaces for the Maxwell problem Before going into the details of the problem, we find it useful here to discuss some of the function spaces that we will use. For the following, Ω will denote a bounded, Lipschitz domain in R3 . We recall: Definition 2.1. A bounded domain Ω ⊂ R3 is called a Lipschitz domain if for each point p ∈ ∂Ω, there exists an open set O ⊂ R3 such that p ∈ O, and an orthogonal coordinate system with coordinates ξ = (ξ1 , ξ2 , ξ3 ) having the following property: There exists a vector b ∈ R3 so that O = {ξ : −bj < ξj < bj , 1 ≤ j ≤ 3} and a Lipschitz continuous function φ defined on the set O = {ξ  ∈ R2 : −bj < ξj < bj , 1 ≤ j ≤ 2} such that Ω ∩ O = {ξ : ξ3 < φ(ξ  ), ξ  ∈ O } and ∂Ω ∩ O = {ξ : ξ3 = φ(ξ  ), ξ  ∈ O }  3 Recall that L2 (Ω) denotes the Lebesgue space of square integrable functions on Ω, and L2 (Ω) denotes the space of square integrable vectors f = (f1 , f2 , f3 ). Further recall that the L2 scalar product of U = (u1 , u2 , u3 ) and V = (v1 , v2 , v3 ) is given by (U, V )L2 (Ω)3 :=

  3

uj v j dx

Ω j=1

1

Such parameters frequently arise in applications, e.g. in crystal optics [14].

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where the bar denotes complex conjugate. The main Hilbert space we work in will be  3  3 H := L2 (Ω) × L2 (Ω)

(1)

For an element U ∈ L2 (Ω), any derivative of U must be understood in the sense of distributions. Recall next the standard Sobolev spaces W s,p (Ω) for s ∈ Z+ and 1 ≤ p < ∞, are defined by W s,p (Ω) = {φ ∈ Lp (Ω) : Dα φ ∈ Lp (Ω) ∀ |α| ≤ s} One way of defining the spaces H s (Ω) is by restriction:   H s (Ω) = u ∈ D (Ω) : u = U |Ω for some U ∈ W s,2 (R3 ) where D (Ω) denotes the space of distributions on Ω. We next define the fractional Sobolev spaces on Ω. Let s ∈ R+ such that s = m + σ, with m ∈ Z+ and 0 < σ < 1. Then H s (Ω) is defined to be the space of distributions U ∈ D (Ω) such that U ∈ W m,2 (Ω) and   Ω Ω

|Dα U (x) − Dα U (y)|2 dxdy < +∞ |x − y|3+2σ

for all |α| = m. For s ∈ R+ , the space H s (Ω) is a Hilbert space. Then, for s > 1, define the normed space  H s (∂Ω) = u ∈ L2 (∂Ω) : u = U |∂Ω for some U ∈ H s+1/2 (Ω) with norm ||u||H s (∂Ω) =

inf

U ∈H s+1/2 (Ω),u=U |∂Ω

||U ||H s+1/2 (Ω)

For 0 ≤ s < 1, we define the space H s (∂Ω) as follows. Definition 2.2. Let Ω ⊂ R3 be a bounded, Lipschitz domain. A distribution U defined on ∂Ω is said to belong to the space W s,p (∂Ω) for 0 < s < 1 if   U ◦ φ ∈ W s,p O ∩ φ−1 (∂Ω ∩ O) for all possible O, φ as in Definition 2.1. When p = 2 we have H s (∂Ω) = W s,2 (∂Ω) for 0 ≤ s ≤ 1. We next define the space   3 H(div, Ω) := U ∈ L2 (Ω) : ∇ · U ∈ L2 (Ω) When endowed with the graph norm

1/2 ||U ||H(div,Ω) = ||U ||2L2 (Ω)3 + ||∇ · U ||2L2 (Ω) the space H(div, Ω) becomes a Hilbert space. Next we define H0 (div, Ω) := completion of (C0∞ (Ω))

3

in the H(div, Ω) norm

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as well as   3  3 H(curl, Ω) = U ∈ L2 (Ω) : ∇ × U ∈ L2 (Ω) and H0 (curl, Ω) = completion of (C0∞ (Ω)) in the H(curl, Ω) norm 3

where the norm in H(curl, Ω) is the graph norm. We need the following key lemma: Lemma 2.3 ([7]). Let Ω ⊂ R3 be a bounded Lipschitz domain, and let ν denote the outer unit normal to the boundary ∂Ω. Then: 1. The following equality holds: 3 ||·||H(curl,Ω)  H(curl, Ω) = C ∞ (Ω) 2. The trace map for w ∈ (C ∞ (Ω))3 γt (w) = ν × w|∂Ω extends to a continuous linear map from H(curl, Ω) into (H −1/2 (∂Ω))3 .   3 3. H0 (curl, Ω) = U ∈ H(curl, Ω) : (U, ∇ × φ)L2 = (∇ × U, φ)L2 ∀ φ ∈ C ∞ (Ω) 4. The space H0 (curl, Ω) can also be characterized by those functions in H(curl, Ω) having tangential trace zero: H0 (curl, Ω) = {U ∈ H(curl, Ω) : ν × U = 0 on ∂Ω} The space H(curl, Ω) is of central importance to the Maxwell system since this corresponds to the space of finite energy solutions. 3. Formulation of the boundary value problem Let E(x, t) = (E1 (x, t), E2 (x, t), E3 (x, t)) and H(x, t) = (H1 (x, t), H2 (x, t), H3 (x, t)) denote the (time dependent) electric and magnetic fields, where x ∈ R3 and t > 0. The equations which govern classical electromagnetism in R3 are the Maxwell equations, which describe the evolution of the electromagnetic field: ∂E =∇×H−J ∂t

(2a)

∂H = −∇ × E ∂t

(2b)

ε μ

∇ · (εE) = ρ

(2c)

∇ · (μH) = 0

(2d)

where: • ε(x) is the material permittivity and μ(x) is the magnetic permeability

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• ρ(x) is the total charge density • J(x, t) is the current density We will focus on general material parameters ε = ε(x, y, z) and μ = μ(x, y, z), each 3 × 3 matrices with elements depending on position, and what degree of regularity is required on these coefficients to obtain a existence and uniqueness of solutions to the time dependent Maxwell system. Our first objective is to prove global existence of C 1 in time solutions to the Maxwell system with general assumptions on ε, μ. We first impose that the electric field has vanishing tangential trace, and eventually allow for nonzero prescribed tangential trace. To do so, we begin by reviewing an approach due to R. Picard [16] and R. Leis [10] on how to formulate the problem as an abstract differential equation in H . Let U = (E, H) and V = (E , H ). The space H will be endowed with the inner product (U, V )H := (U, M V )L2 = (E, εE )L2 + (H, μH )L2 We have written M for the matrix



ε 0

M :=

0 μ

(3)

We will impose conditions on ε, μ so that the matrices M and M −1 are positive definite. Define the unbounded Maxwell operator A : D (A) → H with domain D (A) := H0 (curl, Ω) × H(curl, Ω) ⊂ H

(4)

by A := −iM

−1

0 ∇×

−∇ × 0

(5)

Remark 3.1. Here we point out specifically the equations (2c) and (2d) with ρ = 0. These conditions are somehow swept under the rug in the following analysis. Mostly this is due to the fact that in the relevant inverse problems literature, the focus is on the equations (2a) and (2b); however, the other two equations can be incorporated as follows. Let W denote the subspace of the Hilbert space H consisting of pairs (U1 , U2 ) such that ∇ · (εU1 ) = ∇ · (μU2 ) = 0 Then we can define the Maxwell operator (A, D (A)) on this subspace; it will still be self-adjoint. In particular, Lemma 3.2 over W still holds. So whenever D (A) is encountered, if one wishes to take into account (2c) and (2d) as well, the space D (A) should be replaced by D (A) ∩ W . Now, the initial value problem for the Maxwell system becomes U  (t) + iAU (t) = 0,

U (0) = U0 = (E0 , H0 ) ∈ H

(6)

In fact, R. Leis [10] proves existence and uniqueness of weak solutions to (6) using the Spectral Theorem. We will show a stronger result, namely, that there is a solution U (t) which is C 1 in time, provided the initial data belongs to the domain of the underlying operator. When Ω is smooth and zero boundary condition is imposed, this was shown by R. Picard in [16]. We will improve this by relaxing boundary regularity and taking nonzero boundary conditions. Now we can state our assumptions on ε, μ. For x ∈ Ω we assume that ε, μ are 3 × 3 matrices, such that:

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(H1) εjk (x) = εkj (x) and μjk (x) = μkj (x) for 1 ≤ j, k ≤ 3; (H2) εjk (x) and μjk (x) are bounded, measurable, for 1 ≤ j, k ≤ 3; (H3) There exists C1 > 0 so that for each 1 ≤ j, k ≤ 3, x ∈ Ω, ξ ∈ R3 , ξj εjk (x)ξk ≥ C|ξ|2 (H4) There exists C2 > 0 so that for each 1 ≤ j, k ≤ 3, x ∈ Ω, ξ ∈ R3 , ξj μjk (x)ξk ≥ C2 |ξ|2 A key result used to solve the homogeneous problem is the following. Lemma 3.2 ([10]). Suppose ε, μ satisfy (H1)–(H4). The operator A defined in (5) with domain D (A) = H0 (curl, Ω) × H(curl, Ω)

is self-adjoint on the Hilbert space H . We will use the above result plus semigroup theory to obtain existence and uniqueness of classical solutions when J = 0, and then for nonzero J. 4. The boundary value problem with perfect conductor boundary condition Before studying the case of nonzero boundary data, we first need to solve the corresponding homogeneous problem with and without current density. 4.1. The case of no current density The main result in this section is the following. Theorem 4.1. Let Ω ⊂ R3 be a bounded, Lipschitz domain. Let ε, μ satisfy (H1)–(H4), and write U = (E, H)t . Then for each U0 ∈ D (A) = H0 (curl, Ω) × H(curl, Ω) the problem ⎧  ⎪ ⎪ ⎨U (t) + iAU (t) = 0

U (x, 0) = U0 ∈ D (A) ⎪ ⎪ ⎩ν × E = 0 on ∂Ω has a unique solution U (t) ∈ C 1 ([0, ∞); D (A)). This is an improvement of the result in [10, Chapter 8] as we obtain global existence of classical (i.e. C 1 in time) solutions, rather than just weak solutions. Proof. By Lemma 3.2, we have that A is self-adjoint on D (A). Hence by Stone’s Theorem [18], we have that −iA is the infinitesimal generator of a C0 -semigroup T (t) = e−iAt . Semigroup theory implies we have U (t) = e−iAt U0 is the unique D (A) valued, C 1 solution on [0, ∞). 2

E. Stachura / J. Math. Anal. Appl. 452 (2017) 941–956

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4.2. The case of non-zero current density Assume again that the hypotheses (H1)–(H4) are satisfied by ε, μ. We define the operator A exactly as before, with D (A) = H0 (curl, Ω) × H(curl, Ω)

Now, when J = 0, the abstract initial boundary value problem we wish to solve takes the form, again with U = (E, H), ⎧  ⎪ ⎪ ⎨U (t) + iAU (t) = F (t) (7) U (0) = U0 ∈ D (A) ⎪ ⎪ ⎩ν × E = 0 on ∂Ω where now  F (t) =

−ε−1 (x)J(x, t) 0



can be viewed as a map F : [0, T ) → H . Since A is self-adjoint on D (A), we again have that −iA is the infinitesimal generator of a C0 -semigroup {T (t) = e−iAt : t ≥ 0}. We shall prove existence of classical solutions: Theorem 4.2. Suppose the assumptions (H1)–(H4) hold in a bounded, Lipschitz domain Ω. Then, if J ∈ C 1 ([0, T ); H ), there exists a unique classical solution U ∈ C 1 ([0, T ); D (A)) to (7) given by −iAt

U (t) = e

t U0 +

e−iA(t−s) F (s)ds

(8)

0

Proof. First note that since ε and its inverse are time independent, it is enough to study the continuity and differentiability of the current density J. Let T (t) = e−iAt be the semigroup which gives the solution to the homogeneous problem U  (t) + iAU (t) = 0, so that U (t) = T (t)U0 . Then [15, Theorem 4.2.4] shows that U (t) defined by (8) is a (uniquely determined) solution of (7). 2 Remark 4.3. In principle one could allow the material parameters to depend on time as well, ε = ε(t) and μ = μ(t) to allow for time dependent media. As this is a rather unusual situation, we have not pursued ∂ε ∂μ and , so the this direction. In particular, the Maxwell system would now have terms of the form ∂t ∂t parameters would have to be sufficiently regular in time to ensure existence and uniqueness of solutions. Additionally, the Maxwell operator A would now depend on time. The following section is the main focus of this article, where we impose time independent boundary conditions on a less regular domain. 5. Nonzero general boundary conditions In this section the distinction between Lipschitz domains and smooth domains becomes more prominent. To this end first suppose that Ω ⊂ R3 is a bounded, smooth domain. Recall that the space H0 (curl, Ω) can be characterized by those functions having tangential trace equal to zero on the boundary of Ω. But this

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 3 characterization is not always useful because the trace mapping γt : H(curl, Ω) → H −1/2 (∂Ω) that sends w → ν × w|∂Ω is not surjective. Thus, we follow [4] and introduce the trace space Y (∂Ω) as  

3 −1/2 Y (∂Ω) = f ∈ H (∂Ω) : ∃ U ∈ H(curl, Ω) such that γt (U ) = f

(9)

When it is endowed with the norm ||f ||Y (∂Ω) :=

inf

U ∈H(curl,Ω);γt (U )=f

||U ||H(curl,Ω)

then Y (∂Ω) becomes a Banach space. In fact, the following is true. Theorem 5.1 ([2]). Let Ω ⊂ R3 be a bounded, smooth domain. Then the space Y (∂Ω) as above is a Hilbert space. The trace map γt : H(curl, Ω) → Y (∂Ω) is surjective. Even if Ω is Lipschitz a similar result is true; the proof is more involved, but the space Y (∂Ω) can be characterized fully. For smooth domains the space Y (∂Ω) has a nice characterization. To this end we must introduce a few function spaces on ∂Ω. Define   3 L2t (∂Ω) := U ∈ L2 (Ω) : ν · U = 0 a.e. on ∂Ω

(10)

and −1/2 Ht (∂Ω)

 :=



3 −1/2 f∈ H (∂Ω) : f · ν = 0 a.e. on ∂Ω

(11)

Now, for a given function p which is differentiable on a neighborhood of the boundary ∂Ω, one can define the surface gradient of p, denoted ∇∂Ω p, by ∇∂Ω p = (ν × ∇p) × ν where as usual, ν denotes the outer unit normal to Ω. This formula also holds when Ω is Lipschitz [12]. The surface gradient can also be extended for functions p ∈ H 1 (∂Ω); we will also denote this operator by ∇∂Ω . Once this operator is defined we can define the surface divergence operator ∇∂Ω · : L2t (∂Ω) → H 1 (Ω) by duality. It turns out that when Ω is smooth, the following is true. Proposition 5.2 ([4]). If Ω ⊂ R3 is a smooth, bounded domain, then  −1/2 Y (∂Ω) = U ∈ Ht (∂Ω) : ∇∂Ω · U ∈ H −1/2 (∂Ω)

(12)

For more details on this characterization the reader is referred to [1] (in particular Theorem 3.1, p. 165). When Ω is Lipschitz, see [2] for a detailed discussion of the previous surface differential operators. 5.1. The boundary condition ν × E = f Suppose now that Ω is a Lipschitz domain. In this case there is not such a nice characterization of the space Y (∂Ω); however, there is a characterization which is due to L. Tartar [19]. In fact, we define the Tartar trace space on a bounded, Lipschitz domain Ω by

E. Stachura / J. Math. Anal. Appl. 452 (2017) 941–956

 T(∂Ω) =

949

3 ξ ∈ H −1/2 (∂Ω) : ∃ η ∈ H −1/2 (∂Ω) s.t. ∀ φ ∈ H 2 (Ω), ξ, γ(∇φ)H 1/2 (∂Ω) = η, γ(φ)H 1/2 (∂Ω)



where γ denotes the scalar trace mapping γ : f → f |∂Ω . Then Tartar [19] showed that γt : H(curl, Ω) → T(∂Ω) is surjective. We impose now the general boundary condition ν × E|∂Ω = f ∈ T(∂Ω)

(13)

where f is assumed to be time independent. Since γt : H(curl, Ω) → T(∂Ω) is onto, there exists a right  and H  as follows: inverse, call it R. Then we define the adjusted fields E  := E − Rf E

 := H H

and

 and H:  We have the Maxwell equations for E ⎧  ∂E ⎪ ⎪  −J =∇×H ⎪ ε ⎪ ⎪ ⎪ ∂t ⎪ ⎪  ⎪ ⎪ ∂H ⎪  − ∇ × Rf ⎨μ = −∇ × E ∂t ⎪  = 0 on ∂Ω ⎪ ν×E ⎪ ⎪ ⎪ ⎪ ⎪   0 (x) ⎪ H(0) =H ⎪ ⎪ ⎪ ⎩  0 (x) E(0) = E

(14)

We can (formally) transform this into the following abstract evolution problem: ⎧   ⎪ ⎪ ⎨U (t) + iAU (t) = F (t)  (0) = U 0 U ⎪ ⎪ ⎩ν × E  = 0 on ∂Ω

(15)

 and  H)  = (E, where U  F (t) = −

ε−1 (x)J(x, t) μ−1 (x)∇ × Rf (x)



Notice that  0 = U

0 E 0 H



 =

E0 − Rf H0



where U0 = (E0 , H0 ) is the initial condition for the Maxwell equations in (E, H). Thus we are reduced to studying the inhomogeneous problem with zero boundary data, which is solvable. In particular, suppose 0 ∈ D (A) = H0 (curl, Ω) × H(curl, Ω) and J ∈ C 1 ([0, T ); H ). Then by Theorem 4.2, there exists a unique U  on [0, T ) to (15). classical solution U With regards to the solution (E, H) the above result tells us  U (t) =

 E(t) H(t)

 =

 + Rf E(t)  H(t)



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In particular, U (t) is continuously differentiable in time. Indeed, this follows from the fact that f, Rf are  is continuously differentiable. We also have that Rf ∈ H(curl, Ω), so time independent and the fact that U 1 the solution U (t) ∈ C ([0, T ); H(curl, Ω) × H(curl, Ω)). Hence we’ve shown: Theorem 5.3. Suppose (H1)–(H4) hold in a bounded, Lipschitz domain Ω. Let  F (t) =

−ε−1 (x)J(x, t) 0



and let J ∈ C 1 ([0, T ); H ). Then the Maxwell system for U (t) = (E(t), H(t)) ⎧  ⎪ ⎪ ⎨U (t) + iAU (t) = F (t) U (0) = U0 ∈ D (A) ⎪ ⎪ ⎩ν × E = f ∈ T(∂Ω)

(16)

has a unique classical solution U (t) ∈ C 1 ([0, T ); H(curl, Ω) × H(curl, Ω)). 6. Higher order Sobolev regularity Using the spaces H s (Ω), we can define the higher order H(curl) spaces for s ≥ 0 by  3 3 Hs (curl, Ω) := U ∈ (H s (Ω)) : ∇ × U ∈ (H s (Ω))

(17)

The norm on Hs (curl, Ω) is given by ||U ||2Hs (curl,Ω) := ||U ||2H s (Ω) + ||∇ × U ||2H s (Ω) It is clear what is meant by this space if s ∈ Z+ . Now, if s ∈ R+ , write s = m + σ, where m ∈ Z+ and 0 < σ < 1. The norm on Hs (curl, Ω) then becomes ||U ||2Hs (curl,Ω) = ||U ||2H m (Ω) + ||∇ × U ||2H m (Ω) + ⎞ ⎛   α α 2    |Dα U (x) − Dα U (y)|2 |D (∇ × U )(x) − D (∇ × U )(y)| ⎝ dxdy + dxdy ⎠ |x − y|3+2σ |x − y|3+2σ |α|=m

Ω Ω

Ω Ω

From now on we will assume s ≥ 1 is an integer. Then we can define the space Hs0 (curl, Ω) = closure of (C0∞ (Ω))

3

in Hs (curl, Ω)

We next require the following technical lemma, which is adapted for higher order Sobolev spaces from [8, Lemma 2.4], where the result is proved with s = 0. Lemma 6.1. Let Ω be bounded and Lipschitz. Take s ∈ Z+ , and let U ∈ Hs (curl, Ω) be such that for each  3 φ ∈ C ∞ (Ω) , there holds (∇ × U, φ)H s − (U, ∇ × φ)H s = 0 Then, U ∈ Hs0 (curl, Ω).

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Proof. Let U ∈ Hs (curl, Ω). Since Ω ⊂ R3 is a bounded, Lipschitz domain, we can find a finite collection of open sets Uj such that Ω ⊂ ∪N j=1 Uj and such that each Ωj := Uj ∩ Ω is a bounded, starlike Lipschitz ∞ domain. Then we can find a partition of unity {αj }N j=1 such that each αj ∈ C0 (Uj ) and 0 ≤ αj (x) ≤ 1 for N j = 1, ..., N , and with j=1 αj (x) = 1 for each x ∈ Ω.  denote the extension of U by zero outside of Ω. Then in particular U  ∈ Hs (curl, R3 ). We can then Let U write, using our partition of unity, = U

N 

j U

j=1

j ) ⊂ Ωj . For 0 < t < 1, define U j (x/t) which converges j = αj U  ∈ Hs (curl, R3 ) with supp(U  t (x) := U where U j s 3 t j in H (curl, R ) as t → 1. We also have that supp(U  ) ⊂ Ωj for 0 < t < 1. to U j

Let now ρ ∈ C0∞ (R3 ) be such that ρ ≥ 0, ρ(x) = 0 if |x| ≥ 1, and  ρ(x)dx = 1 R3

Then for > 0 define the family of functions ρ by ρ (x) =

ρ(x/ ) 3

For any V ∈ L2 (R3 ) the convolution ρ ∗ V is defined by  ρ ∗ V (x) =

ρ (x − y)V (y)dy R3

It is well known then that ρ ∗ V → V in L2 (R3 ), and each ρ ∗ V ∈ C0∞ (R3 ). Now, the key part here is that, via the differentiability properties of the convolution, we have for a distribution V , ∇ × (ρ ∗ V ) = ρ ∗ (∇ × V ) t → U  t as Note that the convolution with a vector is defined component wise. We next claim that ρ ∗ U j j s 3 → 0 in H (curl, R ). Indeed,  t ||2 s t 2 t 2 ||ρ ∗ U j H (curl,R3 ) = ||ρ ∗ Uj ||H s + ||∇ × (ρ ∗ Uj )||H s := I + II Now, the first term is easily treated: jt ||2H s = I = ||ρ ∗ U



jt )||2 2 = ||Dα (ρ ∗ U L

|α|≤s

 |α|≤s

which converges to 

jt ||2 2 = ||U jt ||2H s ||Dα U L

|α|≤s

as → 0. The second term is also straight forward:

jt ||2 2 ||ρ ∗ Dα U L

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952

jt )||2H s = II = ||∇×(ρ ∗U



jt ))||2 2 = ||Dα (∇×(ρ ∗U L

|α|≤s



jt )||2 2 = ||Dα (ρ ∗∇×U L

|α|≤s



jt )||2 2 ||ρ ∗Dα (∇×U L

|α|≤s

which tends to 

 t )||2 2 = ||∇ × U  t ||2 s ||Dα (∇ × U j j H L

|α|≤s

as → 0. Hence bringing everything together we see that →0  t ||2 s  t ||2 s ||ρ ∗ U −−→ ||U j H (curl,R3 ) − j H (curl,R3 )

 t ) ⊂ Ωj and so ρ ∗ U  t ∈ (C ∞ (Ωj ))3 . Thus we can find a Now, if is sufficiently small, then supp(ρ ∗ U 0 j j sequence k ∈ (0, 1) with k → 0 and tk ∈ (0, 1) such that tk → 1, such that  tk −k→∞ j ρk ∗ U −−−→ U j in Hs (curl, Ωj ). Next we define  k := U

N 

 tk ρk ∗ U j

j=1

 k ∈ (C ∞ (Ω))3 for each k and U  k → U in Hs (curl, Ω). Hence U ∈ Hs (curl, Ω) as Then it is clear that U 0 0 desired. 2 We shall next show the following, which is similar to Lemma 2.3. Lemma 6.2. Let Ω ⊂ R3 be a bounded, Lipschitz domain with unit normal ν, and let s ∈ Z+ .  3 ||·||Hs (curl,Ω) 1. Hs (curl, Ω) = C ∞ (Ω)  3 2. The trace map for W ∈ C ∞ (Ω) γt (W ) = ν × W |∂Ω  3 extends to a continuous linear map from Hs (curl, Ω) into H −(s−1/2) (∂Ω) .   3 3. Hs0 (curl, Ω) = U ∈ Hs (curl, Ω) : (U, ∇ × φ)H s = (∇ × U, φ)H s ∀φ ∈ C ∞ (Ω) 4. Hs0 (curl, Ω) = {U ∈ Hs (curl, Ω) : ν × U = 0 on ∂Ω} We mention here that if 0 < s < 1/2, there is no corresponding trace theorem. Namely, there is no continuous trace operator T from H s (Ω) onto any Lebesgue space, no matter how smooth Ω is. This in particular means that a boundary condition such as u = g on ∂Ω is not well-defined for 0 < s < 1/2. Proof. As before, to prove (1) we write  3  3 ⊥ Hs (curl, Ω) = C ∞ (Ω) ⊕ C ∞ (Ω)  3 Then we take U ∈ Hs (curl, Ω) such that U ⊥ C ∞ (Ω) , i.e.

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953

 3 (U, φ)H s + (∇ × U, ∇ × φ)H s = 0 ∀ φ ∈ C ∞ (Ω) 3

Since U ∈ Hs (curl, Ω) there exists g ∈ (H s (Ω)) such that ∀φ ∈ (C0∞ (Ω))

3

(g, φ)H s = (U, ∇ × φ)H s Then we see that, using the orthogonality condition,

(U, φ)H s = −(g, ∇ × φ)H s = −(∇ × g, φ)H s so that U = −∇ × g, and so g ∈ Hs (curl, Ω). We also have that  3 ∀φ ∈ C ∞ (Ω)

(g, ∇ × φ)H s = (∇ × g, φ)H s

Hence Lemma 6.1 implies that g ∈ Hs0 (curl, Ω). Then by density there exists φn ∈ (C0∞ (Ω)) such that 3

n→∞

||φn − g||Hs (curl,Ω) −−−−→ 0 Finally we have ||U ||2Hs (curl,Ω) = (U, U )H s + (∇ × U, ∇ × U )H s = lim [−(U, ∇ × φn ) + (∇ × U, φ)n ] = 0 n→∞

so that U = 0, and (1) is proved. Next we prove (2). Our starting point is the Divergence Theorem; taking F = Dα U × Dα φ for |α| ≤ s  3 and U, φ ∈ C ∞ (Ω) , we obtain the following identity: 

 D φ · ∇ × D U dx − α

Ω

 D U · ∇ × D φdx =

α

α

ν × Dα U · Dα φdx

α

Ω

∂Ω

Then by summing on |α| ≤ s and exchanging the order of differentiation we arrive at the following identity: (φ, ∇ × U )H s − (∇ × φ, U )H s = ν × U, φH s (∂Ω)

(18)

 3 3 for U, φ ∈ C ∞ (Ω) . By density [13] the formula holds for φ ∈ (H s (Ω)) . Then by Cauchy–Schwarz, there holds |ν × U, φH s (∂Ω) | ≤ ||U ||Hs (curl,Ω) ||φ||H s (Ω) Next let f ∈ H s−1/2 (∂Ω), and choose φ ∈ H s (Ω) to be the weak solution of  Δφ = φ φ=f

in Ω on ∂Ω

along with the estimate ||φ||H s (Ω)3 ≤ C||f ||H s−1/2 (∂Ω) Note that the boundary condition makes sense because the trace operator γ : u → u|∂Ω extends to a continuous map γ : H s (Ω) → H s−1/2 (∂Ω) for s > 1/2, see [11, Theorem 3.37].

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954

Hence we have |ν × U, φH s (∂Ω) | ≤ C||U ||Hs (curl,Ω) ||f ||H s−1/2 (∂Ω)  3 for all U ∈ C ∞ (Ω) and f ∈ H s−1/2 (∂Ω). Then we have ||ν × U ||H −(s−1/2) (∂Ω) =

sup f ∈H s−1/2 (∂Ω)

|ν × U, f H s (∂Ω) | ≤ C||U ||Hs (curl,Ω) ||f ||H s−1/2 (∂Ω)

3 3   Hence γt on C ∞ (Ω) is continuous as a map from Hs (curl, Ω) to H −(s−1/2) (∂Ω). By density of C ∞ (Ω) in Hs (curl, Ω), we have shown (2). To see (3) and (4), notice that the inclusion 

 3 U ∈ Hs (curl, Ω) : (U, ∇ × φ)H s = (∇ × U, φ)H s ∀ φ ∈ C ∞ (Ω) ⊆ Hs0 (curl, Ω)

follows from Lemma 6.1. Now, appealing to the identity (18) with γt (U ) = 0, we see that {U ∈ Hs (curl, Ω) : γt (U ) = 0} ⊆   3 U ∈ Hs (curl, Ω) : (U, ∇ × φ)H s = (∇ × U, φ)H s ∀ φ ∈ C ∞ (Ω) Since (C0∞ (Ω)) ⊂ {U ∈ Hs (curl, Ω) : γt (U ) = 0}, and the latter set is closed by continuity of the trace map, we conclude that Hs0 (curl, Ω) ⊂ {U ∈ Hs (curl, Ω) : γt (U ) = 0} which proves (3) and (4). 2 3

Next we proceed to solve the Maxwell equations with higher order Sobolev regularity. To this end let ε, μ be 3 × 3 constant, symmetric, positive definite matrices. Let s ≥ 1 be an integer, and we consider 3

3

H = (H s (Ω)) × (H s (Ω))

with the inner product (U, V )H = (U1 , εV1 )H s (Ω) + (U2 , μV2 )H s (Ω) where U = (U1 , U2 ) and U1 = (U11 , U12 , U13 ), and similar for V . We take J = 0, and begin with the perfect conductor boundary condition ν × E = 0 on ∂Ω. Define the domain of the Maxwell operator A now by D(A) = Hs0 (curl, Ω) × Hs (curl, Ω) It is clear by the definitions that D(A) = H . The following lemma is key for all that follows. Lemma 6.3. With the above setup, the operator A : D(A) → H is self-adjoint. Proof. We begin by showing that A is symmetric, i.e. for each U, V ∈ D(A), there holds (AU, V )H = (U, AV )H Note that (AU, V )H = ((AU )1 , εV1 )H s (Ω) + ((AU )2 , μV2 )H s (Ω) = (iε−1 ∇ × U2 , εV1 )H s + (−iμ−1 ∇ × U1 , μV2 )H s

E. Stachura / J. Math. Anal. Appl. 452 (2017) 941–956

955

We have, using the fact that U, V ∈ D(A), (ε−1 ∇ × U2 , εV1 )H s =



(Dα ε−1 ∇ × U2 , Dα εV1 )L2

|α|≤s

=



(ε−1 Dα ∇ × U2 , εDα V1 )L2

|α|≤s

=



(Dα U2 , Dα ∇ × V1 )L2

|α|≤s

where we have used that ε has constant entries to pull it outside of the derivative. Clearly a similar formulation is true for the inner product with μ. We also have that (U, AV )H = (U1 , ε(AV )1 )H s + (U2 , μ(AV )2 )H s and (U2 , μ(AV )2 )H s = (U2 , μμ−1 ∇ × V1 )H s  = (Dα U2 , Dα ∇ × V1 )L2 |α|≤s

which is exactly what appeared in the first computation. Hence by computing all four inner products and comparing we see that (U, AV )H = (A, U V )H , and so A is symmetric. To see that A is self-adjoint, we use the fact that D(A∗ ) = {V ∈ H : ∃ F ∈ H , ∀ U ∈ D(A), (AU, V ) = (U, F )} to show that D(A∗ ) ⊂ D(A) directly, and conclude that D(A∗ ) = D(A).

2

Having shown that A is self-adjoint, the methods used in the case s = 0 give us the following result, at least when the material parameters are constant matrices. Theorem 6.4. Let Ω be a bounded, Lipschitz domain, and let ε, μ be constant, symmetric, positive definite matrices, and write U = (E, H)t . Then for each U0 ∈ D(A) = Hs0 (curl, Ω) × Hs (curl, Ω) the Maxwell equations ⎧  ⎪ ⎪ ⎨U (t) + iAU (t) = 0 U (x, 0) = U0 ∈ D(A) ⎪ ⎪ ⎩ν × E = 0 on ∂Ω

have a unique solution U ∈ C 1 ([0, ∞); D(A)). When there is a nonzero current density J, the corresponding version of Theorem 4.2 holds as well, now with ε, μ constant, symmetric, positive definite, via semigroup theory. Future work entails imposing the nonzero boundary condition ν × E = f . A number of trace results will have to be shown before this problem can be solved. In particular, surjectivity of the tangential trace map in a higher order Sobolev space on a Lipschitz domain will need to be proved. Remark 6.5. The results in this section will be useful not only to study higher order regularity in electromagnetic problems, but also to problems related to the Navier–Stokes system [8] and linear acoustic problems (which, incidentally, come from linearizing the Navier–Stokes system).

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7. Conclusions and future work For the anisotropic Maxwell equations, we have established existence and uniqueness of global in time solutions to boundary value problems with general time independent boundary conditions, in the setting of bounded, Lipschitz domains in R3 . This problem was initially motivated by an inverse problem, similar to the classical Calderón problem in Electrical Impedance Tomography. We can now proceed to study the time dependent inverse problem, first by constructing the Dirichlet-to-Neumann map. The fact that the boundary data is assumed to be time independent is restrictive, but a starting point nonetheless for time dependent problems in non-smooth domains. The idea to use single and double layer potentials was derived by Costabel in [6], and we expect this to apply to the time dependent Maxwell system, as there exists a rich literature on layer potentials in Lipschitz domains, see e.g. [5]. Finally, trace theorems for higher order Sobolev spaces on Lipschitz domains will be developed in order to study related problems on the Hs (curl, Ω) spaces. References [1] A. Alonso, A. Valli, Some remarks on the characterization of the space of tangential traces of H(rot; Ω) and the construction of an extension operator, Manuscripta Math. 89 (1) (1996) 159–178. [2] A. Buffa, M. Costabel, D. Sheen, On traces for H(curl, Ω) in Lipschitz domains, J. Math. Anal. Appl. 276 (2) (2002) 845–867. [3] A.P. Calderón, On an inverse boundary value problem, in: Seminar on Numerical Analysis and Its Applications to Continuum Physics, 1980, pp. 65–73, Rio de Janeiro. [4] Z. Chen, Q. Du, J. Zou, Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients, SIAM J. Numer. Anal. 37 (5) (2000) 1542–1570. [5] M. Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (3) (1988) 613–626. [6] M. Costabel, Time-Dependent Problems with the Boundary Integral Equation Method, Encyclopedia of Computational Mechanics, 2004. [7] R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications, Springer, 2000. [8] V. Girault, P. Raviart, Finite Element Approximation of the Navier–Stokes Equations, vol. 749, 1979. [9] I.W. Kay, M. Kline, Electromagnetic Theory and Geometrical Optics, John Wiley & Sons, 1965. [10] R. Leis, Initial Boundary Value Problems in Mathematical Physics, John Wiley & Sons, 1986. [11] W.C.H. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. [12] P. Monk, Finite Element Methods for Maxwell’s Equations, Oxford University Press, 2003. [13] J. Nečas, Les méthodes directes en théorie des équations elliptiques, Academia, 1967. [14] J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, 1985. [15] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. [16] R. Picard, On a structural observation in generalized electromagnetic theory, J. Math. Anal. Appl. 110 (1) (1985) 247–264. [17] F. Poupaud, M. Remaki, Existence et unicité des solutions du système de Maxwell pour des milieux hétérogènes non régulier, C. R. Acad. Sci. 330 (2000) 99–103. [18] M. Reed, B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1980. [19] L. Tartar, On the characterization of traces of a Sobolev space used for Maxwell’s equations, in: Proceedings, Bordeaux, November 6–7, 1997.