W1,p Regularity of Weak Solutions to Maxwell’s Equations

In this paper, we study the steady-state Maxwell’s equations.)e weak solution defined in weak formulation is considered, and the global existence is obtained in general bounded open domain. )e interior W1,p(2≤p<∞) estimates of the weak solution are obtained, where the coefficient matrix is assumed to be BMO with small seminorm. )e main analytical tools are the Vitali covering lemma, the maximal function technique, and the compactness method. We also consider the time-harmonic Maxwell’s equations and obtain the interior W1,p estimates.


Introduction
It is well known that the classical Maxwell's equations can be written in differential form as follows: where ε is the permittivity of the electric field, μ the permeability of the magnetic field, and σ the conductivity of the material. When the material is conductive, the current displacement εE t can be ignored since it is very small compared with the eddy current σE. en, we have the following evolution system: If H is assumed to be time independent, we can obtain where A(x) � σ − 1 . System (3) is an important mathematical model for the study of the penetration of magnetic field in materials. Yin [1] pointed out that this system is degenerated by the classical definition (see [2]). us, it has a different structure with general elliptic equations, and the regularity should be restudied. e existence of a unique weak solution can be found in [3,4]. By using Campanato theory, this system has been studied in [1,3,5]. ey showed that the weak solution is Hölder continuous with the assumption that A is a positive bounded scalar function. In [1,5], they got the local Hölder continuity. Afterwards, Kang and Kim [3] obtained the global Hölder continuity on the Lipschitz domain. For the higher regularity, the interior C 1,α estimate has been given in [5]. e W 1,p estimate can be found in [6], in which A − 1 is assumed to be in the VMO space and the domain is assumed to be C 1 .
In this paper, we establish the existence theorem of weak solution of (3) in general bounded domain and study the W 1,p (2 ≤ p < ∞) regularity with the assumptions that A(x) is defined on R 3 and has the small BMO seminorm (see Definition 2).
Another goal of this paper is to establish the W 1,p (2 ≤ p < ∞) regularity of the following system: which can describe time-harmonic electromagnetic field. We prove that if the matrix A is uniformly positive definite and has the small BMO seminorm, then the weak solution of system (4) belongs to W 1,p . We weaken the assumption in [7] that A is Lipschitz continuous and also generalize the assumption in [8] that A is a bounded scalar function and the real part of A has a positive lower bound. e remaining sections are organized in the following way. In Section 2, we introduce the relevant concepts and lemmas. In Section 3, we state our main theorems and give some remarks concerning them. In Section 4, the proofs of our main results are given.

Preliminaries
We introduce some notations and lemmas here.
(1) B r � y ∈ R 3 : |x| < r is an open ball centered at origin with radius r and Ω r � Ω∩B r (2) For two vector fields a ∈ R 3 and b ∈ R 3 , a · b and a × b define the scalar product and the cross product, respectively (3) ∇g is the gradient of g; ∇ · u is the divergence of u; and ∇ × u is the curl of u. (4) For a locally integrable function f, is the average of f over B r (x).
is called the Hardy-Littlewood maximal function of f. We also use M Ω (f), if f is not defined outside of Ω.
Definition 1. We say that the matrix A(x) � (a ij (x)) 3×3 is uniformly positive definite if there exists λ > 0: where A B r is the average of A over B r (x).
(a) If f ∈ L 1 (R n ), then for every α > 0, where C is a constant which depends only on the dimension n. and where C depends only on p and the dimension n.
Lemma 3 (see [11]). Assume that C and D are measurable sets of R n , C ⊂ D ⊂ B 1 , and that there exists an ε > 0 such that |C| < ε|B 1 |, and for all x ∈ B 1 and for all r Lemma 4 (see [7]).
where C depends only on p and Ω.

Main Theorems
In the following, we assume that B 6 ⊂ B R ⊂ Ω and δ is a small positive constant. Our first theorem is the well-posedness in H 1 (Ω; R 3 ). Considering the weak solution defined in weak formulation (see Definition 3), we have the following theorem.

Theorem 1. Let Ω be a bounded open domain and
has a unique weak solution u ∈ H 1 0 (Ω; where C depends on λ. Remark 1. We point out that this theorem holds in general bounded open domain, which may not be Lipschitz, like the Reifenberg flat domain (see [11,12]). When zΩ ∈ C 0,1 , this theorem has been proved in many papers (see [1,3,4,13]) because for the Lipschitz domain, the following identity (19) can be easily obtained by performing integration by parts.
Here, we will give a proof of (19) in general bounded open domain by density.
where the constant C is independent of u and f.

Remark 2.
We remark that our assumption that A is (δ, R)-vanishing weakens the assumption in [6] that A − 1 belongs to VMO. Since the linear system (13) is degenerate (see [1]), the regularity theory of elliptic systems cannot be applied directly. We established some useful lemmas to handle the difficulty. Our basic tools are the Vitali covering lemma, the Hardy-Littlewood maximal function, and the compactness method, which have been used in [11] to deal with elliptic equations. then where the constant C is independent of u and f.
}. e existence of weak solution of (16) in H(curl, Ω) can be found in [8]. We point out that our assumption that A is (δ, R)-vanishing weakens the assumption in [7] that A is Lipschitz continuous.
Remark 4. If p > 3, we will have the following interior Hölder estimate: where α � 1 − 3/p. We should remark that the Hölder estimate has been established in [8], but they did not give the concrete value of the Hölder exponent.

Proofs of Main Theorems
4.1. Proof of eorem 1. Let us first prove the following important equality.

Lemma 5.
Let Ω ⊂ R 3 be bounded and u ∈ H 1 0 (Ω; R 3 ). We then have the following identity: We multiply this identity by u and integrate Since u ∈ H 1 0 (Ω; R 3 ), by using the definition of weak derivative, we have Similarly, Hence, we obtain en, if we take ϕ ⟶ u in H 1 0 (Ω; R 3 ), (19) can be proved by the density.

Remark 5.
In the proof of (19), we do not use integration by parts. us, we do not need the domain to be Lipschitz.

Remark 6.
e identity (19) implies that the norm of H 1 0 (Ω; R 3 ) is equivalent to the right hand of it raised to the power 1/2. is also means we can consider the weak solution of (13) in H 0 (div, Ω) with the norm ‖∇ × u‖ L 2 (Ω) instead of us, we can define the weak solution of (13) as follows.

Definition 3.
A vector field u ∈ H 0 (div, Ω) is said to be a weak solution of (13), if the following identity holds: for any v ∈ H 0 (div, Ω). Now, we give the proof of eorem 1.
Proof of eorem 1. In order to prove the existence and uniqueness of weak solution, we define the bilinear form as follows: for any u, v ∈ H 0 (div, Ω). Since A(x) is uniformly positive definite with λ, we have

Journal of Mathematics 3 is means that the bilinear form B(u, v) is coercive on
Now fix f ∈ L 2 (Ω; R 3 ) and set is is a bounded linear functional on H 0 (div, Ω). us, we can apply Lax-Milgram theorem (see [14]) to find a unique function u ∈ H 0 (div, Ω) satisfying for all v ∈ H 0 (div, Ω); u is consequently the weak solution of (13). Moreover, we can choose u as a test function to get where C depends on λ and the theorem is proved.

Proof of eorem 2.
For simplicity, we take R > 6 and assume B 6 ⊂ B R ⊂ Ω. We will locally approximate solution (13) by a function satisfying a suitable homogeneous problem. We need some lemmas here. e first one is the following energy estimate.

Lemma 6.
Assume that u is a weak solution of (13) in B 1 . en for any ϕ ∈ C ∞ 0 (B 1 ), where C depends on λ.
Proof. First note that ϕ 2 u ∈ H 1 0 (B 1 ; R 3 ), so we have (33) where we used the Hölder inequality, and C depends on λ.
Moreover, we know that ϕu ∈ H 1 0 (B 1 ; R 3 ). By Lemma 5, we have where we have used the fact that ∇ · u � 0 in B 1 .

Lemma 7.
For any ε > 0, there is a small δ � δ(ε) > 0 such that for any weak solution u of (13) there exists a weak solution v of Proof. Firstly, we claim that, for any η > 0, there is a small δ � δ(η) > 0 and a weak solution v of (38), such that 1 Suppose it is false. en, we can find η 0 > 0 and se- But for any weak solution v k of we have By (42) and Poincaré inequality, is a bounded sequence of constant matrices, there exist a constant matrix A 0 and a subsequence, still denoted as A k , such that A k ⟶ A 0 . Combining (42), we know that A k ∞ k�1 has a subsequence, denoted also as A k , such that A k ⟶ A 0 strongly in L 2 (B 4 ).
us, u 0 satisfies the following system: Take ) satisfies the following system: Using eorem 1, we have Moreover, v k satisfies system (43), and is means But this is a contradiction to (44), and thus, (40) holds and the claim is proved. Now, we give the proof of (39). It is easy to see that u − v satisfies the following system: By Lemma 6, we have Here, we used the interior W 1,∞ regularity of v (see eorem 2.2 of [5]). Combining (37) and (40), we conclude 1 by taking η and δ satisfying the last identity. is finishes the proof of this lemma.

Journal of Mathematics
Note that v satisfies system (38). We can find a constant N such that Take M 2 : � max 3 3 , 4N 2 . Now suppose that When r ≤ 1, then B r (x 1 ) ⊂ B 2 . Hence, we have When r > 1, then B r (x 1 ) ⊂ B 3r (x 0 ). Hence, by (55), we have e above two inequalities show that Combining (59) and (62), we have Consequently, by taking η satisfying the last inequality above. is finishes the proof.
Let us consider the functions u(x) � u(rx) r , with x ∈ B 1 . en, it is easy to check that u, A, f satisfy the conditions of Lemma 8, and x ∈ Ω: M |∇u| 2 > M 2 ∩B 1 < ε B 1 .
Scaling back in the above estimate yields which is contradiction to (65).
Now take M, ε, and the corresponding δ > 0 given by Lemma 8.

Lemma 10.
Assume that A is uniformly positive definite and (δ, R)-vanishing. Suppose that u is a weak solution of (13) in Ω B R B 6 and Let k be a positive integer and set ε 1 � 10 3 ε. en, we have Proof. We intend to prove by induction on k. For the case k � 1, let en, in view of (71), Lemmas 9 and 3, we see that |C| ≤ ε 1 |D|, and our conclusion is valid for k � 1.
Assume that the conclusion is valid for some positive integer k ≥ 2. Let u 1 � u/M and corresponding f 1 � f/M. en u 1 is the weak solution of and the following inequality holds: x ∈ B 1 : M ∇u 1 2 > M 2 < ε B 1 .
us, we have