Regularity of Weak Solutions to Maxwell’s Equations

Abstract

In this paper, we study the steady-state Maxwell’s equations. The weak solution defined in weak formulation is considered, and the global existence is obtained in general bounded open domain. The interior estimates of the weak solution are obtained, where the coefficient matrix is assumed to be BMO with small seminorm. The 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 estimates.

1. Introduction

It is well known that the classical Maxwell’s equations can be written in differential form as follows:where is the permittivity of the electric field, the permeability of the magnetic field, and the conductivity of the material.

When the material is conductive, the current displacement can be ignored since it is very small compared with the eddy current . Then, we have the following evolution system:

If is assumed to be time independent, we can obtainwhere .

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]). Thus, it has a different structure with general elliptic equations, and the regularity should be restudied. The 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]. They showed that the weak solution is Hölder continuous with the assumption that 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 estimate has been given in [5]. The estimate can be found in [6], in which is assumed to be in the VMO space and the domain is assumed to be .

In this paper, we establish the existence theorem of weak solution of (3) in general bounded domain and study the regularity with the assumptions that is defined on and has the small BMO seminorm (see Definition 2).

Another goal of this paper is to establish the regularity of the following system:which can describe time-harmonic electromagnetic field. We prove that if the matrix is uniformly positive definite and has the small BMO seminorm, then the weak solution of system (4) belongs to . We weaken the assumption in [7] that is Lipschitz continuous and also generalize the assumption in [8] that is a bounded scalar function and the real part of has a positive lower bound.

The 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.

2. Preliminaries

We introduce some notations and lemmas here.(1) is an open ball centered at origin with radius and (2)For two vector fields and , and define the scalar product and the cross product, respectively(3) is the gradient of ; is the divergence of ; and is the curl of .(4)For a locally integrable function , is the average of over . is called the Hardy–Littlewood maximal function of . We also use , if is not defined outside of .(5).(6).(7).

Definition 1. We say that the matrix is uniformly positive definite if there exists :

Definition 2. The matrix is called -vanishing ifwhere is the average of over .

Lemma 1 (see [9]). Let be a measurable function in , and be constants. Then, for any ,where is a constant depending only on , and .

Lemma 2 (see [10]). (a)If , then for every , where is a constant which depends only on the dimension .(b)If , with , then andwhere depends only on and the dimension .

Lemma 3 (see [11]). Assume that and are measurable sets of , , and that there exists an such that , and for all and for all with implying that . Then

Lemma 4 (see [7]). Let be a bounded domain with connected boundary. Let with for some . Then, there exists such that . Moreover, in and , where depends only on and .

3. Main Theorems

In the following, we assume that and is a small positive constant. Our first theorem is the well-posedness in . Considering the weak solution defined in weak formulation (see Definition 3), we have the following theorem.

Theorem 1. Let be a bounded open domain and be uniformly positive definite. Then, for , the Dirichlet problemhas a unique weak solution withwhere depends on .

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 , 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.

Theorem 2. Let and . There is a small such that for all with uniformly positive definite and -vanishing, and for all with , if is a weak solution of (13) in , then belongs to with the estimatewhere the constant is independent of and .

Remark 2. We remark that our assumption that is -vanishing weakens the assumption in [6] that 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.

Theorem 3. Let and . There is a small such that for all with uniformly positive definite and -vanishing, and for all with , if is a weak solution ofthenwhere the constant is independent of and .

Remark 3. Here, . The existence of weak solution of (16) in can be found in [8]. We point out that our assumption that is -vanishing weakens the assumption in [7] that is Lipschitz continuous.

Remark 4. If , we will have the following interior Hölder estimate:where . We should remark that the Hölder estimate has been established in [8], but they did not give the concrete value of the Hölder exponent.

4. Proofs of Main Theorems

4.1. Proof of Theorem 1

Let us first prove the following important equality.

Lemma 5. Let be bounded and . We then have the following identity:

Proof. Let , then we haveWe multiply this identity by and integrateSince , by using the definition of weak derivative, we haveSimilarly,Hence, we obtainThen, if we take in , (19) can be proved by the density.

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

Remark 6. The identity (19) implies that the norm of is equivalent to the right hand of it raised to the power . This also means we can consider the weak solution of (13) in with the norm instead of , where .

Thus, we can define the weak solution of (13) as follows.

Definition 3. A vector field is said to be a weak solution of (13), if the following identity holds:for any .

Now, we give the proof of Theorem 1.

Proof of Theorem 1. In order to prove the existence and uniqueness of weak solution, we define the bilinear form as follows:for any . Since is uniformly positive definite with , we haveThis means that the bilinear form is coercive on . Moreover,Now fix and setThis is a bounded linear functional on .
Thus, we can apply Lax–Milgram theorem (see [14]) to find a unique function satisfyingfor all ; is consequently the weak solution of (13).
Moreover, we can choose as a test function to getwhere depends on and the theorem is proved.

4.2. Proof of Theorem 2

For simplicity, we take and assume . We will locally approximate solution (13) by a function satisfying a suitable homogeneous problem. We need some lemmas here. The first one is the following energy estimate.

Lemma 6. Assume that is a weak solution of (13) in . Thenfor any , where depends on .

Proof. First note that , so we haveUsing the identity , we obtainwhere we used the Hölder inequality, and depends on .
Moreover, we know that . By Lemma 5, we haveTherefore,where we have used the fact that in .

Lemma 7. For any , there is a small such that for any weak solution of (13) in withthere exists a weak solution ofsuch that

Proof. Firstly, we claim that, for any , there is a small and a weak solution of (38), such thatSuppose it is false. Then, we can find and sequences , and , such that is a weak solution ofwithBut for any weak solution ofwe haveBy (42) and Poincaré inequality, is bounded in . So there exist and a subsequence, still denoted as , such that strongly in and weakly in . Since is a bounded sequence of constant matrices, there exist a constant matrix and a subsequence, still denoted as , such that . Combining (42), we know that has a subsequence, denoted also as , such that strongly in . Thus, satisfies the following system:Take , where satisfies the following system:Using Theorem 1, we haveMoreover, satisfies system (43), andThis meansBut 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 satisfies the following system:By Lemma 6, we haveHere, we used the interior regularity of (see Theorem 2.2 of [5]). Combining (37) and (40), we concludeby taking and satisfying the last identity. This finishes the proof of this lemma.

Lemma 8. Let be a weak solution of (13) in . There exists a constant such that for any , there exists a small , ifthen

Proof. By assumption (53), there is a point such that for all ,Since , we conclude thatApplying Lemma 7 to and , we haveNote that satisfies system (38). We can find a constant such thatTake . Now suppose thatWhen , then . Hence, we haveWhen , then . Hence, by (55), we haveThe above two inequalities show thatCombining (59) and (62), we haveConsequently,by taking satisfying the last inequality above. This finishes the proof.

Lemma 9. Let be a weak solution of (13) in . There exists a constant such that for any and , there exists a small , ifthen

Proof. We argue by contradiction. If conclusion (66) is false, thenLet us consider the functionswith . Then, it is easy to check that satisfy the conditions of Lemma 8, andScaling back in the above estimate yieldswhich is contradiction to (65).

Now take , and the corresponding given by Lemma 8.

Lemma 10. Assume that is uniformly positive definite and -vanishing. Suppose that is a weak solution of (13) in andLet be a positive integer and set . Then, we have

Proof. We intend to prove by induction on . For the case , letThen, in view of (71), Lemmas 9 and 3, we see that , and our conclusion is valid for .
Assume that the conclusion is valid for some positive integer . Let and corresponding . Then is the weak solution ofand the following inequality holds:Thus, we haveThis estimate in turn completes the induction on .

Finally, we give the proof of Theorem 2.

Proof of Theorem 2. We will consider the case only. The case is Theorem 1. Since , then by Lemma 2. In view of Lemma 1, there is a constant depending only on such thatLet us suppose that and are small enough such thatThese assumptions are reasonable, since we can multiply system (13) by a small constant depending on and .
Let us computewhere we used Lemma 10 and selected such that . Then by Lemma 1, we get , and this gives . The proof is now completed.

4.3. Proof of Theorem 3

Now we give the proof of Theorem 3

Proof of Theorem 3. Since in , then, by Lemma 4, there exists satisfying and . Thus, we can rewrite system (16) asThus, we can use Theorem 2 and getThis completes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this work.

Acknowledgments

The authors are sincerely grateful to Prof. Dongsheng Li for very helpful suggestions and comments.