UNIFORM A PRIORI ESTIMATES FOR ELLIPTIC AND STATIC MAXWELL INTERFACE PROBLEMS

We present some new a priori estimates of the solutions to three-dimensional 
elliptic interface problems and static Maxwell interface system with variable 
coefficients. Different from the classical a priori estimates, the physical 
coefficients of the interface problems appear in these new estimates 
explicitly.


1.
Introduction.Interface problems arise in many application areas, such as material sciences, fluid dynamics and electromagnetics.It is the case when different materials, fluids and media with different physical properties are involved.In this paper we are interested in the interface problems which may be modeled by the second-order elliptic equation or by the following static Maxwell interface system (cf.[9,10]) ∇ • (µ(x)H) = 0 in Ω (1.5) where Ω is an open simply-connected bounded domain in R 3 , and is assumed to be either a convex polyhedron or a domain with C 2 -smooth boundary.We shall consider that Ω is occupied by two different physical media (or materials) Ω 1 and Ω 2 , see Fig. 1 for a two-dimensional sample.The coefficient β(x) in (1.1) may represent physical parameters (e.g., diffusion, heat conductivity) of different materials in different applications, hence is only piecewise smooth in Ω.The coefficients ε(x) and µ(x) in (1.4) and (1.5) represent the electric permittivity and magnetic permeability of two different media occupying the physical domain Ω, thus may have large jumps across the interface Γ between two different media Ω 1 and Ω 2 (cf.Fig. 1).Due to the importance of interface problems in applications, extensive studies have been devoted to the mathematical behaviors and numerical solutions of the systems (1.1) and (1.2)-(1.5) in the past few decades.The regularities of the solutions to the problem (1.1) and various a priori estimates of the solutions have been widely investigated (cf.[4,17,20,23]), while regularities and edge/corner singularities of solutions were also studied, for instance, in [8,28] for time-dependent and time-harmonic Maxwell equations, and in [2,12,14] for the static Maxwell system.The primary interest of this paper is to study the mathematical behaviors of solutions to the interface systems (1.1) and (1.2)- (1.5) in terms of the discontinuous physical coefficients and how the solutions depend on the coefficients and their jumps across the interfaces, especially when the jumps are large.We shall achieve the goal through establishing some new a priori estimates of the interface solutions, in which the physical coefficients of the PDEs and their jumps appear explicitly, called uniform a priori estimates (with respect to the coefficients).Such uniform a priori estimates are essentially different from the existing estimates where physical coefficients are hidden and no any effects of coefficients on the solutions can be seen from the estimates.The new uniform a priori estimates are not only interesting mathematically, but may also provide more insights into physical behaviors of interface solutions.Moreover, a priori estimates are needed in convergence analysis of every numerical method [4,6,7,23], so the new a priori estimates may help establish more accurate error estimates where one can see clearly how the convergence of numerical methods depends on and is affected by the physical coefficients and their jumps.It may further help construct more effective numerical methods for interface problems.Very little has been done in the literature about this topic.To our knowledge, the first such uniform a priori estimates were established in [15] for the elliptic interface problem (1.1) with piecewise constant coefficients.
In this paper, we shall establish the uniform a priori estimates for the system (1.1) with general variable coefficients, and this is much more difficult than the piecewise constant case treated in our early work [15].The uniform a priori estimates are achieved by using some novel techniques, or an elegant combination of the theory of single and double layer potentials, Sobolev theory, maximum principle for elliptic equations, integral representation of piecewise harmonic functions and "formal" asymptotic expansions.The basic idea is to first reduce the a priori estimates of solutions to interface problems with variable coefficients into the estimates of piecewise harmonic solutions which are incorporated with the interface conditions of the original interface system; then the piecewise harmonic solutions will be represented by the single and double layer potentials through some integral interface equations.It is this new and elegant representation that enables us to trace closely Fig. 1. domain Ω, its subregions Ω 1 (shaded region), Ω 2 (white region) and interface Γ (boundary of the shaded region) the changes of coefficients from PDEs in every step of our estimates.Due to variable coefficients, the justification of representation of piecewise harmonic solutions by single and double layer potentials is done through two newly established uniqueness theorems for piecewise harmonic functions satisfying interface conditions; The well-posedness of the integral interface equation is demonstrated through Fredholm index theory, potential and Sobolev embedding theory as well as Hopf's maximum principles.The uniform a priori estimates obtained up to this stage are not yet optimal in terms of the jumps of coefficients in PDEs.These estimates are further improved by means of a new and powerful "formal" asymptotic expansions in terms of the jumps of coefficients.We emphasize that the final a priori estimates derived here are not only valid for variable coefficients, but also have greatly improved the results we obtained earlier in [15] for piecewise constant cases.
The new uniform a priori estimates for elliptic interface problems will be then applied to establish similar uniform estimates for the solutions to the static Maxwell interface system (1.2)- (1.5).
It is very interesting for us to notice a recent related work.Some uniform a priori estimates were obtained in [23] for the H 2 -smooth part of the solution to elliptic interface problems with most general interfaces allowed but piecewise constant coefficients.The methodology there is completely different from ours, and can not be extended to deal with piecewise variable coefficients as the cases treated in the current work.
For the ease of exposition, we will frequently use the notation " • • • " to denote "≤ C • • • " for some generic constant C > 0 which depends only on the geometric properties of Ω 1 and Ω 2 .
2. Interface problems.In this section we shall introduce a three-dimensional elliptic interface problem and static Maxwell interface system, which will be studied.

Elliptic interface problems.
The elliptic interface problem to be considered is of the form where Ω is an open simply-connected bounded domain in R 3 , and is assumed to be either a convex polyhedron or a domain with C 2 -smooth boundary.We will consider that Ω is occupied by two different materials Ω 1 and Ω 2 , with Ω 1 strictly lying inside Ω, see Fig. 1.The major case of our interest is that the coefficient β(x) represents physical parameters of two different materials in Ω 1 and Ω 2 , so will be only piecewise smooth and possibly may have large jumps across the interface Γ between Ω 1 and Ω 2 .With such background, we assume that β i ∈ C 1 ( Ωi ) for i = 1, 2, and satisfies the conditions where β1 and β2 are two positive constants measuring the magnitude of β 1 (x) and β 2 (x) in Ω 1 and Ω 2 respectively.For the interface problem, one is often more interested in the case that the magnitudes of β 1 (x) and β 2 (x) are of different scales, so it is reasonable to assume that β 1 (x) = β 2 (x) for all x ∈ Γ.
The interface Γ = ∂Ω 1 can be of arbitrary shape but is assumed to be C 2smooth.For any vector-valued function v in Ω, we shall use v 1 and v 2 to denote its restrictions to Ω 1 and Ω 2 respectively.The same convention is adopted for a scalar function v.And for definiteness, we shall define Physically, the solution u needs to satisfy certain interface conditions.The frequently encountered interface conditions are of the form: where n is the unit outward normal to ∂Ω 1 .On the exterior boundary ∂Ω, we shall consider both the Dirichlet boundary condition and the Neumann boundary condition (with ν being the unit outward normal to ∂Ω) ∂ ν u(x) = 0 on ∂Ω .
(2.5) To ensure the solvability of the problem (2.1)-(2.3)with Neumann boundary condition (2.5), the prescribed functions f and g must satisfy the consistency condition 2.2.Static Maxwell interface system.Another system to be studied in this work is the following static Maxwell interface problem: where Ω is an open simply-connected bounded domain in R 3 , and is assumed to be either a convex polyhedron or a domain with C 2 -smooth boundary.We will consider that Ω is occupied by two dielectric materials Ω 1 and Ω 2 , with Ω 1 strictly lying inside Ω, see Fig. 1.E and H are the electric and magnetic fields, and J and ρ the current and charge density.The major case of our interest is that the magnetic permeability µ(x) and the electric permittivity ε(x) of the medium are discontinuous across the interface Γ between medium Ω 1 and medium Ω 2 .Hence, we may write where ε i (x) and µ i (x) are C 1 -smooth in the individual subregion Ωi (i = 1, 2).As illustrated for the elliptic interface problem in Subsection 2.1, we may assume the following conditions for the parameters ε i (x) and µ i (x) (i = 1, 2): where εi and μi are positive constants, and It is well-known (cf.[9,10]) that the electric and magnetic fields E and H should satisfy the jump conditions across the interface Γ : ) where ρ Γ is the surface charge density.We supplement the system (2.6)-(2.9)with the perfect conductor boundary conditions where ν is the unit outward normal to ∂Ω.

3.
Preliminaries.In this section, we shall present some fundamental results from the theory of single and double layer potentials, uniqueness theorems on piecewise harmonic functions and integral representations of piecewise harmonic functions with different boundary conditions.These serve important tools in our subsequent efforts of establishing uniform a priori estimates for the solution to the elliptic interface problem (2.1)-(2.3).
3.1.Some fundamental results about single and double layer potentials.
We begin with some basic results on single and double layer potentials.Given a simply connected domain D with Lipschitz continuous boundary ∂D, let n x be the unit outward normal to ∂D at x. Then the single and double layer potentials of any density function q are respectively defined by where E(x) is the fundamental solution associated with the Laplacian: and dσ y the surface measure.Note that S D q (resp.D D q) is defined in the entire space R 3 , but we will also frequently use S D q (resp.D D q), restricted on ∂D, as a boundary integral operator on ∂D when there is no confusion caused.For a function v defined in R 3 and any x ∈ ∂D, we shall adopt whenever the limits exist.We have the following classical trace formulas (cf.[5,18,26]): where K D is the integral operator given by and The following two lemmas collect some properties about the operators S D , D D and K * D : Lemma 3.1.If D is a bounded domain with Lipschitz boundary, we have , and has a bounded inverse (cf.[18, p. 56]).2. For any q ∈ L 2 (∂D), there holds (cf.[5, p. 259 and p. 280]) We shall need the boundedness of the single and double layer potentials as stated in the following two lemmas.
Proof.We first consider the case where D is a bounded domain with a boundary of class C 2 .For any q ∈ H 1/2 (∂D), it follows from Lemma 3.2 that S D q ∈ H 3/2 (∂D) and hence continuous on ∂D by the Sobolev embedding theorem (cf.[1,3]).Since S D q is harmonic in D, this implies the boundedness of S D q in D by the maximum principle on harmonic functions (cf.[11], [24, p. 64]).To see the boundedness of S D q in R 3 \ D, it suffices to show its boundedness in D\ D with D being a suitably large domain (containing D), due to the fact that lim |x|→+∞ S D q(x) = O 1 |x| .But the conclusion follows directly from the infinite differentiability of S D q on ∂ D and the harmonicity of S D q in D\ D.
We now consider the case where D is a bounded polyhedron.We start to show the single layer S D q is well-defined for each x ∈ ∂D.For any ε such that 0 Then for all ε 1 and ε 2 satisfying 0 < ε 1 < ε 2 ≪ 1, we easily see (3.5) Noting q ∈ H 1/2 (∂D), we know q ∈ L 4 (∂D) by the Sobolev embedding theorem.Thus we derive by the Hölder inequality that As D is a polyhedron, there are only three possible locations for the point x: in the interior of a face, or on an edge or at a vertex of D. In all three cases, we can easily show by direct computations that Applying this to the inequality (3.6), we obtain Now this, along with (3.5), ensures the existence of the limit lim ε→0+ ∂D\Γε E(x − z)q(z)dσ z , i.e., S D q is well-defined for every x ∈ ∂D.Finally, applying the technique in [5, p. 226] and the estimate (3.7), we immediately get This shows S D q is continuous in R 3 , which with the decay property of S D q (cf.Lemma 3.1) leads to the boundedness of S D q in R 3 .2 Lemma 3.4.If D is a bounded (not necessarily convex) polyhedron or a bounded domain with a boundary of class C 2 , then D D q is a bounded function in R 3 for any q, which is the restriction of some function v ∈ H 2 (D) on ∂D.
Proof.We first consider the case that D is a bounded domain with a C 2 -smooth boundary.By the trace theorem, it is clear that q ∈ H 3/2 (∂D) and thus embedded in W 1,4 (∂D).Noting that W 1,4 (∂D) ֒→ W 3/4,4 (∂D) and the last statement of Lemma 3.2, we know K D q is in W 3/4,4 (∂D), thus continuous on ∂D by Sobolev embedding theorem.This further implies the continuity of (D D q) − (x) on ∂D using the evaluation formula (3.3).Then using the harmonicity of D D q(x) in D, D D q(x) must be bounded in D by the maximum principle on harmonic functions.Applying the same argument in the domain R 3 \ D, and noting the evaluation formula (3.3) and the last statement of Lemma 3.1, we can show the boundedness of D D q in R 3 \D.
Now consider the domain D to be a bounded polyhedron.As v ∈ H 2 (D), we know v ∈ C 0,1/2 ( D) by the Sobolev embedding theorem (cf.[1]).Here C 0,1/2 ( D) (cf.[11]) consists of all continuous functions such that for all x, y ∈ D, Using a similar argument as for proving Theorem 6.5.2 in [5, pp.231-236], we find that each function on the right-hand side of (3.8) is continuous in R 3 , so is the function K D q.Now the desired boundedness follows from the same argument as used in the first part.2 3.2.Uniqueness about piecewise harmonic functions.In this subsection, we present two uniqueness results, which will play an important role in the justification of an integral representation of piecewise harmonic functions.
Theorem 3.1.Let v be a function in R 3 with ṽ1 and ṽ2 being its restrictions to Ω 1 and R 3 \ Ω1 respectively.Assume ṽ1 ∈ H 2 (Ω 1 ), and ṽ1 , ṽ2 solve the problem: Then v is identically zero in R 3 .
Then by the Sobolev embedding theorem again we know with r * = 1−3/s * .In the same manner, we can show that ṽ2 = S Ω1 q ∈ C 1,r * ( D\ Ω1 ) for any domain D such that Ω 1 ⊂⊂ D by noting the infinite differentiability of S Ω1 q in D\ Ω1 .
We are now ready to show the desired result, using the classical maximum principle.We first assume that both functions ṽ1 and ṽ2 are not constant functions.By the maximum principle on harmonic functions, both ṽ1 and ṽ2 must take their maxima at a common point x 0 on Γ.But by the Hopf's maximum principle on harmonic functions (cf.[11], [24, p. 65]), we further have which contradicts with the second interface condition in (3.9).Hence it is possible that either ṽ1 or ṽ2 is a constant function.
On the other hand, if ṽ2 is constant, then ṽ2 ≡ 0 in R 3 \ Ω1 by the decay condition (3.10).Clearly we also have ṽ1 = 0 on Γ from the first interface condition, then ṽ1 must be identically zero as it is harmonic in Ω 1 .2 Borrowing the proof of Theorem 3.1, we can now show the unique solvability of the following integral equation, which will be essential to our subsequent analysis: Lemma 3.5.The integral equation Proof.By Lemma 3.2 and noting that Ω1 is a compact operator on L 2 (Γ) and H 1/2 (Γ) respectively.Hence, by the Fredholm theory for linear operators (cf.[5, p. 111 is a Fredholm operator with zero index.Therefore, for Lemma 3.5 it suffices to prove Ker Assume that q is an element in the above kernel and let v = S Ω1 q, then one can derive v = 0 following the same argument as used in the proof of Theorem 3.1 (see the derivations there starting from (3.11)).Now, q = 0 is a consequence of the isomorphism of S Ω1 (cf.Lemma 3.2). 2 We end up this subsection with another uniqueness result about piecewise harmonic functions, which is a direct consequence of Theorem 3.1.
4. Uniform a priori estimates for elliptic interface problems.With preparations in Subsections 3.1-3.3,we are now ready to establish the main results of this paper, i.e., uniform a priori estimates for the solutions to the elliptic interface problem (2.1)-(2.3)with both Dirichlet and Neumann boundary conditions (2.4) and (2.5).We shall derive the a priori H 1 -estimates in the next subsection and the H 2 -estimates in Subsection 4.2.
4.1.H 1 -estimates.We first present an auxiliary lemma which will be important to our subsequent analysis.
Lemma 4.1.Let v be a function in H 1 (Ω) such that ∂Ω vdσ = 0 and it satisfies for i = 1, 2, then it holds that Proof.When the coefficient β(x) is piecewise constant and v ∈ H 1 0 (Ω), the estimate (4.2) follows immediately from the basic harmonic extension property (cf.[27]).For the general case, since the function v has zero integral average over ∂Ω, we have by the trace theorem and the Poincaré inequality (cf.[1]) that On the other hand, using the inverse trace inequality (cf.[1]), we can find a function v1 ∈ H 1 (Ω 1 ) such that it equals v 1 on Γ and admits the estimate Noting (4.1) for v 1 , we see v 1 − v1 ∈ H 1 0 (Ω 1 ) and solves Taking e 1 = v 1 − v1 in (4.5), then using (2.2) and (4.4), we obtain β1 The combination of this, (4.3) and the fact that It is important to remark that inequality (4.2) does not hold when v 1 and v 2 are swapped.This can be verified from the following simple example: let v be a function in H 1 0 (Ω) such that v 1 (x) ≡ 1 in Ω 1 and v 2 is the solution of the following problem: We are now ready to establish the desired a priori estimates in H 1 -norm.For this, we introduce a constant k(β) = β2 / β1 .Clearly, k(β) measures the discrepancy between the coefficients β 1 (x) and β 2 (x).When no confusion is caused, we shall write k for k(β), and for any w ∈ H −1 (Ω) or w ∈ (H 1 (Ω)) ′ , we may write the norm of w simply as w −1,Ω .Theorem 4.1.Assume that u is a solution to the interface problem (2.1)-(2.3)with Dirichlet boundary condition (2.4) or Neumann boundary condition (2.5), g ∈ H −1/2 (Γ), and f ∈ H −1 (Ω) when (2.4) holds and f ∈ (H 1 (Ω)) ′ when (2.5) holds.
Proof.We first prove for Neumann boundary condition (2.5).Observing that the solution u is unique up to an additive constant, it suffices to derive the estimates (4.6) and (4.7) for the solution u with vanishing average on ∂Ω: By the Poincaré inequality, we have for all v 2 ∈ H 1 (Ω 2 ) satisfying (4.8) that Next, we assume f ∈ L 2 (Ω) and introduce two auxiliary functions ũi [11,13]) and there hold Ωi Taking v i = ũi above and noting the assumption (2.2), we have It is easy to show from (4.10) that [ū] = 0, [β∂ n ū] = g + g 1 on Γ, (4.13) where . The variational form of (4.12)-(4.14) is (4.15) Taking v = ū in the above equation and noting the fact that ∂Ω ū2 dσ = 0, we have by (4.9) and the definition of norms of linear functionals that To estimate the last two terms in (4.16), for any η ∈ H 1/2 (Γ) we introduce v η to be a function in Owing to the Green's formula (2.17) in [12, p. 28] and (4.10) we have Using (4.11) and the basic estimate v η 1,Ω η 1/2,Γ (cf.[27]), we are further led to Similarly, we can deduce which, together with (4.11), proves (4.7).Now, the desired estimates for the general f ∈ (H 1 (Ω)) ′ can be obtained by the established results (4.6) and (4.7) for f ∈ L 2 (Ω) and the usual density argument (cf.[1]).
The situation with Dirichlet boundary condition (2.4) can be handled in a same way as for the Neumann case.We first assume f ∈ L 2 (Ω) and introduce two auxiliary functions ũi ∈ H 1 0 (Ω i ) (i = 1, 2) satisfying (4.10) and then define ūi = u i − ũi (i = 1, 2).Then following the same derivations as for getting (4.15), we can show that ū ∈ H 1 0 (Ω) satisfies the variational equation Letting v = ū and using (4.9) we know β2 ∇ū 2 0,Ω2 g −1/2,Γ + g to measure the relative oscillation of the coefficient β(x) in each individual subregion, Ω 1 and Ω 2 .For the ease of exposition, we shall assume that where c1 and c2 are two positive constants independent of β 1 (x) and β 2 (x).By the mean value theorem, the assumption (4.20) implies So the relative oscillation of β(x) in each subregion Ω 1 and Ω 2 is bounded independent of β(x).Assumption (4.20)only helps avoid some unnecessary technical complications in the subsequent estimates.In fact, by slightly more careful derivations, one can achieve explicit dependence on d 1 (β) and d 2 (β) in the a priori estimates which follow.
The next estimates will be frequently used and can be checked directly according to the definition of fractional Sobolev norms: , and there hold the estimates: Now we start with the H 2 -estimates for the interface system (2.1)-(2.3)with Neumann boundary condition (2.5).For the purpose, we introduce two auxiliary functions w 1 and w 2 such that w 1 ∈ H 1 0 (Ω 1 ) and satisfies while where u i and f i are restrictions of solution u and function f to Ω i (i = 1, 2) respectively.By the standard a priori estimates for elliptic problems, we have [ w] = 0, [β∂ n w] = g + g on Γ .We shall need the next estimate for g, which follows from Lemma 4.2 and the trace theorem: Theorem 4.2.Assume that f ∈ L 2 (Ω), g ∈ H 1/2 (Γ), and u is the solution to the interface problem (2.1)-(2.3)with Neumann boundary condition (2.5), then the following a priori estimates hold: Proof.By Theorem 3.4, we can represent w1 and w2 in (4.24)-(4.25)as where ψ = w on ∂Ω, and φ ∈ H 1/2 (Γ) solves the integral equation Noting that w is harmonic in both Ω 1 and Ω 2 , we drive from (4.30) that But for S Ω1 φ we get by Lemma 3.2 that To estimate D Ω ψ, we note for any C 2 -smooth surface Γ ′ ⊂⊂ Ω and x ∈ Γ ′ , the kernel function ∂ ny E(x − y) of the operator D Ω is C ∞ -smooth.So it is easy to see from the definition of D Ω that Noting the fact that ψ = w on ∂Ω, we have by (4.32)-(4.34) that It remains to bound φ 1/2,Γ .We rewrite (4.31) as By direct computations and (4.20) we derive On the other hand, since D Ω ψ is harmonic and H 2 -smooth in any bounded domain Ω2 , with its interior boundary being Γ and an exterior boundary being a C ∞smooth surface Γ 1 such that Ω2 strictly lies in Ω 2 .Then by the regularity estimates for harmonic functions, and the same argument as used for deriving (4.34) we obtain We have by Lemma 3.2 and the Sobolev intermediate inequality (cf.[1,3]) that But noting that ψ = w on ∂Ω, by Lemma 3.2 and the relation (4.30) we derive To estimate w2 1,Ω2 , we know by the definition of w 2 that w2 = u 2 − w 2 ∈ H 1 (Ω 2 ) satisfies −∆ w2 = 0 in Ω 2 ; w2 = u 2 on Γ; ∂ ν w2 = 0 on ∂Ω.Multiplying −∆ w2 = 0 by w2 and u 2 respectively and integrating over Ω 2 yield (4.45) Combining these estimates with (4.27) yields Following the proof of Theorem 4.2 but with f = 0, w 1 = w 2 = 0 and g = 0, we come immediately to the following simple results: Theorem 4.3.Assume that f = 0, g ∈ H 1/2 (Γ) and the coefficient β(x) in (2.1) is equal to constant β1 in Ω 1 and constant β2 in Ω 2 , then the solution u to the interface problem (2.1)-(2.3)with Neumann boundary condition (2.5) admits the a priori estimates: For the Dirichlet boundary condition (2.4), we have the following similar results.
Theorem 4.4.Assume that that f ∈ L 2 (Ω), g ∈ H 1/2 (Γ), and u is the solution to the interface problem (2.1)-(2.3)with Dirichlet boundary condition (2.4), then the following a priori estimates hold: where ψ = ∂ ν w on ∂Ω, and φ ∈ H 1/2 (Γ) solves the integral equation Noting that w2 = 0 on ∂Ω, it follows from (4.49) that ), we find that the factor k−1 (β) appears necessary as long as the norms used contain the L 2 -norm part, and unnecessary when the H 1 and H 2 semi-norms are considered.The removal of this factor is of essential importance as it can be very large if β 1 (x) is much larger than β 2 (x) in magnitude.This may happen often in applications and is in fact more interesting from the physical point of view.Indeed, as we shall demonstrate, the factor k−1 appearing in the a priori estimates of Theorems 4.1-4.4can be removed.The improvements will be achieved based on the established a priori estimates in Theorems 4.1-4.4 and by means of a novel technique, which mimics the standard asymptotic analysis (cf.[21]) but is in fact not an actual asymptotic expansion.
Dirichlet boundary condition.We start with the analysis on Dirichlet boundary condition case.For simplicity, we shall write k for k(β), and di for di (β) below.Clearly for our purpose, we need only to consider the case where k(β) < 1.
5. Uniform a priori estimates for static Maxwell interface problems.In this section, we will present some uniform a priori estimates for the electric and magnetic field E and H to the static Maxwell system.In this case, the electric and magnetic fields E and H are uniquely determined by two independent systems, and thus their estimates can be obtained separately.As showed in Subsection 2.2, we assume that conditions (2.10) and (2.11) hold for the material parameters ε(x) and µ(x).

5.1.
Electric field E. It follows from the equations (2.6), (2.8), and the interface and boundary conditions (2.12)-(2.14) that the electric field E satisfies the condition n × E = 0 on ∂Ω and is governed by the following system: Using ∇ × E = 0, we know that there exists a scalar potential u ∈ H 1 0 (Ω) such that (cf.[12, p. 31]) E = −∇u.
(5.3) Substituting this into the first equation in (5.1) yields On the other hand, by direct computations we find using (5.2) and (5.In general, the magnetic field H can be described by introducing a vector field which satisfying some gauge conditions (cf.[9,10]).But we shall use a different way to represent H (cf. [19]).Noting the first equation in (5.6), we should have the following consistency condition: ∇ • J =0.
Since the domain Ω is a simply-connected convex polyhedron or a domain with a smooth boundary, we know by Theorem 3.12 and Theorem 2.17 in [2] that there exists a vector potential W in (H 1 (Ω)) 3 satisfying that W • ν = 0 on ∂Ω and with the stability estimate W 1,Ω J 0,Ω . (5.9) Combining (5.8) with (5.6) yields ∇ × (H − W) = 0 , hence there exists (cf.Theorem 2.9, [12, p. 31]) a scalar potential ω ∈ H 1 (Ω) such that H = W+∇ω in Ω. (5.10) Substituting this into the second equation of (5.6) and noting ∇ • W = 0, we obtain (5.12) On the other hand, using (5.10), the interface and boundary conditions on H and W, we can see that the potential function ω satisfies the boundary condition ∂ ν ω = 0 on ∂Ω and the interface conditions Then applying Theorem 4.6 and (5.9)-(5.10),we come immediately to the following conclusion.
Proof.For μ2 > μ1 , we apply Theorem 4.1 (with f = 0) to the system (5.11)-(5.12) to obtain that With these estimates, the results for the case that μ2 > μ1 comes readily from (5.10).
To treat the case with μ1 > μ2 , we apply Theorem 4.6 to the system (5.11)-(5.12) to obtain that now the desired estimate follows immediately from these and (5.10). 2
19))Proof.It is easy to see that the right-hand side of(3.19)lies in H 1/2 (Γ).By Lemma 3.5, there exists a unique solution φ ∈ H 1/2 (Γ) to the integral equation(3.19).Applying Theorem 3.2, we know that the following interface problem has at most one bounded solution in H 1 ∇u 2 dx,