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On the Coefficients of the Singularities of the Solution of Maxwell's Equations near Polyhedral Edges.

1. Introduction

Unlike the regularity analysis for elliptic boundary value problems in domains with geometric singularities, where there exists a unified theory based on the shift theorem (see [1-3]), the regularity analysis of the solution of Maxwell's equations has several interpretations. Most papers are concerned with the [H.sup.1]-regularity for nonconvex domains, and it is shown that the main singularity is the gradient of singular functions associated with the Laplace equation; see, for example, Birman and Solomyak [4], Bonnet-Ben Dhia et al. [5], Moussaoui [6], Hazard and Lohrengel [7], Hazard and Lenoir [8], and Lohrengel [9]. Costabel in [10] went further to address the [H.sup.1/2]-regularity for Lipschitz domains.

The [H.sup.2]-regularity of the solution of Maxwell's equations has been considered, for example, by Nkemzi [11-14]. To be more precise, the asymptotic behaviour of the solution near axisymmetric edges and their efficient numerical treatment by means of the Fourier-finite-element method on graded meshes were analyzed in [11, 14]. In [12] the problem was considered in axisymmetric domains with conical points and the asymptotic behaviour of the solution near the conical points was analyzed. Here explicit representation formulas for the coefficients of the singularities were derived. In [13] Maxwell's equations in polygonal domains were considered and formulas for the coefficients of the singularities were derived. The present paper considers Maxwell's equations in three-dimensional domains with polyhedral edges and the main focus is on the explicit description of the coefficients of the singularities. Unlike in two-dimensional case and the case of conical points where the space of the singular solutions is finite dimensional and the coefficients of the singularities are some real numbers, in three-dimensional domains with edges the space of the singular solutions is infinite dimensional and the coefficients of the singularities are functions defined along the edges. Here the space of the singular solutions along polyhedral edges is completely described.

It should be noted that a more rigorous regularity analysis based on the shift theorem for the Maxwell equations in plane domains with corners and in polyhedral domains has been carried out by Costabel et al. in [15, 16]. They showed, using the classical Mellin analysis, a technique due to Kondratev [17], that, for a given current density in [H.sup.s-1], s [greater than or equal to] 1, the electromagnetic fields either belong to the space [H.sup.s+1] or else can be split into a regular part in [H.sup.s+1] and an explicitly defined singular part up to some unknown coefficients. They represented the singular fields along the edge in a tensor product form. However, it has been shown (see, e.g., Heinrich [18]) that the numerical treatment of boundary value problems in three-dimensional domains with edges is more efficient if the singular solutions along the edges are expressed in nontensor product forms.

Asymptotic analysis of solutions of boundary value problems near geometric singularities is usually carried out by considering only the principal part of the differential operator with frozen coefficients; see, for example, [16, 17]. This is due to the fact that the qualitative behaviour of the solutions near geometric singularities depends only on the principal part of the operator, the geometry of the domain near the singularity, and the nature of the boundary conditions. On the other hand, it is known that the coefficients in the asymptotic expansion are linear continuous functions of the right hand side datum of the differential equation; see [2, 3]. Thus, formulas for the coefficients are derived by taking into account the exact differential operator under consideration. These coefficients, especially the leading coefficients, determine the actual strength of the singularities and are the principal indicators for material damage analysis, for example, fracture analysis. Hence, their computation is not only important for the numerical treatment of the boundary value problem but also important for practical applications.

In this paper we extend the [H.sup.2]-regularity analysis of the solution of time-harmonic Maxwell equations in plane domains with corners (see [13]) to three-dimensional domains with polyhedral edges. The main concern is to derive explicit computable formulas for the coefficients of the singularities. The singular functions along polyhedral edges are expressed in Fourier series and in nontensor product forms which are more suitable for constructing accurate finite-element solutions. The results presented here can be used to construct postprocessing procedures for the [H.sup.1]-nodal finite-element treatment of Maxwell's equations in domains with geometric singularities and with optimal accuracy.

This paper is organised as follows. In Section 2 we introduce the boundary value problem and related function spaces and address the issue of existence and uniqueness of the weak solution. In Section 3 we formulate in Theorems 5 and 6 the main results concerning the regularity properties of Maxwell's equations in two-dimensional domains with corners and three-dimensional domains with polyhedral edges, respectively. Sections 4 and 5 contain detailed proofs of Theorems 5 and 6, respectively.

2. The Boundary Value Problem and Functional Tools

We consider as model problem the electromagnetic fields {E, B} (E the electric field and B the magnetic field) of time-harmonic Maxwell's equations in a simply connected and bounded domain [OMEGA] [subset] [R.sup.3] with Lipschitz boundary [GAMMA] containing an isotropic and homogeneous medium subject to perfect conductor boundary conditions [5, 19, 20]:

[mathematical expression not reproducible], (1)

where the domain related parameters [epsilon] > 0, [mu] > 0, and [sigma] > 0 are, respectively, the dielectric permittivity, magnetic permeability, and electric conductivity, J is a given divergence-free current density, that is, divJ = 0, [omega] [not equal to] 0 is the pulsation of the electromagnetic fields, i is the imaginary unity, and n denotes the unit outward normal on the boundary r.

If we suppose temporarily that the vector function J is sufficiently smooth, then system (1) can be written as two decoupled systems in terms of the magnetic field B and the electric field E as follows:

curl curl B - [[alpha].sup.2]B = h in [OMEGA],

B x n = 0 on [GAMMA], (2)

curl curl E - [[alpha].sup.2]E = f in [OMEGA],

E [conjunction] n = 0 on [GAMMA], (3)

where h = [mu] curl J, [[alpha].sup.2] = [[omega].sup.2][mu]([epsilon] - i[sigma]/[omega]), and f = -i[omega][mu]J.

It follows directly from (1), (2), and (3) that the solution of problem (2) can be derived from the solution of problem (3) and vice versa. Thus it suffices to solve or analyze only one of the problems. Subsequently we will dedicate our analysis to the boundary value problem (3). For the Hilbert space formulation of (3) we introduce the function spaces; see [19, 21]:

[mathematical expression not reproducible], (4)

equipped with the norm

[mathematical expression not reproducible]. (5)

The variational formulation of problem (3) is as follows.

Find u [member of] [H.sub.0](curl, [OMEGA]) such that

a(u, v) := (curl u, curl v) - [[alpha].sup.2](u, v) = (f, v) =: f(v),

[for all]v [member of] [H.sub.0] (curl, [OMEGA]), (6)

where (*, *) denotes the usual scalar product in the Hilbert space [L.sub.2]([OMEGA]) of complex-valued functions. The well-posedness of problem (6) is addressed by the following theorem; see also [19, 22, 23].

Theorem 1. Let [OMEGA] [subset] [R.sup.3] be a bounded domain with at least Lipschitz continuous boundary [GAMMA]. Suppose [alpha] [not equal to] 0 and [[alpha].sup.2] is not an eigenvalue of the operator curl curl with electric boundary condition. Then, for any f [member of] [([L.sub.2]([OMEGA])).sup.3], there exists a unique solution u [member of] [H.sub.0](curl, [OMEGA]) of the variational problem (6). Moreover, the solution satisfies the estimate

[mathematical expression not reproducible]. (7)

We will always assume where necessary, without explicitly stating so, that the conditions of Theorem 1 are satisfied.

We observe that the operator u [??] curl curl u - [[alpha].sup.2]u from (2) and (3) is not elliptic. However, a widely used alternative formulation of the boundary value problem (3) is the so-called regularized formulation of the Maxwell equations; see [5, 7-9]. In fact, it is easily seen that the boundary value problem (3) is equivalent to the boundary value problem

curl curl E - grad div E - [[alpha].sup.2]E = f in [OMEGA],

E [conjunction] n = 0 on [GAMMA],

div E = 0 on [GAMMA], (8)

in the sense that the solution of (3) solves (8) and vice versa. We notice that the operator E [??] curl curl E-graddiv E-[[alpha].sup.2]E is elliptic. The associated Hilbert space formulation for the boundary value problem (8) is as follows.

Find u [member of] [H.sub.0](curl, div, [OMEGA]) such that

a(u, v) = f(v), [for all]v [member of] [H.sub.0](curl, div, [OMEGA]), (9)

where

[mathematical expression not reproducible]. (10)

The following theorem addresses the question of well-posedness of problem (9); see also [5, 9].

Theorem 2. Let [OMEGA] [subset] [R.sup.3] be a bounded domain with at least Lipschitz continuous boundary [GAMMA]. Suppose [alpha] [not equal to] 0 and [[alpha].sup.2] is not an eigenvalue of the Dirichlet-Laplace operator on [OMEGA]. Then, for any f [member of] [([L.sub.2]([OMEGA])).sup.3], there exists a unique solution u [member of] [H.sub.0](curl, div, [OMEGA]) of the variational problem (9). Moreover, the solution satisfies the estimate

[mathematical expression not reproducible]. (11)

The rest of this paper is dedicated to a rigorous regularity analysis of the solution of time-harmonic Maxwell equations in simply connected and bounded domains [OMEGA] [subset] [R.sup.d], d = 2, 3, with Lipschitz boundary r containing an isotropic and homogeneous medium subject to perfect conductor boundary conditions. We will systemically use the regularized formulation (8) and all derivatives should always be understood in the sense of distributions. First we state here one regularity result that is frequently quoted in the literature; see [21]. We will use the notation

[mathematical expression not reproducible]. (12)

Theorem 3. Let [OMEGA] [subset] [R.sup.d] be a bounded domain with boundary [GAMMA]. If [GAMMA] is of class [C.sup.1,1] or if [OMEGA] is a convex polygon in [R.sup.2] or a convex polyhedron in [R.sup.3], then the relation

[H.sub.N]([OMEGA]) = [H.sub.0](curl, div, [OMEGA]) (13)

holds and on these spaces the norms [mathematical expression not reproducible] and [parallel]*[[parallel].sub.H(curl,div,[OMEGA])] are equivalent.

On the other hand, if [OMEGA] [subset] [R.sup.d] is a nonconvex polygon or polyhedron, then the space [H.sub.N]([OMEGA]) is a proper closed subset of the space [H.sub.0](curl, div, [OMEGA]) and the following result holds; see [4, 6].

Theorem 4. The space [H.sub.0](curl, div, [OMEGA]) can be split as a direct sum of the form

[H.sub.0] (curl, div, [OMEGA]) = [H.sub.N]([OMEGA]) [direct sum] grad S, (14)

where S denotes the space of functions spanned by the singular functions associated with the Dirichlet boundary value problem for the Laplace equation in [OMEGA] [subset] [R.sup.d].

An immediate consequence of the regularity Theorems 3 and 4 is that, in nonconvex polygons or polyhedrons, the weak solution of the boundary value problem (8) does not belong to the space [H.sup.1] and can therefore not be approximated by means of the usual [H.sup.1]-conforming nodal finite-element method.

3. Corner and Edge Singularities for Maxwell's Equations

In this section we formulate the main results of this paper. The proofs which are very lengthy in nature will be carried out in subsequent sections.

3.1. Corner Singularities for Maxwell's Equations. Here we consider the electric field u of time-harmonic Maxwell's equations in a simply connected and bounded domain [OMEGA] [subset] [R.sup.2] with Lipschitz boundary [GAMMA], formally the variational solution of the boundary value problem

curl curl u - graddiv u - [[alpha].sup.2]u = f in [OMEGA],

u [conjunction] n = 0 on [GAMMA],

div u = 0 on [GAMMA], (15)

where f [member of] [([L.sub.2]([OMEGA])).sup.2] and the parameter [alpha] [not equal to] 0 are given.

Now, suppose that the boundary [GAMMA] of [OMEGA] consists of finitely many disjoint analytic arcs [[GAMMA].sub.j], j = 1, ..., J, such that [GAMMA] = [[union].sup.J.sub.j=1] [[bar.[GAMMA]].sub.j], where the segments [[GAMMA].sub.j] are numbered according to the positive orientation, that is, in anticlockwise direction. Let the endpoints of each [[GAMMA].sub.j] be denoted by [A.sub.j] and let the solid angle at [A.sub.j] be denoted by [[omega].sub.j], where 0 < [[omega].sub.j] < 2[pi]. We denote by [r.sub.j] and [[theta].sub.j] (resp., [x.sub.j] and [y.sub.j]) local polar (resp., Cartesian) coordinates attached to the vertex [A.sub.j], such that

[x.sub.j] = [r.sub.j] cos([[theta].sub.j]),

[y.sub.j] = [r.sub.j] sin([[theta].sub.j]); (16)

that is, [[GAMMA].sub.j] is supported by the line [[theta].sub.j] = [[omega].sub.j] and [[GAMMA].sub.j+1] is on the line [[theta].sub.j] = 0. Suppose that the domain [OMEGA] coincides near each singular point [A.sub.j] with a circular sector [[??].sub.j] with radius [R.sub.j] and angle [[omega].sub.j] that is,

[mathematical expression not reproducible]. (17)

The boundary [partial derivative][[??].sub.j] will be represented subsequently as [mathematical expression not reproducible], where

[mathematical expression not reproducible]. (18)

We define with respect to the vertex [A.sub.j] a smooth truncation function [[eta].sub.j] [member of] D([bar.[OMEGA]]) which depends only on the distance [r.sub.j] from [A.sub.j] by

[mathematical expression not reproducible], (19)

where [R.sub.j] is taken from (17); that is, supp([[eta].sub.j]) [subset] [[??].sub.j]. Furthermore, we define on each sector neighbourhood [[??].sub.j] of the vertex [A.sub.j] the functions

[mathematical expression not reproducible], (20)

where f = ([f.sub.1], [f.sub.2]), u = ([u.sub.1], [u.sub.2]) and the parameter a are taken from (15), [[eta].sub.j] is as defined in (19), and

[mathematical expression not reproducible]. (21)

Our main result on corner singularity is the following.

Theorem 5. For each f = ([f.sub.1], [f.sub.2]) [member of] [([L.sub.2]([OMEGA])).sup.2] and [alpha] [not equal to] 0, let u [member of] [H.sub.0](curl, div, [OMEGA]) be the variational solution of the boundary value problem (15). Let [mathematical expression not reproducible], then there exist coefficients [[gamma].sub.jk] such that the solution u can be split as a sum in the form

[mathematical expression not reproducible] (22)

with w = ([w.sub.1], [w.sub.2]) [member of] [([H.sup.2]([OMEGA])).sup.2]. The coefficients [[gamma].sub.jk] of the asymptotic expansion (22) are given explicitly by the formula

[mathematical expression not reproducible], (23)

where the function [f.sub.j] = ([f.sub.1j], [f.sub.2j]) is defined in (20). The constants [R.sub.j] and [[omega].sub.j] and the local Cartesian and polar coordinates [x.sub.j], [y.sub.j] and [r.sub.j], [[theta].sub.j] are as specified in (17) and (16). Moreover, there exists a constant C > 0 independent of f and u such that

[mathematical expression not reproducible]. (24)

The proof of Theorem 5 will be carried out in Section 4; see Theorem 9.

3.2. Edge Singularities for Maxwell's Equations. Let Q [subset] [R.sup.3] be a simply connected and bounded domain with Lipschitz boundary [partial derivative]Q. For f [member of] [([L.sub.2](Q)).sup.3] and [alpha] [not equal to] 0, we consider the variational solution u = ([u.sub.1], [u.sub.2], [u.sub.3]) [member of] [H.sub.0](curl, div, Q) of the boundary value problem

curl curl u - graddiv u - [[alpha].sup.2]u = f in Q,

u [conjunction] n = 0 on [partial derivative]Q,

div u =0 on [partial derivative]Q. (25)

Since we are interested only in the asymptotic behaviour of the solution u near straight edges of the domain Q, we can assume, without loss of generality, that the domain Q is a prismatic cylinder; that is, it has the form Q = [OMEGA] x (0, l) with a real constant l > 0 and a bounded domain Q [subset] [R.sup.2] with piecewise smooth boundary [GAMMA] and such that for each ([x.sub.1], [x.sub.2], [x.sub.3]) [member of] Q, ([x.sub.1], [x.sub.2]) [member of] [OMEGA], and [x.sub.3] [member of] (0, l). In this way we can use the same notations as in Section 3.1 for [OMEGA]. In particular, the edges of the domain Q are [E.sub.j] = [A.sub.j] x (0, l) and the measure of the interior angle along the edge [E.sub.j] is [[omega].sub.j], j = 1, ..., J. We associate with each edge [E.sub.j] a wedge [G.sub.j] = [[??].sub.j] x (0, l), where [[??].sub.j] [subset] [OMEGA] is as defined in (17). Further we introduce on [G.sub.j] the functions

[mathematical expression not reproducible], (26)

where the functions f = ([f.sub.1], [f.sub.2], [f.sub.3]), u = ([u.sub.1], [u.sub.2], [u.sub.3]), and the parameter [alpha] are taken from (25), [[eta].sub.j] is from (19), and [mathematical expression not reproducible] are as defined in (21). Obviously [f.sup.*.sub.j] = ([f.sup.*.sub.1j], [f.sup.*.sub.2j], [f.sup.*.sub.3j]) [member of] [([L.sub.2]([G.sub.j])).sup.3].

Our main result on edge singularity is the following.

Theorem 6. For f [member of] [([L.sub.2](Q)).sup.3] and [alpha] [not equal to] 0, let u [member of] [H.sub.0](curl, div, Q) be the variational solution of the boundary value problem (25). Let [[lambda].sub.jk] := k[pi]/[[omega].sub.j], k [member of] N, [[omega].sub.j] [not equal to] [pi], j = 1, ..., J. If [[lambda].sub.jk] [not equal to] 2, k [member of] N, j = 1, ..., J, then there exist unique functions [mathematical expression not reproducible] and [mathematical expression not reproducible] such that the solution u [member of] [H.sub.0](curl, div, Q) can be split into a regular and a singular part in the form

[mathematical expression not reproducible]. (27)

The functions [T.sub.j] and [T.sub.3j] are fixed kernels defined by

[mathematical expression not reproducible]. (28)

The coefficients [[PSI].sub.jk] and [[PSI].sub.j] of the asymptotic expansion (27) can be expressed in Fourier series in the form

[mathematical expression not reproducible], (29)

where the Fourier coefficients [[gamma].sub.jkn] and [[gamma].sub.jn] are given explicitly by the formulas

[mathematical expression not reproducible]. (30)

Here, the function [f.sup.*.sub.j] = ([f.sup.*.sub.1j], [f.sup.*.sub.2j], [f.sup.*.sub.3j]) is as defined in (26). The constants [R.sub.j] and [[omega].sub.j] and the local Cartesian and polar coordinates [x.sub.j], [y.sub.j], [[theta].sub.j], and are as defined in (17) and (16). In (27) the symbol "*" denotes convolution product in the variable [x.sub.3]; that is,

[mathematical expression not reproducible]. (31)

The proof of Theorem 6 will be carried out in Section 5; see Theorem 24.

4. Maxwell's Equations in Two-Dimensional Domains with Corners

In this section, we consider in greater detail the structure of the solution of the Maxwell equations (15) in two-dimensional domains with corners and show how the results of Theorem 5 are derived.

4.1. Maxwell's Equations in a Bounded Sector. For purely mathematical reasons we consider first a slightly modified boundary value problem for the Maxwell equations in a circular sector; see Figure 1. The results of this subsection are largely found in Nkemzi [13] and will be kept very brief.

Let [??] denote a circular sector in [R.sup.2] with radius R, interior angle [omega] [not equal to] [pi], and boundary [mathematical expression not reproducible]; see Figure 1. We assume that the Cartesian coordinate system of [x.sub.1] and [x.sub.2] is positioned such that the vertex A of [??] coincides with the origin and the side [[GAMMA].sub.2] is supported by the [x.sub.1]-axis.

For [??] = ([f.sub.1], [f.sub.2]) [member of] [L.sub.2][([??]).sup.2], we consider the unique weak solution [??] = ([u.sub.1], [u.sub.2]) [member of] [H.sub.0](curl, div, [??]) of the boundary value problem

[mathematical expression not reproducible]. (32)

Local polar coordinates r and [theta] in [??] with respect to the vertex A are related to the Cartesian coordinates [x.sub.1] and [x.sub.2]; namely,

[x.sub.1] = r cos [theta],

[x.sub.2] = r sin [theta],

0 < r < R, 0 < [theta] < [omega]. (33)

Accordingly, the sector domain [??] is transformed by the one-to-one mapping into the rectangle

K = {(r, [theta]) : 0 < r < [R.sub.0], 0 < [theta] < [omega]} (34)

in the polar coordinate system. By the transformation (33), each function [??] = [??]([x.sub.1], [x.sub.2]) defined on [??] is mapped uniquely to some function u = u(r, [theta]) defined on K by

u(r, [theta]) = [??](r cos [theta], r sin [theta]). (35)

Similarly, each vector field [??] = ([u.sub.1]([x.sub.1], [x.sub.2]), [u.sub.2]([x.sub.1], [x.sub.2])) defined on K is mapped uniquely to some vector field u = ([u.sub.r](r, [theta]), [u.sub.[theta]](r, [theta])) defined on K by

[mathematical expression not reproducible]. (36)

The boundary value problem (32) can be solved explicitly. In fact, we have the following result which can be verified by direct substitution.

Lemma 7. The weak solution [??] = u = ([u.sub.r](r, [theta]), [u.sub.[theta]](r, [theta])) of the boundary value problem (32) can be represented in Fourier series in the form

[mathematical expression not reproducible], (37)

where the Fourier coefficients {[u.sub.k] = ([u.sub.rk], [u.sub.[theta]k]) : k [member of] N} satisfy the relations

[mathematical expression not reproducible]. (38)

Using the explicit representation formulas (37)-(38) for the solution of the boundary value problem (32) and taking into account relation (36) one can derive various regularity properties for the solution.

The main result of this subsection is the following; see, for example, [13, 15, 16], for the proof.

Theorem 8. Let [??] be a circular sector with angle [omega] [member of] (0, 2[pi]), [omega] [not equal to] [pi]. Let [[lambda].sub.k] = k[pi]/[omega], k [member of] N. Then for each [??] [member of] [([L.sub.2]([??])).sup.2] the solution [??] [member of] [H.sub.0](curl, div, [??]) of the boundary value problem (32) has the following additional regularity properties:

(a) There exists a constant C > 0 independent of [??] and [??] such that the

[mathematical expression not reproducible]. (39)

That is, the condition div [??] [member of] [L.sub.2]([??]) is always satisfied.

(b) If [[lambda].sub.k] > 1, k [member of] N (i.e., [??] is convex), then [??] [member of] [([H.sup.1]([??])).sup.2] and there exists a constant C > 0 independent off and [??] such that

[mathematical expression not reproducible]. (40)

(c) If [[lambda].sub.k] > 2, k [member of] N (i.e., 0 < [omega] < [pi]/2), then [??] [member of] [([H.sup.2]([??])).sup.2] and there exists a constant C > 0 independent of [??] and [??] such that

[mathematical expression not reproducible]. (41)

(d) If [[lambda].sub.k] [not equal to] 2, k [member of] N, then [??] can be split into a regular and a singular part in the form

[mathematical expression not reproducible]. (42)

The coefficients [[gamma].sub.k] are given explicitly by the formula

[mathematical expression not reproducible]. (43)

Moreover, there exists a constant C > 0 independent of [??] such that

[mathematical expression not reproducible]. (44)

4.2. Maxwell's Equations in Plane Domains with Corners. We can now make definite statements on the regularity properties of the solution u [member of] [H.sub.0](curl, div, [OMEGA]) of the Maxwell boundary value problem (15) in bounded domains [OMEGA] [subset] [R.sup.2] with piecewise smooth boundary r. We will use the same notations as in Section 3.1.

Let u [member of] [H.sub.0](curl, div, [OMEGA]) be the solution of (15). Then the function [u.sub.j] := [[eta].sub.j]u, where [[eta].sub.j] is the smooth truncation function from (19), belongs to the space [H.sub.0](curl, div, [[??].sub.j]) and is the unique weak solution of the boundary value problem

[mathematical expression not reproducible], (45)

where the function [f.sub.j] = ([f.sub.1j]([x.sub.j], [y.sub.j]), [f.sub.2j]([x.sub.j], [y.sub.j])) [member of] [([L.sub.2]([[??].sub.j])).sup.2] is as defined in (20).

We observe that problems (45) and (32) are similar and therefore their solutions have the same regularity properties as described in Theorem 8. On the other hand, the solution [u.sub.j] [member of] [H.sub.0](curl, div, [[??].sub.j]) of problem (45) coincides near the vertex [A.sub.j] of [OMEGA] to the solution u [member of] [H.sub.0](curl, div, [OMEGA]) of problem (15). Thus the two solutions have the same asymptotic behaviour near the vertex [A.sub.j]. Taking into consideration the fact that singularity is a local property and the technique for coupling local and global regularity properties (see [2, 18]), we obtain directly from Theorem 8 the following properties for the solution u [member of] [H.sub.0](curl, div, [OMEGA]) of problem (15).

Theorem 9. For each f [member of] [([L.sub.2]([OMEGA])).sup.2], let u [member of] [H.sub.0](curl, div, [OMEGA]) be the solution of the boundary value problem (15). Let [[lambda].sub.jk] = k[pi]/[[omega].sub.j], k [member of] N, j = 1, ..., J, and [[omega].sub.j] [not equal to] [pi]. Then the solution u has the following additional regularity properties:

(a) div u [member of] [L.sub.2]([OMEGA]) and there exists a constant C > 0 independent off and u such that

[mathematical expression not reproducible]. (46)

(b) If [[lambda].sub.jk] > 1, k [member of] N, j = 1, ..., J (i.e., [OMEGA] is convex), then u [member of] [H.sup.1.sub.N]([OMEGA]) and there exists a constant C > 0 independent of f and u such that

[mathematical expression not reproducible]. (47)

(c) If [[lambda].sub.jk] > 2, k [member of] N, j = 1, ..., J (i.e., 0 < [[omega].sub.j] < [pi]/2 for all j), then u [member of] [([H.sup.2]([OMEGA])).sup.2] and there exists a constant C > 0 independent of f and u such that

[mathematical expression not reproducible]. (48)

(d) If [[lambda].sub.jk] = 2, k [member of] N, j = 1, ..., J, then the solution u can be split into a regular and a singular part in the form

[mathematical expression not reproducible]. (49)

The coefficients [[gamma].sub.jk] are given explicitly by the formula

[mathematical expression not reproducible], (50)

where [f.sub.j] = ([f.sub.1j]([x.sub.j], [y.sub.j]), [f.sub.2j]([x.sub.j], [y.sub.j])) is defined in (20) and the local Cartesian coordinates [x.sub.j] and [y.sub.j] are defined in (16). Moreover, there exists a constant C > 0 independent off such that

[mathematical expression not reproducible]. (51)

4.3. Some Auxiliary Boundary Value Problems with a Parameter. Let [mathematical expression not reproducible] and [mathematical expression not reproducible] be given functions. We consider in this subsection the following boundary value problems with parameter on the sector domain [??] [subset] [R.sup.2] (see Figure 1), formulated in polar coordinates:

[mathematical expression not reproducible], (52)

[mathematical expression not reproducible], (53)

where the vector field f = ([f.sub.r](r, [theta]), [f.sub.[theta]](r, [theta])) from (52) in polar coordinates is linked to the vector field [??] = ([f.sub.1]([x.sub.1], [x.sub.2]), [f.sub.2]([x.sub.1], [x.sub.2])) in Cartesian coordinates according to relation (36) and the scalar function f = f(r, [theta]) from (53) is linked to the scalar function [??] = [??]([x.sub.1], [x.sub.2]) according to relation (35). The symbol [xi] denotes a positive real parameter. The interest on the boundary value problems (52) and (53) is purely mathematical and there is no evidence that these problems have any practical applications. The results obtained in this subsection will be used for the analysis of the regularity properties of the solution of Maxwell's equations near polyhedral edges.

The main results concerning the regularity properties of the solution of the boundaryvalue problems (52) and (53) are formulated in Theorems 10 and 11.

Theorem 10. Let [??] [member of] [L.sub.2](K), [xi] > 0, and [[lambda].sub.k] = k[pi]/[omega], k [member of] N, with [omega] [not equal to] [pi]. Then the boundary value problem (53) has a unique variational solution u = [??] [member of] [H.sup.1.sub.0]([??]):

(a) If [[lambda].sub.1] > 1, then the solution [??] [member of] [H.sup.2]([??]) and there exists a constant C > 0 independent of [??],[??], and [xi] such that

[mathematical expression not reproducible]. (54)

(b) If 0 < [[lambda].sub.1] < 1 and [[lambda].sub.2] [not equal to] 1, then there exists a coefficient [[gamma].sub.1] = [[gamma].sub.1]([xi]) such that the solution [??] [member of] [H.sup.1.sub.0]([??]) can be split as a sum of a regular and a singular part in the form

[mathematical expression not reproducible]. (55)

The coefficient [[gamma].sub.1] = [[gamma].sub.1]([xi]) is given explicitly by the formula

[mathematical expression not reproducible]. (56)

Furthermore, there exists a constant C > 0 independent of [??] and [xi] such that

[mathematical expression not reproducible]

For the proof of Theorem 10, see [24, pp. 171-174].

Theorem 11. Let [??] = ([f.sub.1], [f.sub.2]) [member of] [([L.sub.2]([??])).sup.2], [xi] > 0, and [[lambda].sub.k] = k[pi]/[omega], k [member of] N, with [omega] [not equal to] [pi]. Then there exists a unique variational solution [??] = ([u.sub.1], [u.sub.2]) [member of] [H.sub.0](curl, div, [??]) of the boundary value problem (52). If [[lambda].sub.k] [not equal to] 2, k [member of] N, then there exist coefficients [[gamma].sub.1k] = [[gamma].sub.1k]([xi]) such that the solution [??] can be split as a sum of a regular and a singular part in the form

[mathematical expression not reproducible]. (57)

The coefficients [[gamma].sub.1k] = [[gamma].sub.1k]([xi]) are given explicitly by the formula

[mathematical expression not reproducible]. (58)

Furthermore, there exists a constant C > 0 independent of [??] and [xi] such that

[mathematical expression not reproducible].

The proof of Theorem 11 is very lengthy and will be broken down into several lemmas as follows.

Lemma 12. The solution [??] = u = ([u.sub.r], [u.sub.[theta]]) of problem (52) can be represented in a Fourier series in the form

[mathematical expression not reproducible], (59)

where the Fourier coefficients [u.sub.k] = ([u.sub.rk], [u.sub.[theta]k]), k [member of] N, are given explicitly by the formulas

[mathematical expression not reproducible]. (60)

Lemma 13. Let

[mathematical expression not reproducible]. (61)

If 0 < [[lambda].sub.k] < 2, then there exists a constant C > 0 independent of [f.sub.k] and [xi] such that

[mathematical expression not reproducible]. (62)

Proof. Application of Cauchy-Schwartz inequality and the substitution [xi][tau] = s lead to the estimates

[mathematical expression not reproducible]. (63)

Lemma 14. Let

[mathematical expression not reproducible]. (64)

If [[lambda].sub.k] > 2, then there exists a constant C > 0 independent of [xi] such that

(i)

[mathematical expression not reproducible], (65)

(ii)

[mathematical expression not reproducible], (66)

(iii)

[mathematical expression not reproducible]. (67)

Proof. Using the substitution [xi][tau] = s one easily verifies the estimates

[mathematical expression not reproducible]. (68)

The following lemmas can be proved by analogy. We omit the proofs for the sake of brevity.

Lemma 15. Let

[mathematical expression not reproducible]. (69)

Then there exists a constant C > 0 independent of [f.sub.k] and [xi] such that

[mathematical expression not reproducible]. (70)

Lemma 16. Let

[mathematical expression not reproducible]. (71)

Then there exists a constant C > 0 independent of [xi] such that

(i)

[mathematical expression not reproducible], (72)

(ii)

[mathematical expression not reproducible], (73)

(iii)

[mathematical expression not reproducible]. (74)

Lemma 17. Let

[mathematical expression not reproducible]. (75)

Then there exists a constant C > 0 independent of [f.sub.k] and [xi] such that

(i)

[mathematical expression not reproducible], (76)

(ii)

[mathematical expression not reproducible], (77)

(iii)

[mathematical expression not reproducible]. (78)

Lemma 18. Let

[mathematical expression not reproducible]. (79)

Then there exists a constant C > 0 independent of [f.sub.k] and [xi] such that

(i)

[mathematical expression not reproducible], (80)

(ii)

[mathematical expression not reproducible], (81)

(iii)

[mathematical expression not reproducible]. (82)

The proof of Theorem 11 can now be summarised as follows.

Proof (Theorem 11). The results of Theorem 11 follow by combining Lemmas 12-18, taking note of the definition of the representation of norms of functions as series of norms of their Fourier coefficients by means of generalized Parseval identities; see, for example, [13, 18].

5. Maxwell's Equations in Domains with Polyhedral Edges

In this section, we consider and analyze the Maxwell equations (25) in three-dimensional domains with polyhedral edges and prove Theorem 6.

5.1. Maxwell's Equations in a Three-Dimensional Wedge. We consider first a three-dimensional domain of the form [OMEGA] = [??] x (0, l), where l > 0 is a real constant and [??] [subset] [R.sup.2]; see Figure 1. We will use the same notations as in Section 4.1 for [??]. Thus the boundary [GAMMA] of [OMEGA] can be represented in the form [gamma] = [[bar.T].sub.0] [union] [[bar.T].sub.1] [union] [[bar.T].sub.2] [union] [[bar.T].sub.3] [union] [[bar.T].sub.4], where [T.sub.3] = [OMEGA] x {0}, [T.sub.4] = [OMEGA] x {l}, and [T.sub.j] = [[GAMMA].sub.j] x (0, l), j = 0, 1, 2; see Figure 2.

For a given vector field f [member of] [([L.sub.2]([OMEGA])).sup.3], let u [member of] [H.sub.0](curl, div, [OMEGA]) be the variational solution of the boundary value problem

[mathematical expression not reproducible]. (83)

We observe that the systems of trigonometric functions {sin(n[pi][x.sub.3]/l) : n [member of] N} and {cos(n[pi][x.sub.3]/l) : n [member of] [N.sub.0]} ([N.sub.0] := {0, 1, 2, ...}) are orthogonal and complete in [L.sub.2](0, l); see [25, 26]. Thus functions v [member of] [([L.sub.2]([OMEGA])).sup.3] can be characterized by their Fourier coefficients as follows.

Lemma 19. (1) Let v = ([v.sub.1], [v.sub.2], [v.sub.3]) [member of] [([L.sub.2]([OMEGA])).sup.3]. Then there exist in [([L.sub.2]([??])).sup.3] Fourier coefficients {[v.sub.n] = ([v.sub.1n], [v.sub.2n], [v.sub.3n]) : n [member of] [N.sub.0]} of v defined by

[mathematical expression not reproducible] (84)

and satisfying the relations

[mathematical expression not reproducible]. (85)

Moreover, Parseval's identity holds in the form

[mathematical expression not reproducible]. (86)

(2) For v [member of] [H.sub.0](curl, div, [OMEGA]), relations (84), (85), and (86) hold and additionally

[mathematical expression not reproducible]. (87)

For the study of the Fourier coefficients {[u.sub.n] = ([u.sub.1n], [u.sub.2n], [u.sub.3n]) : n [member of] [N.sub.0]} of the solution u [member of] [H.sub.0](curl, div, [OMEGA]) of the boundary value problem (83), we introduce on [??] the spaces

[mathematical expression not reproducible], (88)

where n = ([n.sub.1], [n.sub.2]) denotes the unit outward normal on the boundary [[GAMMA].sub.1] [union] [[GAMMA].sub.2]. These spaces are equipped with the norm

[mathematical expression not reproducible]. (89)

Clearly, the spaces Y([??]) and [Y.sub.0]([??]) endowed with the norm (89) are Hilbert spaces. Indeed, we observe the identities

[mathematical expression not reproducible]. (90)

Lemma 20. Let {[v.sub.n] = ([v.sub.1n], [v.sub.2n], [v.sub.3n]) : n [member of] [N.sub.0]} denote the Fourier coefficients of a function v [member of] [H.sub.0] (curl, div, [OMEGA]) defined according to relation (84). Then [v.sub.n] [member of] [Y.sub.0]([??]) for any n [member of] [N.sub.0].

Proof. It follows from (84) that [v.sub.0] = (0, 0, [v.sub.30]). The completeness relationship (87) infers then that [v.sub.30] [member of] [H.sup.1]([??]) and consequently [v.sub.0] [member of] Y([??]). Further, with the help of the triangle inequality and relation (87) we get the estimates

[mathematical expression not reproducible]. (91)

Hence, taking into account the definition of Y([??]) and the completeness relationship (87) we get [v.sub.n] [member of] Y([??]), n [member of] [N.sub.0]. The boundary conditions follow from the boundary conditions in [H.sub.0](curl, div, [OMEGA]).

Lemma 21. For f [member of] [([L.sub.2]([OMEGA])).sup.3], let u [member of] [H.sub.0](curl, div, [OMEGA]) be the solution of the boundary value problem (83). Let {[u.sub.n] = ([u.sub.1n], [u.sub.2n], [u.sub.3n]) =: ([[??].sub.n], [u.sub.3n]) : n [member of] [N.sub.0]} and {[f.sub.n] = ([f.sub.1n], [f.sub.2n], [f.sub.3n]) =: ([[??].sub.n], [f.sub.3n]) : n [member of] [N.sub.0]} denote the Fourier coefficients of u and f, respectively, defined according to (84). Then the functions [u.sub.n] = ([[??].sub.n], [u.sub.3n]), n [member of] [N.sub.0], satisfy the relations [[??].sub.n] [member of] [H.sub.0](curl, div, [??]) and [u.sub.3n] [member of] [H.sup.1.sub.0]([??]) and are the solutions of the boundary value problems

[mathematical expression not reproducible]. (92)

Proof. Problems (92) are obtained directly from (83) by substituting the functions u and f by their respective Fourier series defined according to (85), differentiating term by term and comparing coefficients. The assertions [[??].sub.n] [member of] [H.sub.0](curl, div, [??]) and [u.sub.3n] [member of] [H.sup.1.sub.0]([??]) follow from Lemma 20.

We observe that in local polar coordinates r and [theta] problems (92) take the form

[mathematical expression not reproducible]. (93)

We will need the following notations:

[mathematical expression not reproducible], (94)

where [f.sub.n] = ([f.sub.rn], [f.sub.[theta]n], [f.sub.3n]) and [u.sub.n] = ([u.sub.rn], [u.sub.[theta]n], [u.sub.3n]), n [member of] [N.sub.0] are taken from (93). Obviously [f.sup.*.sub.n] = ([f.sup.*.sub.rn], [f.sup.*.sub.[theta]n], [f.sup.*.sub.3n]) [member of] [([L.sub.2]([??])).sup.3].

Theorem 22. Let [??] be a circular sector with angle [omega] [member of] (0, 2[pi]), [omega] [not equal to] [pi], and let [[lambda].sub.k] := k[pi]/[omega], k [member of] N. For each [f.sub.n] = ([f.sub.1n], [f.sub.2n], [f.sub.3n]) [member of] [([L.sub.2]([??])).sup.3], let [u.sub.n] = ([u.sub.1n], [u.sub.2n], [u.sub.3n]) [member of] [Y.sub.0]([??]), n [member of] [N.sub.0], be the solutions of the boundary value problems (92). If [[lambda].sub.k] = 2, k [member of] N, then there exist coefficients [[gamma].sub.nk] and [[gamma].sub.n] such that the solutions [u.sub.n], n [member of] [N.sub.0], can be represented in the form

[mathematical expression not reproducible]. (95)

The coefficients [[gamma].sub.nk] and [[gamma].sub.n] of the expansion (95) are given explicitly by the formulas

[mathematical expression not reproducible], (96)

where the functions [f.sup.*.sub.n] = ([f.sup.*.sub.rn], [f.sup.*.sub.[theta]n], [f.sup.*.sub.3n]), n [member of] [N.sub.0], are as defined in (94). Moreover, there exists a constant C > 0 independent of [f.sub.n] such that

[mathematical expression not reproducible], (97a)

[mathematical expression not reproducible], (97b)

[mathematical expression not reproducible], (97c)

[mathematical expression not reproducible]. (97d)

Proof. Relations (95), (96), (97a), (97b), (97c), and (97d) are obtained by a straightforward application of Theorems 10 and 11 to the boundary value problems (93) with [xi] = n[pi]/l and taking note of the modified right hand side defined in (94).

Theorem 23. Let [OMEGA] = [??] x (0, l) be a three-dimensional wedge and [[lambda].sub.k] = k[pi]/[omega], k [member of] N, [omega] [not equal to] [pi]. For each f = ([f.sub.1], [f.sub.2], [f.sub.3]) [member of] [([L.sub.2]([OMEGA])).sup.3], let u = ([u.sub.1], [u.sub.2], [u.sub.3]) [member of] [H.sub.0](curl, div, [OMEGA]) be the solution of the boundary value problem (83). If [[lambda].sub.k] [not equal to] 2, k [member of] N, then there exist unique functions [mathematical expression not reproducible] and [[PSI].sub.3] [member of] [H.sup.1-[lambda]](0, l) such that the solution u [member of] [H.sub.0](curl, div, [OMEGA]) can be split in the form

[mathematical expression not reproducible], (98)

where

[mathematical expression not reproducible]. (99)

In (98), the symbol "*" denotes the convolution product in the variable [x.sub.3]; that is,

[mathematical expression not reproducible]. (100)

The coefficients [[gamma].sub.nk] and [[gamma].sub.n] are as defined in Theorem 22.

Proof. The expressions (98), (99), and (100) are direct consequences of Lemma 21 and Theorem 22, taking into consideration relations (35) and (36). The inequalities (97c) and (97d) and Parseval's identity (86) imply that the Fourier coefficients [w.sub.n] = ([w.sub.1n], [w.sub.2n], [w.sub.3n]), n [member of] [N.sub.0], satisfy the estimate

[mathematical expression not reproducible] (101)

Hence, w [member of] [([H.sup.2]([OMEGA])).sup.3]; see [24, Theorem 3.2]. The inequalities (97a) and (97b) lead to the estimates

[mathematical expression not reproducible] (102)

which by the generalized Riesz-Fischer theorem imply the existence of functions [mathematical expression not reproducible] and [mathematical expression not reproducible] whose Fourier coefficients are {[[gamma].sub.nk] : n [member of] N} and {[[gamma].sub.n] : n [member of] [N.sub.0]}, respectively.

5.2. Singularities near Polyhedral Edges. In this subsection, we consider and analyze the regularity properties of the solution of the Maxwell equations (25) in general three-dimensional domains with straight edges bounded by plain faces, that is, polyhedral edges. In fact it would be sufficient for us to consider three-dimensional domains of the form Q = [OMEGA] x (0, l), where l > 0 is a real constant and [OMEGA] [subset] [R.sup.2] is a general bounded domain with piecewise smooth boundary r. We will use the same notations as in Section 3 for [OMEGA] and Q.

Thus the edges of Q are [E.sub.j] = [A.sub.j] x (0, l), j = 1, ..., J, and the boundary [partial derivative]Q is the union of the disjoint faces [T.sub.j] := [[GAMMA].sub.j], x (0, l), j = 1, ..., J, and the two bases [T.sub.j+1] := [omega] x {0} and [T.sub.J+2] := [OMEGA] x {l}. We assume that, for ([x.sub.1], [x.sub.2], [x.sub.3]) [member of] [OMEGA], ([x.sub.1], [x.sub.2]) [member of] [OMEGA] and [x.sub.3] [member of] (0, l). We associate with each edge [E.sub.j] a three-dimensional wedge [G.sub.j] := [[??].sub.j] x (0, l), where [[??].sub.j] [subset] [OMEGA] is a circular sector; see (17) and (18). Thus the boundary [partial derivative][G.sub.j] of [G.sub.j] is the union of the disjoint faces [mathematical expression not reproducible], and [T.sub.j4] := [[??].sub.j] x {l}. Furthermore, we define on each G, a smooth truncation function [[eta].sub.j] = [[eta].sub.j]([x.sub.1], [x.sub.2]) = [[eta].sub.j]([r.sub.j]); see (19).

For f [member of] [([L.sub.2]([OMEGA])).sup.3] and [alpha] [not equal to] 0, let u [member of] [H.sub.0](curl, div, Q) be the variational solution of the Maxwell equations (25). Then the function [u.sub.j] := [[eta].sub.j]u which is defined on the wedge [G.sub.j] belongs to the space [H.sub.0](curl, div, [G.sub.j]) and is the unique weak solution of the boundary value problem

[mathematical expression not reproducible], (103)

where the function [f.sub.j] = ([f.sub.1j], [f.sub.2j], [f.sub.3j]) [member of] [([L.sub.2]([G.sub.j])).sup.3] is given by

[mathematical expression not reproducible]. (104)

We observe that problems (83) and (103) are similar and therefore their regularity properties are described by Theorem 23. On the other hand, since the solution u [member of] [H.sub.0] (curl, div, Q) of problem (25) coincides with the solution [u.sub.j] [member of] [H.sub.0](curl, div, [G.sub.j]) of problem (103) near the edge [E.sub.j] of Q, Theorem 23 also addresses the regularity properties of the solution u of (25). Thus we obtain the following theorem.

Theorem 24. For f [member of] [L.sub.2][([OMEGA]).sup.3] and [alpha] [not equal to] 0, let u [member of] [H.sub.0] (curl, div, Q) be the variational solution of the boundary value problem (25). Let [[lambda].sub.jk] := k[pi]/[[omega].sub.j], k [member of] N, and [[omega].sub.j] [not equal to] [pi], j = 1, ..., J. If [[lambda].sub.jk] [not equal to] 2, k [member of] N, j = 1, ..., J, then there exist functions [mathematical expression not reproducible] and [mathematical expression not reproducible] such that the solution u [member of] [H.sub.0](curl, div, Q) can be represented in the form

[mathematical expression not reproducible]. (105)

The functions [T.sub.j] and [T.sub.3j], are fixed kernels given by

[mathematical expression not reproducible]. (106)

The coefficients [[PSI].sub.jk] and [[PSI].sub.j] are defined explicitly by

[mathematical expression not reproducible], (107)

where the coefficients [[gamma].sub.jnk] and [[gamma].sub.jn] are as defined in (30).

http://dx.doi.org/10.1155/2016/7965642

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Alexander von Humboldt Foundation (AvH), Bonn, Germany, and the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy.

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Boniface Nkemzi

Department of Mathematics, University of Buea, Buea, Cameroon

Correspondence should be addressed to Boniface Nkemzi; nkemzi@yahoo.com

Received 26 August 2015; Accepted 17 December 2015

Academic Editor: Haipeng Peng

Caption: Figure 1: A circular sector.

Caption: Figure 2: A three-dimensional wedge.
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