Homogenization of Maxwell's Equations in Domains with Dense Perfectly Conducting Grids
We consider Maxwell’s equations in domains that are complements to connected, grid-like sets formed by intersecting thin wires. We impose the boundary conditions that correspond to perfectly conducting wires, and study the asymptotic behavior of solutions as grids are becoming thinner and denser. We...
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irk-123456789-1245862017-09-30T03:03:45Z Homogenization of Maxwell's Equations in Domains with Dense Perfectly Conducting Grids Khruslov, E.Ya. We consider Maxwell’s equations in domains that are complements to connected, grid-like sets formed by intersecting thin wires. We impose the boundary conditions that correspond to perfectly conducting wires, and study the asymptotic behavior of solutions as grids are becoming thinner and denser. We derive a homogenized system of equations describing the leading term of the asymptotics. Assuming that a Korn-type inequality holds, we validate the homogenization procedure. 2005 Article Homogenization of Maxwell's Equations in Domains with Dense Perfectly Conducting Grids / E.Ya. Khruslov // Український математичний вісник. — 2005. — Т. 2, № 1. — С. 109-142. — Бібліогр.: 14 назв. — англ. 1810-3200 2000 MSC. 35B27, 78M40 http://dspace.nbuv.gov.ua/handle/123456789/124586 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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We consider Maxwell’s equations in domains that are complements to connected, grid-like sets formed by intersecting thin wires. We impose the boundary conditions that correspond to perfectly conducting wires, and study the asymptotic behavior of solutions as grids are becoming thinner and denser. We derive a homogenized system of equations describing the leading term of the asymptotics. Assuming that a Korn-type inequality holds, we validate the homogenization procedure. |
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Khruslov, E.Ya. |
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Khruslov, E.Ya. Homogenization of Maxwell's Equations in Domains with Dense Perfectly Conducting Grids Український математичний вісник |
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Khruslov, E.Ya. |
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Khruslov, E.Ya. |
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Homogenization of Maxwell's Equations in Domains with Dense Perfectly Conducting Grids |
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Homogenization of Maxwell's Equations in Domains with Dense Perfectly Conducting Grids |
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Homogenization of Maxwell's Equations in Domains with Dense Perfectly Conducting Grids |
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Homogenization of Maxwell's Equations in Domains with Dense Perfectly Conducting Grids |
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Homogenization of Maxwell's Equations in Domains with Dense Perfectly Conducting Grids |
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homogenization of maxwell's equations in domains with dense perfectly conducting grids |
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Інститут прикладної математики і механіки НАН України |
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2005 |
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Homogenization of Maxwell's Equations in Domains with Dense Perfectly Conducting Grids / E.Ya. Khruslov // Український математичний вісник. — 2005. — Т. 2, № 1. — С. 109-142. — Бібліогр.: 14 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT khrusloveya homogenizationofmaxwellsequationsindomainswithdenseperfectlyconductinggrids |
| first_indexed |
2025-07-09T01:40:05Z |
| last_indexed |
2025-07-09T01:40:05Z |
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1837131585288142848 |
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Український математичний вiсник
Том 2 (2005), № 1, 109 – 142
Homogenization of Maxwell’s Equations
in Domains with Dense Perfectly
Conducting Grids
Evgenii Ya. Khruslov
Abstract. We consider Maxwell’s equations in domains that are com-
plements to connected, grid-like sets formed by intersecting thin wires.
We impose the boundary conditions that correspond to perfectly con-
ducting wires, and study the asymptotic behavior of solutions as grids
are becoming thinner and denser. We derive a homogenized system of
equations describing the leading term of the asymptotics. Assuming that
a Korn-type inequality holds, we validate the homogenization procedure.
2000 MSC. 35B27, 78M40.
Key words and phrases. Maxwell’s equations, grid structure, meso-
scopic characteristics, homogenization, effective equations.
Introduction
Grid structures like metal wire gratings with cells of various form are
widely used as elements of radio devices, in particular, antennas, radio re-
lays [7], various devices designed for physical experiments [12], etc. They
provide shielding, polarizing, retarding (accelerating), and other types
of control of electromagnetic fields. In order to study electrodynamic
properties of a grid structure, it is necessary to study a boundary value
problem for Maxwell’s equations in a complex domain that is a comple-
ment to the grid, with certain boundary conditions on the grid surface.
Often, a grid can be thought of (with good accuracy) as perfectly con-
ducting, so that the corresponding boundary conditions must reflect the
fact that the tangential component of the electric field vanishes on the
grid surface.
If a grid is dense, it is practically impossible to solve the problem ex-
actly, because of very complicated behavior of the electromagnetic field
near the grid. On the other hand, an exact knowledge of the field near
Received 2.11.2004
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
110 Homogenization of Maxwell’s Equations...
the grid is often needless, since we are usually interested, in applications,
in certain integral characteristics such as eigenfrequencies, reflection and
transmission coefficients, etc., which are determined by the “mean” field
or by the behavior of the field far from the grid. Therefore, it is reason-
able to assume that if a grid is sufficiently dense, then it acts similarly
to some effective continuous medium (or film), so that its influence can
be described, approximately, by homogenized differential equations (or
homogenized boundary conditions). In order to derive these equations
(boundary conditions), one has to analyze the asymptotic behavior of so-
lutions of Maxwell’s equations in domains with grids that are becoming
denser. This is the main problem of homogenization theory for partial dif-
ferential equations. It was formulated, in a rigorous mathematical form,
by V.A.Marchenko in the middle of 1960s and since that, it has been
attracting the attention of mathematicians thus stimulating the develop-
ment of the homogenization theory as a whole.
The problem consists in the study of the asymptotic behavior of solu-
tions of Maxwell’s equations in a domain Ω ⊂ R3 with perfectly conduct-
ing inclusions Fε ⊂ Ω of arbitrary form; the inclusions depend of a small
parameter ε such that, as ε → 0, they are becoming “rarer” but filling
Ω “denser”. The problem in such a general formulation has appeared to
be very difficult. Therefore, researchers (mathematicians as well as radio
physicists) have being concentrated on two polar particular cases having
prior interest for applications: (i) inclusions Fε are assumed to have a
fine-grained structure, i.e., they are unions of small disjoint components
(grains); such structures are used in the synthesis of artificial dielectrics;
(ii) inclusions are connected domains like grids formed by thin intersect-
ing wires.
For the case of fine-grained periodic inclusions, a rigorous solution of
the problem was obtained by V. V. Zhikov and O. A. Nazarova in [11] and
[14] (see also [13]). They proved that in this case, the homogenized models
are described by the standard Maxwell’s equations for (non-conducting)
continuous media with constant effective parameters: the dielectric per-
meability ε and the magnetic permittivity µ. This result was completely
in accordance with the physical intuition and was not unexpected: the
same results had been obtained earlier for dielectric inclusions [1] and for
perfectly conducting inclusions of small concentration [4].
A different situation occurs in the case of grid-type inclusions. De-
spite of the fact that grid structures are of much higher interest for radio
physicists then the fine-grained ones, at present there is no rigorous math-
ematical solution in this case. Moreover, the type of homogenized equa-
tions is unclear, even intuitively, although radio physicists and engineers
are using, quite effectively, various approximate models [7].
E. Ya. Khruslov 111
In the paper, we propose a conditional solution of the problem. Name-
ly, we derive homogenized equations describing the main term of the
asymptotics as ε→ 0, under the assumption that a Korn-type inequality
holds true. This inequality is to be satisfied (uniformly with respect to
ε) by vector functions of a special class introduced for domains Ω \ Fε.
The question whether there exists a grid structure {Fε} for which this
inequality is satisfied, remains open. But, assuming that this inequality
is satisfied for grids of a certain type, the derived equations are in fact the
effective equations of electrodynamics of a continuous medium equivalent
to the grid structure.
1. Problem Statement and Qualitative
Description of Main Result
Let Ω ⊂ R3 be a bounded domain with smooth connected boundary
∂Ω and let {Fε, ε > 0} be a family of closed sets in Ω depending on a
small parameter ε > 0. We assume that the structure of sets of this
family is as follows:
1. For all ε > 0, the sets Fε are connected and belong to a fixed
subdomain G that is compact in Ω and has a smooth boundary
∂G; the boundary ∂Fε of Fε is also smooth (i.e., it is a smooth
manifold of dimension 2);
2. As ε→ 0, the intersection of Fε and of its complement Ωε = Ω \Fε
with any cube K(x, h) of size h > 0 centered at x ∈ G (i.e. the
sets Fε ∩K(x, h) and Ωε ∩K(x, h)), are becoming non-empty and
connected.
Consider in Ωε = Ω \ Fε the initial boundary value problem for
Maxwell’s equations
∂Hε
∂t
+ rotEε = 0, x ∈ Ωε, t > 0, (1.1)
−∂Eε
∂t
+ rotHε = J, x ∈ Ωε, t > 0, (1.2)
n ∧ Eε = 0, x ∈ ∂Fε ∪ ∂Ω, (1.3)
Eε(x, 0) = E0(x), Hε(x, 0) = H0(x), x ∈ Ωε, (1.4)
where Eε = Eε(x, t) and Hε = Hε(x, t) are the vectors of electric and
magnetic fields, respectively, E0(x) and H0(x) are the given vectors of
the initial distribution of these fields, J = J(x, t) is the given current,
and n is the normal vector to ∂Fε or ∂Ω (the symbol ∧ denotes the vector
112 Homogenization of Maxwell’s Equations...
product). The boundary condition (1.3) means that the tangential com-
ponent of the electric field vanishes on the perfectly conducting grid Fε
as well as on the external boundary ∂Ω. Notice that in this problem, the
boundary conditions on Fε are essential whereas the boundary conditions
on ∂Ω are irrelevant: one can assume any boundary conditions on ∂Ω,
including nonhomogeneous ones.
Remark 1.1. Equations (1.1) and (1.2) yield the following equations
divEε = ρ, (1.5)
∂ρ
∂t
+ divJ = 0, ρ(x, 0) = divE0(x), (1.6)
and, if divH0(x) = 0 (which is naturally assumed), then
divHε = 0. (1.7)
For the sake of simplicity, we will assume that the given E0(x), H0(x),
and J(x, t) are sufficiently smooth, that their supports (with respect to
x) are compact in Ω \ Ḡ, and that |J(x, t)| < C for all t.
It is known (see, e.g., [2]) that there exists a unique solution {Eε(x, t),
Hε(x, t)} of problem (1.1)–(1.4). The present paper aims at describing the
asymptotic behavior of this solution as ε → 0. Notice that assumptions
1 and 2 above (which are assumptions about the geometrical structure
of sets {Fε, ε > 0}) do not guarantee the existence of the asymptotics
of {Eε(x, t), Hε(x, t)}. In Section 2 we will formulate additional (quan-
titative) conditions which do guarantee the existence of the (weak) limit
lim
ε→0
{Eε(x, t), Hε(x, t)} = {E(x, t), H(x, t)}. These conditions have to be
verified in each particular case (for a particularly given system of sets
{Fε, ε > 0}), which would give, simultaneously, the coefficients of the
homogenized equations for the limiting field {E(x, t), H(x, t)}.
In Section 2 we will also formulate the main result of the paper. But
it seems reasonable to give first a qualitative description of the expected
result. Namely, we give the form of the homogenized equations, leaving
the precise formulation of conditions under which these equations are
valid, to Section 3.
It turns out that the limiting field {E(x, t), H(x, t)} satisfies the fol-
lowing system of equations in Ω:
∂H
∂t
+ rotE = 0, x ∈ Ω, t > 0, (1.8)
−∂E
∂t
+rotH−
t∫
0
L(E(·, τ)−gradΦ(·, τ))dτ = J x ∈ Ω, t > 0, (1.9)
E. Ya. Khruslov 113
C
∂2Φ
∂t2
− div(LgradΦ) − div(LE) = 0, x ∈ G, t > 0 (1.10)
and the following boundary and initial conditions:
n ∧ E = 0, x ∈ ∂Ω, t > 0, (1.11)
(LgradΦ)n = (LE)n, x ∈ ∂G, t > 0, (1.12)
E(x, 0) = E0(x), H(x, 0) = H0(x), Φ(x, 0) =
∂Φ
∂t
(x, 0) = 0 (1.13)
(the index n denotes the normal component of a vector). Here C = C(x)
is a positive function and L = L(x) is a symmetric nonnegative tensor;
they are given on G and are extended by zero outside G. The additional
unknown function Φ(x, t) (the potential) can be uniquely determined
only on G ⊂ Ω. For definiteness, we assume that Φ(x, t) is extended
by zero outside G; then the initial boundary value problem (1.8)–(1.13)
has a unique solution {E(x, t), H(x, t),Φ(x, t)} (in an appropriate class
of functions).
The system of equations (1.8)–(1.10) can be written in the following,
more physical, form:
∂H
∂t
+ rotE = 0,
−∂E
∂t
+ rotH = J + Jeff,
divE = ρ+ ρeff,
∂ρeff
∂t
+ div Jeff = 0,
where
Jeff =
t∫
0
L(E − grad Φ)dτ (1.14)
and
ρeff = CΦ. (1.15)
The physical meaning of this system is as follows: perfectly conducting
grids Fε behave, as ε → 0, as a continuous medium, in which a current
and a charge are induced, with the densities Jeff(x, t) and ρeff(x, t), respec-
tively, determined by (1.14) and (1.15). Equalities (1.14) and (1.15) play
the role of the constitutive equations for an effective medium. But, un-
like the classical constitutive equations for electromagnetic media, these
equations involve an additional scalar field, Φ(x, t) (the electric poten-
tial), which cannot be excluded from them. Such a medium can be called
inductive-capacitive.
114 Homogenization of Maxwell’s Equations...
Remark 1.2. The homogenized equations (1.8)–(1.10) describing an
effective continuous medium are relevant in the case of a “bulk” distri-
bution of grids Fε in G ⊂ Ω (see assumption 1). A “surface” distribution
of grids is also of great interest; here the sets Fε are concentrating, as
ε → 0, in an arbitrarily small neighborhood of some surface Γ ⊂ Ω [7].
In this case, grids behave like a continuous film, and the limiting field
can be described by using homogenized boundary conditions on Γ. We
will not consider this case in the present paper.
2. Local Quantitative Characteristics of Grids.
Main Result
Let us introduce needed quantitative characteristics of sets Fε: scalar
quantities C(x, h, ε) characterizing the capability of the grid Fε to con-
centrate on it the electric charge, and tensors L(x, h, ε) characterizing the
capability of the grid to keep the magnetic field coupled with it. These
quantities must characterize the structure of Fε is some small neighbor-
hood of every x ∈ G. We take these neighborhoods in the form of cubes
K(x, h) of size h > 0 centered at x ∈ Ḡ and oriented along the coordinate
axes (a particular orientation is irrelevant but it has to be the same for
all x and all h). The length of edges has to be sufficiently small but it
must be much larger than the characteristic scale ε (which can be, for
example, the size of the grid cells): 0 < ε ≪ h ≪ 1. Therefore, these
characteristics are called mesoscopic.
Denote by H0(x, h, ε) the class of functions in the Sobolev space
W 1
2 (K(x, h)) equal to zero on Fε ∩K(x, h). Set
C(x, h, ε) = inf
vε∈H0(x,h,ε)
∫
K(x,h)
{
|∇vε|2 + h−2−γ |vε − 1|2
}
dx, (2.1)
where γ > 0 is a penalty parameter. In what follows, we will choose it in
the interval 0 < γ < 2.
Obviously, if Fε1 ∩ K(x, h) ⊆ Fε2 ∩ K(x, h), then C(x, h, ε1) ≤
C(x, h, ε2); therefore, C(x, h, ε) characterizes the massiveness (capacity)
of the set Fε ∩K(x, h). The quantities C(x, h, ε) depend on the param-
eter γ but one can show that, as ε → 0 and h → 0, this dependence is
vanishing, so that
C(x, h, ε) ∼ Cap(Fε ∩K(x, h)), ε≪ h→ 0,
where Cap(F ) denotes Newton’s capacity of F [8]. Therefore, C(x, h, ε)
is equivalent to Newton’s capacity of the sets Fε ∩ K(x, h) and thus
E. Ya. Khruslov 115
characterizes the capability of Fε to concentrate a charge on it, in some
neighborhood of x.
Denote by R0(x, h, ε) the closure, with respect to the norm
‖vε‖2 =
∫
K(x,h)
{|rot vε|2 + |vε|2}dx, (2.2)
of the set of 3-component vector functions in (W 1
2 (K(x, h)))3 equal to
zero on Fε ∩K(x, h). It is known [3] that this set coincides with the set
of vector functions, with finite norm (2.2), which equal 0 on Fε ∩K(x, h)
and have vanishing tangential components on ∂Fε ∩K(x, h).
Set
L(x, h, ε, l) = inf
vε∈R0(x,h,ε)
∫
K(x,h)
{|rotvε|2 + h−2−γ |vε − l|2}dx (2.3)
for all l = {l1, l2, l3} ∈ R3 and some 0 < γ < 2. It is easy to show (see
Section 4) that L(x, h, ε, l) is a homogeneous quadratic function with
respect to l, which can be expressed in the form
L(x, h, ε, l) =
3∑
i,j=1
Lij(x, h, ε)lilj , (2.4)
where the system of numbers {Lij(x, h, ε)}3
i,j=1 constitutes a symmet-
ric non-negative tensor in R3 of the second rang. By analogy with the
electrotechnical terminology, this tensor can be called tensor of back in-
duction of the grid Fε ∩K(x, h).
We assume that, for all x ∈ Ḡ, the following limits exist:
(c1) lim
h→0
lim
ε→0
C(x,h,ε)
h3 = lim
h→0
lim
ε→0
C(x,h,ε)
h3 = C(x),
(c2) lim
h→0
lim
ε→0
Lij(x,h,ε)
h3 = lim
h→0
lim
ε→0
Lij(x,h,ε)
h3 = Lij(x),
where C(x) and Lij(x) (i, j = 1, 2, 3) are functions bounded in Ḡ and
continuous in G such that C(x) > 0 and {Lij(x)} is a nonnegative tensor
in G.
Remark 2.1. It suffices that the limits in (c1) and (c2) be finite for
some value of the penalty parameter γ in the interval 0 < γ < 2. Then
one can show that the limits are finite as well for all γ > 0 and that the
limiting functions C(x) and Lij(x) (i, j = 1, 2, 3) are independent of γ.
116 Homogenization of Maxwell’s Equations...
Now we make one more assumption about the structure of sets Fε.
Consider the Hilbert space of real-valued vector functions W0(Ωε) =
(
◦
W 1
2 (Ωε))
3 defined in Ωε = Ω \ Fε, with components uk(x) (k = 1, 2, 3)
in the Sobolev space
◦
W 1
2 (Ωε) which are equal to zero on the boundary
∂Ωε = ∂Fε ∪ ∂Ω. Define an inner product (·, ·)ε (and, consequently, a
norm ‖ · ‖ε = (·, ·)1/2
ε ) in W0(Ωε) by
(u, v)ε =
∫
Ωε
{ 3∑
i,k=1
∂uk
∂xi
∂vk
∂xi
+ λ2
3∑
k=1
ukvk
}
dx,
where λ2 > 0. Let G0(Ωε) be a subspace in W0(Ω) consisting of gradients
of functions in the Sobolev space W 2
2 (Ωε) which are constants on ∂Fε and
∂Ω and the normal derivatives of which vanish on ∂Fε and ∂Ω . Without
loss of generality, we can choose these functions to be equal to zero on
∂Fε, i.e.,
G0(Ωε) =
{
uε(x) = grad ϕε(x), ϕε(x) ∈W 2
2 (Ωε);
ϕε(x) =
∂ϕε
∂n
(x) = 0, x ∈ ∂Fε;
ϕε(x) = const,
∂ϕε
∂n
(x) = 0, x ∈ Ωε
}
.
Denote by W1(Ωε) the orthogonal complement to G0(Ωε) in W0(Ωε):
W1(Ωε) = W0(Ωε)⊖G0(Ωε). We will assume that the following condition
holds true:
(c3) for all uε ∈W1(Ωε),
‖uε‖2
ε < C
∫
Ωε
|rotuε|2dx,
where the constant C is independent of ε.
Now we are at a position to formulate the main result.
Theorem 2.1. Assume that conditions (c1)–(c3) are satisfied. Then
solutions {Eε(x, t), Hε(x, t)} of problem (1.1)–(1.4), being extended by
zero, with respect to x, on Fε, converge, as ε → 0, weakly in (L2(Ω ×
[0, T ]))3 × (L2(Ω× [0, T ]))3 (for all T ) to a solution {E(x, t), H(x, t)} of
problem (1.8)–(1.13).
The proof of Theorem 2.1 will be given in Sects. 3–5, the scheme
of the proof being as follows. By using the Laplace transform, we re-
duce problem (1.1)–(1.4) to two stationary boundary value problems in
E. Ya. Khruslov 117
Ωε = Ω\Fε, which in turn are reduced, for λ > 0, to associated variational
problems. The main part of the proof consists in the study of the asymp-
totic behavior of these variational problems and in the derivation of the
homogenized equations (for λ > 0). Then we study analytical properties
of solutions of the original and the homogenized problems. Finally, the
application of the inverse Laplace transform completes the proof.
3. The Stationary Problem
Let us apply the Laplace transform, with respect to t, to problem
(1.1)–(1.4). Introduce the vector functions
Eε(x, λ) =
∞∫
0
Eε(x, t)e
−λtdt, Hε(x, λ) =
∞∫
0
Hε(x, t)e
−λtdt, (3.1)
which depend on x ∈ Ωε and a complex variable λ with Reλ > 0. For
the sake of simplicity, we keep the original notations: Eε = Eε(x, λ) and
Hε = Hε(x, λ). Then we arrive at the following stationary boundary
value problem in Ωε:
rotEε + λHε = H0, x ∈ Ωε, (3.2)
rotHε − λEε = J − E0, x ∈ Ωε, (3.3)
n ∧ Eε = 0, x ∈ ∂Ωε. (3.4)
Therefore, Eε(x, λ) has to solve the following boundary value problem:
rot rotEε + λ2Eε = J0, x ∈ Ωε, (3.5)
n ∧ Eε = 0, x ∈ ∂Ωε, (3.6)
where
J0(x, λ) = −λ
(
J(x, λ) − E0(x)
)
+ rotH0(x). (3.7)
In turn, Hε(x, λ) is determined by Eε(x, λ):
Hε(x, λ) = − 1
λ
(rotEε(x, λ) −H0(x)). (3.8)
For all λ with Reλ > 0, problem (3.5)–(3.6) has a unique solution
Eε(x, λ).
Our primary goal is to study the asymptotic behavior of this solution
as ε→ 0. Consider first the case of real λ (λ > 0). Denote by W0(Ωε, rot)
118 Homogenization of Maxwell’s Equations...
the closure of the set of vector functions in (
◦
W 1
2 (Ωε))
3 with respect to
the norm
‖vε‖2 =
∫
Ωε
{|rot vε|2 + λ2|vε|2}dx. (3.9)
Here and below, |uε| denotes the Euclidean norm of uε in R3.
It is known [8] that the tangential components of vector functions
with finite norms (3.9) have traces on ∂Ωε as elements of the space
W
−1/2
2 (∂Ωε). Therefore, W0(Ω, rot) consists of vector functions vε(x)
with finite norm (3.9), the tangential component of which equals zero on
∂Ωε, i.e., n ∧ vε = 0.
Let us define in W0(Ωε, rot) the functional
Φε[vε] =
∫
Ωε
{|rot vε|2 + λ2|vε|2 + (J0, vε)}dx, (3.10)
where λ > 0 and (·, ·) denotes the inner product in R3. Consider the
minimization problem for this functional in the class W0(Ωε, rot). In a
standard way (see, e.g., [10]), one can show that there exists a unique
vector function vε = Eε(x, λ) ∈W0(Ωε, rot) such that
Φ[Eε] = min
vε∈W0(Ωε,rot)
Φε[vε] (3.11)
and that this function solves (in the sense of distributions) the boundary
value problem (3.5)–(3.6) for λ > 0. Conversely, the solution of (3.5)–
(3.6) minimizes Φε in the variational problem (3.11).
We want to obtain for Eε(x, λ) an appropriate representation that will
be convenient for the study of its asymptotic behavior as ε→ 0. Introduce
the Hilbert space W (Ωε) as follows: it consists of vector functions vε
defined in Ωε such that the integral
∫
Ωε
{|rot vε|2 + (div vε)
2 + λ|vε|2}dx
is finite, the tangential component vanishes on the boundary of Ωε:
n ∧ vε = 0 for x ∈ ∂Ωε,
and the flux through each connected component of the boundary equals
zero: ∫
∂Fε
(vε)nds =
∫
∂Ω
(vε)nds = 0.
E. Ya. Khruslov 119
Here ds denotes the area of the surface element of ∂Fε or ∂Ω.
We define an inner product in W (Ωε) by
(uε, vε)ε =
∫
Ωε
{(rotuε, rot vε) + div uεdiv vε + λ2(uε, vε)}dx. (3.12)
Introduce in W (Ωε) the following subspaces:
• W0(Ωε) is the subspace of vector functions equal to zero on ∂Fε
and ∂Ω;
• G(Ωε) is the subspace of vector functions such that they are gra-
dients of functions ϕε(x) ∈ W 2
2 (Ωε) equal to constants on ∂Fε and
∂Ω, and that the fluxes through each connected component of ∂Ωε
vanish: ∫
∂Fε
∂ϕε
∂n
ds =
∫
∂Ω
∂ϕε
∂n
ds = 0.
Set
G0(Ωε) = W0(Ωε) ∩G(Ωε)
and introduce the subspaces
W1(Ωε) = W0(Ωε) ∩G⊥
0 (Ωε)
and
G1(Ωε) = G(Ωε) ∩G⊥
0 (Ωε),
where G⊥
0 (Ωε) is the orthogonal complement to G0(Ωε) in W (Ωε) with
respect to the inner product (3.12). It is easily seen that the subspaces
W0(Ωε), G0(Ωε), and W1(Ωε) in W (Ωε) are the same spaces as those
introduced above, in the formulation of condition (c3) (that is why we
use for them the same notations).
Lemma 3.1. Every vε ∈W (Ωε) can be represented in the form
vε(x) = hε(x) + gε(x) + pε(x), (3.13)
where hε(x) ∈W1(Ωε), gε(x) ∈ G1(Ωε), and pε(x) ∈ G0(Ωε).
Proof. First, let us show that the representation (3.13) is unique. Indeed,
if it is not true, then there must exist vectors h′ε ∈W1(Ωε), g
′
ε ∈ G1(Ωε),
and p′ε ∈ G0(Ωε) such that
h′ε + g′ε + p′ε = 0, (3.14)
120 Homogenization of Maxwell’s Equations...
where not all the terms in this equation equal zero. Since W1(Ωε) and
G1(Ωε) are orthogonal toG0(Ωε), it follows that p′ε = 0 and thus h′ε+g
′
ε =
0. The latter equality yields h′ε = −g′ε ∈ W1(Ωε) ∩ G1(Ωε) ⊂ W0(Ωε) ∩
G(Ωε) = G0(Ωε) and thus h′ε = 0 and g′ε = 0. Therefore, all the vectors
in (3.14) equal 0, and we have arrived at a contradiction. Therefore, the
representation (3.13) is unique.
In order to prove (3.13), it suffices to show that [W1(Ωε), G1(Ωε),
G0(Ωε)] = W (Ωε) (here the square brackets denote a linear span of the
corresponding subspaces). From the definitions of the subspaces W1(Ωε)
and G1(Ωε) it follows that
[W1(Ωε), G1(Ωε), G0(Ωε)] = [W0(Ωε), G(Ωε)]
and thus
[W1(Ωε), G1(Ωε), G0(Ωε)]
⊥ = [W0(Ωε), G(Ωε)]
⊥. (3.15)
By the duality principle,
[W0(Ωε), G(Ωε)]
⊥ = W⊥
0 (Ωε) ∩G⊥(Ωε). (3.16)
Now, by the definitions of W (Ωε) and W0(Ωε) ⊂ W (Ωε), we conclude
that W⊥
0 (Ωε) consists of vector functions vε(x) that satisfy in Ωε the
differential equation
rot rot vε − grad divvε + λ2vε = 0, x ∈ Ωε, (3.17)
have on ∂Ωε a vanishing tangential component
n ∧ vε = 0, x ∈ ∂Fε ∪ Ωε, (3.18)
and have vanishing fluxes through each connected component of the
boundary ∫
∂Fε
(vε)nds =
∫
∂Ω
(vε)nds = 0. (3.19)
In a similar way, from the definition of G(Ωε) it follows that G⊥(Ωε)
consists of vector functions vε(x) satisfying in Ωε the differential equation
∆div vε − λ2div vε = 0
and the boundary condition
div vε = const (3.20)
E. Ya. Khruslov 121
on each connected component of ∂Ωε.
Let vε ∈ W⊥
0 (Ωε) ∩ G⊥(Ωε). Multiply (3.17) by vε(x) and integrate
by parts; then, using (3.18) – (3.19), we obtain
∫
Ω
{|rot vε|2 + (div vε)
2 + λ2|vε|2}dx = 0.
Therefore, vε(x) ≡ 0 (x ∈ Ωε). Finally, by (3.16) and (3.15) we have
[W1(Ωε), G1(Ωε), G0(Ωε)]
⊥ = 0
and thus
[W1(Ωε), G1(Ωε), G0(Ωε)] = W (Ωε).
Lemma 3.1 is proved.
Notice that the subspaces W1(Ωε) and G1(Ωε) are orthogonal to the
subspace G0(Ωε) (with respect to the inner product (3.12)) but do not
orthogonal to one another. However, if condition (c3) is satisfied, then
the mutual slope of these subspaces is bounded form below uniformly
with respect to ε. More precisely, the following lemma holds true.
Lemma 3.2. Let condition (c3) be satisfied. Then, for all vε ∈ W1(Ωε)
and all uε ∈ Gε(Ωε), the following inequality holds true:
|(uε, vε)ε| ≤ α‖uε‖ε‖vε‖ε,
where the constant α is independent of ε and α < 1.
Proof. Since uε ∈ G1(Ωε) and thus rotuε = 0, by the Schwartz inequality
we obtain
|(uε, vε)ε| =
∣∣∣∣∣
∫
Ωε
{div uε,div vε + λ2(uε, vε)}dx
∣∣∣∣∣
≤
{∫
Ω
[
(div uε)
2 + λ2|uε|2
]
dx
}1/2{∫
Ω
[
(div vε)
2 + λ2|vε|2
]
dx
}1/2
= ‖uε‖ε
{∫
Ω
[
(div vε)
2 + λ2|vε|2
]
dx
}1/2
. (3.21)
By using condition (c3) for vε ∈W1(Ωε), we can write
∫
Ωε
{
|rot vε|2 + (div vε)
2 + λ2|vε|2
}
dx = ‖vε‖2
ε ≤ C
∫
Ωε
|rot vε|2dx,
122 Homogenization of Maxwell’s Equations...
where C is independent of ε and C > 1. This implies that
∫
Ωε
[
(div vε)
2 + λ2|vε|2
]
dx ≤ C − 1
C
‖vε‖2
ε. (3.22)
Now (3.21) and (3.22) yield the sought inequality, with the constant
α =
(
C−1
C
)1/2
< 1 independent of ε. Lemma 3.2 is proved.
Lemma 3.2 allows us to make the assertion of Lemma 3.1 stronger.
Namely, the following lemma holds true.
Lemma 3.3. Every vε(x) ∈ W (Ωε) can be uniquely represented in the
form
Vε(x) = hε(x) + gε(x) + pε(x),
where pε = P0εvε ∈ G0(Ωε), hε ∈W1(Ωε), and gε ∈ G1(Ωε), with
‖pε‖ε ≤ ‖vε‖, ‖hε‖ε ≤ C‖vε‖ε, ‖gε‖ε ≤ C‖vε‖ε. (3.23)
Here P0ε is the operator of the orthogonal projection on G0(Ωε) ⊂W (Ωε)
and the constant C is independent of ε.
Proof. By the definition of the subspaces G0(Ωε), W1(Ωε), and G1(Ωε)
in W (Ωε) and Lemma 3.1, we can write the following orthogonal decom-
position:
W (Ωε) = [W1(Ωε), G1(Ωε)] ⊕G0(Ωε).
Therefore, in (3.13) we have pε = P0εvε and hε + gε ∈ [W1(Ωε), G1(Ωε)],
with
‖pε‖ε ≤ ‖vε‖ε, ‖hε + gε‖ε ≤ ‖vε‖ε. (3.24)
Hence, we have established the first inequality in (3.23). In order to prove
the other two, we use the second inequality in (3.24) and Lemma 3.2; in
this way we obtain
‖vε‖2
ε ≥ ‖hε + gε‖2
ε = ‖hε‖2
ε + ‖gε‖2
ε + 2(hε, gε)ε
≥ ‖hε‖2
ε + ‖gε‖2
ε − 2|(hε, gε)ε| ≥ ‖hε‖2
ε + ‖gε‖2
ε − 2α‖hε‖ε‖gε‖ε
= (‖hε‖ε − ‖gε‖ε)
2 + 2(α− 1)‖hε‖ε‖gε‖ε.
This implies the inequalities
2‖hε‖ε‖gε‖ε ≤
1
(1 − α)
‖vε‖2
ε
and
‖hε‖2
ε + ‖gε‖2
ε ≤ ‖vε‖2
ε + 2‖hε‖ε‖gε‖2
ε
which in turn yield the remaining two inequalities in (3.23), with the
constant C =
(
2−α
1−α
)1/2
independent of ε. Lemma 3.3 is proved.
E. Ya. Khruslov 123
Now return to the solution Eε(x, λ) of the variational problem (3.11)
or, equivalently, the boundary value problem (3.5)–(3.6) for λ > 0. Since
(by (3.11)) Φε[Eε] ≤ Φε[0] = 0, using (3.10) and (3.7) gives
∫
Ωε
{|rotEε|2 + λ2|Eε|2}dx < C1.
From (3.5) and (3.7) it follows that
∫
Ωε
(divEε)
2dx < C2,
where C1 and C2 are independent of ε.
These two inequalities mean that Eε = Eε(x, λ) belongs to the space
W (Ωε) and that the norm of Eε is bounded uniformly with respect to ε:
‖Eε‖ε < C.
By Lemma 3.3, Eε can be uniquely represented in the form
Eε(x) = uε(x) + gradϕε(x), (3.25)
where uε(x) ∈ W1(Ωε) and ϕε(x) is a function in W 2
2 (Ωε) equal to con-
stants on ∂Fε and ∂Ω and having vanishing fluxes through ∂Fε and ∂Ω
(i.e., gradϕε ∈ G1(Ωε) ⊕G0(Ωε) = G(Ωε)); moreover, the following esti-
mates hold: ∫
Ωε
{|rotuε|2 + (div uε)
2 + λ2u2
ε}dx < C, (3.26)
∫
Ωε
{(∆ϕε)
2 + λ2|gradϕε|2}dx < C, (3.27)
with C independent of ε. Without loss of generality, we may assume that
ϕε(x) equals zero on ∂Ω.
From (3.25) and (3.5)–(3.6) it follows that the vector function uε(x)
and the function ϕε(x), being considered together, solve the following
boundary value problem (I)–(II) in Ωε:
rot rotuε + λ2uε = J0 − λ2gradϕε, x ∈ Ωε
n ∧ uε = 0, x ∈ ∂Ωε
}
(I)
124 Homogenization of Maxwell’s Equations...
∆ϕε = 1
λ2 div J0 − div uε, x ∈ Ωε
ϕε = const(= Aε), x ∈ ∂Fε
ϕε = 0, x ∈ ∂Ω
∫
∂Fε
∂ϕε
∂n ds = 0.
(II)
Due to the calibration uε → uε + ∇ϕ′
ε, ϕε → ϕε − ϕ′
ε, with ϕ′
ε ∈
G0(Ωε), the boundary value problem (I)–(II) has many solutions, but we
can single out the unique solution {uε, ϕε} specified by the representation
(3.25). Our aim is to study the asymptotic behavior of this solution as
ε→ 0.
Let us extend uε(x) ∈ W1(Ωε) = (
◦
W 1
2 (Ωε))
3 on Fε by zero and let
us extend ϕε(x) on Fε by the constants Aε. Then the inequality (3.26)
implies that the set of extended vector functions {uε(x) ∈ (
◦
W 1
2 (Ω))3,
ε > 0} is weakly compact in (
◦
W 1
2 (Ω))3. Since ϕε(x) = 0 on ∂Ω, from
(3.27) it follows that the set of extended functions {ϕε(x) ∈
◦
W 1
2 (Ω), ε >
0} is weakly compact in
◦
W 1
2 (Ω). Extract a subsequence {ε = εk → 0}
such that the corresponding subsequences of vector functions {uεk
(x) ∈
(
◦
W 1
2 (Ω))3} and of functions {ϕεk
(x) ∈
◦
W 1
2 (Ω)} converge weakly to u(x) ∈
(
◦
W 1
2 (Ω))3 and ϕ(x) ∈
◦
W 1
2 (Ω), respectively.
In order to describe the limiting pair {u(x), ϕ(x)}, we proceed as
follows. Consider the boundary value problem (I) with respect to uε(x),
assuming that ϕε(x) is known. Then the following theorem holds.
Theorem 3.1. Assume that conditions (c1) and (c2) are satisfied. Then
solutions uε(x) of problem (I) (extended by zero on Fε) converge weakly
in (
◦
W 1
2 (Ω))3, on a subsequence {ε = εk → 0}, to a vector function u(x),
which solves the following boundary value problem:
rot rotu+ λ2u+ L(x)u = J0 − λ2gradϕ, x ∈ Ω,
n ∧ uε = 0, x ∈ ∂Ω,
(I ′)
where ϕ(x) is the weak limit of ϕε(x) on the same subsequence and L =
{Lik(x)}3
i,k=1 is the limiting tensor in condition (c2) extended by zero
outside G ⊂ Ω.
E. Ya. Khruslov 125
The proof of this theorem will be given in the next section.
Now consider the boundary value problem (II) with respect to the
function ϕε(x), assuming that the vector function uε(x) is known.
Theorem 3.2. Assume that condition (c1) is satisfied. Then solutions
ϕε(x) of problem (II) (extended on Fε by the constants Aε) converge
weakly in
◦
W 1
2 (Ω), on a subsequence {ε = εk → 0}, to a function ϕ(x),
which solves the boundary value problem
∆ϕ− C(x)
ϕ(x) −
(∫
Ω
C(x)dx
)−1 ∫
Ω
C(x)ϕ(x)dx
=
1
λ2
div J0 − div u, x ∈ Ω,
ϕ(x) = 0, x ∈ ∂Ω,
(II ′)
where u(x) is the weak limit of uε(x) on the same subsequence and C(x)
is the limiting function in condition (c1) extended by zero outside G ⊂ Ω.
The proof of this theorem is, in fact, given in [5]; only minor modifi-
cations are required, so we will not dwell on this.
By using Theorems 3.1 and 3.2, it is easy now to obtain the main result
concerning solutions Eε(x, λ) of problem (3.5)–(3.6). These solutions are
assumed to be extended, with respect to x ∈ Ω, by zero on Fε ⊂ Ω; we
will keep the same notation Eε for these extensions.
Theorem 3.3. Assume that conditions (c1)–(c3) are satisfied. Then
Eε(x, λ) converge weakly in (L2(Ω))3 to a vector function E(x, λ), which,
being considered together with Φ(x,λ), solves the following boundary value
problem:
rot rotE + λ2(E − gradΦ) = J0, x ∈ Ω, (3.28)
n ∧ E = 0, x ∈ ∂Ω, (3.29)
div (Lgrad Φ) − λ2CΦ = div (LE), x ∈ G, (3.30)
(Lgrad Φ)n = (LE)n, x ∈ ∂G, (3.31)
Proof. From Theorems 3.1 and 3.2 and the representation (3.25) it fol-
lows that, for λ > 0, Eε(x, λ) converge weakly in (L2(Ω))3, by a subse-
quence {ε = εk → 0}, to a vector function
E(x, λ) = u(x, λ) + gradϕ(x, λ) ∈ (W 1
2 (Ω))3.
126 Homogenization of Maxwell’s Equations...
Introduce in the subdomain G ⊂ Ω (outside of which C(x) ≡ 0 and
L(x) ≡ 0) the function
Φ(x, λ) = ϕ(x, λ) −
(∫
Ω
C(x) dx
)−1 ∫
Ω
C(x)ϕ(x) dx ∈W 1
2 (G).
Then, by (I ′) and (II ′), we see that E(x, λ) and Φ(x, λ) have to satisfy
the differential equations (3.28) and (3.30) and the boundary conditions
(3.29) and (3.31).
Let us show that Eε(x, λ) converge weakly in (L2(Ω))3 to E(x, λ)
as ε → 0 (and not only on a subsequence {ε = εk → 0}). Notice,
first of all, that the set {Eε(x, λ), ε > 0} is bounded and, therefore, is
weakly compact in (L2(Ω))3. Hence, it suffices to show that the boundary
value problem (3.22)–(3.25), describing the limits on subsequences, has
a unique solution.
Consider the corresponding homogeneous system of equations (i.e.,
the system (3.28)–(3.31) with J0 = 0). Multiply the homogeneous ana-
logue of (3.28) by E (in the sense of the inner product in R3) and integrate
the resulting equation over Ω. Then, integration by parts gives (taking
into account (3.29) and the fact that the tensor L is symmetric)
∫
Ω
{(rotE)2 + λ2(E,E) + (LE,E) − (LE, gradΦ)}dx = 0. (3.32)
Now multiply (3.28) by Φ and integrate the result over G. Then, inte-
grating by parts and using the boundary condition (3.31), we obtain
∫
G
{(Lgrad Φ, gradΦ) + λ2CΦ2 − (LE, gradΦ)}dx = 0. (3.33)
Since L is symmetric and nonnegative, it follow that
2|(LE, gradΦ)| ≤ (LE,E) + (Lgrad Φ, gradΦ).
Using this inequality and the fact that L(x) = 0 and C(x) = 0 outside
G, by (3.32) and (3.33) we conclude that
∫
Ω
{|rotE|2 + λ2|E|2 + λ2CΦ2}dx ≤ 0.
This implies that E(x) = 0 for x ∈ Ω and, since C(x) > 0 for x ∈ G,
Φ(x) = 0 for x ∈ G. We have shown that the homogeneous problem
has only trivial solution; hence, the solution of problem (3.28)–(3.31) is
unique. This completes the proof of Theorem 3.3 for λ > 0.
E. Ya. Khruslov 127
Now consider problem (3.5)–(3.6) for complex λ. Using standard
technique of the perturbation theory for the operator equations (see, e.g.,
[6]), it is easy to show that if arg λ2 6= π, then this problem has the unique
solution Eε(x, λ), which is an analytic function of λ in the half-plane
Reλ > 0.
Multiply (3.5) by Eε(x, λ) and integrate over Ωε. Then, integrating
by parts, using (3.6), and separating the real and imaginary parts, we
obtain two equations:
(µ2 − ν2)
∫
Ωε
|Eε|2dx+
∫
Ωε
|rotEε|2dx
= µRe
∫
Ωε
(g1, Eε) dx− νIm
∫
Ωε
(g1, Eε) dx+ Re
∫
Ωε
(g2, Eε) dx,
2µν
∫
Ωε
|Eε|2dx = µIm
∫
Ωε
(g1, Eε) dx+νRe
∫
Ωε
(g1, Eε) dx+Im
∫
Ωε
(g2, Eε) dx,
where the following notations have been introduced: λ = µ + iν, g1 =
J(x, λ) − E0(x), and g2 = rotH0(x).
These equalities yield the following estimates for Eε(x.λ) (which are
uniform with respect to ε):
∫
Ωε
|Eε(x, λ)|2dx < C1
Reλ
(3.34)
and ∫
Ωε
|rotEε|2dx < C2, (3.35)
where Reλ ≥ µ0 > 0. In a similar way one can show that if arg λ2 6= π,
then problem (3.28)–(3.31) has the unique solution (E(x, λ),Φ(x, λ)),
which is analytic with respect to λ in the half-plane Reλ > 0, and that
the following inequality holds true:
∫
Ω
|E(x, λ)|2dx < C
Reλ
. (3.36)
Now, since the convergence of Eε(x, λ) to E(x, λ) for λ > 0 has al-
ready been proved, by the estimate (3.34) and the Vitalli theorem [9]
we conclude that Eε(x, λ) converge to E(x, λ) as ε → 0 in the whole
right half-plane λ (Reλ > 0) uniformly in every its compact subdomain.
Theorem 3.3 is proved.
128 Homogenization of Maxwell’s Equations...
4. Proof of Theorem 3.1
First, we give the brief scheme of the proof based on the variational
principle. Namely, for all λ > 0 and the given J0(x, λ) ∈ (L2(Ωε))
3 and
ϕε(x, λ) ∈ W 1
2 (Ω), the solution uε(x, λ) of problem (I) minimizes the
functional
Jε[wε] =
∫
Ω
{rotwε|2 + λ2|wε|2 + 2(J0 − λ2gradϕε, wε)} dx (4.1)
in the class R0(Ω, Fε)={wε(x) :Ω → R3; wε(x) ≡ 0, x ∈ Fε; n ∧ wε =0,
x ∈ ∂Ωε; wε(x) ∈ (L2(Ω))3; rotwε ∈ (L2(Ω))3}. This means that for all
wε(x) ∈ R0(Ω, Fε), the following inequality holds true:
Jε[uε] ≤ Jε[wε]. (4.2)
In Section 4.2 we will construct special test vector functions wεh(x)
(which depend on w(x) ∈ (C2
0 (Ω))3, the problem parameter ε > 0, and
an auxiliary parameter h > 0) and show that
lim
h→0
lim
ε=εk→0
Jε[wεh] ≤ J [w], (4.3)
where the functional J [w] is defined by
J [w] =
∫
Ω
{|rotw|2 + λ2|w|2 + (Lw,w) + 2(J0 + λ gradϕ,w)} dx. (4.4)
Here ϕ(x) is the weak limit in W 1
2 (Ω) of the functions ϕε(x) on a subse-
quence {ε = εk → 0}, and the tensor L = L(x) is defined in condition
(c2).
From (4.2) and (4.3) it follows that
lim
ε=εk→0
Jε[uε] ≤ J [w], (4.5)
for all w(x) ∈ (C2
0 (Ω))3. Since (C2
0 (Ω))3 is dense in (
◦
W 1
2 (Ω))3, this
inequality holds true for all w ∈ (
◦
W 1
2 (Ω))3.
In Section 4.3, by using the fact that uε(x, λ) converge to u(x, λ)
weakly in (
◦
W 1
2 (Ω))3 on a subsequence {ε = εk → 0}, we will establish
the “converse” inequality
lim
ε=εk→0
Jε[uε] ≥ J [u]. (4.6)
E. Ya. Khruslov 129
It follows from (4.5) and (4.6) that
J [u] ≤ J [w] for all w ∈ (
◦
W 1
2 (Ω))3;
hence, u∈(
◦
W 1
2 (Ω))3 minimizes the functional (4.4) in the class (
◦
W 1
2 (Ω))3.
This, by standard arguments, implies that u(x, λ) is a (generalized) so-
lution of problem (I ′).
To implement this scheme, we will need special (“coordinate”) vector
functions vi
εh(x) (i = 1, 2, 3) satisfying certain estimates.
4.1. Coordinate Vector Functions vix
εh(ξ).
Let vi
εh ≡ vi
εh(ξ), i = 1, 2, 3, be vector functions giving the infimum
in (2.3) when l coincides with the unit vector ei of the coordinate axis xi.
It is easily seen that for an arbitrary l = {l1, l2, l3} ∈ R3, the minimizer
v
l(x)
εh (ξ) of (2.3) is a (generalized) solution of the following boundary value
problem in Ωε ∩K(x, h):
rot rot v
l(x)
εh (ξ) + h−2−γv
l(x)
εh (ξ) = h−2−γ l, ξ ∈ Ωε ∩K(x, h),
n ∧ vl(x)
εh (ξ) = 0, ξ ∈ ∂Fε ∩K(x, h) ∪ ∂K(x, h),
v
l(x)
εh (ξ) = 0, x ∈ Fε ∩K(x, h).
Since this problem is linear, it follows that
v
l(x)
εh (ξ) =
3∑
i=1
liv
ix
εh(ξ),
from which, by (2.3), we obtain the representation (2.4) with Lij(x, h, ε)
defined by
Lij(x, h, ε)=
∫
K(x,h)
{(rot vix
εh, rot vjx
εh)+h−2−γ(vix
εh−ei, vjx
εh−ej)} dx, (4.7)
i, j = 1, 2, 3.
Let us derive some estimates for vix
εh(ξ), which will be used in what
follows.
Lemma 4.1. Let Kx
h1
= K(x, h1) be a cube of size h1 = h−2r (r = o(h))
centered at x ∈ Ω and oriented as the cube K(x, h) = Kx
h .
130 Homogenization of Maxwell’s Equations...
For sufficiently small ε < ε̂(h, x), the following estimates hold true
as h→ 0: ∫
Kx
h
|rot vix
εh(ξ)|2dξ = O(h3), (4.8)
∫
Kx
h
|vix
εh − ei|2dξ = O(h5+γ), (4.9)
∫
(Kx
h\Kx
h1
)
|rot vix
εh(ξ)|2dξ = o(h3), (4.10)
∫
(Kx
h\Kx
h1
)
|vix
εh − ei|2dξ = o(h5+γ). (4.11)
Moreover, if condition (c2) is satisfied uniformly with respect to x ∈ G,
then these estimates are also satisfied uniformly with respect to x (i.e.,
for ε < ε̂(h) for all x).
Proof. The estimates (4.8) and (4.9) follow immediately from condition
(c2). By (4.9), we have
∫
(Kx
h\Kx
h1
)
{|rot vix
εh(ξ)|2 + h−2−γ |vix
εh − ei|2} dξ
=
∫
Kx
h
{|rot vix
εh(ξ)|2 + h−2−γ |vix
εh − ei|2} dξ
−
∫
Kx
h1
{|rot vix
εh(ξ)|2 + h−2−γ
1 |vix
εh − ei|2} dξ +O(rh2). (4.12)
According to (2.3), the first term in the right-hand side of (4.12) equals
L(x, h, ε, ei), and the second term is not less than L(x, h1, ε, e
i). Hence,
∫
(Kx
h\Kx
h1
)
{|rot vix
εh(ξ)|2 + h−2−γ |vix
εh − ei|2} dξ
≤ L(x, h, ε, ei) − L(x, h1, ε, e
i) +O(rh2),
from which, taking into account condition (c2) and the fact that h1 =
h− 2r with r = o(h), we obtain the estimate
∫
(Kx
h\Kx
h1
)
{|rot vix
εh(ξ)|2+h−2−γ |vix
εh−ei|2} dξ = o(h3), h→ 0, ε < ε̂(h, x).
E. Ya. Khruslov 131
Finally, this estimate yields the required estimates (4.10) and (4.11),
which completes the proof of the lemma.
4.2. Construction of Test Vector Functions wεh and
Proof of Inequality (4.3).
Consider the covering of Ω by cubes Kα
h =K(xα, h) of size h>0 cen-
tered at points xα forming a cubic lattice of period h−r with r = h1+γ/2 =
o(h) and oriented along the coordinate axes. Construct a partition of
unity associated with this covering, i.e., a system of functions {ϕα(x)}
satisfying the following conditions: ϕα(x) ∈ C2
0 (R3); 0 ≤ ϕα(x) ≤ 1 and∑
α ϕα = 1; ϕα(x) = 1 for x ∈ Kα
h \ ⋃β 6=αK
β
h ; ϕα(x) = 0 for x 6∈ Kα
h ;
|∇ϕα(x)| ≤ Cr−1.
Let w(x) be an arbitrary vector function with components wi(x) ∈
C2
0 (Ω) (i = 1, 2, 3). Set
wεh(x) =
∑
α
∑
i
wi(x)v
iα
εh(x)ϕα(x), (4.13)
where viα
εh(x) denote “coordinate” vector functions of cubesKα
h introduced
in Section 4.1.
Taking into account specific properties of the vector functions viα
εh(x)
and the functions {ϕα(x)}, it is easy to see that wεh(x) belongs to
R0(Ω, Fε) and thus can be taken as a test vector function for the es-
timation of the functional (4.1).
Let us estimate Jε(wεh). First, we notice that, by the properties of
the partition of unity, formula (4.13) can be transformed into the form
wεh(x) = w(x) +
∑
α
3∑
i=1
wi(x)[v
iα
εh(x) − ei]ϕα(x), (4.14)
which yields
rotwεh(x) = rotw(x) +
∑
α
3∑
i=1
wi(x)rot viα
εh(x)ϕα(x)
+
∑
α
3∑
i=1
gradwi(x) ∧ [viα
εh(x) − ei]ϕα(x)
+
∑
α
3∑
i=1
wi(x)[v
iα
εh(x) − ei] ∧ gradϕα(x). (4.15)
The second term in the right-hand side of (4.14) (the sum with respect
to α), as well as the third and the fourth terms in the right-hand side
132 Homogenization of Maxwell’s Equations...
of (4.15) give a vanishing (as ε → 0 and h → 0) contribution to the
functional (4.1). Indeed, setting in (4.1) wε = wεh(x), by (4.14), (4.15),
and the estimates (4.9) and (4.11) we obtain
Jε[wεh] =
∫
Ω
{|rotw|2 + λ2|w|2 + 2(J0 + λ2gradϕ,w)}dx
+ I
(1)
εh + I
(2)
εh +O(h2+γ) + o(r−2h2+γ) (4.16)
for ε = εk ≤ ε̂(h) and h→ 0, where
I
(1)
εh =
∑
α,β
∑
i,j
∫
Kα
h ∩Kβ
h
wiwjϕαϕβ rot viα
εh rot vjβ
εh dx,
I
(2)
εh =
∫
Ω
(
rotw,
∑
α
3∑
i=1
wiϕα rot viα
εh
)
dx.
We have taken into account also that w ∈ (C2
0 (Ω))3 and that gradϕε
converge weakly in (W 1
2 (Ω))3, on a subsequence ε = εk → 0, to gradϕ.
The term I
(2)
εh can be transformed (by integration by parts) to the
form
I
(2)
εh =
∫
Ω
(
rotw,
∑
α
3∑
i=1
wiϕα rot viα
εh
)
dx
=
∑
α
3∑
i=1
∫
Kα
h
(rot rotw + rotw ∧ grad (wiϕα), [viα
εh − ei]) dx.
This, by the estimates (4.9) and (4.11), gives
I
(2)
εh = O(h2+γ) + o(r−2h2+γ). (4.17)
Now consider the term I
(1)
εh . As above, one can transform this term
(using the construction of the partition of unity, the smoothness of w(x),
and the estimates (4.8) and (4.10)) to the form
I
(1)
εh =
∑
α
3∑
i=1
wi(x
α)wj(x
α)
∫
Kα
h
(rot viα
εh, rot vjα
εh ) dx+ o(1), h→ 0,
which, by the definition of the tensor L = {Lij(x, h, ε)}3
i,j=1 (see (4.7)),
gives
I
(1)
εh ≤
∑
α
3∑
i=1
wi(x
α)wj(x
α)Lij(x
α, h, ε) + o(1), h→ 0.
E. Ya. Khruslov 133
In view of the smoothness of w(x) and condition (c2), it follows that
lim
h→0
lim
ε→0
I
(1)
εh ≤
∫
Ω
(Lw,w) dx. (4.18)
Now, combining (4.16), (4.17), and (4.18) and taking into account that
r = h1+γ/2, we obtain the sought inequality (4.3).
4.3. Proof of Inequality (4.6).
First, we prove an auxiliary statement. Denote by R(Ω) the Hilbert
space of vector functions w(x) : Ω → R3 which is the closure of the set of
vector functions in (C2
0 (Ω))3 with respect to the norm ‖w‖R = (w,w)
1/2
R
generated by the inner product
(w, v)R =
∫
Ω
{(rotw, rot v) + (w, v)} dx.
It is known [3] that the tangential components of vectors of R(Ω) vanish
on the boundary of Ω.
It turns out that the class R0(Ω, Fε) introduced in Section 3, is weakly
compact in R(Ω) and strongly dense in R(Ω) with respect to the norm
of L2(Ω). More precisely, the following lemma holds true:
Lemma 4.2. Assume that condition (c2) is satisfied. Then, for any
w(x) ∈ R(Ω) one can construct vector functions wε(x) ∈ R0(Ω, Fε),
depending of the parameter ε (0 < ε < ε0(w)), which converge, as ε→ 0,
to w(x) weakly in R(Ω) and strongly in L2(Ω) and such that
lim
ε→0
‖wε‖2
R ≤ C‖w‖2
R,
where the constant C is independent of w.
Proof. Since the class (C2
0 (Ω))3 is dense in R(Ω), it suffices to prove the
assertion of the lemma for w(x) ∈ (C2
0 (Ω))3.
Given w(x) ∈ (C2
0 (Ω))3, we construct wεh(x) by (4.13). By the same
arguments as in Section 4.2 (by using (4.14), (4.15), Lemma 4.1, the
properties of functions constituting the partition of unity {ϕα(x)}, and
condition (c2)) we conclude that
‖wεh‖2
R ≤ ‖w‖2
R +
∫
Ω
(Lw,w)dx+ o(1) (4.19)
134 Homogenization of Maxwell’s Equations...
for sufficiently small h (h ≤ ĥ(w)), r = h1+γ/2 and sufficiently small ε
(ε < ε̂(h)). In this way, we define a monotone decreasing, as h → 0,
function ε̂(h): (0, ĥ(w)) → R+. Introduce a step-like monotone function
h(ε) by setting h(ε) = 1
k when ε is in the interval ε̂( 1
k+1) ≤ ε ≤ ε̂( 1
k ), k ∈
N, and define the sought vector function wε(x) by wε(x) = wεh(x)|h=h(ε)
(0 < ε < ε̂(ĥ(w))). Then from (4.19) it follows that
lim
ε→0
‖wε‖2
R ≤ C‖w‖2
R,
with C = 1 + max
x∈Ω
‖L(x)‖.
It remains to show that wε(x) converge, as ε → 0, to w(x) weakly
in R(Ω) and strongly in L2(Ω). But this can be achieved by using the
same arguments as in Section 4.2, with the help of formulas (4.14), (4.15),
lemma 4.1, and taking into account the properties of the partition of unity
{ϕα(x)}. Lemma 4.2 is proved.
Now we are at a position to prove the inequality (4.6). Denote by
u(x) ∈
◦
W 1
2 (Ω) ⊂ R(Ω) the weak limit in
◦
W 1
2 (Ω), on a subsequence {ε =
εk → 0}, of solutions uε(x) ∈
◦
W 1
2 (Ωε) ⊂ R0(Ω, Fε) of problem (I). Let
uδ(x) be a vector function in (C2
0 (Ω))3 such that
‖uδ − u‖R < δ, δ > 0. (4.20)
Since uδ − u ∈ R(Ω and uε ∈ R0(Ω, Fε), we can construct, by using
Lemma 4.2, a sequence of vector functions {uδ
ε ∈ R0(Ω, Fε), ε = εk → 0}
such that it converges to uδ(x) weakly in R(Ω) and strongly in L2(Ω)
and that the following inequality holds true:
lim
ε=εk→0
‖uδ
ε‖2
R ≤ C‖uδ − u‖2
R, (4.21)
where C is independent of δ.
Split Ω into disjoint (i.e., having no common interior points) cubes
Kα
h = K(xα, h) of sufficiently small size h (0 < h≪ δ). Let ϕα
εh(x) be the
function minimizing the functional (2.1) in Kα
h . In what follows, we will
use some properties of this function, namely: ϕα
εh(x) = 0 for x ∈ Fε ∩Kα
h
and
∫
Kα
h
|∇ϕα
εh(x)|2dx ≤ Ch3,
∫
Kα
h
|∇ϕα
εh − 1|2dx ≤ Ch5+γ
for sufficiently small ε (ε < ε̂(h)).
E. Ya. Khruslov 135
The last two inequality follow immediately from condition (c1). In
each cube Kα
h , consider the vector function
vδ
εh = uδ
ε(x) − [uδ(x) − uδ(xα)]ϕα
εh(x). (4.22)
By the properties of uδ
ε(x) ∈ R0(Ω, Fε) and the function ϕα
εh(x) (ϕα
εh(x) =
0 for x ∈ Fε ∩Kα
h ), vδ
εh(x) belongs to the class R0(x
α, h, ε), in which the
infimum in (2.3) is sought. Therefore, by (2.3), we have the inequality
∫
Kα
h
{|rot vδ
εh|2 + h−2−γ |vδ
εh − l|2} dx ≥
3∑
i,j=1
Lij(x
α, h, ε)lilj ,
where l ∈ R3. Setting l = uδ(xα), we obtain
∫
Kα
h
{|rot vδ
εh|2 + h−2−γ |vδ
εh − uδ(xα)|2 dx
≥
3∑
i,j=1
Lij(x
α, h, ε)uδ
i (x
α)uδ
j(x
α). (4.23)
By (4.22), we can write
rot vδ
εh = rotuδ
ε − rotuδ + rotuδ(ϕα
εh − 1)
+ (uδ − uδ(xα)) ∧ gradϕα
εh.
Substituting this expression into (4.23), we obtain the inequality
∫
Kα
h
|rotuδ
ε|2 dx ≥
∫
Kα
h
{2(rotuδ
ε, rotuδ) − |rotuδ|2} dx
+
3∑
i,j=1
Lij(x
α, h, ε)uδ
i (x
α)uδ
j(x
α) − Iα
1 (ε, h, δ) + h−2−γIα
2 (ε, h, δ),
(4.24)
where
Iα
1 (ε, h, δ) = 2
∫
Kα
h
(rot [uδ
ε − uδ], rotuδ)(ϕα
εh − 1) dx
+ 2
∫
Kα
h
(rot [uδ
ε − uδ], [uδ − uδ(xα)] ∧ gradϕα
εh) dx
+ 2
∫
Kα
h
(rotuδ, [uδ − uδ(xα)] ∧ gradϕα
εh)(ϕα
εh − 1)dx, (4.25)
136 Homogenization of Maxwell’s Equations...
Iα
2 (ε, h, δ) = 2
∫
Kα
h
(uδ
ε − uδ, uδ − uδ(xα))(ϕα
εh − 1) dx
−
∫
Kα
h
|uδ
ε − uδ|2 dx−
∫
Kα
h
|uδ − uδ(xα)|2(ϕα
εh − 1)2 dx. (4.26)
Let us estimate from below the value of the functional Jε, defined by
(4.1), on the vector function uδ
ε(x). Representing this functional in the
form
Jε[u
δ
ε] =
∑
α
∫
Kα
h
|rotuδ
ε|2dx+ λ2
∫
Ω
|uδ
ε|2dx
+
∫
Ω
(J0 + λ2gradϕε, u
δ
ε) dx,
by (4.24) we obtain
Jε[u
δ
ε] ≥
∫
Ω
{2(rotuδ
ε, rotuδ) − |rotuδ
ε|2} dx
+ λ2
∫
Ω
|uδ
ε|2dx+
∫
Ω
(J0 − λ2gradϕε, u
δ
ε) dx
+
∑
α
3∑
i,j=1
Lij(x
α, h, ε)uδ
i (x
α)uδ
j(x
α)
−
∑
α
|Iα
1 (ε, h, δ)| − h−2−γ
∑
α
|Iα
2 (ε, h, δ)|. (4.27)
Since the norms of uδ
ε and uδ in R(Ω) are bounded uniformly with respect
to ε and δ, and |uδ(x)−uδ(xα)| ≤ C(δ) for x ∈ Kα
h , it follows from (4.25),
(4.26), and the properties of ϕα
εh(x) that
lim
ε→0
∑
α
|Iα
1 (ε, h, δ)| ≤ C1(δ)h (4.28)
and
lim
ε→0
∑
α
|Iα
2 (ε, h, δ)|
≤ C2(δ)h
2+γ/2‖uδ
ε − uδ‖L2(Ω) + ‖uδ
ε − uδ‖2
L2(Ω) + C3(δ)h
4+γ , (4.29)
where the constants Ci(δ) (i = 1, 2, 3) depend only on u(x) and δ.
E. Ya. Khruslov 137
Now, taking into account that, as ε = εk → 0, uδ
ε(x) converge to
uδ(x) weakly in R(Ω) and strongly in L2(Ω), and gradϕε(x) converge to
gradϕ(x) weakly in L2(Ω), by (4.27), (4.28), (4.29), and condition (c2)
we conclude that
lim
h→0
lim
ε=εk→0
Jε[u
δ
ε] ≥ J [uδ]
for any fixed δ > 0, where the functional J [w] is defined by (4.4). Passing
in this inequality to the limit as δ → 0 and taking into account (4.20)
and (4.21), we obtain the sought inequality (4.6), which completes the
proof of Theorem 3.1.
5. Completion of Proof of Theorem 2.1
First of all, we notice that, in view of the estimates (3.34) and (3.35)
and Theorem 3.3, solutions Eε(x, λ) of problem (3.5)–(3.6) (the electric
field in problem (3.2)–(3.4)) converge, for Reλ > 0, to E(x, λ) (the com-
ponent of the solution (E,Φ) of problem (3.28)–(3.31)) weakly in R(Ω).
This, in view of (3.8), implies that the magnetic fields Hε(x, λ) converge
weakly in L2(Ω) to the vector function
H(x, λ) = − 1
λ
(rotE(x, λ) −H0(x)).
By this, problem (3.28)–(3.31) can be rewritten in the form
rotH − λE − 1
λ
L(E − grad Φ) = J − E0, x ∈ Ω,
rotE + λH = H0, x ∈ Ω,
div (Lgrad Φ) − λ2CΦ = div (LE), x ∈ G,
n ∧ E = 0, x ∈ ∂Ω,
(Lgrad Φ)n = (LE)n, x ∈ ∂G.
(5.1)
Therefore, solutions (Eε, Hε) of problem (3.2)–(3.4) with Reλ > 0 con-
verge weakly in (L2(Ω))3 × (L2(Ω))3 to a pair of functions (E,H) such
that the triple (E,H,Φ) solves problem (5.1). Moreover, the convergence
is uniform in every compact part of the half-plane Reλ > 0.
Now consider the solution (Eε(x, t), Hε(x, t)) of the original (non-
stationary) problem (1.1)–(1.4). By standard arguments, one can easily
obtain the estimate
T∫
0
∫
Ω
{|Eε(x, t)|2 + |Hε(x, t)|2}dxdt < C, (5.2)
138 Homogenization of Maxwell’s Equations...
where the constant C is independent of ε (it depends on J , E0, H0,
and T ). This estimate implies that the set of solutions {Eε, Hε, ε > 0}
of problem (1.1)–(1.4) is weakly compact in (L2(Ω × [0, T ]))3 × (L2(Ω ×
[0, T ]))3 for all T > 0. By (3.1), these solutions can be expressed in terms
of {Eε(x, λ), Hε(x, λ)} by using the inverse Laplace transform
Eε(x, t) =
1
2πi
σ+i∞∫
σ−i∞
Eε(x, λ)eλtdλ, (5.3)
Hε(x, t) =
1
2πi
σ+i∞∫
σ−i∞
Hε(x, λ)eλtdλ, σ > 0.
The integrals here converge in the sense of distributionsD′((L2(Ω))3;(0,T ))
for all T .
Let ϕ(x) ∈ (L2(Ω))3 and ψ ∈ C∞
0 (0, T ). Using (5.3) and changing
the order of integration, we can write
T∫
0
∫
Ω
Eε(x, t)ϕ(x) dxψ(t) dt
=
1
2πi
σ+i∞∫
σ−i∞
(∫
Ω
Eε(x, λ)ϕ(x) dx
T∫
0
eλtψ(t) dt
)
dλ. (5.4)
Here the change of order of integration is allowed, since, by (3.34),
∣∣∣∣∣
∫
Ω
Eε(x, λ)ϕ(x) dx
∣∣∣∣∣ < C, Reλ = σ
and
T∫
0
eλtψ(t)dt = O
(
1
|λ|2
)
.
Using these estimates and taking into account that {Eε(x, λ), Hε(x, λ)}
converge weakly to {E(x, λ,H(x, λ)}, from (5.4) we obtain
lim
ε→0
∫
Ω
Eε(x, t)ϕ(x)ψ(t) dx dt
=
1
2πi
σ+i∞∫
σ−i∞
(∫
Ω
E(x, λ)ϕ(x) dx
T∫
0
eλtψ(t) dt
)
dλ
E. Ya. Khruslov 139
=
T∫
0
∫
Ω
E(x, t)ϕ(x)ψ(t) dx dt.
where
E(x, t) =
1
2πi
σ+i∞∫
σ−i∞
E(x, λ)eλtdλ. (5.5)
Similarly,
lim
ε→0
T∫
0
∫
Ω
Hε(x, t)ϕ(x)ψ(t) dx dt =
T∫
0
∫
Ω
H(x, t)ϕ(x)ψ(t) dt,
where
H(x, t) =
1
2πi
σ+i∞∫
σ−i∞
H(x, λ)eλtdλ. (5.6)
Here the integrals (5.5) and (5.6) converge in the sense of distributions
in D′(L2(Ω))3; [0, T ]).
Since the set of vector functions of the form
∑
k ϕk(x)ψk(t), with
ϕk(x) ∈ (L2(Ω))3 and ψk(t) ∈ C∞
0 (0, T ), is dense in (L2(Ω) × (0, T ))3,
and the estimate (5.2) holds uniformly with respect to ε, it follows that
{Eε(x, t), Hε(x, t)} converge, as ε → 0, weakly in (L2(Ω) × [0, T ])3 ×
(L2(Ω) × [0, T ])3 to {E(x, t), H(x, t)}, where E(x, t) and H(x, t) are de-
fined by (5.5) and (5.6), respectively. Finally, introducing the function
Φ(x, t) =
1
2πi
σ+i∞∫
σ−i∞
Φ(x, λ)eλtdλ (5.7)
and taking into account that (E(x, λ), H(x, λ),Φ(x, λ)) solves the bound-
ary value problem (5.1), by (5.5)–(5.7) we conclude that {E(x, t), H(x, t),
Φ(x, t)} solves the initial boundary value problem (1.8)–(1.13), which
completes the proof of Theorem 2.1.
Remark 5.1. Theorems 2.1 and 3.1 (the main results of the paper)
are conditional, since we assume that conditions (c1)–(c3) are satisfied;
moreover, at present we know no grid structure {Fε, ε > 0} for which
condition (c3) is established to be true. As for conditions (c1) and (c2),
one can verify whether they are true or not, for every particular case.
For instance, let sets Fε are formed by thin wires of diameter dε ≪ ε, the
140 Homogenization of Maxwell’s Equations...
axes of which form a periodic lattice in R3, with periods hiε (0 < hi ≤ 1,
i = 1, 2, 3), and let the following limit exists:
d = lim
ε→0
1
ε2 ln ε
dε
.
Then conditions (c1) and (c2) are satisfied; moreover, the limiting den-
sity of capacity C(x) and the limiting tensor of back induction L(x) are
constants:
C(x) ≡ C =
2πd(h1 + h2 + h3)
h1h2h3
,
L(x) ≡ L =
2πd
h1h2h3
diag{hi, i = 1, 2, 3}.
The proof of this fact as well as the discussion of condition (c3) will be
presented elsewhere.
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Contact information
Evgenii Ya.
Khruslov
B. Verkin Institute for Low Temperature
Physics and Engineering, NAS of Ukraine
Lenin Ave. 47,
61103, Kharkov,
Ukraine
E-Mail: KHRUSLOV@ilt.kharkov.ua
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