Consider the scattering of time-harmonic electromagnetic plane waves from an infinite cylinder with a mixture of an open crack
and a bounded domain
in
as cross section. For further considerations, we suppose that
has smooth boundary
(e.g.,
), and the crack
(smooth) can be extended to an arbitrary smooth, simply connected, closed curve
enclosing a bounded domain
such that the normal vector
on
coincides with the outward normal vector on
which we again denote by
. The bounded domain
is located inside the domain
, and
.

In the whole paper, we assume that
and
.

Suppose that

where
is an injective piecewise
function. We denote the outside of
with respect to the chosen orientation by
and the inside by
. Here we suppose that the
is divided into two parts
and
and consider the electromagnetic field E-polarized. Different boundary conditions on
,
and
lead to the following problem:

where
for
and
for
. The total field
is decomposed into the given incident field
, and the unknown scattered field
which is required to satisfy the Sommerfeld radiation condition

uniformly in
with
.

We recall some usual Sobolev spaces and some trace spaces on
in the following.

Let
be a piece of the boundary. Use
and
to denote the usual Sobolev spaces,
is the trace space, and we define

Just consider the scattered field
, then (2.2) and (2.3) are a special case of the following problem.

Given
,
,
and
find
such that

and
is required to satisfy the Sommerfeld radiation condition (2.3). For simplicity, we assume that
and
.

Theorem 2.1.

The problems (2.5) and (2.3) have at most one solution.

Proof.

Let
be a solution to the problem (2.5) with
, we want to show that
in
.

Suppose that
(with boundary
) is a sufficiently large ball which contains the domain
. Obviously, to the Helmholtz equation in (2.5), the solution
satisfies the following transmission boundary conditions on the complementary part
of
:

where "
" denote the limit approaching
from outside and inside
, respectively. Applying Green's formula for
and
in
and
, we have

where
is directed into the exterior of the corresponding domain.

Using boundary conditions on
,
and the above transmission boundary condition (2.6), we have

Hence

So, from [13, Theorem
] and a unique continuation argument we obtain that
in
.

We use
and
to denote the jump of
and
across the crack
, respectively. Then we have the following.

Lemma 2.2.

If
is a solution of (2.5) and (2.3), then
and
.

The proof of this lemma can be found in [11].

We are now ready to prove the existence of a solution to the above scattering problem by using an integral equation approaching. For
, by Green representation formula

and for

where

is the fundamental solution to the Helmholtz equation in
, and
is a Hankel function of the first kind of order zero.

By making use of the known jump relationships of the single- and double-layer potentials across the boundary
(see [5, 11]) and approaching the boundary
from inside
, we obtain (for
)

where
,
,
, and
are boundary integral operators:

defined by (for
)

Similarly, approaching the boundary
from inside
we obtain (for
)

From (2.13)–(2.18), we have

Restrict
on
, from (2.19) we have

where
means a restriction to
.

Define

Then zero extend
,
, and
to the whole
in the following:

By using the boundary conditions in (2.5), we rewrite (2.21) as

where

Furthermore, we modify (2.24) as

where the operator
is the operator applied to a function with
and evaluated on
, with analogous definition for
,
, and
. We have mapping properties (see [5, 11])

Again from (2.13)–(2.18), restricting
to boundary
we have

or

Like previous, define

Then we can rewrite (2.29) as

where

Similar to (2.26), we modify (2.31) as

and we have mapping properties:

Combining (2.13) and (2.14),

Using (2.17) and (2.18),

Then using (2.36),

From (2.29), we have

Restricting (2.38) to
and using (2.39), we modify (2.38) as

where

for
.

Define

and using the notation in previous, we can rewrite (2.40) as

or

where the operators
,
, and
are restriction operators (see (2.29)). As before, we have mapping properties:

By using Green formula and approaching the boundary
from inside
we obtain (for
)

The last term in (2.46) can be reformulated as

Since
and
in (2.47), we have the following result (see [13]).

Lemma 2.3.

By using Green formula and the Sommerfeld radiation condition (2.3), one obtains

Proof.

Denote by
a sufficiently large ball with radius
containing
and use Green formula inside
. Furthermore noticing
,
, and the Sommerfeld radiation condition (2.3), we can prove this lemma.

Combining (2.46), (2.47), and Lemma 2.3 and restricting
to
we have

where

Define

and then we can rewrite (2.49) as

Similarly,
and
are restriction operators as before, and we have mapping properties:

Combining (2.52), (2.26), (2.33), and (2.44), we have

If we define

then (2.54) can be rewritten as a boundary integral system:

Remark 2.4.

If the above system (2.56) has a unique solution, our problem (2.5) with (2.3) will have a unique solution (see [13, 14]).