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]).