1 Introduction
This paper develops a new and efficient approach for maximizing the transmission signal between two points at a chosen frequency through changes to specific eigenmodes of the cavity. These changes are achieved by changing the boundary conditions. The eigenmodes and the associated eigenfrequencies of a cavity are sensitively dependant on the geometric properties of the domains, as well as the location of Dirichlet and Neumann boundary conditions. Many recent works have been devoted to the understanding of the effect of changing the boundary condition on the eigenmodes and the eigenfrequencies [1, 2, 4, 5, 11, 13, 14, 16].
Through the use of a tunable reflecting metasurface, the boundary condition can be switched from Dirichlet to Neumann at some specific resonant frequencies [3]. In [3, Part I], the physical mechanism underlying the concept of tunable metasurfaces is modeled both mathematically and numerically. It is shown that an array of coupled pairs of Helmholtz resonators behaves as an equivalent surface with Neumann boundary condition at some specific subwavelength resonant frequencies, where the size of one pair of Helmholtz resonators is much smaller than the wavelengths at the resonant frequencies. The Green’s function of a cavity with mixed (Dirichlet and Neumann) boundary conditions (called also the Zaremba function) is also characterized. In [3, Part II], a oneshot optimization algorithm is proposed and used to obtain a good initial guess for the positions around which the boundary conditions should be switched from Dirichlet to Neumann.
In this paper, we present a new methodology for maximizing the Zaremba function between two points at a chosen frequency through specific eigemodes of the cavity. The paper is organized as follows. In Section 2, we first recall some useful results on the eigenvalues of the mixed boundary value problem (called Zaremba eigenvalue problem). Of particular interest is their monotonicity property with respect to the size of the Neumann part proven in [14]. Then we reformulate the eigenvalue problem using boundary integral operators. Based on this nonlinear formulation and the use of the generalized argument principle for the characterization of the characteristic values of finitely meromorphic operatorvalued functions of Fredholmtype, we derive an accurate asymptotic formula of the changes of eigenfrequencies of a cavity with mixed boundary conditions in terms of the size of the part of the cavity boundary where the boundary condition is switched from Dirichlet to Neumann. Finally, we recall the asymptotic expansion of the Zaremba function in terms of the size of the Neumann part. The problem of changing a portion of a Dirichlet boundary to Neumann is more delicate than the converse. If a portion of the boundary is changed from having Neumann conditions to Dirichlet, the reverse consideration than in this paper, then an asymptotic expansion of the eigenvalues is easier to derive [5, 16]. The perturbation theory for the introduction of Neumann boundaries requires a careful consideration of the asymptotic behaviour of the Zaremba near the perturbation [3, Part II]. In Section 3, we derive a spectral decomposition of the Zaremba function. In Section 4, we consider the problem where we have a source in a bounded domain operating at a given frequency, and we want to determine, by exploiting the monotonicity property of the eigenvalues of the mixed boundary value problem, which part of the boundary to choose to be reflecting such that an eigenvalue of the mixed boundary value problem gets close enough to the operating frequency. In order to significantly enhance the signal at a given receiving point, both the emitter and the receiver should not belong to the nodal set corresponding to the eigenmode associated with the eigenvalue of the mixed boundary value problem.
There are two distinct issues: where to place the Neumann boundary condition, and how long it should be, to achieve the twin objectives of maximizing gain between a fixed sourcereceiver pair, and at a frequency close to a desired one.
Our main idea is to first nucleate the Neumann boundary conditions in order to maximize gain of the Zaremba function by making use of an asymptotic expansion of the Zaremba function in terms of the size of the Neumann part. Then the size of the Neumann part is changed in such away that an eigenvalue of the mixed boundary value problem gets close to the operating frequency by using the monotonicity property of the eigenvalues of the mixed eigenvalue problem. The optimization needs the highaccuracy evaluation of certain boundary integral operators, and this is done using techniques from [1, 2].
We present in Section 5 some numerical experiments to show the applicability and the accuracy of the proposed methodology.
2 Preliminaries
2.1 Laplace Eigenvalue with Mixed Boundary Conditions
Let be an open, bounded domain with a smooth boundary. We define as the topological closure of . We decompose the boundary into two parts, , where and are finite unions of open boundary sets. We define to be a partition of . Let and . The Zaremba function is the Green’s function to the Zaremba problem, also known as the fundamental Helmholtz equation with mixed boundary conditions,
(2.1) 
Here denotes the outer normal at and the normal derivative at . It is clear that we can write
where is the fundamental solution of the Helmholtz problem with wavenumber , and is a smooth function satisfying the boundary value problem
(2.2) 
In Section 3, we will see that exists for all but countably many values of , which are related to the unique solvability of the problem for . These exceptional values of are the eigenvalues to the associated Laplace eigenvalue problem with mixed boundary conditions
(2.3) 
Equation (2.3) has a nontrivial solution for a countable set of real values of [15, Theorem 4.10], which we refer to as , so that . We know that and that for all partitions of .
We denote by the pure Dirichlet eigenvalues for , corresponding to the case . We let denote the Neumann eigenvalues associated to the case . Then we have
In [9], it is shown that , for all , for a very general class of domains .
Remark
Let be the unit circle, we have that is (up to sorting) equal to
where is a Bessel function of the first kind and order . The eigenvalues corresponding to the roots of appear as simple Dirichlet eigenvalues; all others have multiplicity two. and is (up to sorting) equal to
Again, the eigenvalues corresponding to the roots of appear as simple Neumann eigenvalues; all others have multiplicity two. We refer to [10].
Recently, Lotoreichik and Rohleder [14, Proposition 2.3] showed the following monotonicity statement.
Proposition
Let be two partitions of , such that . If has a nonempty interior then
With Proposition 2.1, we can readily infer that if , and , then
2.2 Boundary Integral Formulation of the Eigenvalue Problem
The solution of the eigenvalue(2.3) can be represented by a single layer potential
(2.4) 
with surface density .
We define then the operators , , and by
where the ’p.v.’ stands for the principle value integral. This actually is the standard (Lebesgue) integral for a smooth curved , since is a bounded and sufficiently smooth integral operator kernel. From [17, Chapter 11] we have that is a Fredholm operator with index 0, we also readily infer that , , and are compact operator.
We then define in terms of these integral operators through
(2.5) 
We readily see that is an analytic Fredholm operator of index in .
To locate the Zaremba eigenvalues, we have the following statement:
“The real positive characteristics values of the operatorvalued function  
(2.6) 
In [2, Section 3] and [1], it is shown that every square root of a Zaremba eigenvalue is a real positive characteristic value of and every real positive characteristic value of is the square root of a Zaremba eigenvalue.
We see that is invertible for not a square root of a Zaremba eigenvalue.
We remark that the nonreal characteristic values of cannot correspond to eigenvalues to the Laplace equation. This yields the undesirable, but avoidable, difficulty in choosing a neighbourhood to apply Proposition 2.3 in our algorithm, see also Section 4, comment on Line 13.
The Statement (2.2) allows for a discretization and thus a numerical approximation of the value . We will use this further on. For these facts, we refer to [2, Sections 3 and 5].
Let us also consider the regularity of the solution and the density near a DirichletNeumann junction. The following result can be found in [1, Theorems 4.2 and 4.3].
Proposition
Let be nonempty. Let and satisfy the Statement (2.2). Let . Then there exists a neighborhood around such that for all and for all
where is the complexification of , that is with being the imaginary unit, and being its conjugate value, and where are polynomial functions of their respective arguments and of a degree such that none of their terms can be included in their respective error terms.
2.3 Approximation of the Zaremba Eigenvalue using the Generalized Argument Principle
In this section we derive asymptotic expressions for the perturbation of the Zaremba eigenvalues, when a small portion of the boundary is changed from Dirichlet to Neumann.
Let be a boundary interval of length . Let be a partition of . We associate the operator , defined via (2.5), to that partition. This corresponds to having a Dirichlet boundary condition. Then we define , also by obvious changes in the integrals in (2.5), to be the operator associated to the partition . This in turn corresponds to being a Neumann part. For ease of notation, we define and for all , and call those characteristic values to their respective operators. From [6, Lemma 3.8] we then have the following Lemma:
Lemma
Let be a simple characteristic value. Let be a neighbourhood of , such that . Assume further that no other square root of Zaremba eigenvalue to the partition of is in . Then is given by the contour integral
Here denotes the variation of the operator in the wavenumber parameter . This expression is exact. Unfortunately, its use in a practical algorithm is limited, since it would entail inverting the operator for each used in an optimization. It is useful, therefore, to locate an expression in which this inverse is approximated by instead.
From [6, Theorem 3.12] we get the approximation
(2.7) 
where we expect the error to be in . We can, in fact, obtain a faster and even more accurate approximation, which we describe in the following proposition.
Proposition
Let be the (sorted) characteristic value of corresponding to the decomposition and assume it is simple Then one can find a and a neighbourhood containing so that

the characteristic value of the operator (obtained by changing to a Neumann boundary condition) is contained ;

no other square root of the Laplace eigenvalues to the partition of are in .

The characteristic value of the perturbed operator is given by
Here is the identity operator.
Proof
We first observe from Proposition 2.1 together with the fact that is a Fredholm analytic operator of index in , we can see that for . We now examine the perturbed operator . Its characteristic value is . Provided is sufficiently close to , we have the following Taylor expansion:
(2.8) 
where . This expansion holds only in a neighborhood of , and so must be small enough such that .
Then consider that we have in the operatornorm
for close enough to both and , because then the Taylor remainder . If is small enough, then . Then by the Generalization of Rouché’s Theorem [6, Theorem 1.15] we have that since and are close in operator norm, they both have the same number of characteristic values in . Thus has a simple characteristic value in . Now we can use Lemma 2.3, but replacing by :
to get
and hence,
Moreover, by a standard perturbation argument [6, Section 5.2.4], we have at the leadingorder term
where is the root function associated with the characteristic value evaluated at . Thus,
and therefore, Proposition 2.3 holds.
We remark on the significance of this result from the point of view of computation, and which makes it a key ingredient in our algorithm. If one seeks a highaccuracy approximation of the characteristic value of , and one already has a good approximation of , the approximation in Proposition 2.5 allows us to proceed by assembling only one matrix, that corresponding to . The contour integrals can be effectively computed using the trapezoidal rule, making this an inexpensive but very accurate approximation of .
2.4 Approximation of the Zaremba Function
Let , and let be a boundary interval of length with center . Let be the partition of , and with it we associate the Zaremba function , for , defined via (2.1). This corresponds to having a Dirichlet boundary condition. Then we define , , also defined via (2.1), to be the Zaremba function associated to the partition . This in turn corresponds to having a Neumann boundary condition. We then have the following lemma.
Lemma
Let and be defined as described above. Let be small enough. Let , such that , and , for all . Then for all ,
Numerical experiments confirm that is of order of , as long as is far enough away from the boundary.
3 Spectral Decomposition of the Zaremba Function
Let us again consider the more general setup at the beginning of Section 2, that is let be a partition of , let be the Zaremba eigenvalues and let be an
orthonormal basis of associated eigenfunctions. Then we have the following statement about the Zaremba function
, , defined by (2.1).Theorem
For all , and for all which are not in the spectrum, ie, of the Zaremba eigenvalue problem, the Zaremba function , given by (2.1), exists and is in . Furthermore, we can write it as
Next, we will consider the proof of Theorem 3. To this end, we define
Consider that the solution to the Laplace eigenvalueequation is element of .
The operator is selfadjoint in , which we readily see using Green’s identity, and it has thus a discrete spectrum. Moreover, corresponds to the sesquilinear form with domain , since for all , see [7, 12, 18] for more details on semibounded selfadjoint operators and corresponding quadratic forms. And the form is closed, nonnegative and symmetric. This allows us to use the minmax principle. Thus we can write for all ,
(3.1) 
This leads us to the following lemma.
Lemma
For all , we have that
(3.2) 
where , that is the linear subset spanned by eigenfunctions of the Laplace eigenvalueequation with mixed boundary conditions (2.3) is dense in .
Proof
Let . Then for all , we have that
where we used Green’s identity and the fact that . Next, we want to show that
(3.3) 
To this end, consider the minmax principle (3.1), it tells us that
Thus, we can infer from , which in turn is given by an induction argument, whose induction basis follows trivially from the minmax principle (3.1). Using the definition of , we have that
Thus, using (3.3), we have that
(3.4) 
Since , is bounded. Using the fact that , we have that . This completes the proof of Lemma 3.
Proof (Theorem 3)
To show the existence of the Zaremba function , we write , for all as
(3.5) 
where is the fundamental solution to the Helmholtz equation, and satisfies
(3.6) 
The solution to (3.6) does exist, for those values of specified in the theorem, and it is in , see [15, Theorem 4.10]. Using that , we have that . Thus from Lemma 3 and the density of in , we have that for all , ,
for some , depending on . Let us give an expression for the . Using Green’s identity, we have that
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