Integral and Variational Formulations for the Helmholtz Equation Inverse Source Problem

Abstract

The purpose of this paper is to explore the Hilbert space functional structure of the Helmholtz equation inverse source problem. An integral equation for the sources reconstruction based on the composition of the trace and Green's function operators is introduced and compared with the reciprocity source reconstruction methodologies. An equivalence theorem comparing the integral inverse source equation with the variational weak reciprocity gap functional equation is then demonstrated. Some examples on applications to the unitary disk are presented.

1. Introduction

The inverse source problem for the Helmholtz Dirichlet equation is a basic tool for the investigation of transient source problems [1–4]. In order to investigate this class of problems, let Ω⊂ℝ^𝑁 be a bounded domain with smooth boundary Γ. Let 𝜅 be a complex number, 𝑔∈𝐻^1/2(Γ), and 𝑓∈𝐿²(Ω). The direct problem with the Helmholtz operator: to find a regular field 𝑢 that satisfy the system −Δ𝑢−𝜅²𝛾𝑢=𝑓inΩ,₀𝑢=𝑔onΓ, (1.1) is well posed and has a unique solution 𝑢∈𝐻¹(Ω) when 𝜅² is not an eigenvalue of the Laplacian. In this paper we consider 𝛾₀∶𝐻¹(Ω)→𝐻^1/2(Γ) the trace operator and simplify the notation by calling the boundary data 𝑔 of problem (1.1) 𝛾₀𝑢. The trace theorem [5] assures the existence of a function 𝑔_𝜈𝑥∈𝐻^−1/2(Γ) which is the normal trace, that is, 𝛾₁𝑢=𝜕_𝜈𝑥𝑢. (1.2)

When 𝜅 is a real positive or an imaginary number we have, respectively, the proper or the modified Helmholtz equation. When 𝜅=0, we obtain the Laplace equation. For the complete setting of complex values, we consider the problem as the Helmholtz equation direct problem.

The inverse source problem consist, in knowing the Cauchy data in the boundary Γ, that is, the Dirichlet to Neumann map in at least one Dirichlet datum 𝑔, to recover the source 𝑓. It may be formally posed as follows: givenaCauchydataset𝑔,𝑔_𝜈𝑥⊂𝐻^1/2(Γ)×𝐻^−1/2(Γ),withcompatibilitycondition𝜕𝑔𝜕𝜈_𝑥=𝑔_𝜈𝑥(),onΓ,fortheequationmodelinthesystem1.1tofind𝑢,𝑓∈𝐻¹(Ω)×𝐿²(Ω). (1.3)

The two problems, direct and inverse, can also be formulated with only one system of equations: to find (𝑢,𝑓)∈𝐻¹(Ω)×𝐿²(Ω) such that −Δ𝑢−𝑘²𝛾𝑢=𝑓inΩ,₀𝛾𝑢=𝑔onΓ,₁𝑢=𝑔_𝜈𝑥onΓ. (1.4)

Since in the inverse problem the Cauchy data are known, we may associate these data with a fourth-order Dirichlet problem with the Bilaplacian operator Δ²𝑢−𝜅⁴𝛾𝑢=ℎinΩ,₀𝛾𝑢=𝑔onΓ,₁𝑢=𝑔_𝜈𝑥onΓ, (1.5) where ℎ∈𝐻⁻²(Ω) is an arbitrary given function. This problem is well posed and has a unique solution 𝑢∈𝐻²(Ω) when 𝜅⁴ is not an eigenvalue of the Bilaplacian. This motivates the following naive existence result

Remark 1.1. Suppose that a Cauchy data pair (𝑔,𝑔_𝜈𝑥)∈𝐻^3/2(Γ)×𝐻^1/2(Γ) is more regular than normal case and 𝜅⁴∉Σ₄ is not an eigenvalue of the Bilaplacian, then there exists a solution to inverse source problem (1.3).

Proof. Since data is regular, we may consider the fourth-order direct Dirichlet problem (1.5) with the given Cauchy data and some ℎ∈𝐻⁻²(Ω). The inverse source solution will be 𝑓=−Δ𝑢+𝜅²𝑢∈𝐿²(Ω).

To obtain the existence of global solution for problems (1.1), and (1.5), please see [6]. For more information about linear integral equations, see [7]. For the inverse source problems, see [8]. For functional analysis, please see [9].

In Section 2 we develop a Hilbert space functional framework to the problem based on special 𝐿²(Ω) decomposition. The analysis of the Dirichlet to Newman map and of the Source to Neumann maps is done in Sections 2.1 and 2.2, respectively. It is based only on the analysis of the direct problem structure. The analysis of the Adjoint Source to Neumann map is done in Section 2.3. In Section 3 we use the Green function operator to put together the results found in Section 2. There in Section 3.1 we present an integral equation for the inverse problem based on the relative Dirichlet to Newman map. The reciprocity gap functional is introduced in Section 4, where an equivalence theorem between this and the integral formulation is proved. In Section 5 some particular results for the unitary disk in ℝ² are presented. Finally, we conclude the paper in Section 6.

2. The Dirichlet and the Source to Neumann Map

For 𝑙∈ℝ, the space 𝐻^𝑙(Ω) is the Sobolev class of the functions of the spatial variable 𝑥. For more information, see [5]. Let us consider for future use the following sets of eigenvalues: Σ₂∶=𝜆∈ℂ∶−Δ𝑢=𝜆𝑢in𝐻²(Ω)∩𝐻¹₀(Ω,Σ₄)∶=𝜆∈ℂ∶Δ²𝑢=𝜆𝑢in𝐻⁴(Ω)∩𝐻²₀(Ω). (2.1)

Definition 2.1. One says that a function is metaharmonic when it is in the set 𝐻_{−Δ−𝜅2}(Ω)=𝑣∈𝐿²(Ω);−Δ𝑣−𝜅²𝑣=0, (2.2) where 𝜅²∉Σ₂ and 𝜅⁴∉Σ₄.

2.1. The Dirichlet to Neumann Map

Definition 2.2. Consider problem (1.1) with zero source, that is, 𝑓=0 and 𝑔∈𝐻^1/2(Γ). This problem has a solution 𝑤⁰∈𝐻¹(Ω). One defines the Dirichlet to Neumann map for the Helmholtz equation as the operator Λ⁰∶𝐻^1/2(Γ)⟶𝐻^−1/2(Γ),Λ⁰𝑔(𝑥)=𝜕𝑤⁰𝜕𝜈_𝑥(𝑥),𝑥∈Γ. (2.3) By the trace theorem [5], it is well defined, linear, and continuous.

Remark 2.3. We alternatively can define the Dirichlet to Neumann map (2.2) for the problem (1.1) as an operator with a nonzero source problem Λ^𝑓∶𝐻^1/2(Γ)→𝐻^−1/2(Γ) defined by the graph (𝑔,Λ^𝑓𝑔)∶=(𝑢,𝜕_𝜈𝑢)∣_𝜕Ω.

Definition 2.4. One has ℳ∶=𝑔,𝑔_𝜈𝑥∈𝐻^1/2(Γ)×𝐻^−1/2(Γ);𝑔_𝜈𝑥−Λ⁰𝑔∈𝐻^1/2(Γ), (2.4) where Λ⁰ is a Dirichlet to Neumann map.

Theorem 2.5. Let (𝑔,𝑔_𝜈𝑥)∈𝐻^1/2(Γ)×𝐻^−1/2(Γ). A function 𝑓∈𝐻_−Δ+𝜅2(Ω)is a solution to the inverse source problem (1.3) if and only if (𝑔,𝑔_𝜈𝑥)∈ℳ

Proof. Suppose (𝑔,𝑔_𝜈𝑥)∈ℳ. Consider the following fourth-order problem: Δ²−𝑘⁴𝛾𝑤=0inΩ,₀𝛾𝑤=0onΓ,₁𝑤=𝑔_𝜈𝑥−Λ⁰𝑔onΓ. (2.5) Let us use this problem to define a source 𝑓₀=(−Δ−𝑘²)𝑤, where 𝑤∈𝐻²(Ω) is solution of problem (2.5). Note that 𝑓₀∈𝐻_−Δ+𝜅2(Ω). Let (𝑢,𝑓)∈𝐻¹(Ω)×𝐿²(Ω) be a solution of the inverse source problem (1.4). Then 𝑢=𝑤⁰+𝑤, where 𝑤⁰ is the solution of the homogeneous source problem in the definition of the Dirichlet to Neumann map (2.2) and 𝑓=𝑓₀. The sufficiency is proved.
To proof necessity, suppose that there exists a (𝑢,𝑓)∈𝐻¹(Ω)×𝐻_−Δ+𝜅2(Ω) which is solution of the inverse problem (1.4).
Consider the second-order problem with homogeneous boundary Dirichlet data −Δ−𝑘²𝑤₀𝛾=𝑓,𝑥∈Ω,₀𝑤₀=0,𝑥∈Γ, (2.6) with a unique solution 𝑤₀∈𝐻²(Ω)∩𝐻¹₀(Ω). By the trace theorem, we have 𝛾₁𝑤₀∈𝐻^1/2(Γ). Note that 𝑔_𝜈𝑥=𝛾₁𝑤₀+𝛾₁𝑤⁰, where 𝛾₁𝑤⁰=Λ⁰𝑔. So, 𝛾₁𝑤₀=𝑔_𝜈𝑥−Λ⁰𝑔.

Remark 2.6. We have proved the existence and uniqueness of solution to (1.4) in 𝐻¹(Ω)×𝐻_−Δ+𝜅2(Ω). However, this does not mean that when we do the search in a larger space 𝐻¹(Ω)×𝐿²(Ω), we will continue having uniqueness. In fact, we will prove in the next proposition that 𝑓−𝑓₀⟂𝑣∈𝐻_{−Δ−𝜅2}(Ω) and consequently 𝑓∈𝑓₀+𝐻_{−Δ−𝜅2}(Ω)^⟂, (2.7) where 𝑓∈𝐿²(Ω) and 𝑓₀∈𝐻_−Δ+𝜅2(Ω). This leads us to conclude that the solution of (1.4) is actually a class of functions, where we can “reconstruct” or “observe” only part of the solution in 𝐻_−Δ+𝜅2(Ω).

Proposition 2.7. If 𝜅²∉Σ₂ and 𝜅⁴∉Σ₄, then 𝐿²(Ω)=𝐻_{−Δ−𝜅2}(Ω)^⟂⊕𝐻_−Δ+𝜅2(Ω). (2.8)Proof. Consequence of Lemmas 2.8 and 2.9.From now on, we are supposing always that 𝜅²∉Σ₂ and 𝜅⁴∉Σ₄.

Lemma 2.8. One has 𝐿²(Ω)=𝐻_−Δ+𝜅2(Ω)⊕(−Δ−𝜅²)[𝐻²₀(Ω)].

Proof. For an arbitrarily given 𝑓∈𝐿²(Ω), the problem (1.1) with zero Dirichlet datum 𝑔=0 is well posed and has an 𝑤₀∈𝐻²(Ω) with normal derivative trace Λ^𝑓=𝜕_𝜈𝑤₀∈𝐻^1/2(Γ). Also, since 𝜅⁴∉Σ₄, the fourth-order problem (1.5) with Cauchy datum (0,Λ^𝑓)∈𝐻^3/2(Γ)×𝐻^1/2(Γ) and zero source has solution 𝑣 and is well posed in 𝐻²(Ω). These two 𝐻²(Ω) solutions may be used to define a function 𝑤∶=𝑤₀−𝑣∈𝐻²₀(Ω), (2.9) and since by problem (1.1) 𝑓=−Δ−𝜅²𝑤₀=−Δ−𝜅²𝑣+−Δ−𝜅²𝑤, (2.10) we obtain that arbitrary 𝐿²(Ω) function 𝑓 is the sum of a function in 𝐻_{−Δ−𝜅2}(Ω) and a function in (−Δ−𝜅²)[𝐻²₀(Ω)]. If we show that 𝐻_−Δ+𝜅2(Ω)∩−Δ−𝜅²𝐻²₀(Ω)={0}, (2.11) we prove that 𝐿²(Ω)=𝐻_−Δ+𝜅2⊕(−Δ−𝜅²)[𝐻²₀(Ω)]. For this, take a 𝑢 in the intersection 𝐻_{−Δ−𝜅2}∩(−Δ+𝜅²)[𝐻²₀(Ω)]. Then −Δ−𝜅²𝑢=0,𝑢=−Δ+𝜅²𝑣,forsome𝑣∈𝐻²₀(Ω) (2.12) which means that 𝑣 is a solution of completely homogeneous fourth-order problem (1.5), with no source and zero Cauchy datum. Since 𝜅⁴∉Σ₄, the unique solution is trivial 𝑣=0, and 𝑢=(−Δ+𝜅²)𝑣=0. We have proved that 𝐿²(Ω)=𝐻_{−Δ−𝜅2}(Ω)⊕(−Δ+𝜅²)[𝐻²₀(Ω)].

Lemma 2.9. One has (−Δ−𝜅²)[𝐻²₀(Ω)]=[𝐻_{−Δ−𝜅2}(Ω)]^⟂.

Proof. Using the second Green theorem with 𝑓∈𝐻_{−Δ−𝜅2}(Ω) and 𝑣=(−Δ−𝜅²)𝑤 for some 𝑤∈𝐻²₀(Ω), we obtain _Ω𝑓𝑣𝑑𝑥=_Ω𝑓−Δ−𝜅²𝑤𝑑𝑥=_Ω𝑓(−Δ𝑤)𝑑𝑥−𝜅²_Ω𝑓𝑤𝑑𝑥=_Ω−Δ𝑓𝑤𝑑𝑥−𝜅²_Ω𝑓𝑤𝑑𝑥=0, (2.13) and since the scalar product with arbitrary ℎ∈(−Δ−𝜅²)[𝐻²₀(Ω)] is zero, the inclusion (−Δ−𝜅²)[𝐻²₀(Ω)]⊂𝐻^⟂_{−Δ−𝜅2}(Ω) follows.
To prove the other inclusion, let 𝑓∈𝐿²(Ω), suppose 𝑓 is orthogonal to 𝑣 for all 𝑣∈𝐻_{−Δ−𝜅2}(Ω), and take a function 𝑤∈𝐻²(Ω) such that 𝑤 is solution of (1.1) with source 𝑓 and Dirichlet datum 𝑔=0.
By using the second Green formulas for 𝑣∈𝐻_{−Δ−𝜅2}(Ω), we have 0=_Ω𝑓𝑣𝑑𝑥=_Ω−Δ−𝜅²𝑤𝑣𝑑𝑥=_Ω𝑤−Δ−𝜅²𝑣𝑑𝑥=_Γ𝑣𝜕_𝜈𝑥𝑤𝑑𝜎_𝑥. (2.14) Note that, since the test functions are dense on 𝐻_{−Δ−𝜅2}(Ω), we may consider a variational formulation using the dual system (𝐷(Ω),𝐷(Ω)^∗). Because 𝑣∈𝐿²(Ω), by trace theorem 𝑣|_𝜕Ω∈𝐻^−1/2(Γ) and _Γ𝑣𝜕_𝜈𝑥𝑤𝑑𝜎_𝑥=𝑣,𝜕_𝜈𝑥𝑤_{(𝐷(Ω),[𝐷(Ω)]∗)}=0⟹𝜕_𝜈𝑥𝑤=0on𝐻^1/2(Γ). (2.15) So, if an arbitrary 𝐿²(Ω) function is orthogonal to all functions in 𝐻_{−Δ−𝜅2}(Ω), then it is in (−Δ−𝜅²)[𝐻²₀(Ω)] and the reverse inclusion follows.

2.2. The Source to Neumann Map

Let us consider the simultaneous solution of the direct and inverse source problem (1.4) and search for a solution (𝑢,𝑓)∈𝐻¹(Ω)×𝐿²(Ω). It follows from Theorem 2.5 that if we restrict the source search to the subspace 𝐻_−Δ+𝜅2(Ω)⊂𝐿²(Ω), we will find a unique solution.

Definition 2.10. Consider problem (1.1) with zero Dirichlet data, that is, 𝑔=0 and 𝑓∈𝐻_−Δ+𝜅2(Ω)⊂𝐿²(Ω) and solution 𝑤^𝑓₀∈𝐻²(Ω)∩𝐻¹₀(Ω). One defines the Source to Neumann map for the Helmholtz equation as the operator Λ₀∶𝐻_−Δ+𝜅2(Ω)⟶𝐻^1/2(Γ),Λ₀𝑓(𝑥)=𝜕𝑤^𝑓₀𝜕𝜈_𝑥(𝑥)=𝛾₁𝑤^𝑓₀(𝑥),𝑥∈Γ. (2.16) By the trace theorem [5], it is well defined, linear, and continuous.

Theorem 2.11. Λ₀∶𝐻_−Δ+𝜅2(Ω)→𝐻^1/2(Γ),  is an isomorphism.

Remark 2.12. If we consider problem (1.1) with Dirichlet data 𝑔∈𝐻^3/2(Γ) and 𝑓∈𝐻_−Δ+𝜅2(Ω)⊂𝐿²(Ω) and solution 𝑤^𝑓_𝑔∈𝐻²(Ω)∩𝐻¹₀(Ω), we can define the Source to Neumann map for the Helmholtz equation as the operator Λ_𝑔∶𝐻_−Δ+𝜅2(Ω)⟶𝐻^−1/2(Γ),Λ_𝑔𝑓(𝑥)=𝜕𝑤^𝑓_𝑔𝜕𝜈_𝑥(𝑥)=𝛾₁𝑤^𝑓_𝑔(𝑥),𝑥∈Γ. (2.17)
Note that this more general situation will produce results such as Λ_𝑔[0](𝑥)=Λ⁰𝑔(𝑥)=𝛾₁𝑔=𝜕𝑤⁰_𝑔𝜕𝜈_𝑥(𝑥). (2.18) As consequence,(i)a functional such as a Source-Dirichlet to Neumann map may be defined Λ[⋅,⋅]∶𝐻_−Δ+𝜅2(Ω)×𝐻^1/2(Γ)⟶𝐻^−1/2(Γ),𝑓,𝑔⟼Λ𝑓,𝑔=𝛾₁𝑤^𝑓_𝑔, (2.19)(ii)and restrieted to be a functional such as a Dirichlet to Neumann map Λ[0,⋅]∶{0}×𝐻^1/2(Γ)⟶𝐻^−1/2(Γ),𝑔⟼Λ0,𝑔≡Λ⁰𝑔=𝛾₁𝑤⁰_𝑔, (2.20)(iii)or to be a functional such as a Source to Neumann map Λ[⋅,0]∶𝐻_−Δ+𝜅2(Ω)×{0}⟶𝐻^−1/2(Γ),𝑓⟶Λ0,𝑓≡Λ₀𝑓=𝛾₁𝑤^𝑓₀. (2.21) It is important to note that in this more general definition it is not possible to prove that Λ[⋅,⋅] is an isomorphism, since the fact that the trace 𝛾₀[𝑤^𝑓_𝑔]=0 is used in the proof.

Lemma 2.13. 𝐻_−Δ+𝜅2(Ω) is a Hilbert space with 𝐿²(Ω) norm.

Proof. Let us consider the canonical projection 𝜋₂∶𝐿²(Ω)=𝐻_−Δ+𝜅2(Ω)⊕𝐻^⟂_{−Δ−𝜅2}(Ω)⟶𝐻^⟂_{−Δ−𝜅2}(Ω)𝑓,𝑓^⟂⟼𝑓^⟂. (2.22) Note that 𝜋₂ is continuous, 𝜋₂⁻¹[0] is closed. Since 𝜋₂⁻¹[0]=𝑓,0 (2.23) for all 𝑓∈𝐻_−Δ+𝜅2(Ω), it follows that {0}×𝐻_Δ+𝜅2(Ω) is closed. Consequently 𝐻_−Δ+𝜅2(Ω) is closed subspace of the Banach space 𝐿²(Ω). So, it is Banach and, consequently, is a Hilbert space with the scalar product induced by 𝐿²(Ω).

Proof of Theorem 2.11. (i) Λ₀∶𝐻_−Δ+𝜅2(Ω)→𝐻^1/2(Γ) is continuous.
Note that Λ₀=𝛾₁∘𝑖 is a composition of the normal trace 𝛾₁∶𝐻²(Ω)→𝐻^1/2(Γ) and the canonical embedding 𝑖∶𝐻_−Δ+𝜅2(Ω)→𝐻²(Ω). The normal trace is continuous by trace theorem. The canonical embedding is also continuous since 𝐻_−Δ+𝜅2(Ω) is closed by Lemma 2.13. So, Λ₀ is continuous.(ii) Λ₀∶𝐻_−Δ+𝜅2(Ω)→𝐻^1/2(Γ) is one to one.
Take some arbitrary 𝑓∈Ker(Λ₀). Then Λ₀[𝑓]=𝛾₁[𝑤₀]=0 is the normal derivative of the problem (1.1) with 𝑔=0. By hypotheses, 𝑓∈𝐻_−Δ+𝜅2(Ω) and, consequently, (−Δ+𝜅²)𝑓=(−Δ+𝜅²)(−Δ−𝜅²)𝑤^𝑓₀=0. The fourth-order problem Δ²−𝑘⁴𝑤^𝑓₀𝛾=0inΩ,₀𝑤^𝑓₀𝛾=0onΓ,₁𝑤^𝑓₀=0onΓ, (2.24) is well posed and has a unique 𝑤^𝑓₀=0. Then 𝑓=(−Δ−𝜅²)𝑤^𝑓₀=0. Since 𝑓 is arbitrary, Ker(Λ₀)={0} and the injectivity is proved.(iii) Λ₀∶𝐻_−Δ+𝜅2(Ω)→𝐻^1/2(Γ) is onto.
Consider an arbitrary 𝑔_𝜈𝑥∈𝐻^1/2(Γ) and 𝜅≠0, where 𝑔_𝜈𝑥 does not necessarily satisfy the compatibility condition. Let 𝑤_𝑔𝜈𝑥∈𝐻_−Δ+𝜅2(Ω) be a solution of the well-posed Neumann data problem, −Δ+𝜅²𝑤_𝑔𝜈𝑥𝛾=0onΩ,₁𝑤_𝑔𝜈𝑥=𝑔_𝜈𝑥inΓ. (2.25) Note that with 𝑓=𝑤_𝑔𝜈𝑥 we obtain Λ₀[𝑓]=𝛾₁[𝑤_𝑔𝜈𝑥]=𝑔_𝜈𝑥. So, Λ₀ is surjective.
It remains to prove the following.(iv) Λ₀⁻¹∶𝐻^1/2(Γ)→𝐻_−Δ+𝜅2(Ω) is continuous.
In fact, this is a consequence of the Banach open map theorem, since. Λ₀∶𝐻_−Δ+𝜅2(Ω)⟶𝐻^1/2(Γ) (2.26) is a linear continuous bijective application between Banach spaces.

2.3. The Adjoint Source to Neumann Operator

Definition 2.14. Consider again the problem (1.1) with zero Dirichlet data, that is, 𝑔=0 and 𝑓∈𝐿²(Ω)and  solution 𝑤^𝑓₀∈𝐻²(Ω)∩𝐻¹₀(Ω). One defines the extension of the Source to Neumann map for the Helmholtz equation as the operator Λ₀∶𝐿²(Ω)⟶𝐻^1/2(Γ),Λ₀𝑓(𝑥)=𝜕𝑤^𝑓₀𝜕𝜈_𝑥(𝑥)=𝛾₁𝑤^𝑓₀(𝑥),𝑥∈Γ. (2.27)

Remark 2.15. By the trace theorem [5], it is well defined, linear, and continuous. As an extension of Λ₀, surjectivity is preserved. Also Λ₀=Λ₀∘𝜋₁.

Corollary 2.16. The quotient of 𝐿²(Ω) by 𝐻^⟂_{−Δ−𝜅2}(Ω) is a copy of 𝐻^1/2(Γ).

Proof. Consider the following chain of embeddings: (2.28) where 𝑐₀ is a canonic embedding and 𝑐₁ is an isomorphism by the Banach isomorphism theorem. Since Λ₀ also is an isomorphism by Theorem 2.11, the corollary is proved, that is, 𝐿²(Ω)/𝐻^⟂_−Δ+𝜅2(Ω)≅𝐻^1/2(Γ).

Corollary 2.17. Ker(Λ₀)=𝐻^⟂_{−Δ−𝜅2}(Ω) is a closed subspace of 𝐿²(Ω).

Remark 2.18. Since Λ₀∈ℒ(𝐿²(Ω);𝐻^1/2(Γ)) is bounded, then its adjoint operator Λ₀^∗∈ℒ(𝐻^−1/2(Γ);𝐿²(Ω)) is well defined and continuous and one to one.

Corollary 2.19. Λ₀^∗∶𝐻^−1/2(Γ)→𝐻_{−Δ−𝜅2}(Ω)  is an isomorphism.

Proof. (i) Λ₀^∗∶𝐻^−1/2(Γ)→𝐻_{−Δ−𝜅2}(Ω) is well defined.
We know that 𝐻^⟂_{−Δ−𝑘2}(Ω)=KerΛ₀=RgΛ₀^∗,_⟂RgΛ₀^∗⟂=𝐻_{−Δ−𝑘2}(Ω), (2.29) where Rg and Ker denote the operator range and kernel, respectively. It is well known that if 𝑋 is a Banach space and 𝑁, that is, a subspace of its dual 𝑋^∗, then [^⟂𝑁]^⟂=𝑁^𝑤∗ is the weak star closure of 𝑁 in 𝑋^∗. Applying this result to our case, we have 𝐻_{−Δ−𝜅2}(Ω)=Rg(Λ₀^∗)^𝑤∗ in 𝐿²(Ω). Particularly, Rg(Λ₀^∗)⊂𝐻_{−Δ−𝜅2}(Ω) assures that Λ₀^∗ is well defined.(ii) Λ₀^∗∶𝐻^−1/2(Γ)→𝐻_{−Δ−𝜅2}(Ω) is one to one.
Note that, Λ₀∶𝐻_−Δ+𝜅2(Ω)→𝐻^1/2(Γ) is onto and 𝐻_{−Δ−𝜅2}(Ω) is a closed subspace of 𝐿²(Ω),(iii) Λ₀^∗∶𝐻^−1/2(Γ)→𝐻_{−Δ−𝜅2}(Ω) is onto.
Note that since, Rg(Λ₀) is closed in 𝐿²(Ω), KerΛ₀=𝐻^⟂_{−Δ−𝜅2}(Ω)⟹𝐻_{−Δ−𝜅2}(Ω)=KerΛ₀^⟂⊂RgΛ₀^∗=RgΛ₀^∗𝑤∗=𝐻_{−Δ−𝜅2}(Ω). (2.30) Since Λ₀^∗ is linear, continuous, and bijective from 𝐻^−1/2(Γ) to 𝐻_{−Δ−𝜅2}(Ω), by the open mapping Banach theorem, Λ₀^∗ is an isomorphism.

Proposition 2.20. If 𝜅₁≠𝜅₂, then 𝐻_−Δ−𝜅²₁(Ω)∩𝐻_−Δ−𝜅²₂(Ω)={0}.

Proof. Suppose that 𝑣∈𝐻_−Δ−𝜅²₁(Ω)∩𝐻_−Δ−𝜅²₂(Ω). Then −Δ−𝜅²₁𝑣=0,if𝑣∈𝐻_−Δ−𝜅²₁(Ω),−Δ−𝜅²₂𝑣=0,if𝑣∈𝐻_−Δ−𝜅²₂(Ω), (2.31) and, consequently, 𝑣=0.

Remark 2.21. If we substitute 𝜅 by 𝑖𝜅 in problem (1.1) and use the same argument already used in the precedent proofs, then(i)𝐿²(Ω)=𝐻_{−Δ−𝜅2}(Ω)⊕(−Δ+𝜅²)[𝐻²₀(Ω)],(ii)𝐻^⟂_−Δ+𝜅2(Ω)=(−Δ+𝜅²)[𝐻²₀(Ω)],(iii)𝐻_−Δ+𝜅2(Ω) is a closed subspace of 𝐿²(Ω),(iv)𝐿²(Ω)=𝐻_{−Δ−𝜅2}(Ω)⊕𝐻^⟂_−Δ+𝜅2(Ω),(v)if 𝜅₁≠𝜅₂, 𝐻_−Δ−𝜅²₁(Ω)∩𝐻_−Δ−𝜅²₂(Ω)={0}.

3. Integral Representation

Definition 3.1. The Dirichlet Green function 𝐺(𝑥,𝜁) for the problem (1.1) is its solution with source 𝛿(𝑥−𝜁), 𝑥,𝜁∈Ω, and homogeneous Dirichlet data, that is, 𝐺(𝑥,𝜁)=0 for 𝑥 on Γ.

Definition 3.2. Let 𝐺(𝑥,𝜁) be the Green function for problem (1.1) with homogeneous Dirichlet boundary conditions. Then, the solution 𝑆∶𝐿²(Ω)×𝐻^1/2(Ω)⟶𝐻¹(Ω),,𝑆𝑥)=𝑓,𝑔⟼𝑢=𝑆𝑓,𝑔𝑓,𝑔(_Ω𝑓(𝜁)𝐺(𝑥,𝜁)𝑑𝜁+_Γ𝑔(𝜁)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝜁𝑑𝜎_𝜁for𝑥∈Ω. (3.1)

Remark 3.3. By using problem (1.1) linearity, we formally decompose the solution in two additive parts 𝑢=𝑆𝑓,𝑔∶=𝑆𝑓,0+𝑆0,𝑔∶𝐿²(Ω)×𝐻^1/2(Γ)⟶𝐻¹(Ω), (3.2) where 𝑆[𝑓,0]∶𝐿²(Ω)×{0}→𝐻¹(Ω) is the homogeneous Dirichlet source auxiliary problem solution and 𝑆[0,𝑔]∶{0}×𝐻^1/2(Γ)→𝐻¹(Ω) is the zero source auxiliary Dirichlet problem solution. For simplicity, for a fixed 𝑓∈𝐿²(Ω) or 𝑔∈𝐻^1/2(Γ), we will denote 𝑆^𝑓=𝑆𝑓,⋅∶𝐻^1/2(Γ)⟶𝐻¹(Ω),𝑆_𝑔=𝑆⋅,𝑔∶𝐿²(Ω)⟶𝐻¹(Ω). (3.3) By taking the normal trace of the solution (3.1), we obtain Λ𝑓,𝑔(𝑥)=𝛾₁∘𝑆𝑓,𝑔=_Ω𝑓(𝜁)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝑑𝜁+_Γ𝑔(𝜁)𝜕²𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝜕𝜈_𝜁𝑑𝜎_𝜁for𝑥∈Γ (3.4) which is an explicit representation to the Dirichlet to Newman map with arbitrary 𝑓 and 𝑔.
By using the same notation adopted for the additive decomposition of the solution map, fixed 𝑓∈𝐿²(Ω) or 𝑔∈𝐻^1/2(Γ), we will denote Λ^𝑓=Λ𝑓,⋅∶𝐻^1/2(Γ)⟶𝐻¹(Ω),Λ_𝑔=Λ⋅,𝑔∶𝐿²(Ω)⟶𝐻¹(Ω). (3.5)

Remark 3.4. Note that 𝑆₀𝑓(𝑥)=𝑆𝑓,0(𝑥)=_Ω𝑓(𝜁)𝐺()𝑆𝑥,𝜁𝑑𝜁,𝑥∈Ω,₀𝑔(𝑥)=𝑆0,𝑔(𝑥)=_Γ𝑔(𝜁)𝜕G(𝑥,𝜁)𝜕𝜈_𝜁𝑑𝜎(𝜁),𝑥∈Ω. (3.6)
With this decomposition, we obtain the following explicit representation to operators in Section 2:(1)Λ𝑓,𝑔=𝛾₁∘𝑆𝑓,𝑔(𝑥)=𝜕𝑆𝑓,𝑔𝜕𝜈_𝑥(𝑥)=_Ω𝑓(𝜁)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝑑𝜁+_Γ𝑔(𝜁)𝜕²𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝜕𝜈_𝜁𝑑𝜎(𝜁), (3.7) with 𝑥∈Γ;(2)Λ⁰𝑔(𝑥)=Λ0,𝑔(𝑥)=𝛾₁∘𝑆0,𝑔(𝑥)=_Γ𝑔(𝜁)𝜕²𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝜕𝜈_𝜁𝑑𝜎(𝜁),𝑥∈Γ, (3.8) is an explicit representation to the Dirichlet to Neumann map;(3)Λ₀𝑓=(𝑥)Λ𝑓,0(𝑥)=𝛾₁∘𝑆𝑓,0(𝑥)=_Ω𝑓(𝜁)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝑑𝜁,𝑥∈Γ, (3.9) is an explicit representation to the Source to Neumann map.

3.1. The Inverse Source Integral Equation

Lemma 3.5. Let 𝑢_𝑗, 𝑗=1,2, be two solutions of problem (1.1) with the same source 𝑓 and different Dirichlet data 𝑔_𝑗, 𝑗=1,2, respectively. Then(i)Λ^𝑓[𝑔₁]−Λ⁰[𝑔₁]=Λ^𝑓[𝑔₂]−Λ⁰[𝑔₂]𝑜𝑛Γ, that is, the relative Dirichlet to Newman operator Λ^𝑓−Λ⁰∶𝐻^1/2(Γ)→𝐻^1/2(Γ) is constant operator whose functional value is independent of the Dirichlet datum 𝑔 and depends only on the source function 𝑓;(ii)∫_Ω𝑓(𝜁)(𝜕𝐺(𝑥,𝜁)/𝜕𝜈_𝑥)𝑑𝜁=Λ^𝑓[𝑔_𝑗]−Λ⁰[𝑔_𝑗] for all solutions of (1.1) with arbitrary Dirichlet data but the same source, that is, the integral is the function given by the relative Dirichlet to Newman map.

Proof. The equality ∫_Ω𝑓(𝜁)(𝜕𝐺(𝑥,𝜁)/𝜕𝜈_𝑥)𝑑𝜁=Λ^𝑓[𝑔₁]−Λ⁰[𝑔₁]=Λ^𝑓[𝑔₂]−Λ⁰[𝑔₂] in both (i) and (ii) is a trivial consequence of (3.4). Note that in this case the unique information available for source reconstruction is given by only one measurement, say that Neumann boundary measurement 𝜕_𝜈𝑥𝑢=Λ^𝑓𝑔, (3.10) corresponding to some specific Dirichlet datum 𝑔, which without loss of generality can be assumed zero. Note also that problem (1.1) with Dirichlet datum 𝑔=0 and source 𝑓∈𝐿²(Ω) has solution 𝑢∈𝐻²(Ω). The normal trace of this regular solution is in 𝐻^1/2(Γ). So we have proved that the range of Λ^𝑓−Λ⁰ is in 𝐻^1/2(Γ). The domain of Λ^𝑓−Λ⁰ is 𝐻^1/2(Ω) since this is the set of nonzero Dirichlet data that gives the same function in the range.

Definition 3.6 (strong integral equation problem). Since in the inverse source problem the exact Cauchy data pair (𝑔,𝑔_𝜈𝑥) is given, the relative Dirichlet to Newman map value for the source problem (1.1) Λ=Λ^𝑓−Λ⁰𝑔=𝑔_𝜈𝑥(𝑥)−_Γ𝑔(𝜁)𝜕²𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝜕𝜈_𝜁𝑑𝜎_𝜁for𝑥∈Γ, (3.11) is known and Lemma 3.5 suggests the following integral equation formulation for the source reconstruction problem: to find 𝑓∈𝐿²(Ω) such that 𝐹𝑓=ΛonΓ, (3.12) where 𝐹∶=Λ₀=Λ[⋅,0]∶𝐿²(Ω)→𝐻^1/2(Γ), 𝐹𝑓(𝑥)∶=_Ω𝑓(𝜁)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝑑𝜁, (3.13) for 𝑥∈Γ.

Remark 3.7. Note that we introduce here 𝐹 as a simplified notation to the extended Source to Neumann map Λ₀. This notation is more usual.

The following corollary resumes all that has been discussed.

Corollary 3.8. Supposed that 𝜅²∉Σ₂ and 𝜅⁴∉Σ₄. Then, (i)for a Cauchy datum (𝑣,𝜕_𝜈𝑣)∣_Γ=(0,Λ)∈𝐻^3/2(Γ)×𝐻^1/2(Γ), there exists a unique function 𝑓∈𝐻_−Δ+𝜅2(Ω) solution of the inverse source problem (1.3) for the Helmholtz equation (1.1),(ii)the associated mapping 𝐹⁻¹∶𝐻^1/2(Γ)⟶𝐻_−Δ+𝜅2(Ω) (3.14) defines a linear homeomorphism between these spaces, (iii)and is a right inverse of the mapping 𝐹∶𝐿²(Ω)→𝐻^1/2(Ω) defined by the strong inverse source equation (3.12),(iv)the projection 𝑄∶𝐿²(Ω)→𝐿²(Ω)⧵𝐻_−Δ+𝜅2(Ω) is well defined and constant in the level set 𝐶_ℎ=𝑓∈𝐿²∶−Δ+𝜅²𝑓=ℎ, (3.15)(v)if the source 𝑓 is known to be in the class 𝐶_ℎ, then a single boundary measurement (0,Λ) is sufficient to identify 𝑓,(vi)for a Cauchy datum (𝑣,𝜕_𝜈𝑣)∣_Γ=(0,Λ)∈𝐻^3/2(Γ)×𝐻^1/2(Γ), there are many functions 𝑓=𝑓+ℎ, where 𝑓∈𝐻_−Δ+𝜅2(Ω) is an observed consequence of (ii) and ℎ∈(−Δ+𝜅²)[𝐻²₀(Ω)]=𝐿²(Ω)⧵𝐻_−Δ+𝜅2(Ω) is an arbitrary nonobserved function.

Proof. The items are trivial consequences of the results already proved.

Remark 3.9. Given ℎ∈𝐶_ℎ, the unique solution referred to in Section 3.1 is the unique solution of the fourth-order direct problem [1]: Δ²−𝜅⁴𝑤^𝑓𝑤=ℎinΩ,_𝑓𝜕=0onΓ,_𝜈𝑤^𝑓=0onΓ. (3.16)

Remark 3.10. The adopted Hilbert space framework for solution of the problem may be understood as an a priori information about the criteria for selecting the observable and the nonobservable part of the source. Other Sobolev spaces that induced partitions of the pivot space 𝐿^𝑝(Ω),1≤𝑝≤∞ (in this work 𝑝=2) will modify this observability relation.

Remark 3.11 (relation between star-shaped and metaharmonic functions). Let us define the set𝑈_Type=𝜒_𝜔∶𝜔⊂ΩisoftypeType. (3.17) The 𝑈_squares⊂𝑈_starshape⊂𝑈_{characteristic}. Note that 𝑈_squares is dense in 𝐿²(Ω). If, for all 𝜔⊂Ω, there exists a family of metaharmonic functions in 𝐻_−Δ+𝜅2(Ω) that approach 𝜒_𝜔, then 𝐻_−Δ+𝜅2(Ω) is dense in 𝐿²(Ω).

Remark 3.12. The most important classes of sources that may be reconstructed uniquely from boundary data occur when 𝑓 is metaharmonic or when 𝑓≡𝑓𝜒_𝜔, where 𝜒_𝜔 is the characteristic function of an open star-shaped set 𝜔⊂Ω with 𝐶² being boundary and 𝑓 a 𝐶²(Ω) function. We will discuss these classes when establishing uniqueness.

Remark 3.13 (the adjoint integral equation). This equation may be used for the source reconstruction independent of solution of the direct problem (1.1). By substituting the explicit integral definition of 𝐹 in the duality definition of adjoint 𝐹[𝑓],𝜓_{𝐻1/2×𝐻−1/2}=𝑓,𝐹^∗𝜓_𝐿2×𝐿2, (3.18) we obtain that 𝐹^∗∶𝐻^−1/2(Γ)→𝐿²(Ω) is explicitly given by ℎ(𝜁)=𝐹^∗𝜓(𝜁)=_Γ𝜓(𝑥)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝑑𝜎_𝑥. (3.19)

Remark 3.14. From these formulas, we can deduce that, for a fixed 𝑓∈𝐿²(Ω), the operator Λ^𝑓_𝑔(𝑥)−Λ⁰_𝑔(𝑥)=𝐹𝑓(𝑥),𝑥∈Γ, (3.20) for any 𝑔∈𝐻²(Γ), we will call this operator Extended Dirichlet to Neumann map.

Remark 3.15. Once we know the integral formulation to 𝐹, we can determine the integral formulation to 𝐹^∗. In fact, 𝐹𝑓,𝜓_{𝐻1/2×𝐻−1/2}=𝑓,𝐹^∗𝜓_𝐿2×𝐿2, (3.21) and from 𝐹𝑓(𝑥)=_Ω(𝜁)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝑑𝜁,𝑥∈Γ, (3.22) it follows that 𝐹^∗𝜓(𝜁)=_Ω𝜓(𝑥)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝑑𝜎(𝑥),𝑥∈Γ. (3.23)

Remark 3.16. Consider the following direct problem −Δ𝑤+𝑘²𝛾𝑤=0inΩ,₀𝑤=𝜓onΓ, (3.24) with 𝜓∈𝐻^−1/2(Γ). This problem has a unique solution 𝑤∈𝐿²(Ω). Let 𝑤(𝜁)=𝑆_1,𝜓(𝜁)=_Ω𝜓(𝑥)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝑑𝜎(𝑥)=𝐹^∗𝜓(𝜁), (3.25) where 𝐺 is the associated Green function. This happens for each 𝜓∈𝐻^−1/2(Γ) since 𝐹^∗[𝜓](𝜁)=𝑆_1,𝜓(𝜁). From this we deduce that the integral 𝑆_1,𝜓 inherits all good properties from 𝐹^∗.

4. Integral and Variational Solutions: The Equivalence Theorem

4.1. Integral Formulation

With the integral formulas (3.8), (3.2), Definition (2.2), and supposing that the compatibility condition has been verified, we obtain that 𝐹𝑓(𝑥)=Λ^𝑓_𝑔(𝑥)−Λ⁰𝑔(𝑥) (4.1) is the integral equation _Ω𝑓(𝜁)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝑑𝜁=𝑔_𝜈𝑥(𝑥)−_Γ𝑔(𝜁)𝜕𝐺(𝑥,𝜁)(𝑥)(𝜁)(𝜁)𝜕𝜂𝜕𝜂𝑑𝜎. (4.2)

4.2. Variational Formulation

Definition 4.1 (reciprocity gap functional problem). One may use the second Green theorem _Ω(𝑢Δ𝑣−𝑣Δ𝑢)𝑑𝑥=_Γ𝑢𝜕𝑣𝜕𝜈_𝑥−𝑣𝜕𝑢𝜕𝜈_𝑥𝑑𝜎_𝑥, (4.3) valid for all 𝑢,𝑣∈𝐻²(Ω) with 𝑢 a solution of problem (1.1) to formulate the reciprocity gap functional inverse problem: to find 𝑓∈𝐿²(Ω)_Ω𝑓𝑣𝑑𝑥=_Γ𝑔_𝜈𝑥𝑣−𝑔𝜕𝑣𝜕𝜈_𝑥𝑑𝜎_𝑥, (4.4) for all 𝑣∈𝐻_{−Δ−𝜅2}(Ω).
Note that the Lax-Milgram theorem assures the existence of a solution in this case.

4.3. The Equivalence Theorem

Theorem 4.2. Let one consider the two inverse source problems related with problem (1.1) with relative Dirichlet to Newman map Λ∈𝐿²(Ω): (i)integral equation problem given by (3.12);(ii)reciprocity gap functional problem given by (4.4). Then (i)⇒(ii). Suppose additionally that the relative Dirichlet to Newman data Λ∈𝐻^1/2(Γ). Then (ii)⇒(i).

Proof. Let us consider the inverse source problem: to find (𝑢,𝑓)∈𝐻¹(Ω)×𝐿²(Ω) such that −Δ𝑢−𝑘²𝛾𝑢=𝑓inΩ,₀𝛾𝑢=𝑔onΓ,₁𝑢=𝑔_𝜈𝑥onΓ, (4.5) where 𝑔∈𝐻^1/2(Γ), 𝑔_𝜈𝑥∈𝐻^−1/2(Γ) with compatibility condition 𝑔_𝜈=𝜕𝑔/𝜕𝜈.
(i) ⇒ (ii).
We start the demonstration by supposing that (i) is true; that is, there exists 𝑓∈𝐿²(Ω) such that _Ω𝑓(𝜁)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝑑𝜁=𝑔_𝜈𝑥−_Γ𝑔(𝜁)𝜕²𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝜕𝜈_𝜁𝑑𝜎(𝜁), (4.6) where 𝐺 is the Green function associated with the Helmholtz operator in Ω.
Let 𝑣∈𝐻_Δ−𝜅2(Ω) be extended to the boundary of Ω. We then have the following integral representation for 𝑣: 𝑣(𝜁)=_Γ𝑣(𝑥)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥(𝜁)𝑑𝜎(𝑥),𝜁∈Γ. (4.7) By taking the normal trace 𝜕𝑣𝜕𝜈_𝜁(𝜁)=_Γ𝑣(𝑥)𝜕²𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝜕𝜈_𝜁𝑑𝜎(𝑥),𝜁∈Γ. (4.8)
We now multiply (4.2) by 𝑣 and integrate on Γ to obtain _Γ𝑣(𝑥)_Ω𝑓(𝜁)𝜕𝐺𝜕𝜈_𝑥()(𝑥)=𝑥,𝜁𝑑𝜁𝑑𝜎_Γ𝑣(𝑥)𝜑(𝑥)−_Γ𝑔(𝜁)𝜕²𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝜕𝜈_𝜁(𝜁(𝑥)=𝑑𝜎)𝑑𝜎_Γ𝑣(𝑥)𝜑(𝑥)𝑑𝜎(𝑥)−_Γ𝑣(𝑥)_Γ𝑔(𝜁)𝜕²𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝜕𝜈_𝜁𝑑𝜎(𝜁)𝑑𝜎(𝑥). (4.9)
Now applying the Fubini theorem, we obtain _Ω𝑓(𝜁)^{𝑣(𝑥)0𝑥0𝑓𝑒37}_Γ𝑣(𝑥)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥(𝑥=𝑑𝜎)𝑑𝜁_Γ𝑣(𝑥)𝜑(𝑥)^{(𝑥)𝜕𝑔/𝜕𝜈𝑥0𝑥0𝑓𝑒37}𝑑𝜎(𝑥)−_Γ𝑔(𝜁)^{(𝜁)𝜕𝑣/𝜕𝜈𝜁0𝑥0𝑓𝑒37}_Γ𝑣(𝑥)𝜕²𝐺(𝑥,𝜁)𝜕𝜈_𝑥(𝜁)(𝑥(𝜁),𝜕𝜂𝑑𝜎)𝑑𝜎 (4.10) which implies _Ω𝑓(𝜁)𝑣(𝜁)𝑑𝜁=_Γ𝑣(𝜁)𝜕𝑔(𝜁)(𝜁)(𝜁)(𝜁)𝜕𝜂−𝑔𝜕𝑣(𝜁)(𝜁)𝜕𝜂𝑑𝜎. (4.11) Since 𝑣∈𝐻_−Δ+𝜅2(Ω) is arbitrary, we obtain the weak formulation. Note that this is almost expected, since the integral formulation is stronger than the variational, and, as usual, strong ⇒ weak.
(ii) ⇒ (i).
Let us now suppose that for all test functions 𝑣∈𝐻_−Δ+𝜅2(Ω) we have the weak reciprocity integral equation (4.4) _Ω𝑓(𝑥)𝑣(𝑥)𝑑𝑥=_Γ𝑔_𝜈(𝑥)𝑣(𝑥)𝑑𝜎(𝑥)−_Γ𝑔(𝑥)𝜕𝑣(𝑥)(𝑥)𝜕𝜈𝑑𝜎. (4.12) Consider the substitution of the Green function integral representation 𝑣(𝜁)=_Γ𝑣(𝑥)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝑑𝜎(𝑥), (4.13) whose normal derivative is 𝜕𝑣(𝜁)𝜕𝜈_𝜁=_Γ𝑣(𝑥)𝜕²𝐺(𝑥,𝜁)𝜕𝜈_𝜁𝜕𝜈_𝑥𝑑𝜎(𝑥), (4.14) in (4.4) and apply Fubini’s theorem to obtain _Γ𝑣(𝑥)_Ω𝑓(𝜁)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥(𝑥)=𝑑𝜁𝑑𝜎_Γ𝑣(𝑥)_Γ𝑔_𝜈(𝜁)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝑑𝜎(𝑥)−_Γ𝑔(𝜁)𝜕²𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝜕𝜈_𝜁𝑑𝜎(𝜁)𝑑𝜎(𝑥). (4.15)
Note that since(i)_Γ𝑔(𝜁)𝜕²𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝜕𝜈_𝜁𝑑𝜎(𝜁)=Λ⁰𝑔, (4.16) is the Dirichlet to Neumann map,(ii)and _Γ𝑔_𝜈(𝜁)𝜕𝐺(𝑥,𝜁)𝜕𝜈_𝑥𝑑𝜎(𝑥)=𝑔_𝜈(𝜁), (4.17) we obtain _Γ𝑣(𝑥)_Ω𝑓(𝜁)𝜕𝐺(𝑥,𝜁)𝜈_𝑥𝑑𝜁−𝑔_𝜈+Λ⁰𝑔𝑑𝜎(𝑥)=0, (4.18) for all 𝑣∈𝐻^1/2(Γ)≅𝐻_−Δ+𝜅2(Ω). By property of the integral, we obtain (3.12).

5. Examples on the Unitary Disk

5.1. The Green Function for the Helmholtz Equation Dirichlet Problem in Circular Domains

In this section we will consider the Green function determination when the domain Ω⊂ℝ² in problem (1.1) is circular with respect to the polar coordinate system.

A Green function to problem (1.1) is a solution of−Δ𝐺(𝜁,𝑥)−𝜅²𝐺()()𝐺()𝜁,𝑥=𝛿𝜁−𝑥,𝜁∈Ω,𝜁,𝑥=0,𝜁∈𝜕Ω, (5.1) where 𝑥∈Ω is the localization of the delta Dirac source. We may use the linearity of the problem for decomposing the solution in two additive parts 𝐺()()+1𝜁,𝑥=𝐹𝜁,𝑥2𝐺_𝐹(𝜁,𝑥)+𝐺_𝐹(𝑥,𝜁). (5.2) Here 𝐹 is the fundamental solution for the free space Helmholtz equation −Δ𝐹(𝜁,𝑥)−𝜅²𝐹(𝜁,𝑥)=𝛿(𝜁−𝑥), (5.3) and 𝐺_𝐹 is a homogeneous source regular solution of the Helmholtz equation −Δ𝐺_𝐹(𝜁,𝑥)−𝜅²𝐺_𝐹()𝐺()()𝜁,𝑥=0,𝜁∈Ω,𝜁,𝑥=−𝐹𝜁,𝑥,𝜁∈𝜕Ω, (5.4) and 𝑥∈Ω. In polar coordinates with 𝑟=|𝜁−𝑥|−1𝑟𝜕𝑟(𝑟)𝜕𝑟𝜕𝐹𝜕𝑟−𝜅²𝐹(𝑟)=𝛿(𝑟)for𝑟≥0, (5.5) has at 𝑟=0 singular behavior solution 𝐹()1𝜅𝑟=−4𝑌₀(𝜅𝑟), (5.6) which is a Bessel function of second kind.

Remark 5.1. When 𝜅 is small, this solution has a singularity that has the same behavior of the logarithmic function, that is, 𝑌₀()2𝜅𝑟≈−𝜋log𝑟,lim_𝜅→0+Δ+𝜅²=Δ⟹lim_𝜅→0+14𝑌₀()1𝜅𝑟=−2log𝑟. (5.7)

Remark 5.2 (addition theorem). Let 𝑥=(𝜌cos(𝜃),𝜌sin(𝜃)) and 𝜁=(𝜎cos(𝛽),𝜎sin(𝛽)). Then 𝑟=𝜎²+𝜌²,𝑌−2𝜎𝜌cos𝛽−𝜃₀(𝜅𝑟)=^+∞_𝑛=−∞𝑌_𝑛(𝜅𝜎)𝐽_𝑛𝜅𝜌cos𝑛𝛽−𝜃,0=^+∞_𝑛=−∞𝑌_𝑛(𝜅𝜎)𝐽_𝑛𝑛,𝑌𝜅𝜌sin𝛽−𝜃₀(𝜅𝑟)=^+∞_𝑛=−∞𝑌_𝑛(𝜅𝜎)𝐽_𝑛𝜅𝜌𝑒^{𝑖𝑛(𝛽−𝜃)}. (5.8)(5.9)
The nonhomogeneous Dirichlet boundary condition in (5.4) is 𝐺()=1𝜁,𝑥4𝑌₀𝜅1+𝜌²−2𝜌cos𝛽−𝜃,𝜁∈𝜕Ω. (5.10) The regular solution of (5.4) is 𝐺_𝐹𝜎,𝛽;𝜌,𝜃=^+∞_𝑛=−∞𝑐_𝑛𝜌,𝜃𝐽_𝑛(𝜅𝜎)𝑒^𝑖𝑛𝛽, (5.11) where the coefficients 𝑐_𝑛=−𝑐_−𝑛 are determined by the Dirichlet boundary condition: 𝐺_𝐹=11,𝛽;𝜌,𝜃4𝑌₀𝜅1+𝜌²=1−2𝜌cos𝛽−𝜃4^+∞_𝑛=−∞𝑌_𝑛(𝜅)𝐽_𝑛𝜅𝜌𝑒^{𝑖𝑛(𝛽−𝜃)}=14^+∞_𝑛=−∞𝑌_𝑛𝜅𝜌𝐽_𝑛(𝜅)𝑒^{𝑖𝑛(𝛽−𝜃)}=^+∞_𝑛=−∞𝑐_𝑛𝜌,𝜃𝐽_𝑛(𝜅)𝑒^𝑖𝑛𝛽. (5.12) Since the solutions {𝑒^𝑖𝑛𝛽,𝑛=−∞,+∞} are linearly independent, we obtain 𝐽_𝑛(𝜅)𝑐_𝑛=14𝑌_𝑛(𝜅)𝐽_𝑛𝜅𝜌𝑒^{−𝑖𝑛𝜃}=14𝑌_𝑛𝜅𝜌𝐽_𝑛(𝜅)𝑒^{−𝑖𝑛𝜃}, (5.13) and when 𝜅 is not a root of the Bessel function 𝐽_𝑛, 𝑐_𝑛=1𝜌,𝜃4𝑌_𝑛(𝜅)𝐽_𝑛𝜅𝜌𝐽_𝑛(𝜅)𝑒^{−𝑖𝑛𝜃}=14𝑌_𝑛𝜅𝜌𝐽_𝑛(𝜅)𝐽_𝑛(𝜅)𝑒^{−𝑖𝑛𝜃}, (5.14)𝐺_𝐹=1𝜎,𝛽;𝜌,𝜃4^+∞_𝑛=−∞𝑌_𝑛(𝜅)𝐽_𝑛𝜅𝜌𝐽_𝑛(𝜅)𝐽_𝑛(𝜅𝜎)𝑒^{𝑖𝑛(𝛽−𝜃)}=14^+∞_𝑛=−∞𝑌_𝑛𝜅𝜌𝐽_𝑛(𝜅)𝐽_𝑛(𝜅)𝐽_𝑛(𝜅𝜎)𝑒^{𝑖𝑛(𝛽−𝜃)}, (5.15)𝐺_𝐹=1𝜌,𝜃;𝜎,𝛽4^+∞_𝑛=−∞𝑌_𝑛(𝜅)𝐽_𝑛(𝜅𝜎)𝐽_𝑛(𝜅)𝐽_𝑛𝜅𝜌𝑒^{𝑖𝑛(𝛽−𝜃)}=14^+∞_𝑛=−∞𝑌_𝑛(𝜅𝜎)𝐽_𝑛(𝜅)𝐽_𝑛(𝜅)𝐽_𝑛𝜅𝜌𝑒^{𝑖𝑛(𝛽−𝜃)}, (5.16) by noting that by addition theorem 𝐹()1𝜁,𝑥=−4^+∞_𝑛=−∞𝑌_𝑛(𝜅𝜎)𝐽_𝑛𝜅𝜌𝑒^{𝑖𝑛(𝛽−𝜃)}1=−4^+∞_𝑛=−∞𝑌_𝑛𝜅𝜌𝐽_𝑛(𝜅𝜎)𝑒^{𝑖𝑛(𝛽−𝜃)}. (5.17)
By substituting (5.6) and (5.15) and using the addition theorem (5.9) in (5.4), we finally obtain the Green function for the circular domain Helmholtz equation problem (5.1) 𝐺1𝜎,𝛽;𝜌,𝜃=−8^+∞_𝑛=−∞𝐽_𝑛(𝜅)𝑌_𝑛(𝜅𝜎)−𝑌_𝑛(𝜅)𝐽_𝑛(𝜅𝜎)𝐽_𝑛(𝜅)𝐽_𝑛𝜅𝜌𝑒^{𝑖𝑛(𝛽−𝜃)}−18^+∞_𝑛=−∞𝐽_𝑛(𝜅)𝑌_𝑛𝜅𝜌−𝑌_𝑛(𝜅)𝐽_𝑛𝜅𝜌𝐽_𝑛(𝜅)𝐽_𝑛(𝜅𝜎)𝑒^{𝑖𝑛(𝜃−𝛽)}, (5.18) if we define 𝑟>=max𝜎,𝜌 and 𝑟<=min𝜎,𝜌, which can also be rewritten as 𝐺1𝜎,𝛽;𝜌,𝜃=−4^+∞_𝑛=−∞𝐽_𝑛(𝜅)𝑌_𝑛(𝜅𝑟>)−𝑌_𝑛(𝜅)𝐽_𝑛(𝜅𝑟>)𝐽_𝑛(𝜅)𝐽_𝑛(𝜅𝑟<)𝑒^{𝑖𝑛(𝛽−𝜃)}. (5.19)

5.2. The Integral Equation Kernel for the Unitary Disk

The kernel of the integral equation is obtain by taking the normal derivative trace of the Green function (5.18). By using the Wronskian identity for the Bessel functions 𝐽_𝑛(𝑧)𝑌^′_𝑛(𝑧)−𝐽^′_𝑛(𝑧)𝑌_𝑛(𝑧)=2𝜋𝑧, (5.20) we obtain 𝑃_𝜅=()𝜌,𝛽−𝜃𝜕𝐺𝑥,𝜁𝜕𝜈_𝑥=𝜕𝐺𝜎,𝛽;𝜌,𝜃∣𝜕𝜎^𝜎=1=12𝜋^+∞_𝑛=−∞𝐽_𝑛𝜅𝜌𝐽_𝑛(𝜅)𝑒^{𝑖𝑛(𝛽−𝜃)}, (5.21) which is the modified Poisson kernel for Dirichlet Helmholtz equation problem in the disk. Note that, when 𝜅→0, it tends to the classical Poisson kernel for the disk.

Note that, when 𝜅≡𝑖𝜅 is substitute in (1.1), it becomes a modified Helmholtz equation for which extensions of the Maximum Modulo Principle and the Strong Maximum Principle for metaharmonics functions are applicable. In this case the Poisson Kernel may be rewrit with modified Bessel function of first kind 𝐼_𝑛(𝑟)=𝑒^{−𝑖𝑛(𝜋/2)}𝐽_𝑛(),𝑃𝑖𝑟_𝑖𝜅=1𝜌,𝛽−𝜃2𝜋^+∞_𝑛=−∞𝐼_|𝑛|𝜅𝜌𝐼_|𝑛|(𝜅)𝑒^{𝑖𝑛(𝛽−𝜃)}. (5.22)(5.23)

Remark 5.3. Substituting the kernel (5.23) in (3.12), we obtain the strong inverse source equation (3.12) for unitary disk ₀^2𝜋1₀12𝜋^+∞_𝑛=−∞𝐼_|𝑛|𝜅𝜌𝐼_|𝑛|(𝜅)𝑒^{𝑖𝑛(𝛽−𝜃)}𝑓𝜌,𝜃𝜌𝑑𝜌𝑑𝜃=Λ𝛽,𝛽∈[0,2𝜋). (5.24)

Remark 5.4. Another important class of sources to be reconstructed is the characteristic source with star-shaped boundary with parametrization given by (𝑥,𝑦)=(𝑟(𝜃)cos(𝜃),𝑟(𝜃)sin(𝜃)), with 𝑟∈𝐻¹(0,2𝜋). The strong inverse star-shaped characteristic source equation (3.12) for unitary disk is ₀^2𝜋₀^𝑟(𝜃)12𝜋^+∞_𝑛=−∞𝐼_|𝑛|𝜅𝜌𝐼_|𝑛|(𝜅)𝑒^{𝑖𝑛(𝛽−𝜃)}𝜌𝑑𝜌𝑑𝜃=Λ𝛽,𝛽∈[0,2𝜋). (5.25)

Remark 5.5. Note that the kernel with finite sum 𝑃^𝑁_𝑖𝜅=1𝜌,𝛽−𝜃2𝜋^+𝑁_𝑛=−𝑁𝐼_|𝑛|𝜅𝜌𝐼_|𝑛|(𝜅)𝑒^{𝑖𝑛(𝛽−𝜃)}, (5.26) converges with respect to (𝜌,𝜃) pointwise to 𝑃_𝑖𝜅(𝜌,𝛽−𝜃). Since 𝑃_𝑖𝜅=1𝜌,𝛽−𝜃2𝜋^+𝑁_𝑛=−𝑁𝐼_|𝑛|𝜅𝜌𝐼_|𝑛|(𝜅)=12𝜋^+𝑁_𝑛=−𝑁𝜌^|𝑛|1+∑^∞_𝑚=1𝜅²/4^𝑚|𝑛|!𝜌^2𝑚/𝑚!(|𝑛|+𝑚)!1+∑^∞_𝑚=1𝜅²/4^𝑚|𝑛|𝑛|)!≤1!/𝑚!(|+𝑚2𝜋^+𝑁_𝑛=−𝑁𝜌^|𝑛|, (5.27) it is bounded by the function 𝜌𝑑𝜌 integrable 2/(1−𝜌), in such a way that the limit and the integral can be transposed.

Remark 5.6. The transposed equation 12𝜋^+∞_𝑛=−∞0^2𝜋₀^𝑟(𝜃)𝐼_|𝑛|𝜅𝜌𝐼_|𝑛|(𝜅)𝑒^{−𝑖𝑛𝜃}𝜌𝑑𝜌𝑑𝜃𝑒^𝑖𝑛𝛽=Λ𝛽,𝛽∈[0,2𝜋), (5.28) can be Fourier transformed giving ₀^2𝜋₀^𝑟(𝜃)𝐼_|𝑛|𝜅𝜌𝐼_|𝑛|(𝜅)𝑒^{−𝑖𝑛𝜃}𝜌𝑑𝜌𝑑𝜃=Λ(𝑛),𝑛∈𝒵. (5.29)

5.3. The Variational Formulation for the Unitary Disk

Take Ω=𝐷 as the unitary disk. The weak equation (4.4) can also be specialized for the characteristic star-shaped source 𝜒_𝜔=𝜒_𝑟 inside the unitary disk Modified Helmholtz problem _𝐷𝑣𝜌,𝜃𝜒_𝜔𝜌,𝜃𝜌𝑑𝜌𝑑𝜃=₀^2𝜋₀^𝑟(𝜃)𝑣=𝜌,𝜃𝑑𝜌𝑑𝜃_02𝜋𝑔_𝜈𝑥(𝜃)𝑣(1,𝜃)−𝑔(𝜃)𝜕𝑣𝜕𝜈_𝑥(𝜃)𝑑𝜃, (5.30) for all 𝑣∈𝐻_−Δ+𝜅2(𝐷). As we have proved in Lemma 3.5, without loss of generality we can consider the data from the homogeneous Dirichlet problem 𝑔=0. Note that the Modified Poisson Dirichlet kernel 𝑃_𝑖𝜅=1𝜌,𝛽−𝜃2𝜋^+∞_𝑛=−∞𝐼_|𝑛|𝜅𝜌𝐼_|𝑛|(𝜅)𝑒^{𝑖𝑛(𝛽−𝜃)}, (5.31) is a function inside 𝐻_−Δ+𝜅2(𝐷) and can be substituted in this equation giving ₀^2𝜋₀^𝑟(𝜃)12𝜋^+∞_𝑛=−∞𝐼_|𝑛|𝜅𝜌𝐼_|𝑛|(𝜅)𝑒^{𝑖𝑛(𝛽−𝜃)}𝜌𝑑𝜌𝑑𝜃=₀^2𝜋𝑔_𝜈𝑥(𝜃)12𝜋^+∞_𝑛=−∞𝑒^{𝑖𝑛(𝛽−𝜃)}𝑑𝜃. (5.32)

Proposition 5.7. If 𝑔_𝜈𝑥∈𝐻^𝑙(0,2𝜋) for 𝑙≥1/2, then ₀^2𝜋𝑔_𝜈𝑥(𝜃)12𝜋^+∞_𝑛=−∞𝑒^{𝑖𝑛(𝛽−𝜃)}𝑑𝜃=𝑔_𝜈𝑥𝛽. (5.33)

Proof. This is consequence from the fact that 𝐷_𝑁=∑^+𝑁_𝑛=−𝑁𝑒^{𝑖𝑛(𝛽−𝜃)} is the Dirichlet kernel of order 𝑁. When it acts on a continuous function, the fast oscillations of 𝐷_𝑁 far from 𝜃=𝛽 do not contribute to 𝑁-truncated Fourier series 𝑔^𝑁_𝜈𝑥𝛽=12𝜋₀^2𝜋𝑔_𝜈𝑥(𝜃)𝐷_𝑁𝜃−𝛽𝑑𝜃. (5.34) As 𝑁 grows, the dominant contribution to this integral comes from an arbitrarily small neighborhood of 𝜃−𝛽. This behavior gives the Dirac delta distribution character to the series 𝛿𝜃−𝛽=lim𝑁⟶∞𝐷_𝑁𝜃−𝛽, (5.35) and the integral is well defined only when 𝑔_𝜈𝑥 is a continuous function. By Sobolev embedding, the minimal admissible index is 1/2.

Remark 5.8. Note that this result is in agreement with the sufficient condition for (ii) ⇒ (i) in the equivalence theorem Section 4.3.

5.4. The Integral Operator for Star-Shaped Characteristic Source

Let us consider the nonlinear mapping ℱ[𝑟]=𝐹𝜒_𝑟, (5.36) where 𝐹 is the function in the formulation of problem given by (3.12). Note that the strong integral equation (5.25) can be formally stated as the nonlinear problem: to find 𝑟(𝜃) such that ℱ[𝑟]=Λ. (5.37) We can investigate the operator ℱ∶𝐻¹(0,2𝜋)→𝐻^𝑙(0,2𝜋) with respect to a possible set of values 𝑙∈ℝ for which the regularity of the source boundary 𝑟 influences the regularity in the range of the functional.

Proposition 5.9. One considers two possible cases for the star-shaped source inside the unitary disk in which the source boundary can do or do not touch the disk border (i)if 𝑟(𝜃)≤1, then ℱ[𝑟]∈𝐻^𝑙(0,2𝜋) with 𝑙<1/2;(ii)when 𝑟(𝜃)<1,ℱ[𝑟]∈𝐶^∞(0,2𝜋).

Proof. We will estimate the functional norm in 𝐻^𝑙(0,2𝜋) by using its Fourier transform (5.29). Note that 𝐼_|𝑛|𝜅𝜌𝐼_|𝑛|(𝜅)=𝜌^|𝑛|1+∑^∞_𝑚=1𝜅²/4^𝑚|𝑛|!𝜌^2𝑚/𝑚!(|𝑛|+𝑚)!1+∑^∞_𝑚=1𝜅²/4^𝑚|𝑛|!/𝑚!(|𝑛|+𝑚)!≤𝜌^|𝑛|,_0𝑟(𝜃)𝐼_|𝑛|𝜅𝜌𝐼_|𝑛|(𝜅)𝑟(𝜃)𝜌𝑑𝜌≤^|𝑛|+2|𝑛|.+2 (5.38) Since, for 𝑟≤1, ∫₀^𝑟(𝜃)(𝐼_|𝑛|(𝜅𝜌)/𝐼_|𝑛|(𝜅))𝜌𝑑𝜌≤1/(|𝑛|+2) and (1+𝑛²)^𝑙/(|𝑛|+2)²≤2^𝑙|𝑛|^2𝑙−2, we obtain that ‖ℱ[𝑟]‖_{𝐻𝑙(0,2𝜋)}=12𝜋^∞_−∞1+𝑛^2𝑙₀^2𝜋₀^𝑟(𝜃)𝐼_|𝑛|𝜅𝜌𝐼_|𝑛|(𝜅)𝜌𝑑𝜌𝑒^{−𝑖𝑛𝜃}𝑑𝜃²≤^∞_−∞1+𝑛^2𝑙₀^𝑟(𝜃)𝐼_|𝑛|𝜅𝜌𝐼_|𝑛|(𝜅)𝜌𝑑𝜌²≤^∞_−∞1+𝑛^2𝑙(|𝑛|+2)²≤^∞_−∞2^𝑙|𝑛|^2𝑙−2, (5.39) converges when 2𝑙−2<−1, which proves (i). If 2𝑙−2=−1, that is, 𝑙=1/2, the upper bound series results in the divergent harmonic series. Note that the result (i) in the proposition gives only a sufficient condition, since ‖ℱ[𝑟]‖_{𝐻𝑙(0,2𝜋)}=12𝜋^∞_−∞1+𝑛^2𝑙₀^2𝜋₀^𝑟(𝜃)𝜌^|𝑛|1+∑^∞_𝑚=1𝜅²/4^𝑚|𝑛|!𝜌^2𝑚/𝑚!(|𝑛|+𝑚)!1+∑^∞_𝑚=1𝜅²/4^𝑚|𝑛|𝑛|)!𝑒!/𝑚!(|+𝑚𝜌𝑑𝜌^{−𝑖𝑛𝜃}𝑑𝜃²=^∞_−∞1+𝑛^2𝑙×₀^2𝜋𝑟(𝜃)^|𝑛|+2|𝑛|∑+21+^∞_𝑚=1((|𝑛|+2)/(|𝑛|+2𝑚+2))𝜅²/4^𝑚|𝑛|!𝑟(𝜃)^2𝑚)/𝑚!(|𝑛|+𝑚)!1+∑^∞_𝑚=1𝜅²/4^𝑚|𝑛|𝑛|)!𝑒!/𝑚!(|+𝑚^{−𝑖𝑛𝜃}𝑑𝜃², (5.40) if 𝑟(𝜃)=1 for all 𝜃∈[0,2𝜋), then ‖ℱ[𝑟]‖_{𝐻𝑙(0,2𝜋)}=1∀𝑙∈ℝ. (5.41) For (ii) we note that 𝑟∈𝐻¹(0,2𝜋)⊂𝐶(0,2𝜋) and that sup_𝜃𝑟(𝜃)=‖𝑟‖_∞<1. In this case, for each 𝑚∈ℕ, we have lim_𝑛→∞|𝑛|^𝑚ℱ[𝑟](𝑛)=lim_𝑛→∞|𝑛|^𝑚∣‖𝑟‖_∞^|𝑛|+2|𝑛|+2=0, (5.42) which proves that ℱ[𝑟]∈𝐶^∞(0,2𝜋).

Proposition 5.10. Suppose that the source star-shaped boundaries do not touch the unitary circumference. Then ℱ∶𝒩[𝑟]⊂𝐻¹(0,2𝜋)⟶𝐻^𝑙(0,2𝜋), (5.43) is locally Lipschitz continuous for every 𝑙∈ℝ; that is, for all 𝑟, there exists a neighborhood 𝒩(𝑟) of 𝑟 in 𝐻¹(0,2𝜋) such that ℱ is Lipschitz.

Proof. Let 𝑟₁ and 𝑟₂ be two star-shaped boundaries in the same neighborhood. Then ℱ𝑟₁−ℱ𝑟_{2𝐻𝑙(0,2𝜋)}=12𝜋^∞_−∞1+𝑛^2𝑙₀^2𝜋𝐹𝑟₁−𝐹𝑟₂𝑒^{−𝑖𝑛𝜃}𝑑𝜃^21/2, (5.44) where ℱ𝑟₁−ℱ𝑟₂=𝑟₁(𝜃)^|𝑛|+2−𝑟₂(𝜃)^|𝑛|+2/(|𝑛|+2)1+∑^∞_𝑚=1𝜅²/4^𝑚|𝑛|𝑛|)!+∑!/𝑚!(|+𝑚_∞𝑚=1(1/(|𝑛|+2𝑚+2))𝜅²/4^𝑚|𝑛|!𝑟₁(𝜃)^{|𝑛|+2+2𝑚}−𝑟₂(𝜃)^{|𝑛|+2+2𝑚}/𝑚!(|𝑛|+𝑚)!1+∑^∞_𝑚=1𝜅²/4^𝑚|𝑛|𝑛|)!,!/𝑚!(|+𝑚 (5.45) can be factored in (𝑟₁(𝜃)−𝑟₂(𝜃))ℱ𝑟₁−ℱ𝑟₂=𝑟₁(𝜃)−𝑟₂(𝜃)×𝑓_|𝑛|𝜃,𝑟₁(𝜃),𝑟₂(𝜃), (5.46) where the factor function 𝑓_|𝑛|𝜃,𝑟₁(𝜃),𝑟₂(𝜃)∑∶=^{|𝑛|+1𝑡=0}𝑟₁(𝜃)^{|𝑛|+1−𝑡}𝑟₂(𝜃)^𝑡(|𝑛|+2)1+∑^∞_𝑚=1𝜅²/4^𝑚|𝑛|𝑛|)!+∑!/𝑚!(|+𝑚_∞𝑚=1((|𝑛|+2)/(|𝑛|+2𝑚+2))𝜅²/4^𝑚|𝑛|!/𝑚!(|𝑛|+𝑚)!∑^{|𝑛|+1+2𝑚}_𝑡=0𝑟₁(𝜃)^{|𝑛|+1+2𝑚−𝑡}𝑟₂(𝜃)^𝑡(|𝑛|+2)1+∑^∞_𝑚=1𝜅²/4^𝑚|𝑛|!/𝑚!(|𝑛|+𝑚)!≥0 (5.47) is nonnegative. So ℱ𝑟₁−ℱ𝑟_{2𝐻𝑙(0,2𝜋)}=12𝜋^∞_−∞1+𝑛^2𝑙₀^2𝜋𝑟₁(𝜃)−𝑟₂(𝜃)×𝑓_|𝑛|𝜃,𝑟₁(𝜃),𝑟₂(𝜃)𝑒^{−𝑖𝑛𝜃}𝑑𝜃^21/2ℱ𝑟₁−ℱ𝑟_{2𝐻𝑙(0,2𝜋)}≤12𝜋^∞_−∞1+𝑛^2𝑙₀^2𝜋𝑟₁(𝜃)−𝑟₂(𝜃)²𝑓_|𝑛|𝜃,𝑟₁(𝜃),𝑟₂(𝜃)²𝑑𝜃^1/2, (5.48) or ℱ𝑟₁−ℱ𝑟_{2𝐻𝑙(0,2𝜋)}≤𝐶₀𝑟₁(𝜃)−𝑟₂(𝜃)_{𝐶(0,2𝜋)}, (5.49) where 𝐶₀≥^∞_−∞1+𝑛^2𝑙₀^2𝜋𝑓_|𝑛|𝜃,𝑟₁(𝜃),𝑟₂(𝜃)²𝑑𝜃^1/2 (5.50) is the Lipschitz constant. By denoting 𝑟₀=max_{𝜃∈(0,2𝜋)}{𝑟₁(𝜃),𝑟₂(𝜃)}, we have the following estimate: 𝑓_|𝑛|𝜃,𝑟₀,𝑟₀=𝑟₀^|𝑛|+11+∑^∞_𝑚=1((|𝑛|+2)/(|𝑛|+2𝑚+2))𝜅²/4^𝑚|𝑛|!/𝑚!(|𝑛|+𝑚)!𝑟₀^2𝑚1+∑^∞_𝑚=1𝜅²/4^𝑚|𝑛|!/𝑚!(|𝑛|+𝑚)!≤𝑟₀^|𝑛|+1 (5.51) which gives 𝐶₀=^∞_−∞1+𝑛^2𝑙𝑟₀^{2(|𝑛|+1)1/2}<∞∀𝑙∈ℝ. (5.52) Note that we have used the boundedness of the embedding of 𝐻¹(0,2𝜋) in 𝐶(0,2𝜋) to find a neighborhood 𝒩[𝑟₁] of 𝑟₁<1∈𝐻¹(0,2𝜋) such that sup{‖𝑟₂‖_∞∶𝑟₂∈𝒩[𝑟₁]}<1. This embedding says that there also exists a neighborhood in 𝐻¹(0,2𝜋) such that the nonlinear mapping ℱ[𝑟] is Lipschitz continuous, as enunciated in the proposition.

6. Conclusions

The central question investigated in this paper is nonuniqueness of the inverse source problem, which is related with nonobservability of the source by using only boundary data. The Hilbert space framework constrains the class of functions that can be reconstructed and may be considered a kind of a priory information about the source. For more generic Banach spaces and other optimal formulations, different sources may be obtained. The demonstrated equivalence Theorem 4.2 can be used to investigate questions such as stability and regularization. Further numerical studies for the unitary disk based on the equations presented in Section 5 remain as future work.

Acknowledgments

The authors acknowledge the partial support of the Brazilian agencies CNPq, CAPES, and Coppetec Foundation.