BOUNDARY-DOMAIN INTEGRAL EQUATIONS FOR VARIABLE-COEFFICIENT HELMHOLTZ BVPs IN 2D

Abstract

In this paper, we construct boundary-domain integral equations (BDIEs) of the Dirichlet and mixed boundary value problems for a two-dimensional variable-coefficient Helmholtz equation. Using an appropriate parametrix, these problems are reduced to several BDIE systems. It is shown that the BVPs and the formulated BDIE systems are equivalent. Fredholm properties and unique solvability and invertibility of BDIE systems are investigated in appropriate Sobolev spaces.

1 Introduction

Many problems of mathematical physics and engineering such as the ones associated with steady-state oscillations (mechanical, acoustic, electromagnetic, etc.) lead to the Helmholtz equation. Since the fundamental solution of the constant-coefficient Helmholtz equation is known explicitly, the boundary value problems (BVPs) for this equation can be reduced to the boundary integral equations (BIEs), which have the advantage that the dimension of the problem is reduced by one and the BIEs could be effectively solved numerically.

In applications, such as seismic or medical imaging, the coefficients in the Helmholtz equation become variable [26]. For such partial differential equations (PDEs) with variable coefficients, a fundamental solution is generally not available in explicit form, preventing the reduction of BVPs for such PDEs to explicit BIEs. Instead, one can use a parametrix (Levi function), which is more widely available, to reduce the variable-coefficient BVPs to either segregated or united direct boundary-domain integral or integro-differential equations [19], BDIEs or BDIDEs. These equations are well studied for Dirichlet, Neumann, and mixed (Dirichlet-Neumann) BVPs for variable-coefficient second-order scalar elliptic PDE

$$\begin{aligned} Au(x):=\sum _{i=1}^n \dfrac{\partial }{\partial x_i}\left[ a(x)\dfrac{\partial u(x)}{\partial x_i}\right] =f(x), \quad \quad x\in \Omega \end{aligned}$$

(1.1)

in 3D [6,7,8,9, 20, 23, 24] as well as in 2D [4, 5, 13].

However, this is not the case for the parametrix-based system of BDIEs for variable-coefficient Helmholtz equation

$$\begin{aligned} Au(x) + k^2(x)u(x)=f(x), \quad \quad x\in \Omega \end{aligned}$$

(1.2)

where k(x) is a real function of x, a(x) is a known variable coefficient, u is an unknown function, and $f\in L^2(\Omega )$ is a given function. Note that when $\Omega =\mathbb {R}^n$ and k(x) is constant outside a bounded domain, (1.2) can be reduced to the Lippmann-Schwinger type integral equation; see, e.g., [15, Section 8] for the case when a(x) is a constant in $\mathbb {R}^n$, and [11, 16, 17] for the case when a(x) is a constant only outside a bounded domain in $\mathbb {R}^n$. In both cases, the integral equations can be considered as special cases of BDIEs. We also mention [1], where the numerical solutions of BDIE and BDIDE of the mixed problem for PDE (1.2) are given (without analysis of the equivalence to the BVP or the solution existence and uniqueness).

Applying the previously developed techniques for the operator A in (1.1), in this paper, we shall construct and investigate BDIE systems for the Dirichlet and mixed (Dirichlet-Neumann) BVPs associated with PDE (1.2) in appropriate function spaces in the two-dimensional case. The BDIEs in the $n-$dimensional cases with $n\ge 3$ can also be analyzed in a similar way, although the scaling with the parameter $r_0$ in the parametrix will not be needed in such cases because the invertibility of the standard single layer potential operator will not depend on the domain size then.

2 Preliminaries

Let $\Omega$ be a domain in $\mathbb {R}^2$ bounded by a smooth curve $\partial \Omega$. The set of all infinitely differentiable functions on $\Omega$ with compact support is denoted by $\mathcal {D}(\Omega )$. The function space $\mathcal {D}^\prime (\Omega )$ consists of all continuous linear functionals over $\mathcal {D}(\Omega )$. The space of restrictions to $\Omega$ of functions in $\mathcal {D}(\mathbb {R}^2)$ is denoted by $\mathcal {D}(\overline{\Omega })$. The space $H^s(\mathbb {R}^2)$, $s\in \mathbb {R}$, denotes the Bessel potential space, and $H^{-s}(\mathbb {R}^2)$ is the dual space to $H^s(\mathbb {R}^2)$. We define $H^s(\Omega )=\{u\in \mathcal {D}^\prime (\Omega ): u=U|_{\Omega } \hbox { for some } U\in H^s(\mathbb {R}^2)\}$, and $H^1_0(\Omega )$ is the space of functions in $H^1(\Omega )$ with zero traces on $\partial \Omega$. By $H^s(\partial \Omega )$, we denote the Bessel potential spaces on the boundary $\partial \Omega$ (cf., e.g., [18]).

For the scalar elliptic differential operator A given by

$$\begin{aligned} A=\sum _{i=1}^2 \dfrac{\partial }{\partial x_i}\left[ a(x)\dfrac{\partial }{\partial x_i}\right] , \end{aligned}$$

(2.1)

we consider the Helmholtz equation

$$\begin{aligned} Au(x) + k^2(x)u(x)=f(x), \quad \quad x\in \Omega \end{aligned}$$

where k(x) is a real function of x, a(x) is a known variable coefficient, u is an unknown function, and f is a given function in $\Omega$. We assume that $a, k \in C^\infty (\overline{\Omega })$ and $0<a_0< a(x)< a_1< \infty$ for some constants $a_0$ and $a_1$, for all $x \in \Omega$.

Let us denote $A_k:=A+k^2$. Following the definition given, e.g., in [10, 14, 21], for $s\in \mathbb {R}$ the subspace $H^{s,0}(\Omega ; A_k)$ of $H^s(\Omega )$ is defined as

$$\begin{aligned} H^{s,0}(\Omega ; A_k):= \{g \in H^s(\Omega ):\ A_kg \in L^2(\Omega )\}, \end{aligned}$$

(2.2)

with the norm $\Vert g\Vert ^2_{H^{s,0}(\Omega ; A_k)}:= \Vert g\Vert ^2_{H^s(\Omega )} +\Vert A_kg\Vert ^2_{L^2(\Omega )}$. Since $A_ku-Au=k^2u\in L^2(\Omega )$ for $u\in H^1(\Omega )$, we get $H^{1, 0}(\Omega ; A_k)=H^{1, 0}(\Omega ; A)$. Moreover, if $s_2\le s_1$, then we have the embedding $H^{s_1,0}(\Omega ;A_k)\subset H^{s_2,0}(\Omega ;A_k)$.

For $u\in H^s(\Omega )$, $s>3/2$, the corresponding classical co-normal derivative operator on $\partial \Omega$ in the sense of traces denoted by $T^{c+}$ is given by

$$\begin{aligned} T^{c+} u(x)=\sum _{i=1}^2 a(x)n_i(x)\gamma ^+ \dfrac{\partial u(x)}{\partial x_i}, \end{aligned}$$

(2.3)

where n(x) is the outward (to $\Omega$) unit normal vector at the point $x \in \partial \Omega$, and $\gamma ^+$ is the trace operator.

For $u \in H^2(\Omega )$ and $v \in H^1(\Omega )$, from the Gauss-Ostrogradsky theorem, we get

$$\begin{aligned} \int _\Omega v(x)Au(x)dx=-\sum _{i=1}^2 \int _\Omega a(x)\dfrac{\partial u(x)}{\partial x_i}\dfrac{\partial v(x)}{\partial x_i}dx + \int _{\partial \Omega } T^{c+} u(x)\gamma ^+v(x)dS_x. \end{aligned}$$

From this, we obtain the first Green identity:

$$\begin{aligned} \mathcal {E}_k(u,v)=-\int _\Omega v(x)A_ku(x)dx + \int _{\partial \Omega } T^{c+} u(x)\gamma ^+v(x)dS_x, \end{aligned}$$

(2.4)

where

$$\begin{aligned} \mathcal {E}_k(u,v):=\int _\Omega a(x)\nabla u(x)\cdot \nabla v(x)dx - \int _\Omega k^2(x)u(x)v(x)dx \end{aligned}$$

is the symmetric bilinear form.

Even though the classical co-normal derivative is, generally, not defined for $u\in H^s(\Omega )$, $s<3/2$ (some examples are provided in [23, Appendix A]), there exists the following continuous extension of this definition of the classical co-normal derivative hinted by the first Green identity (2.4), for $u \in H^{s,0}(\Omega ; A_k)$, $1/2<s<3/2$ (see, e.g., [10], [18, Lemma 4.3],[21, 22]).

Definition 2.1

For $u \in H^{s,0}(\Omega ; A_k)$, $1/2<s<3/2$, the (canonical) co-normal derivative $T^+ u \in H^{s-\frac{3}{2}}(\partial \Omega )$ is defined in the following weak form:

$$\begin{aligned} \langle T^+u, w \rangle _{_{\partial \Omega }}&:=\langle A_ku, \gamma ^{-1}w\rangle _{_{\Omega }} + \mathcal {E}_k(u,\gamma ^{-1}w) \nonumber \\&=\langle Au, \gamma ^{-1}w\rangle _{_{\Omega }} + \mathcal {E}_0(u,\gamma ^{-1}w), \quad \quad \forall w \in H^{\frac{3}{2}-s}(\partial \Omega ). \end{aligned}$$

(2.5)

In (2.5) and further on, $\gamma ^{-1}: H^{\frac{3}{2}-s}(\partial \Omega )\rightarrow H^{2-s}(\Omega )$ is a bounded right inverse to the trace operator $\gamma : H^{2-s}(\Omega )\rightarrow H^{\frac{3}{2}-s}(\partial \Omega )$, the notation $\langle \cdot , \cdot \rangle _{_{\partial \Omega }}$ denotes the duality brackets between the spaces $H^{s-\frac{3}{2}}(\partial \Omega )$ and $H^{\frac{3}{2}-s}(\partial \Omega )$, while $\langle \cdot , \cdot \rangle _{\Omega }$ denotes the duality brackets between the spaces $H^{s-1}(\Omega )$ and $H^{1-s}(\Omega )$, extending the usual $L^2$-inner products.

The operator $T^+:H^{s,0}(\Omega ;A_k) \rightarrow H^{s-3/2}(\partial \Omega )$ is continuous for $s>1/2$. Moreover, as we observe from [21, Corollary 3.14],

$$\begin{aligned} T^+u=T^{c+}u \text { for } u\in H^s(\Omega ),\quad s>3/2. \end{aligned}$$

(2.6)

By [10, Lemma 3.4], the first Green identity (2.4) in the form

$$\begin{aligned} \langle T^+ u, \gamma ^+ v \rangle _{_{\partial \Omega }}=\mathcal {E}_k(u,v) + \langle A_ku, v \rangle _{_\Omega }. \end{aligned}$$

(2.7)

holds for $u\in H^{1,0}(\Omega ; A_k)$ and $v\in H^1(\Omega )$.

Interchanging the roles of u and v in the first Green identity (2.7) for $u\in H^1(\Omega )$ and $v\in H^{1, 0}(\Omega ; A_k)$, we obtain the first Green identity for v,

$$\begin{aligned} \langle T^+ v, \gamma ^+ u \rangle _{_{\partial \Omega }}=\mathcal {E}_k(v,u) + \langle A_kv, u \rangle _{_\Omega } . \end{aligned}$$

(2.8)

Then, subtracting (2.8) from (2.7), we obtain the second Green identity for $u, v \in H^{1, 0}(\Omega ; A_k)$,

$$\begin{aligned} \langle A_ku, v\rangle _{_{\Omega }} - \langle A_kv, u\rangle _{_{\Omega }}= \langle T^+u, \gamma ^+ v\rangle _{_{\partial \Omega }} - \langle T^+v, \gamma ^+ u\rangle _{_{\partial \Omega }}. \end{aligned}$$

(2.9)

3 Parametrix-based potential operators

Definition 3.1

A function P(x, y) is a parametrix for the operator $A_k$ if

$$\begin{aligned} (A_k)_xP(x,y)=\delta (x-y) + R_k(x,y), \end{aligned}$$

where $\delta$ is the Dirac-delta distribution, while $R_k(x,y)$ is a remainder possessing at most a weak singularity at $x=y$.

Based on [19], the function

$$\begin{aligned} P(x,y)=\dfrac{1}{a(y)}P_{\Delta }(x,y)=\dfrac{1}{2\pi a(y)}\ln \Big (\dfrac{|x-y|}{r_0}\Big ), \quad x, y \in \mathbb {R}^2, \end{aligned}$$

where $\ r_0>0$ is a constant parameter, is a parametrix for the operator A. Note that

$$\begin{aligned} P_{\Delta }(x,y)=\dfrac{1}{2\pi }\ln \Big (\dfrac{|x-y|}{r_0}\Big ),\quad \ r_0>0, \quad x, y \in \mathbb {R}^2 \end{aligned}$$

(3.1)

is a fundamental solution of the Laplace operator, $\Delta$ (cf., e.g., [18, Theorem 8.1]). We can also take P(x, y) as a parametrix for the operator $A_k$. Then, the corresponding remainder function becomes

$$\begin{aligned} R_k(x,y)=k^2(x)P(x,y) + R(x,y),\quad \quad x,y \in \mathbb {R}^2, \end{aligned}$$

(3.2)

where

$$\begin{aligned} R(x, y)=\sum _{i=1}^2 \dfrac{x_i-y_i}{2\pi a(y) |x-y|^2} \dfrac{\partial a(x)}{\partial x_i},\quad \quad x,y \in \mathbb {R}^2, \end{aligned}$$

is the remainder function for the operator A and is weakly singular due to the smoothness of the function a(x). Hence, $R_k(x,y)$ is also weakly singular, and thus, P(x, y) is, indeed, a parametrix for the operator $A_k$.

3.1 Surface potentials

The single and the double layer surface potential operators corresponding to the parametrix P(x, y) are defined for $y\notin \partial \Omega$ as

$$\begin{aligned} Vg(y):= - \int _{\partial \Omega } P(x,y) g(x)dS_x, \qquad Wg(y):=-\int _{\partial \Omega } \left[ T^+_xP(x,y)\right] g(x)dS_x \end{aligned}$$

where the integrals are understood as the appropriate dual products if the scalar density function g is not integrable.

The corresponding boundary integral (pseudodifferential) operators of direct surface values of the single layer potential $\mathcal {V}$ and of the double layer potential $\mathcal {W}$, and the co-normal derivatives of the single layer potential $\mathcal {W}\,'$, and of the double layer potential $\mathcal {L}^+$, for $y\in \partial \Omega$, are

$$\begin{aligned}&\mathcal {V}g(y):=-\int _{\partial \Omega } P(x,y)g(x)dS_x,{} & {} \mathcal {W}g(y):=-\int _{\partial \Omega } \left[ T^+_xP(x,y)\right] g(x)dS_x, \nonumber \\&\mathcal {W}\,'g(y):= - \int _{\partial \Omega } \left[ T^+_yP(x,y)\right] g(x)dS_x,{} & {} \mathcal {L}^+g(y):=T^+Wg(y). \end{aligned}$$

(3.3)

Let $V_{\Delta }, W_{\Delta }, \mathcal {V}_{\Delta }, \mathcal {W}_{\Delta }$ and $\mathcal {L}^+_\Delta$ denote the potentials and the boundary operators corresponding to the Laplace operator $\Delta$. That is, the subscript $\Delta$ means that the corresponding surface potentials are constructed by means of the fundamental solution (3.1) of the Laplace operator $\Delta$. Then, the following relations hold in 2D (cf. [13]).

$$\begin{aligned} Vg&= \frac{1}{a}V_{\Delta } g, \hspace{1in} Wg=\frac{1}{a}W_{\Delta } (ag) \end{aligned}$$

(3.4)

$$\begin{aligned} \mathcal {V}g&=\frac{1}{a}\mathcal {V}_{\Delta } g, \hspace{1in} \mathcal {W}g=\frac{1}{a}\mathcal {W}_{\Delta } (ag), \end{aligned}$$

(3.5)

$$\begin{aligned} \mathcal {W}\,'g&=\mathcal {W}\,'_{\Delta } g + \Big [a \dfrac{\partial }{\partial n}\Big (\dfrac{1}{a}\Big )\Big ]\mathcal {V}_{\Delta } g, \end{aligned}$$

(3.6)

$$\begin{aligned} \mathcal {L}^+g&=\mathcal {L}_\Delta ^+ (ag) + \Big [a \dfrac{\partial }{\partial n}\Big (\dfrac{1}{a}\Big )\Big ] \gamma ^+W_\Delta (ag). \end{aligned}$$

(3.7)

The following two theorems are proved in [13, Theorem 1 and Theorem 2].

Theorem 3.2

Let $u \in H^{-\frac{1}{2}}(\partial \Omega )$ and $v \in H^{\frac{1}{2}}(\partial \Omega )$. Then, the following relations hold for $y\in \partial \Omega$,

$$\begin{aligned} \gamma ^+ V u(y)&=\mathcal {V}u(y), \end{aligned}$$

(3.8)

$$\begin{aligned} \gamma ^+ W v(y)&=- \dfrac{1}{2}v(y) + \mathcal {W}v(y), \end{aligned}$$

(3.9)

$$\begin{aligned} T^+ V u(y)&= \dfrac{1}{2}u(y) + \mathcal {W}\,' u(y). \end{aligned}$$

(3.10)

Theorem 3.3

For $s \in \mathbb {R}$, the following operators are continuous,

$$\begin{aligned} V&: H^s(\partial \Omega ) \rightarrow H^{s+\frac{3}{2}}(\Omega ),\\ W&: H^s(\partial \Omega ) \rightarrow H^{s + \frac{1}{2}}(\Omega ),\\ \mathcal {V}, \mathcal {W},\mathcal {W}\,'&: H^s(\partial \Omega ) \rightarrow H^{s+1}(\partial \Omega ). \end{aligned}$$

These theorems imply the following assertion.

Corollary 3.4

The following operators are continuous,

$$\begin{aligned} V&: H^s(\partial \Omega ) \rightarrow H^{s+\frac{3}{2},0}(\Omega ;A_k),\qquad s\ge -\frac{1}{2},\\ W&: H^s(\partial \Omega ) \rightarrow H^{s + \frac{1}{2},0}(\Omega ;A_k), \qquad s\ge \frac{1}{2}. \end{aligned}$$

Proof

For $g\in H^s(\partial \Omega )$, from Theorem 3.3, we get $Vg\in H^{s+\frac{3}{2}}(\Omega )$. Then,

$$\begin{aligned} A(Vg)&=\Delta (aVg)-\sum _{i=1}^2\partial _i(Vg\partial _ia)\\&=\Delta (V_\Delta g)-\sum _{i=1}^2\partial _i(Vg\partial _ia) =- \sum _{i=1}^2\partial _i(Vg\partial _ia) \end{aligned}$$

belongs to $L^2(\Omega )$ if $s\ge -\frac{1}{2}$. A similar proof holds for the operator W as well.$\square$

The compactness of the following surface potential operators in Corollary 3.5 follows directly from Theorem 3.3 and Rellich compact embedding theorem, see, e.g., [18, Theorem 3.27].

Corollary 3.5

For $s\in \mathbb {R}$, the following operators are compact,

$$\begin{aligned} \mathcal {V}, \mathcal {W}, \mathcal {W}\,': H^s(\partial \Omega ) \rightarrow H^s(\partial \Omega ) . \end{aligned}$$

For $s\in \mathbb {R}$, $\Gamma _1 \subset \partial \Omega$, let us define the following subspaces of the space $H^s(\partial \Omega )$, (see, e.g., [27, pp 147):

$$\begin{aligned} \widetilde{H}^s(\Gamma _1)&:=\{\psi \in {H}^s(\partial \Omega ): \mathrm{supp\,} \psi \subset \overline{\Gamma }_1\},\\ H^s_{**}(\partial \Omega )&:= \{\psi \in H^s(\partial \Omega ): \langle \psi , 1\rangle _{_{\partial \Omega }}=0\}, \\ \widetilde{H}^s_{**}(\Gamma _1)&:=\{\psi \in \widetilde{H}^s(\Gamma _1): \langle \psi , 1 \rangle _{_{\partial \Omega }}=0\}. \end{aligned}$$

Corollary 3.5 implies the following assertion.

Theorem 3.6

Let $\Gamma _1$ and $\Gamma _2$ be non-empty smooth pieces of a curve $\partial \Omega$. Then the operators

$$\begin{aligned} r_{_{\Gamma _2}}\mathcal {V}, r_{_{\Gamma _2}}\mathcal {W}, r_{_{\Gamma _2}}\mathcal {W}\,': \widetilde{H}^s(\Gamma _1)\longrightarrow H^s(\Gamma _2). \end{aligned}$$

(3.11)

are compact for $s\in \mathbb {R}$.

In (3.11) and further on, $r_{_{\Gamma _1}}$, $r_{_{\Gamma _2}}$, etc. denote the corresponding restriction operators.

3.1.1 Invertibility of single layer potential operator on $\partial \Omega$

It is well known that the kernel of the operator

$$\begin{aligned} \mathcal {V}_\Delta : H^{-\frac{1}{2}}(\partial \Omega )\rightarrow H^{\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

(3.12)

with the parameter $r_0=1$ in (3.1), is non-zero for some domains in 2D (see, e.g., [27, Theorem 6.22 proof]). Then, the first relation in (3.5) and scaling imply a non-zero kernel also for $\mathcal {V}$ with $r_0>0$, for some domains $\Omega$.

The following result is proved in [13, Theorem 4].

Theorem 3.7

Let $\psi \in H_{**}^{-1/2}(\partial \Omega )$. If $\mathcal {V}\psi =0$ on $\partial \Omega$, then $\psi =0$.

On the other hand, choosing for a given $\Omega$ an appropriate parameter $r_0$, one can get the zero kernel for $\mathcal {V}$ not only on the subspace $H_{**}^{-1/2}(\partial \Omega )$ but also on the entire space $H^{-1/2}(\partial \Omega )$ and then prove the following invertibility assertion.

Theorem 3.8

Let $\Omega \subset \mathbb {R}^2$ with $r_0>\textrm{diam}(\Omega )$. Then, the operator

$$\begin{aligned} \mathcal {V}: H^{-\frac{1}{2}}(\partial \Omega )\rightarrow H^{\frac{1}{2}}(\partial \Omega ) \end{aligned}$$

(3.13)

is invertible.

Proof

For $r_0=1$, the assertion is available in [13, Theorem 5]. For arbitrary $r_0>\textrm{diam}(\Omega )$, the invertibility of operator (3.12) can be obtained by scaling the result for $r_0=1$, e.g., from Theorem 6.23 and reasoning following it in [27]. Then, the first relation in (3.5) implies the invertibility of operator (3.13) as well. (Cf. also [2, Theorem 5.2] and [3, Theorem 6].)$\square$

Similarly to [5, Corollary 2.7], we obtain the following assertion.

Corollary 3.9

Let $\Gamma _1$ be non-empty relatively open connected part of a curve $\partial \Omega$. Then, the operator

$$\begin{aligned} r_{_{\Gamma _1}} \mathcal {V}:\ \widetilde{H}^{-\frac{1}{2}}(\Gamma _1) \rightarrow H^{\frac{1}{2}}(\Gamma _1) \end{aligned}$$

is bounded and Fredholm of index zero.

Theorem 3.10

Let $\Gamma _1$ be a non-empty relatively open connected part of the boundary curve $\partial \Omega$ with $r_0>\textrm{diam}(\Gamma _1)$. Then, the operator $r_{_{\Gamma _1}}\mathcal {V}:\ \widetilde{H}^{-\frac{1}{2}}(\Gamma _1) \rightarrow H^{\frac{1}{2}}(\Gamma _1)$ has a bounded inverse.

Proof

Taking into account the condition $r_0>\textrm{diam}(\Gamma _1)$, we can follow the proof of [5, Corollary 2.9]. $\square$

Due to (3.9) and the second relation in (3.4), relation (3.7) can also be written as

$$\begin{aligned} \widehat{\mathcal {L}}g=\Big [\mathcal {L}^+ + \dfrac{\partial a}{\partial n}\Big (-\dfrac{1}{2}I + \mathcal {W}\Big )\Big ]g, \quad \text {on} \ \partial \Omega , \end{aligned}$$

(3.14)

where $\widehat{\mathcal {L}}g{:=}\mathcal {L}_\Delta ^+ (ag)$.

The following assertion is available, e.g., in [5, Theorem 2.10] (cf. [6, Theorem 3.6] in the 3D case).

Theorem 3.11

Let $\Gamma _1$ be a non-empty open smooth part of $\partial \Omega$.

• (i) Then, the operator

  $$\begin{aligned} r_{_{\Gamma _1}}\widehat{\mathcal {L}}: \widetilde{H}^{\frac{1}{2}}(\Gamma _1) \rightarrow H^{-\frac{1}{2}}(\Gamma _1) \end{aligned}$$

  is continuously invertible.

• (ii) Moreover, the operator

  $$\begin{aligned} r_{_{\Gamma _1}}(\mathcal {L}^+-\widehat{\mathcal {L}}): \widetilde{H}^{\frac{1}{2}}(\Gamma _1) \rightarrow H^{\frac{1}{2}}(\Gamma _1) \end{aligned}$$

  is bounded, and the operator

  $$\begin{aligned} r_{_{\Gamma _1}}(\mathcal {L}^+-\widehat{\mathcal {L}}): \widetilde{H}^{\frac{1}{2}}(\Gamma _1) \rightarrow H^{-\frac{1}{2}}(\Gamma _1) \end{aligned}$$

  is compact.

3.2 Volume potentials

Similar to [4, 6, 19], we define the parametrix-based logarithmic and remainder volume potential operators, respectively, as

$$\begin{aligned} \mathcal {P} g(y):=\int _\Omega P(x,y)g(x)dx, \qquad \mathcal {R}_kg(y):= \int _\Omega R_k(x,y) g(x)dx, \quad y \in \mathbb {R}^2. \end{aligned}$$

Remark 3.12

As for the layer potentials, let $\mathcal {P}_\Delta$ denote the logarithmic potential for the operator $\Delta$, that is,

$$\begin{aligned} \mathcal {P}_\Delta g(y):=\int _\Omega P_\Delta (x,y)g(x)dx, \quad x, y \in \mathbb {R}^2, \end{aligned}$$

where $P_\Delta$ is the fundamental solution (3.1). Then,

$$\begin{aligned} \mathcal {P}g=\dfrac{1}{a}\mathcal {P}_\Delta g, \quad \mathcal {R}_kg=\mathcal {P}(k^2g)+\mathcal {R}g, \end{aligned}$$

(3.15)

where $\mathcal {R}$ is the parametrix-based remainder volume potential operator for the remainder function R(x, y) and, see [13, 19],

$$\begin{aligned} \mathcal {R}g=-\frac{1}{a}\sum _{i=1}^2 \partial _i [\mathcal {P}_{\Delta } \left( g\partial _i a\right) ], \end{aligned}$$

where $\partial _i=\partial /\partial x_i$.

Theorem 3.13

Let $\Omega$ be a bounded open region in $\mathbb {R}^2$ with closed, infinitely smooth boundary $\partial \Omega$. The following operators are continuous.

$$\begin{aligned} \mathcal {P}&: H^s(\Omega ) \longrightarrow H^{s+2}(\Omega ), \quad \qquad s>-\frac{1}{2}; \end{aligned}$$

(3.16)

$$\begin{aligned} \mathcal {R}&: H^s(\Omega ) \longrightarrow H^{s+1}(\Omega ), \quad \qquad s>-\frac{1}{2}; \end{aligned}$$

(3.17)

$$\begin{aligned} \mathcal {R}_k&: H^s(\Omega ) \longrightarrow H^{s+1}(\Omega ), \quad \qquad s>{-\frac{1}{2}}; \end{aligned}$$

(3.18)

$$\begin{aligned} \gamma ^+\mathcal {R}_k&: H^s(\Omega ) \longrightarrow H^{s+\frac{1}{2}}(\partial \Omega ), \quad \quad s>{-\frac{1}{2}}; \end{aligned}$$

(3.19)

$$\begin{aligned} T^+\mathcal {R}_k&: H^s(\Omega ) \longrightarrow H^{s-\frac{1}{2}}(\partial \Omega ), \quad \quad s>{\frac{1}{2}}. \end{aligned}$$

(3.20)

Proof

For (3.16) and (3.17), we refer to [13, Theorem 3]. From the second relation in (3.15), together with (3.16) and (3.17), we obtain the continuity of (3.18). The continuity of the operators (3.19) and (3.20) is the direct consequence of the trace theorem, Definition 2.1 of the co-normal derivative and relation (2.6).$\square$

Corollary 3.14

The following operators are continuous.

$$\begin{aligned} \mathcal {P}&: H^s(\Omega ) \longrightarrow H^{s+2,0}(\Omega ;A_k), \qquad s\ge 0;\end{aligned}$$

(3.21)

$$\begin{aligned} \mathcal {R}&: H^s(\Omega ) \longrightarrow H^{s+1,0}(\Omega ;A_k), \qquad s\ge 1;\end{aligned}$$

(3.22)

$$\begin{aligned} \mathcal {R}_k&: H^s(\Omega ) \longrightarrow H^{s+1,0}(\Omega ;A_k), \qquad s\ge 1. \end{aligned}$$

(3.23)

Proof

Using the continuity of operators (3.16), (3.17), and (3.18) and the space definition (2.2), we obtain the continuity of operators (3.21), (3.22), and (3.23). $\square$

Corollary 3.15

The following operators are compact.

$$\begin{aligned} \mathcal {R}_k&: H^s(\Omega ) \longrightarrow H^s(\Omega ),\quad s>-\frac{1}{2};\end{aligned}$$

(3.24)

$$\begin{aligned} \gamma ^+ \mathcal {R}_k&: H^s(\Omega ) \longrightarrow H^{s-\frac{1}{2}}(\partial \Omega ),\quad s>-\frac{1}{2};\end{aligned}$$

(3.25)

$$\begin{aligned} T^+ \mathcal {R}_k&: H^s(\Omega ) \longrightarrow H^{s-\frac{3}{2}}(\partial \Omega )\quad s>\frac{1}{2}. \end{aligned}$$

(3.26)

Proof

The compactness of operators (3.24), (3.25), and (3.26) follows from (3.18), (3.19), and (3.20) and the Rellich compact embedding theorem.$\square$

Corollary 3.16

The operator

$$\begin{aligned} \mathcal {R}_k-\mathcal {R}:H^s(\Omega ) \longrightarrow H^{s,0}(\Omega ;A_k),\quad s>0, \end{aligned}$$

(3.27)

is compact.

Proof

From the second equation in (3.15), we see that $\mathcal {R}_kg-\mathcal {R}g=\mathcal {P}(k^2g)$. Then, by (3.16) for $s>-1/2$, the operator $\mathcal {R}_k-\mathcal {R}: H^s(\Omega ) \rightarrow H^{s+2}(\Omega )$ is continuous, and the operator $\mathcal {R}_k-\mathcal {R}: H^s(\Omega ) \rightarrow H^{s}(\Omega )$ is compact. Hence, the operator $\Delta (\mathcal {R}_k-\mathcal {R}):H^s(\Omega )\rightarrow H^s(\Omega )$ is also continuous for $s>-1/2$, and the operator $\Delta (\mathcal {R}_k-\mathcal {R}):H^s(\Omega )\rightarrow L^2(\Omega )$ is compact for $s>0$.

Further, $A_k(\mathcal {R}_k-\mathcal {R})=a\Delta (\mathcal {R}_k-\mathcal {R})+\sum _{j=1}^2(\partial _ja)\partial _j(\mathcal {R}_k-\mathcal {R}) + k^2(\mathcal {R}_k-\mathcal {R})$. The operator $\partial _j(\mathcal {R}_k-\mathcal {R}):H^s(\Omega )\rightarrow H^{s+1}(\Omega )$ is continuous, and hence, the operator $\partial _j(\mathcal {R}_k-\mathcal {R}):H^s(\Omega )\rightarrow H^{0}(\Omega )$ is compact for $s>-1/2$. Thus, the operator $A_k(\mathcal {R}_k-\mathcal {R}):H^s(\Omega )\rightarrow L^2(\Omega )$ is compact for the operator $A_k$ with infinitely smooth coefficients, for $s>0$. Hence, the compactness of operator (3.27) follows from the space definition (2.2). $\square$

Corollary 3.17

Let $\Gamma _1$ and $\Gamma _2$ be non-empty, non-intersecting parts of $\partial \Omega$ such that $\partial \Omega =\overline{\Gamma }_1 \cup \overline{\Gamma }_2$. Then, the operators

$$\begin{aligned} r_{_{\Gamma _1}}\gamma ^+ \mathcal {R}, r_{_{\Gamma _1}}\gamma ^+ \mathcal {R}_k&: H^s(\Omega ) \longrightarrow H^{s-\frac{1}{2}}(\Gamma _1),\\ r_{_{\Gamma _1}}T^+\mathcal {R}, r_{_{\Gamma _1}}T^+\mathcal {R}_k&: H^s(\Omega ) \longrightarrow H^{s-\frac{3}{2}}(\Gamma _1), \end{aligned}$$

are compact for $s>\frac{1}{2}$.

Proof

Theorem 3.13 implies that the following operators are continuous for $s>\frac{1}{2}$:

$$\begin{aligned} r_{_{\Gamma _1}}\gamma ^+ \mathcal {R}_k&: H^s(\Omega ) \longrightarrow H^{s+\frac{1}{2}}(\Gamma _1),\\ r_{_{\Gamma _1}}T^+\mathcal {R}_k&: H^s(\Omega ) \longrightarrow H^{s-\frac{1}{2}}(\Gamma _1). \end{aligned}$$

Then, the proof follows from the compactness of the embeddings $H^{s+\frac{1}{2}}(\Gamma _1) \subset H^{s-\frac{1}{2}}(\Gamma _1)$ and $H^{s-\frac{1}{2}}(\Gamma _1) \subset H^{s-\frac{3}{2}}(\Gamma _1)$. The proof holds true also for $k=0$. $\square$

4 The third Green identity

As, e.g., in [4,5,6, 13], for $u\in H^{1,0}(\Omega ; A_k)$, we substitute P(x, y) for v(x) in Green’s second identity (2.9) for $\Omega \setminus \overline{B_\epsilon }(y)$, where $B_\epsilon (y)$ is a disc of radius $\epsilon$ centered at y and take the limit as $\epsilon \rightarrow 0$ to arrive at the parametrix-based third Green identity

$$\begin{aligned} u + \mathcal {R}_ku - VT^+u + W \gamma ^+u = \mathcal {P}{A}_ku \quad \text{ in }\ \ \Omega . \end{aligned}$$

(4.1)

Taking the trace of (4.1) and using relations (3.8) and (3.9), we obtain

$$\begin{aligned} \frac{1}{2}\gamma ^+ u + \gamma ^+\mathcal {R}_ku - \mathcal {V}T^+u + \mathcal {W}\gamma ^+u= \gamma ^+\mathcal {P}{A}_ku \quad \text {on}\ \ \partial \Omega . \end{aligned}$$

(4.2)

From Corollaries 3.4 and 3.14, we see that each term of (4.1) belongs to $H^{1,0}(\Omega ;A_k)$. Now, taking the co-normal derivative of (4.1) and using relation (3.10), we get

$$\begin{aligned} \frac{1}{2}T^+u + T^+\mathcal {R}_ku - \mathcal {W}\,'T^+u + T^+W\gamma ^+u=T^+ \mathcal {P}{A}_ku \quad \text {on}\ \ \partial \Omega . \end{aligned}$$

(4.3)

If $u\in H^1(\Omega )$ is a solution of equation ${A}_ku=f$ in $\Omega$, where $f \in L^2(\Omega )$, then (4.1) becomes

$$\begin{aligned} u + \mathcal {R}_ku - VT^+u + W \gamma ^+u= \mathcal {P}f \quad \text {in}\ \ \Omega . \end{aligned}$$

(4.4)

For some functions $f, \Psi$, and $\Phi$, let us consider a more general indirect integral relation associated with (4.4),

$$\begin{aligned} u + \mathcal {R}_ku- V\Psi + W\Phi =\mathcal {P}f \quad \text {in} \ \ \Omega . \end{aligned}$$

(4.5)

Lemma 4.1

Let $u\in H^1(\Omega ), f\in L^2(\Omega ), \Psi \in H^{-\frac{1}{2}}(\partial \Omega ), \Phi \in H^{\frac{1}{2}}(\partial \Omega )$ satisfy (4.5). Then, u belongs to $H^{1,0}(\Omega ;{A}_k)$ and is a solution of PDE ${A}_ku=f$ in $\Omega$, and

$$\begin{aligned} V(\Psi -T^+u)(y) - W(\Phi - \gamma ^+u)(y)=0 , \quad y\in \Omega . \end{aligned}$$

(4.6)

Proof

As in [6, Lemma 4.1] in the 3D case for $k=0$, from Corollaries 3.4 and 3.14, we conclude that all terms in (4.5) except u belong to $H^{1, 0}(\Omega ; A_k)$. Then, (4.5) implies that u belongs to $H^{1, 0}(\Omega ; A_k)$ as well. Now, let us prove the remaining results.

Subtracting (4.5) from (4.1), we obtain

$$\begin{aligned} V\Psi ^* - W\Phi ^*=\mathcal {P}[A_ku - f] \quad \text{ in } \ \Omega , \end{aligned}$$

(4.7)

where $\Psi ^*:=T^+u - \Psi$ and $\Phi ^*:=\gamma ^+u - \Phi$. Multiplying equality (4.7) by a(y) and using relation (3.4) and (3.15), we get

$$\begin{aligned} V_{\Delta } \Psi ^* - W_{\Delta } (a\Phi ^*) = \mathcal {P}_\Delta [A_ku-f], \quad \text{ in }\ \Omega . \end{aligned}$$

(4.8)

The application of the Laplace operator $\Delta$ to (4.8) gives

$$\begin{aligned} A_ku - f =0 \quad \text{ in }\ \ \Omega . \end{aligned}$$

(4.9)

This shows that u solves the differential equation $A_ku=f$ in $\Omega$.

Substituting (4.9) into (4.7) leads to (4.6). $\square$

Lemma 4.2

• (i) Let $\Psi ^* \in H^{-\frac{1}{2}}(\partial \Omega )$ and $r_0 >\textrm{diam}(\Omega )$. If $V\Psi ^*=0$ in $\Omega$, then $\Psi ^*=0$ on $\partial \Omega$.

• (ii) Let $\Phi ^* \in H^{\frac{1}{2}}(\partial \Omega )$ and $r_0>0$. If $W\Phi ^*=0$ in $\Omega$, then $\Phi ^*=0$ on $\partial \Omega$.

Proof

The assertion was proved in [13, Lemma 2] for $r_0=1$. Taking into account Theorem 3.8, we follow the proof of [13, Lemma 2] almost word for word to obtain the assertion for arbitrary $r_0>0$.$\square$

Lemma 4.3

Let $\partial \Omega =\overline{\Gamma }_1 \cup \overline{\Gamma }_2$, where $\Gamma _1$ and $\Gamma _2$ are non-empty, non-intersecting relatively open parts of the boundary curve $\partial \Omega$. Let $\Phi ^* \in \widetilde{H}^{\frac{1}{2}}(\Gamma _2)$ and $\Psi ^* \in \widetilde{H}^{-\frac{1}{2}}(\Gamma _1)$ with $r_0 > \textrm{diam}(\Gamma _1)$. If

$$\begin{aligned} V \Psi ^*(y) - W \Phi ^*(y)=0, \quad y \in \Omega , \end{aligned}$$

(4.10)

then $\Psi ^*=0$ and $\Phi ^*=0$ on $\partial \Omega$.

Proof

Keeping in mind [18, Theorem 8.16], we follow the proof of [6, Lemma 4.2 (iii)] (See also [5, Lemma 2.12], [2, Lemma 5.8], [3, Lemma 3]).$\square$

Remark 4.4

The results of Lemmas 4.2 and 4.3 with no restriction on the parameter $r_0$ can be similarly obtained if $\Psi ^* \in H_{**}^{-\frac{1}{2}}(\partial \Omega )$ and $\Psi ^* \in \widetilde{H}_{**}^{-\frac{1}{2}}(\Gamma _1)$, respectively.

5 Boundary-domain integral equations of the Dirichlet BVP

Consider the Dirichlet BVP

$$\begin{aligned} A_ku&= f \quad \quad \ \ \text {in}\ \ \Omega , \nonumber \\ \gamma ^+ u&=\varphi _0 \quad \quad \text {on}\ \ \partial \Omega , \end{aligned}$$

(5.1)

for unknown function $u\in H^1(\Omega )$, where $f \in L^2(\Omega )$ and $\varphi _0 \in H^{\frac{1}{2}}(\partial \Omega )$ are given functions. The first equation is understood in the distribution sense.

Let us derive and analyze BDIE systems for the Dirichlet BVP (5.1).

To reduce the variable-coefficient Dirichlet BVP (5.1) to segregated BDIE systems, we denote the unknown co-normal derivative as $\psi :=T^+u$ and further consider $\psi$ as formally independent of u.

5.1 BDIE system (D1)

We substitute $A_ku$ and $\gamma ^+u$ from the Dirichlet BVP (5.1) into (4.1) and into its trace (4.2) to reduce the Dirichlet BVP (5.1) to the BDIE system (D1) with the unknowns u and $\psi$:

$$\begin{aligned} \begin{aligned} u + \mathcal {R}_ku - V \psi&= F_0 \qquad \text {in}\ \ \Omega ,\\ \gamma ^+ \mathcal {R}_ku - \mathcal {V}\psi&= \gamma ^+ F_0 - \varphi _0 \quad \ \ \text {on} \ \ \partial \Omega , \end{aligned} \end{aligned}$$

(D1)

where

$$\begin{aligned} F_0 = \mathcal {P}f - W\varphi _0 \quad \quad \text{ in } \ \ \Omega . \end{aligned}$$

(5.2)

The matrix form of system (D1) is $\mathcal {A}_k^1\mathcal {U}=\mathcal {F}^1$, where $\mathcal {U}=(u, \psi )^t \in H^{1,0}(\Omega ; A_k) \times H^{-\frac{1}{2}}(\partial \Omega )$,

$$\begin{aligned} \mathcal {A}^1_k= \left[ \begin{array}{ll} I + \mathcal {R}_k \quad -V \\ \ \gamma ^+ \mathcal {R}_k \quad \ -\mathcal {V} \end{array} \right] , \qquad \qquad \mathcal {F}^1= \left[ \begin{array}{ll}\quad \quad F_0 \\ \ \gamma ^+F_0 - \varphi _0 \end{array} \right] . \end{aligned}$$

(5.3)

From the mapping properties of $\mathcal {P}$ and W provided in Section 3, we get $F_0 \in H^{1,0}(\Omega ;A_k)$. Moreover, the trace theorem implies that $\gamma ^+F_0 \in H^{\frac{1}{2}}(\partial \Omega )$. Therefore, $\mathcal {F}^1 \in H^{1,0}(\Omega ; A_k) \times H^{\frac{1}{2}}(\partial \Omega )$. Due to the mapping properties of the operators involved in (5.3) (see Section 3), the following operators are bounded:

$$\begin{aligned} \mathcal {A}^1_k&: H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega ) \longrightarrow H^1(\Omega ) \times H^{\frac{1}{2}}(\partial \Omega ),\end{aligned}$$

(5.4)

$$\begin{aligned} \mathcal {A}^1_k&: H^{1,0}(\Omega ; A_k) \times H^{-\frac{1}{2}}(\partial \Omega ) \longrightarrow H^{1,0}(\Omega ; A_k) \times H^{\frac{1}{2}}(\partial \Omega ). \end{aligned}$$

(5.5)

Remark 5.1

$\mathcal {F}^1={\textbf {0}}$ if and only if $(f, \varphi _0)={\textbf {0}}$.

Proof

If $\mathcal {F}^1={\textbf {0}}$, then $F_0=0$ and $\gamma ^+F_0+\varphi _0=0$. Consequently, $\varphi _0=0$ on $\partial \Omega$. From this and $F_0=0$, we obtain that $\mathcal {P}f=0$ in $\Omega$, and hence, $f=0$ in $\Omega$. The reverse implication is trivial. $\square$

5.2 BDIE system (D2)

This system is obtained by substituting $A_ku$ and $\gamma ^+u$ from the Dirichlet BVP (5.1) into (4.1) and into its co-normal derivative (4.3), with the unknowns u and $\psi$:

$$\begin{aligned} \begin{aligned} u + \mathcal {R}_ku - V \psi&= F_0 \quad \quad \quad \quad \ \ \quad \text {in}\ \ \Omega ,\\ \dfrac{1}{2}\psi + T^+ \mathcal {R}_ku - \mathcal {W}\,'\psi&= T^+ F_0 \hspace{0.66in}\text {on} \ \ \partial \Omega , \end{aligned} \end{aligned}$$

(D2)

where $F_0$ is the relation (5.2). The system (D2) can be written in matrix form as

$$\begin{aligned} \mathcal {A}^2_k\mathcal {U}=\mathcal {F}^2, \end{aligned}$$

where

$$\begin{aligned} \mathcal {A}^2_k:= \left[ \begin{array}{ll} I + \mathcal {R}_k \qquad -V \\ \ T^+ \mathcal {R}_k \quad \ \frac{1}{2}I-\mathcal {W}\,' \end{array} \right] , \qquad \mathcal {F}^2= \left[ \begin{array}{ll}\quad F_0 \\ \ T^+F_0 \end{array} \right] , \end{aligned}$$

and $F_0$ is given by (5.2). The following operators are bounded:

$$\begin{aligned} \mathcal {A}^2_k : H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega ) \longrightarrow H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

(5.6)

$$\begin{aligned} \qquad \qquad \qquad \mathcal {A}^2_k : H^{1,0}(\Omega ; A_k) \times H^{-\frac{1}{2}}(\partial \Omega ) \longrightarrow H^{1,0}(\Omega ; A_k) \times H^{-\frac{1}{2}}(\partial \Omega ). \end{aligned}$$

(5.7)

Remark 5.2

$\mathcal {F}^2={\textbf {0}}$ if and only if $(f, \varphi _0)={\textbf {0}}$.

Proof

If $\mathcal {F}^2={\textbf {0}}$, then $F_0=0$. From which we get

$$\begin{aligned} 0=\Delta (aF_0)=\Delta (\mathcal {P}_{\Delta } f) + \Delta W_{\Delta }(\varphi _0)=f \quad \text{ in }\ \ \Omega . \end{aligned}$$

Then, the condition $F_0=0$ gives $W_{\Delta }(\varphi _0)=0$ and Lemma 4.2(ii) implies that $\varphi _0=0$ on $\partial \Omega$. The reverse implication is trivial. $\square$

6 Equivalence, Fredholm properties, and invertibility for BDIEs of the Dirichlet BVP

In this section, we first prove the equivalence of the Dirichlet BVP (5.1) to the BDIE systems (D1) and (D2), and then we show the necessary conditions for the invertibility of the two corresponding operators to the BDIE systems.

Theorem 6.1

Let $\varphi _0 \in H^{\frac{1}{2}}(\partial \Omega )$ and $f \in L^2(\Omega )$.

• (i) If some $u\in H^1(\Omega )$ solves the BVP (5.1), then the pair $(u, \psi )^t$, where

  $$\begin{aligned} \psi =T^+u \in H^{-\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

  (6.1)

  solves BDIE systems (D1) and (D2).

• (ii) Let $r_0 > \textrm{diam}(\Omega )$. If a pair $(u, \psi )^t \in H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega )$ solves BDIE system (D1), then u solves BVP (5.1) and $\psi$ satisfies (6.1).

• (iii) Let $r_0>0$. If a pair $(u, \psi )^t \in H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega )$ solves BDIE system (D2), then u solves BVP (5.1), and $\psi$ satisfies (6.1).

Proof

To prove (i), we let $u\in H^1(\Omega )$ be a solution of the BVP (5.1). Since $A_ku=f \in L^2(\Omega )$, we get $u \in H^{1,0}(\Omega ; A_k)$. Setting $\psi = T^+u$ and recalling how BDIE system (D1) and (D2) are constructed, we obtain that the couple $(u, \psi )^t$ solves the systems.

To prove (ii) and (iii), let us assume first that a pair $(u, \psi )^t \in H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega )$ solves system (D1) or (D2). Due to the first equation in the BDIE systems, the hypotheses of Lemma 4.1 are satisfied implying that u belongs to $H^{1,0}(\Omega ; A_k)$ and solves the PDE in the BVP (5.1) in $\Omega$. Moreover, the equation

$$\begin{aligned} W(\varphi _0 - \gamma ^+u)(y) - V(\psi - T^+u)(y) =0, \quad y\in \Omega , \end{aligned}$$

(6.2)

holds.

To prove the remaining parts of (ii), we let $(u, \psi )^t \in H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega )$ solve system (D1). Taking the trace of the first equation in (D1) and subtracting the second equation from it, we get the Dirichlet boundary condition

$$\begin{aligned} \gamma ^+u=\varphi _0 \quad \text {on}\ \ \partial \Omega , \end{aligned}$$

and substituting this in equation (6.2) we obtain

$$\begin{aligned} V(\psi - T^+u)(y)=0, \quad \ \ y\in \Omega . \end{aligned}$$

Since $r_0 > \textrm{diam}(\Omega )$, from Lemma 4.2 (i), we get $\psi =T^+u$.

To complete (iii), we let $(u, \psi )^t \in H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega )$ solve system (D2). It is already shown that $u\in H^{1,0}(\Omega ; A_k)$. Moreover, all the remaining terms in the first equation of (D2) belong to $H^{1,0}(\Omega ;A_k)$ due to the mapping properties of the operators involved (see Section 3). Then, taking the co-normal derivative of the first equation in (D2) and subtracting the second one from it, we get

$$\begin{aligned} \psi = T^+ u \quad \text {on}\ \ \partial \Omega . \end{aligned}$$

Then, inserting this in (6.2) gives

$$\begin{aligned} W(\varphi _0 - \gamma ^+u)(y)=0, \quad y \in \Omega , \end{aligned}$$

and Lemma 4.2 (ii) implies $\varphi _0=\gamma ^+u$ on $\partial \Omega$.$\square$

Theorem 6.1 implies the following two corollaries.

Corollary 6.2

Let $\varphi _0 \in H^{\frac{1}{2}}(\partial \Omega )$ and $f \in L^2(\Omega )$.

• (i) Let $r_0 > \textrm{diam}(\Omega )$. If a pair $(u, \psi )^t \in H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega )$ solves BDIE system (D1), it solves BDIE system (D2).

• (ii) Let $r_0>0$. If a pair $(u, \psi )^t \in H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega )$ solves BDIE system (D2), it solves BDIE system (D1).

Corollary 6.3

• (i) Let $r_0 > \textrm{diam}(\Omega )$. The homogeneous counterpart of BDIE system (D1) has a non-trivial solution in $H^1\times H^{-\frac{1}{2}}(\partial \Omega )$ if and only if the homogeneous counterpart of the Dirichlet problem (5.1) has a non-trivial solution in $H^1(\Omega )$.

• (ii) Let $r_0>0$. The homogeneous counterpart of BDIE system (D2) has a non-trivial solution in $H^1(\Omega )\times H^{-\frac{1}{2}}(\partial \Omega )$ if and only if the homogeneous counterpart of the Dirichlet problem (5.1) has a non-trivial solution in $H^1(\Omega )$.

Let us now analyze the Fredholm properties of operators (5.4), (5.5), (5.6), and (5.7). As a bi-product, we also prove the invertibility of the corresponding operators for $k=0$.

Theorem 6.4

• (i) If $r_0 > \textrm{diam} (\Omega )$, then operator (5.4) is Fredholm with zero index.

• (ii) If $r_0 > 0$, then operator (5.6) is Fredholm with zero index.

Proof

(i) Let $r_0>\rm{diam}(\Omega)$. Let us consider the auxiliary operator

$$\begin{aligned} \mathcal {A}^1_*:= \left[ \begin{array}{ll} I \quad \ -V \\ \,\,\, 0 \quad \ - \mathcal {V} \end{array} \right] . \end{aligned}$$

Then, the operator $\mathcal {A}_*^1: H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega ) \longrightarrow H^1(\Omega )\times H^{\frac{1}{2}}(\partial \Omega )$ is bounded. It is invertible due to the invertibility of its diagonal operators

$$\begin{aligned} I: H^1(\Omega ) \rightarrow H^1(\Omega )\quad \text {and} \quad \mathcal {V}: H^{-\frac{1}{2}}(\partial \Omega ) \rightarrow H^{\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

see Theorem 3.8. Due to the mapping properties of the operators involved, the operator $\mathcal {A}^1_k- \mathcal {A}^1_*: H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega ) \rightarrow H^1(\Omega ) \times H^{\frac{1}{2}}(\partial \Omega )$ where

$$\begin{aligned} \mathcal {A}^1_k- \mathcal {A}^1_*= \left[ \begin{array}{ll} \ \ \mathcal {R}_k \quad \quad 0 \\ \ \gamma ^+ \mathcal {R}_k \quad \ 0 \end{array} \right] , \end{aligned}$$

is compact. Thus, operator (5.4) is Fredholm with index zero.

(ii) The operator $\mathcal {A}_*^2: H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega ) \longrightarrow H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega ),$ where

$$\begin{aligned} \mathcal {A}^2_*= \left[ \begin{array}{ll} I \quad -V \\ 0 \qquad \frac{1}{2}I \end{array} \right] \end{aligned}$$

is bounded. It is also invertible due to the invertibility of its diagonal operators

$$\begin{aligned} I: H^1(\Omega ) \longrightarrow H^1(\Omega )\quad \text {and} \quad I: H^{-\frac{1}{2}}(\partial \Omega ) \longrightarrow H^{-\frac{1}{2}}(\partial \Omega ). \end{aligned}$$

By Corollaries 3.5 and 3.15, the operator

$$\begin{aligned} \mathcal {A}^2_k - \mathcal {A}_*^2: H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega )\longrightarrow H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

where

$$\begin{aligned} \mathcal {A}^2_k- \mathcal {A}^2_*= \left[ \begin{array}{ll} \ \ \ \mathcal {R}_k \quad 0 \\ \ T^+\mathcal {R}_k \quad -\mathcal {W}\,' \end{array} \right] , \end{aligned}$$

is compact. This implies that operator (5.6) is a Fredholm operator of index zero. $\square$

Let us consider the particular cases of operators (5.4), (5.5), (5.6), and (5.7), for $k=0$, that is,

$$\begin{aligned} \,\,\mathcal {A}^1_0 : H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega ) \longrightarrow H^1(\Omega ) \times H^{\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

(6.3)

$$\begin{aligned} \qquad \qquad \quad&\mathcal {A}^1_0 : H^{1,0}(\Omega ; A) \times H^{-\frac{1}{2}}(\partial \Omega ) \longrightarrow H^{1,0}(\Omega ; A) \times H^{\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

(6.4)

$$\begin{aligned} \qquad \qquad \quad&\mathcal {A}^{2}_0: H^{1}(\Omega )\times H^{-\frac{1}{2}}(\partial \Omega )\rightarrow H^{1}(\Omega )\times H^{-\frac{1}{2}}(\partial \Omega ),\end{aligned}$$

(6.5)

$$\begin{aligned} \qquad \qquad \quad&\mathcal {A}^{2}_0: H^{1,0}(\Omega ;A)\times H^{-\frac{1}{2}}(\partial \Omega )\rightarrow H^{1,0}(\Omega ;A)\times H^{-\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

(6.6)

where

$$\begin{aligned} \mathcal {A}^1_0= \left[ \begin{array}{ll} I + \mathcal {R} \quad -V \\ \ \gamma ^+ \mathcal {R} \quad \ -\mathcal {V} \end{array} \right] , \quad \mathcal {A}^2_0= \left[ \begin{array}{ll} I + \mathcal {R} &{}\ -V \\ \ T^+ \mathcal {R} &{}\ \frac{1}{2}I-\mathcal {W}\,' \end{array} \right] . \end{aligned}$$

Theorem 6.5

• (i) If $r_0>\textrm{diam}(\Omega )$, then operators (6.3) and (6.4) are invertible.

• (ii) If $r_0 > 0$, then operators (6.5) and (6.6) are invertible.

Proof

The theorem for $r_0=1$ was proved in [13, Theorems 7 and 8]. Here, we update the proof for arbitrary $r_0>0$.

It is well known that the homogeneous Dirichlet problem (5.1) with $k=0$, that is, with $A_k=A$, where the operator A is given by (2.1) and $0<a_0< a(x)< a_1< \infty$, has only the trivial solution in $H^{1,0}(\Omega ;A)$ and $H^{1}(\Omega )$. This can be obtained, e.g., from the first Green identity (2.7). Then, the equivalence Theorem 6.1 implies that operators (6.3), (6.4), (6.5), and (6.6) are injective. By Theorem 6.4, operators (6.3) and (6.5) are Fredholm operators with zero index. Then, the injectivity of operators (6.3) and (6.5) implies their invertibility (see, e.g., [18, Theorem 2.27]).

To prove invertibility of operator (6.4), we remark that for any $\mathcal {F}^{1}\in H^{1,0}(\Omega ;A)\times H^{\frac{1}{2}}(\partial \Omega )$, a solution of the equation $\mathcal {A}^1_0\mathcal U=\mathcal {F}^{1}$ can be written as $\mathcal U=(\mathcal {A}^1_0)^{-1}\mathcal {F}^{1}$, where $(\mathcal {A}^1_0)^{-1}:H^{1}(\Omega )\times H^{\frac{1}{2}}(\partial \Omega )\rightarrow H^{1}(\Omega )\times H^{-\frac{1}{2}}(\partial \Omega )$ is the continuous inverse to operator (6.3). But due to Lemma 4.1 the first equation of system (D1) with $k=0$ implies that $\mathcal U=(\mathcal {A}^1_0)^{-1}\mathcal {F}^{1}\in H^{1,0}(\Omega ;A)\times H^{-\frac{1}{2}}(\partial \Omega )$ and moreover, the operator $(\mathcal {A}^1_0)^{-1}:H^{1,0}(\Omega ;A)\times H^{\frac{1}{2}}(\partial \Omega )\rightarrow H^{1,0}(\Omega ;A)\times H^{-\frac{1}{2}}(\partial \Omega )$ is continuous, which implies invertibility of operator (6.4).

The invertibility of operator (6.6) is proved in a similar fashion.$\square$

Now, we are in the position to prove an analog of Theorem 6.4 for operators (5.5) and (5.7).

Theorem 6.6

• (i) If $r_0 > \textrm{diam}(\Omega )$, then operator (5.5) is Fredholm with zero index.

• (ii) If $r_0 > 0$, then operator (5.7) is Fredholm with zero index.

Proof

By Theorem 6.5, we see that operators (6.4) and (6.6) are invertible. Due to Corollary 3.16, the operators

$$\begin{aligned} \mathcal {A}^1_k- \mathcal {A}^1_0&: H^{1,0}(\Omega ;A_k) \times H^{-\frac{1}{2}}(\partial \Omega ) \rightarrow H^{1,0}(\Omega ;A_k) \times H^{\frac{1}{2}}(\partial \Omega ),\\ \mathcal {A}^2_k- \mathcal {A}^2_0&: H^{1,0}(\Omega ;A_k) \times H^{-\frac{1}{2}}(\partial \Omega ) \rightarrow H^{1,0}(\Omega ;A_k) \times H^{-\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

where

$$\mathcal {A}^1_k- \mathcal {A}^1_0= \left[ \begin{array}{ll} \quad \mathcal {R}_k-\mathcal {R} &{} 0 \\ \ \gamma ^+( \mathcal {R}_k-\mathcal {R}) &{} 0 \end{array} \right] , \quad \mathcal {A}^2_k- \mathcal {A}^2_0= \left[ \begin{array}{ll} \quad \mathcal {R}_k-\mathcal {R} &{} 0 \\ \ T^+( \mathcal {R}_k-\mathcal {R}) &{} 0 \end{array} \right] ,$$

are compact, implying that operators (5.5) and (5.7) are Fredholm operators with index zero. $\square$

Corollary 6.7

• (i) Let $r_0 > \textrm{diam}(\Omega )$. The homogeneous counterpart of the Dirichlet problem (5.1) has only the trivial solution in $H^1(\Omega )$ if and only if operators (5.4) and (5.5) are invertible.

• (ii) Let $r_0>0$. The homogeneous counterpart of the Dirichlet problem (5.1) has only the trivial solution in $H^1(\Omega )$ if and only if operators (5.6) and (5.7) are invertible.

Proof

If the homogeneous counterpart of the Dirichlet problem (5.1) has only the trivial solution in $H^1(\Omega )$, by Corollary 6.3(i), the operators (5.4) and (5.5) will be injective. Hence, these operators become invertible due to Theorem 6.4.

Conversely, if the operator (5.4) or (5.5) is invertible, the homogeneous counterpart of BDIE system (D1) can have only the trivial solution in $H^1(\Omega )\times H^{-\frac{1}{2}}(\partial \Omega )$, and hence, the result follows from Corollary 6.3 (i).

For operators (5.6) and (5.7), the proof is similar.$\square$

7 Boundary-domain integral equations of the mixed BVP

Let $\partial \Omega =\overline{\partial \Omega }_D \cup \overline{\partial \Omega }_N$, where $\partial \Omega _D$ and $\partial \Omega _N$ are non-empty, relatively open, non-intersecting parts of $\partial \Omega$. We will derive and analyze the system of BDIEs for the following mixed BVP

$$\begin{aligned} \begin{aligned} A_ku&=f \quad \quad \quad \text {in} \quad \Omega , \\ \gamma ^+ u&=\varphi _0 \quad \quad \ \text {on} \quad \partial \Omega _D,\\ T^+u&= \psi _0 \quad \quad \ \text {on} \quad \partial \Omega _N, \end{aligned} \end{aligned}$$

(7.1)

for unknown function $u \in H^1(\Omega )$, where $f \in L^2(\Omega )$, $\varphi _0 \in H^{\frac{1}{2}}(\partial \Omega _D)$ and $\psi _0 \in H^{-\frac{1}{2}}(\partial \Omega _N)$ are given functions.

Similar to the 3D case in [6] and the 2D case with $k=0$ in [5], we let $\Phi _0 \in H^{\frac{1}{2}}(\partial \Omega )$ and $\Psi _0 \in H^{-\frac{1}{2}}(\partial \Omega )$ be some extensions of the given function $\varphi _0$ from $\partial \Omega _D$ to $\partial \Omega$ and $\psi _0$ from $\partial \Omega _N$ to $\partial \Omega$, respectively. Then, an arbitrary extension $\Phi \in H^{\frac{1}{2}}(\partial \Omega )$ preserving the function space can be represented as $\Phi =\Phi _0 + \varphi$ with $\varphi \in \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$; and $\Psi \in H^{-\frac{1}{2}}(\partial \Omega )$ as $\Psi =\Psi _0 + \psi$ with $\psi \in \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D)$.

Considering (4.1), and restrictions of either (4.2) or (4.3) on the appropriate parts of $\partial \Omega$, we reduce the BVP (7.1) to four different BDIE systems. In each case, we substitute f for $A_ku$, $\Phi =\Phi _0 + \varphi$ for the boundary trace $\gamma ^+u$ and $\Psi =\Psi _0 + \psi$ for the co-normal derivative $T^+u$, where $\Phi _0$ and $\Psi _0$ are considered known while the triple $(u, \psi , \varphi )\in H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ is to be found.

7.1 BDIE system (M11)

This system is obtained by considering the third Green identity (4.1) in $\Omega$, the restriction of its trace (4.2) on $\partial \Omega _D$, and the restriction of its co-normal derivative (4.3) on $\partial \Omega _N$, with respect to the unknowns u, $\psi$, and $\varphi$:

$$\begin{aligned} \begin{aligned} u + \mathcal {R}_ku - V \psi + W \varphi&=F_0, \qquad \text {in}\ \ \ \Omega , \\ \gamma ^+ \mathcal {R}_ku - \mathcal {V}\psi + \mathcal {W}\varphi&= \gamma ^+F_0 - \varphi _0, \quad \text {on}\ \ \ \partial \Omega _D, \\ T^+ \mathcal {R}_ku - \mathcal {W}\,'\psi + \mathcal {L}^+\varphi&= T^+F_0 - \psi _0, \quad \text {on}\ \ \ \partial \Omega _N, \end{aligned} \end{aligned}$$

(M11)

where

$$\begin{aligned} F_0= \mathcal {P}f + V \Psi _0 - W \Phi _0 \quad \quad \text{ in } \ \ \Omega . \end{aligned}$$

(7.2)

The BDIE system (M11) can be rewritten in matrix form as

$$\begin{aligned} \mathcal {M}_k^{11} \mathscr {U}=\mathcal {F}^{11}, \end{aligned}$$

(7.3)

where $\mathscr {U}=(u, \psi , \varphi )^t \in H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ and

$$\begin{aligned} \mathcal {M}_k^{11}= \left[ \begin{array}{lll} \ \ I + \mathcal {R}_k \quad \quad \quad \ -V \quad \quad \quad W \\ r_{_{\partial \Omega _D}} \gamma ^+ \mathcal {R}_k \quad \ -r_{_{\partial \Omega _D}}\mathcal {V} \ \ \quad r_{_{\partial \Omega _D}}\mathcal {W}\\ r_{_{\partial \Omega _N}}T^+\mathcal {R}_k\quad -r_{_{\partial \Omega _N}}\mathcal {W}\,' \quad \ \ r_{_{\partial \Omega _N}}\mathcal {L}^+ \end{array} \right] , \quad \mathcal {F}^{11}= \left[ \begin{array}{lll}\quad \quad \quad F_0 \\ \ r_{_{\partial \Omega _D}}\gamma ^+F_0 - \varphi _0\\ r_{_{\partial \Omega _N}}T^+F_0 - \psi _0 \end{array} \right] . \end{aligned}$$

Due to Corollaries 3.4 and 3.14, we get $F_0 \in H^{1,0}(\Omega ;A_k)$. Then we have $\mathcal {F}^{11} \in H^{1,0}(\Omega ;A_k) \times H^{\frac{1}{2}}(\partial \Omega _D) \times H^{-\frac{1}{2}}(\partial \Omega _N)$ and the operators

$$\begin{aligned} \mathcal {M}_k^{11}&: H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \nonumber \longrightarrow H^1(\Omega ) \times H^{\frac{1}{2}}(\partial \Omega _D) \times H^{-\frac{1}{2}}(\partial \Omega _N), \end{aligned}$$

(7.4)

$$\begin{aligned} \mathcal {M}_k^{11}&: H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \nonumber \longrightarrow H^{1,0}(\Omega ;A_k) \times H^{\frac{1}{2}}(\partial \Omega _D) \times H^{-\frac{1}{2}}(\partial \Omega _N) \end{aligned}$$

(7.5)

are bounded.

Taking into account Lemma 4.3, we prove the following Remark in the same way as [6, Remark 5.1].

Remark 7.1

Let $r_0>\textrm{diam}(\Omega )$. $\mathcal {F}^{11}=\textbf{0}$ if and only if $(f, \Phi _0, \Psi _0)=\textbf{0}$.

7.2 BDIE system (M12)

By taking the third Green identity (4.1) in $\Omega$ and its trace (4.2) on the whole boundary $\partial \Omega$, we arrive at the system (M12):

$$\begin{aligned} \begin{aligned} u + \mathcal {R}_ku - V \psi + W \varphi =F_0 \qquad \text {in}\ \ \ \Omega ,\\ \dfrac{1}{2}\varphi + \gamma ^+ \mathcal {R}_ku - \mathcal {V}\psi + \mathcal {W}\varphi&= \gamma ^+F_0 - \Phi _0, \quad \text {on}\ \ \ \partial \Omega , \end{aligned} \end{aligned}$$

(M12)

where $F_0$ is given by the relation (7.2). System (M12) can be rewritten in matrix form as

$$\begin{aligned} \mathcal {M}_k^{12}\mathscr {U}=\mathcal {F}^{12}, \end{aligned}$$

(7.6)

where $\mathscr {U}=(u, \psi , \varphi )^t \in H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ and

$$\begin{aligned} \mathcal {M}_k^{12}= \left[ \begin{array}{ll} I + \mathcal {R}_k \qquad -V \quad W \\ \ \gamma ^+ \mathcal {R}_k \quad -\mathcal {V} \quad \frac{1}{2}I+ \mathcal {W} \end{array} \right] , \qquad \mathcal {F}^{12}= \left[ \begin{array}{ll}\quad \quad F_0 \\ \ \gamma ^+F_0 - \Phi _0 \end{array} \right] . \end{aligned}$$

Note that $\mathcal {F}^{12} \in H^{1,0}(\Omega ;A_k) \times H^{\frac{1}{2}}(\partial \Omega )$. Due to the mapping properties of the operators involved (see Corollaries 3.4 and 3.14, Theorem 3.13 and [13, Theorem 1]), we see that the operators

$$\begin{aligned} \mathcal {M}_k^{12}&: H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^1(\Omega ) \times H^{\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

(7.7)

$$\begin{aligned} \mathcal {M}_k^{12}&: H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^{1,0}(\Omega ;A_k) \times H^{\frac{1}{2}}(\partial \Omega ) \end{aligned}$$

(7.8)

are bounded.

Remark 7.2

Let $\Psi _0 \in H^{-\frac{1}{2}}(\partial \Omega )$ with $r_0 > \textrm{diam}(\Omega )$. Then, $\mathcal {F}^{12}=\textbf{0}$ if and only if $(f, \Phi _0, \Psi _0)=\textbf{0}$.

Indeed, the latter obviously implies the former. Conversely, let $\mathcal {F}^{12}=(F_0, \gamma ^+F_0-\Phi _0)=\textbf{0}$. From $F_0=0$, we get $f=0$ and $V\Psi _0 - W\Phi _0=0$ in $\Omega$. Again from $\gamma ^+F_0-\Phi _0=0$, we get $\Phi _0=0$ on $\partial \Omega$. Hence, we obtain $V\Psi _0=0$ in $\Omega$, and the result follows from Lemma 4.2 (i).

7.3 BDIE system (M21)

We obtain this system by using the third Green identity (4.1) on $\Omega$ and its co-normal derivative (4.3) on the whole boundary $\partial \Omega$:

$$\begin{aligned} \begin{aligned} u + \mathcal {R}_ku - V \psi + W \varphi&= F_0 \quad \text {in} \ \ \Omega , \\ \frac{1}{2}\psi + T^+ \mathcal {R}_ku - \mathcal {W}\,'\psi + \mathcal {L}^+ \varphi&=T^+ F_0 - \Psi _0 \quad \text {on} \ \ \partial \Omega , \end{aligned} \end{aligned}$$

(M21)

where $F_0$ is given by (7.2). We rewrite the system (M21) in matrix form as

$$\mathcal {M}_k^{21}\mathscr {U}=\mathcal {F}^{21},$$

where $\mathscr {U}=(u, \psi , \varphi )^t \in H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ and

$$\begin{aligned} \mathcal {M}_k^{21}= \left[ \begin{array}{ll} I + \mathcal {R}_k \qquad -V \qquad \qquad W \\ \ T^+ \mathcal {R}_k \qquad \frac{1}{2}I-\mathcal {W}\,' \qquad \mathcal {L}^+ \end{array} \right] , \qquad \mathcal {F}^{21}= \left[ \begin{array}{ll}\quad \quad F_0 \\ \ T^+F_0 - \Psi _0 \end{array} \right] . \end{aligned}$$

Here, $\mathcal {F}^{21} \in H^{1,0}(\Omega ;A_k) \times H^{-\frac{1}{2}}(\partial \Omega )$. Due to the mapping properties of the operators involved in $\mathcal {M}_k^{21}$, the following operators are bounded.

$$\begin{aligned} \mathcal {M}_k^{21}: H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

(7.9)

$$\begin{aligned} \mathcal {M}_k^{21}: H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^{1,0}(\Omega ;A_k) \times H^{-\frac{1}{2}}(\partial \Omega ). \end{aligned}$$

(7.10)

Remark 7.3

Let $r_0>0$. $\mathcal {F}^{21}=\textbf{0}$ if and only if $(f, \Phi _0, \Psi _0)=\textbf{0}$.

We prove this remark in the same way as Remark 7.2.

7.4 BDIE system (M22)

Here, we use the third Green identity (4.1) in $\Omega$, the restriction of its trace (4.2) on $\partial \Omega _N$ and the restriction of its co-normal derivative (4.3) on $\partial \Omega _D$ to get the system (M22),

$$\begin{aligned} \begin{aligned} u + \mathcal {R}_ku - V \psi + W \varphi&=F_0 \qquad \text {in}\ \ \ \Omega , \\ \dfrac{1}{2}\psi + T^+ \mathcal {R}_ku - \mathcal {W}\,'\psi + \mathcal {L}^+\varphi&= T^+F_0 - \Psi _0 \quad \text {on}\ \ \ \partial \Omega _D, \\ \dfrac{1}{2}\varphi + \gamma ^+ \mathcal {R}_ku - \mathcal {V}\psi + \mathcal {W}\varphi&= \gamma ^+F_0 - \Phi _0 \quad \text {on}\ \ \ \partial \Omega _N, \end{aligned} \end{aligned}$$

(M22)

where $F_0$ is given by (7.2). Let us write the system (M22) in matrix form as

$$\mathcal {M}_k^{22}\mathscr {U}=\mathcal {F}^{22},$$

where $\mathscr {U}=(u, \psi , \varphi )^t \in H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$, and

$$\begin{aligned} \mathcal {M}_k^{22}=&\left[ \begin{array}{lll} \ \ I + \mathcal {R}_k \qquad \qquad \quad \ -V \qquad \qquad \quad W \\ r_{_{\partial \Omega _D}}T^+\mathcal {R}_k\quad r_{_{\partial \Omega _D}}\left( \frac{1}{2}I-\mathcal {W}\,'\right) \quad r_{_{\partial \Omega _D}}\mathcal {L}^+ \\ r_{_{\partial \Omega _N}}\gamma ^+ \mathcal {R}_k \quad -r_{_{\partial \Omega _N}}\mathcal {V} \ \ \quad r_{_{\partial \Omega _N}}\left( \frac{1}{2}I +\mathcal {W}\right) \end{array} \right] , \\ \mathcal {F}^{22}=&\left[ \begin{array}{lll}\quad F_0 \\ \ r_{_{\partial \Omega _D}}(T^+F_0 - \Psi _0) \\ r_{_{\partial \Omega _N}} (\gamma ^+F_0 - \Phi _0) \end{array} \right] . \end{aligned}$$

From the mapping properties of the operators involved, $\mathcal {F}^{22}\in H^{1,0}(\Omega ;A_k) \times H^{-\frac{1}{2}}(\partial \Omega _D) \times H^{\frac{1}{2}}(\partial \Omega _N)$ and the following operators are bounded.

$$\begin{aligned} \begin{aligned} \mathcal {M}_k^{22}&: H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega _D) \times H^{\frac{1}{2}}(\partial \Omega _N), \end{aligned} \end{aligned}$$

(7.11)

$$\begin{aligned} \mathcal {M}_k^{22}&: H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)\nonumber \longrightarrow H^{1,0}(\Omega ;A_k) \times H^{-\frac{1}{2}}(\partial \Omega _D) \times H^{\frac{1}{2}}(\partial \Omega _N). \end{aligned}$$

(7.12)

Taking into account Lemma 4.3, we prove the following remark in the same way as [6, Remark 5.11].

Remark 7.4

Let $r_0>\textrm{diam}(\Omega )$. $\mathcal {F}^{22}=\textbf{0}$ if and only if $(f, \Phi _0, \Psi _0)=\textbf{0}$.

8 Equivalence, Fredholm properties, and invertibility for BDIE operators of the mixed BVP

Let us prove that the mixed BVP (7.1) is equivalent to the BDIE systems (M11), (M12), (M21), and (M22).

Theorem 8.1

Let $\Phi _0 \in H^{\frac{1}{2}}(\partial \Omega )$ and $\Psi _0 \in H^{-\frac{1}{2}}(\partial \Omega )$ be some extensions of $\varphi _0 \in H^{\frac{1}{2}}(\partial \Omega _D)$ and $\psi _0 \in H^{-\frac{1}{2}}(\partial \Omega _N)$, respectively, and let $f \in L^2(\Omega )$.

• (i) If some $u \in H^1(\Omega )$ solves the mixed BVP (7.1), then the triple $(u, \psi , \varphi )^t \in H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$, where

  $$\begin{aligned} \psi =T^+u - \Psi _0, \quad \quad \ \ \varphi =\gamma ^+ u - \Phi _0 \quad \quad \text {on}\ \ \partial \Omega , \end{aligned}$$

  (8.1)

  solves the BDIE systems (M11), (M12), (M21) and (M22).

• (ii) Let $r_0 > \textrm{diam} (\Omega )$. If a triple $(u, \psi , \varphi )^t \in H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ solves one of the BDIE systems (M11) or (M12) or (M22), then u solves BVP (7.1), and relations (8.1) hold.

• (iii) Let $r_0>0$. If a triple $(u, \psi , \varphi )^t \in H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ solves the BDIE system (M21), then u solves BVP (7.1), and relations (8.1) hold.

Proof

To prove (i), we let $u \in H^1(\Omega )$ be a solution to BVP (7.1). Then, for $\psi$ and $\varphi$ defined by (8.1), we get $\psi \in \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D)$ and $\varphi \in \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$. Recalling how the four BDIE systems were constructed, the result immediately follows from relations (4.1)–(4.3).

To prove (ii) and (iii), let us first assume that a triple $(u, \psi , \varphi )^t \in H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ solves either the BDIE system (M11) or (M12) or (M21) or (M22). The first equation of each system and Lemma 4.1 with $\Psi =\psi + \Psi _0$ and $\Phi =\varphi + \Phi _0$ imply that u solves the PDE $A_ku=f$ on $\Omega$ and the relation

$$\begin{aligned} V\Psi ^* - W \Phi ^* =0 \quad \quad \text {in}\ \ \Omega \end{aligned}$$

(8.2)

holds for

$$\begin{aligned} \Psi ^*=\Psi _0 + \psi - T^+u \quad \text {and} \quad \Phi ^*=\Phi _0 + \varphi - \gamma ^+ u. \end{aligned}$$

(8.3)

Whenever in the remaining proof we take the trace or co-normal derivative of the first equation of each system, we make use of relations (3.8)–(3.10) and the last equation in (3.3).

Proof for (M11). Let a triple $(u, \psi , \varphi )^t \in H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ solve the BDIE system (M11). Taking the trace of the first equation in (M11) on $\partial \Omega _D$ and subtracting the second equation from it, we obtain

$$\begin{aligned} \gamma ^+ u=\varphi _0 \quad \text {on}\ \ \partial \Omega _D, \end{aligned}$$

(8.4)

i.e., u satisfies the Dirichlet condition in (7.1). We now take the co-normal derivative of the first equation in (M11) on $\partial \Omega _N$ and subtract the third equation from it to get

$$\begin{aligned} T^+u=\psi _0 \quad \text {on}\ \ \partial \Omega _N, \end{aligned}$$

(8.5)

i.e., u satisfies the Neumann condition in (7.1). Taking into account that $\varphi =0,\ \Phi _0=\varphi _0$ on $\partial \Omega _D$ and $\psi =0,\ \Psi _0=\psi _0$ on $\partial \Omega _N$, (8.4) and (8.5) imply that the first equation in (8.1) is satisfied on $\partial \Omega _N$ and the second equation in (8.1) on $\partial \Omega _D$. From this and relation (8.3), we have $\Psi ^* \in \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D),\ \Phi ^* \in \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$. Since relation (8.2) holds and $r_0 > \textrm{diam} (\partial \Omega _D)$, from Lemma 4.3, we get $\Psi ^*=\Phi ^*=0$, which completes the proof of conditions (8.1).

Proof for (M12). Now, let a triple $(u, \psi , \varphi )^t \in H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ solve BDIE system (M12). Taking trace of the first equation in (M12) on $\partial \Omega$ and subtracting the second one from it, we obtain

$$\begin{aligned} \gamma ^+ u= \Phi _0 + \varphi \quad \ \ \text {on} \ \ \partial \Omega , \end{aligned}$$

(8.6)

which means that the second equation in (8.1) holds. Since $\varphi =0$, $\Phi _0=\varphi _0$ on $\partial \Omega _D$, we see that the Dirichlet condition in (7.1) is satisfied.

Due to (8.6), the second term in (8.2) vanishes and by Lemma 4.2(i), we obtain

$$\begin{aligned} \Psi _0 + \psi - T^+ u =0 \quad \ \text {on} \ \ \partial \Omega , \end{aligned}$$

(8.7)

which shows that the first equation of (8.1) is satisfied as well. Since $\psi =0$, $\Psi _0=\psi _0$ on $\partial \Omega _N$, (8.7) implies that u satisfies the Neumann boundary condition in (7.1).

Proof for (M22). Now, let a triple $(u, \psi , \varphi )^t \in H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ solve the BDIE system (M22). Taking the co-normal derivative of the first equation in (M22) on $\partial \Omega _D$ and subtracting it from the second equation, we obtain

$$\begin{aligned} \psi = T^+u- \Psi _0 \quad \text {on}\ \ \partial \Omega _D. \end{aligned}$$

(8.8)

Taking the trace of the first equation in (M22) on $\partial \Omega _N$ and subtracting it from the third equation yields

$$\begin{aligned} \varphi = \gamma ^+ u - \Phi _0 \quad \text {on}\ \ \partial \Omega _N. \end{aligned}$$

(8.9)

Equations 8.8 and 8.9 imply that the first equation in (8.1) is satisfied on $\partial \Omega _D$ and the second one on $\partial \Omega _N$. Due to (8.8) and (8.9), we have $\Psi ^* \in \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _N),\ \Phi ^* \in \widetilde{H}^{\frac{1}{2}}(\partial \Omega _D)$ in (8.2) and (8.3). Then, Lemma (4.3) with $\Gamma _1=\partial \Omega _N$ and $\Gamma _2=\partial \Omega _D$ implies that $\Psi ^*=\Phi ^*=0,$ which completes the proof of conditions (8.1) on the whole boundary $\partial \Omega$. Taking into account that $\varphi =0, \Phi _0=\varphi _0$ on $\partial \Omega _D$ and $\psi =0, \Psi _0=\psi _0$ on $\partial \Omega _N$, (8.1) implies the boundary conditions in the mixed BVP (7.1).

Proof for (M21). Let a triple $(u, \psi , \varphi )^t \in H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ solve the BDIE system (M21). We take the co-normal derivative of the first equation in (M21) on $\partial \Omega$ and subtract the second equation from it to obtain

$$\begin{aligned} \psi + \Psi _0 - T^+u=0 \quad \text {on} \ \ \partial \Omega , \end{aligned}$$

(8.10)

which is the first equation of (8.1). Since $\psi =0$, $\Psi _0=\psi _0$ on $\partial \Omega _N$, we see that u satisfies the Neumann condition in (7.1).

Due to (8.10), the first term in (8.2) vanishes and, by Lemma 4.2(ii), we obtain

$$\begin{aligned} \Phi _0 + \varphi - \gamma ^+ u=0 \quad \text {on} \ \ \partial \Omega , \end{aligned}$$

(8.11)

which means that the second condition in (8.1) holds as well. Since $\varphi =0$, $\Phi _0=\varphi _0$ on $\partial \Omega _D$, from (8.11), we see that u satisfies the Dirichlet boundary condition in (7.1).

Corollary 8.2

Let $\Phi _0 \in H^{\frac{1}{2}}(\partial \Omega )$ and $\Psi _0 \in H^{-\frac{1}{2}}(\partial \Omega )$ be some extensions of $\varphi _0 \in H^{\frac{1}{2}}(\partial \Omega _D)$ and $\psi _0 \in H^{-\frac{1}{2}}(\partial \Omega _N)$, respectively, and let $f \in L^2(\Omega )$.

• (i) Let $r_0 > \textrm{diam} (\Omega )$. If a triple $(u, \psi , \varphi )^t \in H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ solves the BDIE system (M11) or (M12) or (M22), then it solves all the other three BDIE systems.

• (ii) Let $r_0>0$. If a triple $(u, \psi , \varphi )^t \in H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ solves the BDIE system (M21), then it solves (M11), (M12) and (M22).

Corollary 8.3

• (i) Let $r_0 > \textrm{diam}(\Omega )$. The homogeneous counterpart of BDIE system (M11) or (M12) or (M22) has a non-trivial solution in $H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ if and only if the homogeneous counterpart of the mixed problem (7.1) has a non-trivial solution in $H^1(\Omega )$.

• (ii) Let $r_0>0$. The homogeneous counterpart of BDIE system (M21) has a non-trivial solution in $H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ if and only if the homogeneous counterpart of the mixed problem (7.1) has a non-trivial solution in $H^1(\Omega )$.

Now, we prove the Fredholm property of the corresponding operators of the BDIE system (M11), (M12), and (M21).

Theorem 8.4

• (i) If $r_0 > \textrm{diam} (\Omega )$, operators (7.4) and (7.7) are Fredholm with index zero.

• (ii) If $r_0 > 0$, operator (7.9) is Fredholm with index zero.

Proof

Here, we follow the arguments similar to the ones used in [6, for 3D case].

Operator (7.4). To prove the Fredholm property of operator (7.4), let us consider the operator

$$\mathcal {M}^{11}_*:= \left[ \begin{array}{lll} I \quad \quad \ \ -V \ \quad \quad W \\ 0 \quad \ -r_{_{\partial \Omega _D}}\mathcal {V} \ \quad \quad \ 0\\ 0 \quad \quad \quad \quad 0 \quad \quad \ r_{_{\partial \Omega _N}}\widehat{\mathcal {L}} \end{array} \right] ,$$

where $\widehat{\mathcal {L}}$ is given by (3.14).

The operator $\mathcal {M}^{11}_*$ is an upper triangular matrix operator with the following scalar diagonal operators,

$$\begin{aligned} I&: H^1(\Omega ) \longrightarrow H^1(\Omega ), \\ r_{_{\partial \Omega _D}} \mathcal {V}&: \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \longrightarrow H^{\frac{1}{2}}(\partial \Omega _D), \\ r_{_{\partial \Omega _N}} \widehat{\mathcal {L}}&: \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^{-\frac{1}{2}}(\partial \Omega _N), \end{aligned}$$

that are invertible (due to Theorems 3.10 and 3.11(i) for the second and third operators). Along with the mapping properties of the operators V and W (see Theorem 3.3), the operator

$$\mathcal {M}^{11}_*: H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^1(\Omega ) \times H^{\frac{1}{2}}(\partial \Omega _D) \times H^{-\frac{1}{2}}(\partial \Omega _N)$$

is invertible. The operator

$$\mathcal {M}_k^{11}-\mathcal {M}^{11}_*: H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^1(\Omega ) \times H^{\frac{1}{2}}(\partial \Omega _D) \times H^{-\frac{1}{2}}(\partial \Omega _N),$$

where

$$\mathcal {M}_k^{11}- \mathcal {M}^{11}_*:= \left[ \begin{array}{lll} \quad \mathcal {R}_k \qquad \qquad \qquad \quad 0 \ \qquad \qquad \qquad \quad \,\,\,\, 0 \\ r_{_{\partial \Omega _D}} \gamma ^+ \mathcal {R}_k \qquad \qquad \quad 0 \qquad \qquad \qquad r_{_{\partial \Omega _D}}\mathcal {W}\\ r_{_{\partial \Omega _N}}T^+\mathcal {R}_k \quad \quad -r_{_{\partial \Omega _N}}\mathcal {W}\,' \quad \quad \ r_{_{\partial \Omega _N}}\left( \mathcal {L}^+ - \widehat{\mathcal {L}}\right) \end{array} \right] .$$

is compact due to Corollaries 3.15 and 3.17 as well as Theorems 3.6 and 3.11(ii). Hence, (7.4) is a Fredholm operator with zero index.

Operator (7.7). Let us denote

$$\mathcal {M}^{12}_*:= \left[ \begin{array}{ll} I \qquad -V \qquad W \\ 0 \qquad -\mathcal {V} \qquad \frac{1}{2}I \end{array} \right] .$$

Then,

$$\mathcal {M}^{12}_*: H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^1(\Omega ) \times H^{\frac{1}{2}}(\partial \Omega )$$

is bounded. To show the invertibility of $\mathcal {M}^{12}_*$, taking into account Theorem 3.10, we follow the proof for 3D case in [6]. Consider the equation

$$\begin{aligned} \mathcal {M}_*^{12}\mathscr {U}=\widetilde{F} \end{aligned}$$

(8.12)

with an unknown vector $\mathscr {U}=(u, \psi , \varphi )^t \in H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ and a given vector $\widetilde{F}:=(\widetilde{F}_1, \widetilde{F}_2)^t \in H^1(\Omega ) \times H^{\frac{1}{2}}(\partial \Omega )$. Rewrite (7.9) componentwise as

$$\begin{aligned} u- V\psi + W \varphi&= \widetilde{F}_1 \quad \text {in} \ \ \Omega , \end{aligned}$$

(8.13)

$$\begin{aligned} \dfrac{1}{2}\varphi - \mathcal {V}\psi&=\widetilde{F}_2 \quad \text {on} \ \ \partial \Omega . \end{aligned}$$

(8.14)

The restriction of (8.14) on $\partial \Omega _D$ gives

$$\begin{aligned} -r_{_{\partial \Omega _D}}\mathcal {V}\psi =r_{\partial \Omega _D} \widetilde{F}_2. \end{aligned}$$

(8.15)

Due to Theorem 3.10, (8.15) is uniquely solvable, i.e., for arbitrary $\widetilde{F}_2 \in H^{\frac{1}{2}}(\partial \Omega )$ there exists a unique $\psi \in \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D)$ satisfying (8.15). Moreover,

$$\begin{aligned} \left[ \mathcal {V}\psi + \widetilde{F}_2\right] \in \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N). \end{aligned}$$

(8.16)

Then, (8.14) along with (8.16) yields that $\varphi$ is defined also uniquely as

$$\varphi =2\left[ \mathcal {V}\psi + \widetilde{F}_2 \right] \in \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N).$$

Hence, (8.14) with arbitrary $\widetilde{F}_2 \in \widetilde{H}^{\frac{1}{2}}(\partial \Omega )$ defines $\varphi \in \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ and $\psi \in \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D)$ uniquely. Since $V\psi, W\varphi \in H^1(\Omega )$, from (8.13) we obtain that

$$u= V\psi - W \varphi + \widetilde{F}_1 \quad \text{ in } \ \ \Omega ,$$

showing that the function $u\in H^1(\Omega )$ is also defined uniquely. The above arguments show that operator $\mathcal {M}^{12}_*$ is invertible.

Due to Corollaries 3.5 and 3.15, the operator

$$\mathcal {M}_k^{12} - \mathcal {M}^{12}_*: H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^1(\Omega ) \times H^{\frac{1}{2}}(\partial \Omega )$$

where

$$\mathcal {M}_k^{12} - \mathcal {M}^{12}_*:= \left[ \begin{array}{ll} \qquad \mathcal {R}_k \qquad \, 0 \qquad 0 \\ \ \ \gamma ^+ \mathcal {R}_k \qquad 0 \qquad \mathcal {W} \end{array} \right] ,$$

is compact. Then, operator (7.7) is Fredholm of index zero.

Operator (7.9). The proof for operator (7.9) follows by the arguments similar to those in the proof for operator (7.7). Let

$$\mathcal {M}^{21}_*:= \left[ \begin{array}{ll} I \qquad -V \qquad W \\ 0 \quad \qquad \frac{1}{2}I \qquad \widehat{\mathcal {L}} \end{array} \right] .$$

Then,

$$\mathcal {M}^{21}_*: H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega )$$

is bounded. Since the operators $I: H^1(\Omega )\rightarrow H^1(\Omega )$ and $\widehat{\mathcal {L}}: \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)\rightarrow H^{-\frac{1}{2}}(\partial \Omega )$ are invertible, using similar arguments as in the proof of the operator (7.7), we can show that $\mathcal {M}_*^{21}$ is invertible.

Due to the mapping properties of the operators involved, the operator

$$\mathcal {M}_k^{21} - \mathcal {A}^{21}_*: H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega ),$$

where

$$\mathcal {M}_k^{21} - \mathcal {M}^{21}_*:= \left[ \begin{array}{ll} \qquad \mathcal {R}_k \qquad \qquad 0 \qquad \quad \qquad 0 \\ \ \ T^+ \mathcal {R}_k \qquad -\mathcal {W}\,' \qquad \left( \mathcal {L}^+ - \widehat{\mathcal {L}}\right) \end{array} \right]$$

is compact implying that $\mathcal {M}_k^{21}$ is Fredholm operator of index zero.$\square$

Let us consider the particular cases of operators (7.4), (7.5), (7.7), (7.8), (7.9), and (7.10), for $k=0$, that is,

$$\begin{aligned} \begin{aligned} \mathcal {M}_0^{11}&: H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^1(\Omega ) \times H^{\frac{1}{2}}(\partial \Omega _D) \times H^{-\frac{1}{2}}(\partial \Omega _N), \end{aligned} \end{aligned}$$

(8.17)

$$\begin{aligned} \mathcal {M}_0^{11}&: H^{1,0}(\Omega ;A) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^{1,0}(\Omega ;A) \times H^{\frac{1}{2}}(\partial \Omega _D) \times H^{-\frac{1}{2}}(\partial \Omega _N) , \end{aligned}$$

(8.18)

$$\begin{aligned} \mathcal {M}_0^{12}&: H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^1(\Omega ) \times H^{\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

(8.19)

$$\begin{aligned} \mathcal {M}_0^{12}&: H^{1,0}(\Omega ;A) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^{1,0}(\Omega ;A) \times H^{\frac{1}{2}}(\partial \Omega ) ,\end{aligned}$$

(8.20)

$$\begin{aligned} \mathcal {M}_0^{21}&: H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^1(\Omega ) \times H^{-\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

(8.21)

$$\begin{aligned} \mathcal {M}_0^{21}&: H^{1,0}(\Omega ;A) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^{1,0}(\Omega ;A) \times H^{-\frac{1}{2}}(\partial \Omega ). \end{aligned}$$

(8.22)

where

$$\begin{aligned}&\mathcal {M}^{11}_0=\left[ \begin{array}{lll} \ \ I + \mathcal {R} \quad \quad \quad \ -V \quad \quad \quad W \\ r_{_{\partial \Omega _D}} \gamma ^+ \mathcal {R} \quad \ -r_{_{\partial \Omega _D}}\mathcal {V} \ \ \quad r_{_{\partial \Omega _D}}\mathcal {W}\\ r_{_{\partial \Omega _N}}T^+\mathcal {R}\quad -r_{_{\partial \Omega _N}}\mathcal {W}\,' \quad \ \ r_{_{\partial \Omega _N}}\mathcal {L}^+ \end{array}\ \right] ,\\&\mathcal {M}_0^{12}= \left[ \begin{array}{lll} I + \mathcal {R} &{} -V &{} W \\ \ \gamma ^+ \mathcal {R} &{} -\mathcal {V} &{} \frac{1}{2}I+ \mathcal {W} \end{array} \right] ,\quad \mathcal {M}_0^{21}= \left[ \begin{array}{lll} I + \mathcal {R} &{} -V &{} W \\ \ T^+ \mathcal {R} &{} \frac{1}{2}I-\mathcal {W}\,' &{} \mathcal {L}^+ \end{array} \right] . \end{aligned}$$

Theorem 8.5

• (i) If $r_0>\textrm{diam}(\Omega )$, then operators (8.17), (8.18), (8.19), and (8.20) are invertible.

• (ii) If $r_0>0$, then operators (8.21) and (8.22) are invertible.

Proof

This theorem for $r_0=1$ was proved in [12, Theorem 3.25]. Here, we update the proof for arbitrary $r_0>0$ similar to Theorem 6.5 for the BDIE system of the Dirichlet problem.

It is well known that the homogeneous mixed problem (7.1) with $k=0$, that is, with $A_k=A$, where the operator A is given by (2.1) and $0<a_0< a(x)< a_1< \infty$, has only the trivial solution in $H^{1,0}(\Omega ;A)$ and $H^{1}(\Omega )$. This can be obtained, e.g., from the first Green identity (2.7). Then, the equivalence Theorem 8.1 implies that all operators (8.17)–(8.22) are injective. By Theorem 8.4, operators (8.17), (8.19), and (8.21) are Fredholm with zero index. Then, the injectivity of operators (8.17), (8.19), and (8.21) implies their invertibility (see, e.g., [18, Theorem 2.27]).

To prove the invertibility of operator (8.18), we remark that for any $\mathcal {F}^{11}\in H^{1,0}(\Omega ;A)\times H^{\frac{1}{2}}(\partial \Omega _D) \times H^{-\frac{1}{2}}(\partial \Omega _N)$, a solution of the equation $\mathcal {M}^{11}_0\mathcal U=\mathcal {F}^{11}$ can be written as $\mathcal U=(\mathcal {M}^{11}_0)^{-1}\mathcal {F}^{11}$, where $(\mathcal {M}^{11}_0)^{-1}:H^{1}(\Omega ) \times H^{\frac{1}{2}}(\partial \Omega _D) \times H^{-\frac{1}{2}}(\partial \Omega _N)\rightarrow H^1(\Omega ) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ is the continuous inverse to operator (8.17). But due to Lemma 4.1, the first equation of system (M11) with $k=0$ implies that $\mathcal U=(\mathcal {M}^{11}_0)^{-1}\mathcal {F}^{11}\in H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$, and moreover, the operator $(\mathcal {M}^{11}_0)^{-1}:H^{1,0}(\Omega ;A) \times H^{\frac{1}{2}}(\partial \Omega _D) \times H^{-\frac{1}{2}}(\partial \Omega _N)\rightarrow H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N)$ is continuous, which implies the invertibility of operator (8.18).

The invertibility of operators (8.20) and (8.22) is proved in a similar fashion. $\square$

Now, we are in the position to prove an analog of Theorem 8.4 for operators (7.5), (7.8), and (7.10).

Theorem 8.6

• (i) If $r_0 > \textrm{diam} (\Omega )$, operators (7.5) and (7.8) are Fredholm with index zero.

• (ii) If $r_0 > 0$, operator (7.10) is Fredholm with index zero.

Proof

By Theorem 8.5, we see that operators (8.18), (8.20), and (8.22) are invertible. Due to Corollaries 3.16, the operators

$$\begin{aligned} \mathcal {M}_k^{11}-\mathcal {M}_0^{11}&: H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^{1,0}(\Omega ;A_k) \times H^{\frac{1}{2}}(\partial \Omega _D) \times H^{-\frac{1}{2}}(\partial \Omega _N) , \\ \mathcal {M}_k^{12}-\mathcal {M}_0^{12}&: H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^{1,0}(\Omega ;A_k) \times H^{\frac{1}{2}}(\partial \Omega ), \\ \mathcal {M}_k^{21}-\mathcal {M}_0^{21}&: H^{1,0}(\Omega ;A_k) \times \widetilde{H}^{-\frac{1}{2}}(\partial \Omega _D) \times \widetilde{H}^{\frac{1}{2}}(\partial \Omega _N) \longrightarrow H^{1,0}(\Omega ;A_k) \times H^{-\frac{1}{2}}(\partial \Omega ). \end{aligned}$$

where

$$\begin{aligned}&\mathcal {M}^{11}_k-\mathcal {M}^{11}_0=\left[ \begin{array}{lll} \mathcal {R}_k-\mathcal {R} &{} 0 &{} 0 \\ r_{_{\partial \Omega _D}} \gamma ^+( \mathcal {R}_k-\mathcal {R}) &{} 0 &{} 0 \\ r_{_{\partial \Omega _N}}T^+(\mathcal {R}_k-\mathcal {R}) &{} 0 &{} 0 \end{array}\ \right] ,\\&\mathcal {M}_k^{12}-\mathcal {M}_0^{12}= \left[ \begin{array}{lll} \mathcal {R}_k-\mathcal {R} &{} 0 &{} 0 \\ \gamma ^+(\mathcal {R}_k-\mathcal {R}) &{} 0 &{} 0 \end{array} \right] ,\quad \mathcal {M}_k^{21}-\mathcal {M}_0^{21}= \left[ \begin{array}{lll} \mathcal {R}_k-\mathcal {R} &{} 0 &{} 0 \\ \ T^+ (\mathcal {R}_k-\mathcal {R}) &{} 0 &{} 0 \end{array} \right] . \end{aligned}$$

are compact, implying that operators (7.5), (7.8), and (7.10) are Fredholm operators with index zero. $\square$

Due to Corollary 8.3 and Theorem 8.4, we obtain the following assertion.

Corollary 8.7

• (i) Let $r_0 > \textrm{diam}(\Omega )$. The homogeneous counterpart of the mixed problem (7.1) has only the trivial solution in $H^1(\Omega )$ if and only if the operators (7.4), (7.5), (7.7), and (7.8) are invertible.

• (ii) Let $r_0>0$. The homogeneous counterpart of the mixed problem (7.1) has only the trivial solution in $H^1(\Omega )$ if and only if the operators (7.9) and (7.10) are invertible.

Remark 8.8

Equivalence, Fredholm properties, and invertibility for BDIE operators (7.11) and (7.12), for $\mathcal {M}_k^{22}$, are not analyzed in Section 8. Note that they can be considered using a different approach similar to [9, Theorem 7.1], [12, Theorem 3.31], cf. also [6, Theorems 5.15, 5.19].