Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain

Eshetu Seid Ahimed

doi:doi:10.11648/j.ijtam.20241002.12

Research Article |

| Peer-Reviewed

Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain

Eshetu Seid Ahimed^*

Published in International Journal of Theoretical and Applied Mathematics (Volume 10, Issue 2)

Received: 26 July 2024 Accepted: 20 August 2024 Published: 30 August 2024

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this paper, the direct segregated Boundary Domain Integral Equations (BDIEs) for the Mixed Boundary Value Problems (MBVPs) for a scalar second order elliptic Partial Differential Equation (PDE) with variable coefficient in unbounded (exterior) 2D domain is considered. Otar Chkadua, Sergey Mikhailov and David Natroshvili formulated both the interior and exterior 3D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order elliptic PDE with a variable coefficients. On the other hand Sergey Mikhailov and Tamirat Temesgen formulated only the interior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. However, in this paper we formulated the exterior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. The aim of this work is to reduce the MBVPs to some direct segregated BDIEs with the use of an appropriate parametrix (Levi function). We examine the characteristics of corresponding parametrix-based integral volume and layer potentials in some weighted Sobolev spaces, as well as the unique solvability of BDIEs and their equivalence to the original MBVPs. This analysis is based on the corresponding properties of the MBVPs in weighted Sobolev spaces that are proved as well.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 10, Issue 2)
DOI	10.11648/j.ijtam.20241002.12
Page(s)	23-32
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Partial Differential Equation, Variable Coefficient, Unbounded Domain, Weighted Sobolev Spaces, Parametrix (Levi Funtion), Single Layer Potential, Mixed Boundary Value Problem, Boundary-Domain Integral Equations

1. Introduction

Mathematical modeling of inhomogeneous media (such as functionally graded materials or materials with damage-induced inhomogeneity) in solid mechanics, electromagnetics, thermo-conductivity, fluid flows through porous media, and other branches of physics and engineering frequently involves PDEs with variable coefficients. When the PDE coefficients are not constant, there are typically no explicit fundamental solutions available, which makes it impossible to solve BVPs for such PDEs numerically. However, for an extensive range of variable-coefficient PDEs, an explicit parametrix (Levi function) linked to a fundamental solution of corresponding frozen coefficient PDEs can be utilized instead. This reduces BVPs for these PDEs in interior domains to BDIE systems for additional numerical solution of the latter. (e.g.,

[3, 5, 15, 16, 17, 20]

The primary objective of this study is to demonstrate the reduction of mixed problems with variable coefficients in exterior domains to certain systems of BDIEs. Additionally, we aim to explore the distinct solvability of BDIEs and their equivalence to the original BVP in the weighted Sobolev spaces. To achieve this, we extend to exterior domains and weighted spaces, using the techniques developed in

[20]

for interior domains and standard Sobolev (Bessel potential) spaces.

The characteristics of the related boundary value problems are crucial to the BDIE analysis. Today, a lot of research has been done on variable-coefficient BVPs in bounded domains, (e.g.,

[7, 9]

). In particular, the analysis of segregated boundary-domain integral equations for variable-coefficient MBVPs 3D unbounded domains can be found in

[4]

. Nonetheless, due to the logarithmic term in the parametrix of the related partial differential equation, the BDIEs in the 2D case show unique characteristics when compared to the higher dimensions. As a result, in order to guarantee the invertibility of the layer potentials and, consequently, the BDIEs unique solvability, we must impose requirements on the function spaces.

2. Preliminaries

Let

Ω = Ω^{+}

be an unbounded open domain in

R^{2}

such that the complement

Ω^{-} : = R^{2} ∖ \overset{⃐}{Ω}

is bounded open domain. Let the boundary

\partial Ω = \partial Ω^{-}

be closed and infinitely smooth curve. The space of infinitely differentiable functions having compact support in

Ω

is denotes by

D (Ω)

and its dual space, the space of distributions, by

D^{'} (Ω)

, while

D (\overset{⃐}{Ω})

is the set of restrictions on

\overset{⃐}{Ω}

of functions from

D (R^{2})

. The spaces

H^{s} (Ω), H^{s} (\partial Ω)

denote the Sobolev (Bessel potential) spaces. We also denote

{\tilde{H}}^{s} (Γ_{1}) = \{g : g \in H^{s} (Γ)

, supp

g \subset \overset{̅}{Γ_{1}}\}, H^{s} (Γ_{1}) = \{r_{Γ_{1}} g : g \in H^{s} (Γ)\}

, where

Γ_{1}

is a proper submanifold of a closed surface

Γ

and

r_{Γ_{1}}

is the restriction operator on

Γ_{1}

. Moreover for

s = - \frac{1}{2}

we define the subspace

H_{* *}^{- \frac{1}{2}} (Γ_{1})

H^{- \frac{1}{2}} (Γ_{1})

H_{* *}^{- \frac{1}{2}} (Γ_{1}) : = \{r_{Γ_{1}} g : g \in H^{- \frac{1}{2}} (Γ) : ⟨ g, 1 ⟩_{Γ} = 0\}

We shall consider the following second order partial differential equation, with variable coefficient

Au (x) : = \sum_{i = 1}^{2} \frac{\partial}{\partial x_{i}} (a (x) \frac{\partial u (x)}{\partial x_{i}}) = f (x) x \in Ω

(1)

where

u

is unknown function;

f (x)

and

a (x) > a_{0} > 0

are given functions in

Ω

. We will further use the weighted Sobolev spaces.

Let

ρ (x) : = {(1 + | x |^{2})}^{1 / 2} \ln (2 + | x |^{2})

For any real

α

, we denote by

L_{2} (ρ^{α}; Ω)

the weighted Lebesgue space (e.g.,

[8]

) consisting of all measurable functions

g (x)

Ω

such that

g ρ^{α} \in L_{2} (Ω)

, i.e.,

∥ g ∥_{L_{2} (ρ^{α}; Ω)} = {[\int_{Ω} {|g (x) ρ^{α} (x)|}^{2} dx]}^{\frac{1}{2}} < \infty

The space

L_{2} (ρ^{α}; Ω)

, equipped with the norm

∥ \cdot ∥_{L_{2} (ρ^{α}; Ω)}

and appropriate inner product, is a Hilbert space. The weighted Sobolev space

H^{1} (Ω)

is defined by

H^{1} (Ω) : = \{g \in L_{2} (ρ^{- 1}; Ω) : \nabla g \in L_{2} (Ω)\}

(2)

and for its norm we have

∥ g ∥_{H^{1} (Ω)}^{2} : = ∥ g ∥_{L_{2} (ρ^{- 1}; Ω)}^{2} + ∥ \nabla g ∥_{L_{2} (Ω)}^{2}

, while

| g |_{H^{1} (Ω)}^{2} : = \sum_{i = 1}^{2} \int_{Ω} {|\frac{\partial g}{\partial x_{i}}|}^{2} dx = ∥ \nabla g ∥_{L_{2} (Ω)}^{2}

is the square of the semi-norm. The space

D (R^{2})

is dense in

H^{1} (R^{2})

, (e.g., Theorem 7.2 in

[1]

). This implies that the dual space of

H^{1} (R^{2})

, denoted by

H^{- 1} (R^{2})

, is a space of distributions. It is possible to show that the space

D (\overset{⃐}{Ω})

is dense in

H^{1} (Ω)

by using the corresponding property of the space

H^{1} (Ω)

. The trace operator

γ^{+}

\partial Ω

defined on functions from

H^{1} (Ω)

, satisfies the usual trace theorems. This allows to define in particular the subspace

H_{0}^{1} (Ω) = \{g \in H^{1} (Ω) : γ^{+} g = 0\}

It can be proved that

D (Ω)

is dense in

H_{0}^{1} (Ω)

and therefore, its dual space is a space of distributions. Let us denote by

{\tilde{H}}^{1} (Ω)

a completion of

D (Ω)

H^{1} (R^{2})

, and

{\tilde{H}}^{- 1} (Ω) : = {[H^{1} (Ω)]}^{'}, H^{- 1} (Ω) : = {[{\tilde{H}}^{1} (Ω)]}^{'}

are the corresponding dual spaces. The inclusion

L_{2} (ρ; Ω) \subset H^{- 1} (Ω)

holds and a distribution

f

in the dual space

{\tilde{H}}^{- 1} (Ω)

has the form

f = \sum_{i = 1}^{2} \frac{\partial g_{i}}{\partial x_{i}} + f_{0}

, where

g_{i} \in L_{2} (R^{2})

and is zero outside

Ω, f_{0} \in L_{2} (ρ; Ω)

, (e.g., Eq. (2.5.129) in

[13]

). This implies that

D (Ω)

is dense in

{\tilde{H}}^{- 1} (Ω)

and

D (R^{2})

is dense in

H^{- 1} (R^{2})

Lemma 1. The space

H^{1} (Ω)

contains constant functions.

Proof. Let

C \in R

then from Definition 2.2 the result follows.

Note That 1. Lemma 1 implies that, the space of real constants,

R

, is a closed subspace of

H^{1} (Ω)

. Hence we can define the quotient space

H^{1} (Ω) / R

, which is a complete normed space, and its norm is given by

∥ u + R ∥_{H^{1} (Ω) / R} = \inf_{c \in R} ∥ u + c ∥_{H^{1} (Ω)}

. The dual space

{(H^{1} (Ω) / R)}^{'}

is identified with

{\tilde{H}}^{- 1} (Ω) ⊥ R

, i.e.,

{(H^{1} (Ω) / R)}^{'} = {\tilde{H}}^{- 1} (Ω) ⊥ R

since they are isometrically isomorphic (e.g., Lemma 2.12(ii) in

[9]

) Similarly,

{({\tilde{H}}^{1} (Ω) / R)}^{'} = H^{- 1} (Ω) ⊥ R .

The following Poincaré-type inequalities hold (e.g., Theorems 1.1 and 1.2 in

[2]

Theorem 1. (i) The semi-norm

| \cdot |_{H^{1} (Ω)}

defined on

H^{1} (Ω) / R

is a norm equivalent to the quotient norm, i.e., there exist positive constants

k_{1}, K_{1}

such that

k_{1} | v |_{H^{1} (Ω)} \leq ∥ v ∥_{H^{1} (Ω) / R} \leq K_{1} | v |_{H^{1} (Ω)}

(ii) Moreover, the semi-norm

| \cdot |_{H^{1} (Ω)}

is a norm on

H_{0}^{1} (Ω)

equivalent to the norm

∥ \cdot ∥_{H^{1} (Ω)}

, i.e., there exist positive constants

k_{2}, K_{2}

such that

k_{2} | v |_{H^{1} (Ω)} \leq ∥ v ∥_{H_{0}^{1} (Ω)} \leq K_{2} | v |_{H^{1} (Ω)}

For

u \in H^{1} (Ω)

and the coefficient

a (x) \in L_{\infty} (Ω), PDE

(1) is well defined in the sense of distribution as

⟨ Au, v ⟩_{Ω} : = - ⟨ a \nabla u, \nabla v ⟩_{Ω} = - E (u, v)

, for any

v \in D (Ω)

, where

E (u, v) : = \int_{Ω} E (u, v) (x) dx, E (u, v) (x) : = \nabla v (x) \cdot a (x) \nabla u (x)

. From here on, unless specified otherwise, we presume that there exist some constants

a_{0}, a_{1}

such that

a \in L_{\infty} (R^{2}) and 0 < a_{0} < a (x) < a_{1} < \infty for x \in R^{2}

(3)

To obtain boundary-domain integral equations, we will also always consider the coefficient

a

such that

a \in C^{1} (R^{2}) and ρ \nabla a \in L_{\infty} (R^{2})

(4)

(e.g.,

[7]

) for

u \in H^{1} (Ω)

, if

u \in H^{1} (Ω^{+})

, then from the trace theorem it follows that,

γ^{+} u \in H^{\frac{1}{2}} (\partial Ω)

, where

γ^{+} = γ_{\partial Ω}^{+}

is the trace operator on

\partial Ω

from the exterior domain

Ω^{+}

For the operator

A

, similar to

[4]

for the three dimensional case, we introduce the space,

H^{1, 0} (Ω; A) : = \{g \in H^{1} (Ω) : Ag \in L_{2} (ρ; Ω)\}

, where the norm is given by its square,

∥ g ∥_{H^{1, 0} (Ω; A)}^{2} : = ∥ g ∥_{H^{1} (Ω)}^{2} + ∥ Ag ∥_{L_{2} (ρ; Ω)}^{2}

. For

u \in H^{1, 0} (Ω; A)

, as in the 3 D case,

[4]

, we define the canonical co-normal derivative

T^{+} u \in H^{- \frac{1}{2}} (\partial Ω)

similar to, for example in Lemma 3.2 of

[6]

and Lemma 4.3 of

[9]

{⟨T^{+} u, ω⟩}_{\partial Ω} : = \int_{Ω} [(γ_{- 1}^{+} ω) Au + E (u, γ_{- 1}^{+} ω)] dx \forall ω \in H^{\frac{1}{2}} (\partial Ω)

where

γ_{- 1}^{+} : H^{\frac{1}{2}} (\partial Ω) \to H^{1} (Ω)

is a bounded right inverse to the trace operator

γ^{+} : H^{1} (Ω) \to H^{\frac{1}{2}} (\partial Ω)

, and

⟨ \cdot, \cdot ⟩_{\partial Ω}

denotes the duality brackets between the spaces

H^{- \frac{1}{2}} (\partial Ω)

and

H^{\frac{1}{2}} (\partial Ω)

which extends the usual

L_{2} (\partial Ω)

scalar product. The operator

T^{+} : H^{1, 0} (Ω; A) \to H^{- \frac{1}{2}} (\partial Ω)

is continuous and gives the continuous extension to

H^{1, 0} (Ω; A)

of the classical co-normal derivative operator

a \frac{\partial}{\partial n}

, where

\frac{\partial}{\partial n} = γ^{+} \nabla \cdot n

and

n = n^{+}

is normal vector on

\partial Ω

directed outward the exterior domain

Ω

. When

a \equiv 1

, we employ for

T^{+}

the notation

T_{Δ}^{+}

, which is the continuous extension on

H^{1, 0} (Ω; Δ)

of the classical normal derivative operator

\partial_{n}

. Similar to the proofs available in Lemma 3.4 of

[6]

(for the spaces

H^{s, t} (Ω; A)

see also

[11]

), one can show that for

u \in H^{1, 0} (Ω; A)

and

v \in H^{1} (Ω)

the first Green identity

{⟨T^{+} u, γ^{+} v⟩}_{\partial Ω} = \int_{Ω} [vAu + E (u, v)] dx \forall v \in H^{1} (Ω)

(5)

holds true. Then, for any chooses of

u, v \in H^{1, 0} (Ω; A)

we obtain the second Green identity,

\int_{Ω} [vAu - uAv] dx = {⟨T^{+} u, γ^{+} v⟩}_{\partial Ω} - {⟨T^{+} v, γ^{+} u⟩}_{\partial Ω}

(6)

Remark 1. If a satisfies condition (3) and the second condition in (4), then

| | ga ∥_{H^{1} (Ω)} \leq C_{1} g ∥_{H^{1} (Ω)}, {∥g \frac{1}{a}∥}_{H^{1} (Ω)} \leq C_{2} ∥ g ∥_{H^{1} (Ω)}

, where the constant

C_{1}

and

C_{2}

are independent of

g \in H^{1} (Ω)

, which means, a and

1 /

a are multipliers in the space

H^{1} (Ω)

3. Mixed BVP in Exterior Domains

Let

\partial Ω = {\overset{̅}{\partial Ω}}_{D} \cup {\overset{̅}{\partial Ω}}_{N}

, where

\partial Ω_{D}

and

\partial Ω_{N}

are relatively open, non-empty and non-intersecting parts of

\partial Ω

. We will derive and analyze the system of BDIEs for the following mixed BVP: Given

f \in L_{2} (ρ; Ω) ⊥ R, ψ_{0} \in H_{* *}^{- \frac{1}{2}} (\partial Ω_{N})

and

φ_{0} \in H^{\frac{1}{2}} (\partial Ω_{D})

, find a function

u \in H^{1, 0} (Ω; A)

such that:

Au = f in Ω

(7)

γ^{+} u = φ_{0} on \partial Ω_{D}

(8)

T^{+} u = ψ_{0} on \partial Ω_{N}

(9)

Here

\partial Ω = \overset{̅}{\partial Ω_{N}} ⋃ \overset{̅}{\partial Ω_{D}}

, while

\partial Ω_{D} \neq \emptyset

and

\partial Ω_{N} \neq \emptyset

are nonintersecting simply connected sub-manifolds of

\partial Ω

with an infinitely smooth boundary curve

l = \overset{̅}{\partial Ω_{N}} \cap \overset{̅}{\partial Ω_{D}}

Let us denote by

A_{M} = {[A, T^{+}, γ^{+}]}^{T} : H^{1, 0} (Ω; A) \to L_{2} (ρ; Ω) ⊥ R \times H_{* *}^{- \frac{1}{2}} (\partial Ω_{N}) \times H^{\frac{1}{2}} (\partial Ω_{D})

, the operator on the left, which is obviously continuous. As with the three-dimensional case's proof in

[4]

, one can show the following assertion in the 2D case. (e.g; Theorem 8.6 in

[4]

Theorem 2. Under conditions (3), the Mixed problem (7)-(9) is uniquely solvable and its solution can be written as

u = A_{M}^{- 1} {(f, ψ_{0}, φ_{0})}^{T}

, where the operator

A_{M}^{- 1} : L_{2} (ρ; Ω) \times H^{- \frac{1}{2}} (\partial Ω_{N}) \times H^{\frac{1}{2}} (\partial Ω_{D}) \to H^{1, 0} (Ω; A)

is continuous.

4. Parametrix-Based Potentials in Exterior Domain

A function

P (x, y)

is a parametrix (Levi function) for the operator

A

A_{x} P (x, y) = δ (x - y) + R (x, y)

, where

δ

is the Dirac-delta distribution, while

R (x, y)

is a remainder possessing at most a weak (integrable) singularity at

x = y

. In particular, (e.g.,

[10]

) the function

P (x, y) = \frac{\ln | x - y |}{2 πa (y)}, x, y \in R^{2}

(10)

is a parametrix for the operator

A (x, \partial_{x})

given by:

A (x, \partial_{x}) P (x, y) = R (x, y) + δ (x - y)

(11)

Where

R (x, y) = \sum_{i = 1}^{2} \frac{x_{i} - y_{i}}{2 πa (y) | x - y |^{2}} \frac{\partial a (x)}{\partial x_{i}}, x, y \in R^{2}

(12)

Let

u \in D (\overset{⃐}{Ω})

. For any fixed

y \in Ω

, let

B_{ε} (y)

be an open ball centered at

y

with a sufficiently small radius

ε > 0

, and let

B_{r} (0)

be an open ball centered at the origin with a radius

r

large enough to contain

\partial Ω

and the support tabof

u

, put

Ω_{ε} : = (Ω \cap B_{r} (0)) ∖ B_{ε} (y)

, we have

R (\cdot, y) \in L_{2} (ρ; Ω_{ε})

and thus

P (\cdot, y) \in H^{1, 0} (Ω_{ε})

by (11). Applying the second Green identity (6) in

Ω_{ε}

with

v = P (y, \cdot)

and taking usual limits as

ε \to 0

, (eg.,

[12]

), we get the third Green identity in

Ω_{r} : = Ω \cap B_{r} (0)

u + R u - V (T^{+} u) + W (γ^{+} u) = P Au

(13)

for

u \in D (\overset{⃐}{Ω})

Here,

R g (y) ≔ \int_{Ω} R (x, y) g (x) dx,

(14)

P g (y) : = \int_{Ω} P (x, y) g (x) dx, y \in R^{2}

(15)

are, respectively, the remainder potentials and parametrix-based Newtonian, while

Vg (y) ≔ - \int_{\partial Ω} [P (x, y)] g (x) d S_{x},

Wg (y) : = - \int_{\partial Ω} [T_{x} P (x, y)] g (x) d S_{x}, y \in R^{2} ∖ \partial Ω

are the parametrix-based single layer and double layer potentials. Deducing (13) we took into account that

u \equiv 0

Ω ∖ B_{r} (0) \subset Ω ∖

supp

u

. Since no term in (13) depends on

r

r

is sufficiently large, we obtain that (13) is valid in the whole domain

Ω

for any

u \in D (\overset{⃐}{Ω})

From definitions (10) -(12) and (14)-(15) The parametrix-based potential operators can be represented in terms of their corresponding ones for

a = 1

(i.e., associated with the Laplace operator

Δ

), (eg.,

[3, 4]

P g = \frac{1}{a} P_{Δ} g, Vg = \frac{1}{a} V_{Δ} g,

Wg = \frac{1}{a} W_{Δ} (ag), R g = - \frac{1}{a} \sum_{j = 1}^{2} \partial_{j} [P_{Δ} (g \partial_{j} a)]

(16)

The Newtonian and the remainder potential operators given by (14) for

Ω = R^{2}

will be denoted as

P

and

R

, respectively, and the relations similar to (16) hold for them as well.

5. Invertibility of the Single Layer Potential Operator

The boundary integral operator

V_{Δ} : H^{- 1 / 2} (\partial Ω) \to H^{1 / 2} (\partial Ω)

is Fredholm operator of index zero (e.g., Theorem 7.6 in

[9]

). Thus the relation (16), leads to the same result for single layer potential

V

. For the 3-D case, the following holds. For

ψ^{*} \in H^{- 1 / 2} (\partial Ω)

, if

V ψ^{*} (y) = 0, y \in Ω

, then

ψ^{*} = 0

, which implies the invertibility of single layer potential operator mapping from

H^{- 1 / 2} (\partial Ω)

H^{1 / 2} (\partial Ω)

. But it is not true in the two dimensional case. It is well known for some

2 D

domains the kernel of the operator

V_{Δ}

is non-zero, which by (16) also implies that

\ker V \neq {0}

for the same domains. The following example illustrates this fact.

Example 1. Take the density function

ϕ \equiv 1

and

Ω = B (0, R)

to be a disc of radius

R

centered at the origin and

\partial Ω = \partial B (0, R)

be the circular boundary of the disc. We can show that

a (y) Vϕ (y) = V_{Δ} ϕ (y) = \{\begin{matrix} R \ln | y |, & for | y | > R, \\ R \ln R, & for | y | \leq R \end{matrix}

Proof. Let

ϕ \equiv 1

. Then

V_{Δ} ϕ (y) = \frac{1}{2 π} \int_{| x | = R} \ln | y - x | d S_{x}

| y | > R

, then the function

g (x) = \ln | y - x |

is harmonic in the disk

B (0, R)

. Then

g (x)

has the mean value property,

\ln | y | = g (0) = \frac{1}{2 πR} \int_{| x | = R} g (x) d S_{x}

Therefore,

\frac{1}{2 π} \int_{| x | = R} \ln | y - x | d S_{x} = R \ln | y |, for | y | > R

(17)

For

| y | \leq R

, in particular take

y = 0

(V_{Δ} ϕ) (0) = \frac{1}{2 π} \int_{| x | = R} \ln | x | d S_{x} = R \ln R

The relation (17) implies that, the limit of the value of the potential when

| y |

approach the boundary from exterior is given by

\lim_{| y | \to R^{+}} (V_{Δ} ϕ) (y) = R \ln R for | y | = R

Furthermore, since the single layer potential is continuous on

R^{2}

we have

(V_{Δ} ϕ) (y) = R \ln R for | y | = R

To determine the value of the potential inside the disc for

y \neq 0

, we use the maximum/minimum principle. Since the single layer potential is harmonic on

Ω

it has neither maximum nor minimum in the disc. Let

C_{0} = (V_{Δ} ϕ) (y_{0}) for 0 < |y_{0}| < R

If we assume

C_{0} \neq R \ln R

, i.e.,

C_{0}

is different from the value of potential on the boundary, we will arrive contradiction of the maximum principle. Thus

(V_{Δ} ϕ) (y)

is constant on

\overset{⃐}{Ω}

. Therefore,

(V_{Δ} ϕ) (y) = R \ln R,

for

| y | \leq R

Remark 2. In the above example, if we take the value of

R = 1

, and since

a (y) \neq 0

, then

(Vϕ) (y) = 0

\overset{⃐}{Ω}

Example 1 shows that, the kernel of the operator

V : H^{- 1 / 2} (\partial Ω) \to H^{1 / 2} (\partial Ω)

contains non zero element for a unit ball, i.e., ker

V \neq {0}

for

Ω = B (0, 1)

, which means, the operators

V

is not one to one for this particular domain. Consequently, the following question may arise: does the kernel of

V

contain a non-zero element on every bounded domain in

R^{2}

? The answer is no.

Theorem 3. The following spaces are subspaces of

L_{2} (ρ; Ω), H^{1, 0} (Ω; A)

and

{\tilde{H}}^{s} (Γ_{1}), H^{s} (\partial Ω)

, respectively, Where

Γ_{1} \subset \partial Ω

(i)

L_{2} (ρ; Ω) ⊥ R : = \{f \in L_{2} (ρ; Ω) : ⟨ f, 1 ⟩_{Ω} = 0\}

(ii)

H^{1, 0 ⊥} (Ω; A) : = \{g \in H^{1} (Ω) : Ag \in L_{2} (ρ; Ω) ⊥ R\}

(iii)

H_{* *}^{s} (\partial Ω) : = \{ψ \in H^{s} (\partial Ω) : ⟨ ψ, 1 ⟩_{\partial Ω} = 0\}, {\tilde{H}}_{* *}^{s} (Γ_{1}) : = \{ψ \in {\tilde{H}}^{s} (Γ_{1}) : ⟨ ψ, 1 ⟩_{Γ_{1}} = 0\}

Proof. (i) let f and g be in

L_{2} (ρ; Ω) ⊥ R

and

α, β \in R

then

αf + βg \in L_{2} (ρ; Ω)

then

\begin{matrix} ⟨ αf + βg, 1 ⟩_{Ω} & = ⟨ αf, 1 ⟩_{Ω} + ⟨ βg, 1 ⟩_{Ω} \\ = α ⟨ f, 1 ⟩_{Ω} + β ⟨ g, 1 ⟩_{Ω} \\ = 0 \end{matrix}

(ii) let f and g be in

H^{1, 0 ⊥} (Ω; A)

and

α, β \in R

then

αf + βg \in H^{1, 0} (Ω; A)

then by linearity of an operator A and

L_{2} (ρ; Ω) ⊥ R

above,

\begin{matrix} A (αf + βg) & = A (αf) + A (βg) \\ = αAf + βAg \\ \in L_{2} (ρ; Ω) ⊥ R \end{matrix}

(iii) let

ψ

and

φ

be in

H_{* *}^{s} (\partial Ω)

and

α, β \in R

then

αψ + βφ \in H^{s} (\partial Ω)

then

\begin{matrix} ⟨ αψ + βφ, 1 ⟩_{\partial Ω} & = ⟨ αψ, 1 ⟩_{\partial Ω} + ⟨ βφ, 1 ⟩_{\partial Ω} \\ = α ⟨ ψ, 1 ⟩_{\partial Ω} + β ⟨ φ, 1 ⟩_{\partial Ω} \\ = 0 \end{matrix}

Similarly the right hand side of (iii) follows from the proof of item (iii).

In order to have invertibility for the single layer potential operator in

2 D

, we consider the following theorem.

Theorem 4. If

ψ \in H_{* *}^{- \frac{1}{2}} (\partial Ω)

satisfies

V ψ = 0

\partial Ω

, then

ψ = 0

Proof. The theorem holds for the operator

V_{Δ}

(e.g, corollary 8.11(ii) in

[9]

\begin{matrix} V_{ψ} = 0 \\ \Rightarrow \frac{1}{a (y)} V_{Δ} ψ = 0 \\ \Rightarrow ψ = 0, (since a (y) \neq 0, \Rightarrow V_{Δ} \neq 0) \end{matrix}

Lemma 2. If

u \in H^{1, 0 ⊥} (Ω; A)

then

T^{+} u \in H_{* *}^{- \frac{1}{2}} (\partial Ω)

Proof. Employing the first Green identity (5) with

v = 1

, we have:

\begin{matrix} {⟨T^{+} u, 1⟩}_{\partial Ω} & = \int_{Ω} 1 Audx \\ = ⟨ Au, 1 ⟩_{Ω} \\ = 0; since Au \in L_{2} (ρ; Ω) ⊥ R \end{matrix}

In addition to conditions (3) and (4) on the coefficient

a

, we will sometimes also need the condition

ρ^{2} Δ a \in L_{\infty} (R^{2})

(18)

Employing that the corresponding mapping properties hold true for the potentials associated with the Laplace operator

Δ

, (eg. Section 8 in

[14]

) and references therein, relations (16) lead to the following assertion. (e.g., Theorem 4.1 in

[4]

and Theorem 3 in

[19]

Theorem 5. The following operators are continuous under conditions (4).

\begin{matrix} P : H^{- 1} (R^{2}) ⊥ R \to H^{1} (R^{2}) \\ P : {\tilde{H}}^{- 1} (Ω) ⊥ R \to H^{1} (R^{2}) \\ R : L_{2} (ρ^{- 1}; R^{2}) \to H^{1} (R^{2}) \\ V : H_{* *}^{- \frac{1}{2}} (\partial Ω) \to H^{1} (Ω) \\ W : H^{\frac{1}{2}} (\partial Ω) \to H^{1} (Ω) \end{matrix}

while The following operators are continuous under conditions (4) and (18).

\begin{matrix} P : L_{2} (ρ; Ω) ⊥ R \to H^{1, 0} (R^{2}; A) \\ R : H^{1} (Ω) \to H^{1, 0} (Ω; A) \\ V : H_{* *}^{- \frac{1}{2}} (\partial Ω) \to H^{1, 0} (Ω; A) \\ W : H^{\frac{1}{2}} (\partial Ω) \to H^{1, 0} (Ω; A) \end{matrix}

Remark 3. Similar to Theorem 3.12

[11]

one can prove that

D (\overset{⃐}{Ω})

is dense in

H^{1, 0} (Ω; A)

also in

H^{1, 0 ⊥} (Ω; A)

which then implies by theorem 5 and lemma 2, (13) holds for any

u \in H^{1, 0 ⊥} (Ω; A)

The boundary integral (pseudo-differential) operators of the direct values and of the co-normal derivatives of the single and double layer potentials are defined by

V g (y) ≔ - \int_{Γ} P (x, y) g (x) d s_{x};

W g (y) : = - \int_{Γ} T_{x} P (x, y) g (x) d s_{x} y \in Γ

(19)

W^{'} g (y) ≔ - \int_{Γ} T_{y} P (x, y) g (x) d s_{x};

L^{\pm} g (y) : = T_{y}^{\pm} Wg (y) y \in Γ

(20)

The mapping and jump properties of the operators (19)- (20) follow from relations (16) and are described in details in

[18]

. Particularly, their jump relations are given by the following theorem presented in Theorem 2,

[18]

Theorem 6. let

g_{1} \in H^{- \frac{1}{2}} (Γ), g_{2} \in H^{\frac{1}{2}} (Γ)

and

a \in C^{1} (R^{2})

. Then

\begin{matrix} γ^{\pm} V g_{1} (y) & : = V g_{1} (y) \\ γ^{\pm} W g_{2} (y) & : = \mp \frac{1}{2} g_{2} (y) + W g_{2} (y) \\ T^{\pm} V g_{1} (y) & : = \pm \frac{1}{2} g_{1} (y) + W^{'} g_{1} (y) \end{matrix}

where

y \in \partial Ω

employing the co-normal derivative and trace operators to the third Green identity (13), and using the jump relations for the potential operators we obtain for

u \in H^{1, 0 ⊥} (Ω; A)

\frac{1}{2} γ^{+} u + γ^{+} R u - V T^{+} u + W γ^{+} u = γ^{+} P Au on \partial Ω

(21)

\frac{1}{2} T^{+} u + T^{+} R u - W^{'} T^{+} u + L^{+} γ^{+} u = T^{+} P Au on \partial Ω

(22)

Conditions (4) are assumed to hold for (21) and conditions (4) and (18) for (22). For some functions

f, Ψ

and

Φ

let us consider a more general indirect integral relation associated with equation (13).

u + R u - V Ψ + W Φ = P_{f} in Ω

(23)

Lemma 3. Let

u \in H^{1, 0 ⊥} (Ω), f \in L_{2} (ρ; Ω) ⊥ R, Ψ \in H_{* *}^{- \frac{1}{2}} (\partial Ω)

, and

Φ \in H^{\frac{1}{2}} (\partial Ω)

satisfy equation (23) and let conditions (4) and (18) hold. Then,

u

is a solution of the equation

Au = f in Ω

(24)

While

V (Ψ - T^{+} u) - W (Φ - γ^{+} u) = 0, in Ω

(25)

Proof. Since

u \in H^{1, 0 ⊥} (Ω; A)

, by Remark 3 we can write the third Green identity (13) for the function

u

. Then subtracting (23) from it, we obtain

- V Ψ^{*} + W Φ^{*} = P [Au - f] in Ω

(26)

where

Ψ^{*} : = T^{+} u - Ψ

and

Φ^{*} : = γ^{+} u - Φ

. Multiplying equality (26) by

a (y)

we get

- V_{Δ} Ψ^{*} + W_{Δ} (a Φ^{*}) = P_{Δ} [Au - f] in Ω

Applying the Laplace operator

Δ

to the last equation and taking into consideration that both functions in the left-hand side are harmonic potentials, while the right-hand side function is the classical Newtonian potential, we arrive at Eq. (24) Substituting (24) back into (26) leads to (25).

Lemma 4. Let conditions (4) and (18) hold.

(i) If

Ψ^{*} \in H_{* *}^{- \frac{1}{2}} (\partial Ω)

and

V Ψ^{*} = 0

Ω

, then

Ψ^{*} = 0

(ii) If

Φ^{*} \in H^{\frac{1}{2}} (\partial Ω)

and

W Φ^{*} (y) = 0

Ω

, then

Φ^{*} (x) = C / a (x)

, where

C

is a constant.

(iii) let

\partial Ω = \overset{̅}{Γ_{1}} ⋃ \overset{̅}{Γ_{2}}

, where

Γ_{1}

and

Γ_{2}

are nonempty non intersecting simply connected submanifolds of

\partial Ω

with infinitely smooth boundaries.

Ψ^{*} \in {\tilde{H}}_{* *}^{- \frac{1}{2}} (Γ_{1}), Φ^{*} \in {\tilde{H}}^{\frac{1}{2}} (Γ_{2})

and

V Ψ^{*} (y) - W Ψ^{*} (y) = 0

Ω

, then

Ψ^{*} = 0

and

Φ^{*} = 0

\partial Ω

Proof. The proof of item (i) follows from theorem 4, while the proof of item (iii) is similar to the proof of Lemma 2.12

[20]

To prove item (ii), from the first Green identity (5) for the interior domain

Ω^{-}

employing for

v (x) = C, A = Δ, u = \frac{\ln | x - y |}{2 π}

and for any

y \in Ω

, the function

Φ_{Δ} = C

satisfies the equation

W_{Δ} Φ_{Δ} = 0

in the exterior domain

Ω

for any

C =

const. Now let us check there is no other solution of the equation in

Ω

H^{\frac{1}{2}} (\partial Ω)

. By the Lyapunov-Tauber theorem

T_{Δ}^{+} W_{Δ} Φ_{Δ} = T_{Δ}^{-} W_{Δ} Φ_{Δ} = 0

\partial Ω

, which implies

W_{Δ} Φ_{Δ} =

const inthe interior domain

Ω^{-}

due to the uniqueness up to a constant of the solution of the Neumann problem in

H^{\frac{1}{2}} (Ω^{-})

. Then by the jump property of the double layer

Φ_{Δ} =

const. Applying the relation

Wg = \frac{1}{a} W_{Δ} (ag)

completes the proof of item (ii).

6. BDIEs for Exterior Mixed BVP

To reduce the variable-coefficient Mixed BVP (7)-(9) to a segregated boundary domain integral equation systems, Let us fix an extension

Φ_{0} \in H^{\frac{1}{2}} (\partial Ω)

of the given function

φ_{0}

in the condition (8) from

\partial Ω_{D}

to the whole of

\partial Ω

and an extension

Ψ_{0} \in H_{* *}^{- \frac{1}{2}} (\partial Ω)

of the given function

ψ_{0}

in the condition (9) from

\partial Ω_{N}

to the whole of

\partial Ω

. moreover

Φ_{0}

and

Ψ_{0}

are considered as known.

For a given function

f

L_{2} (ρ; Ω) ⊥ R

, assume that the function

u

satisfies the

PDE Au = f

Ω

. Then, we can reduce the BVP (7)-(9) to a system of Boundary-Domain Integral Equations (BDIEs) and in all of them we represent in (13), (21) and (22) the trace of the function

u

and in its co-normal derivative as

γ^{+} u = Φ_{0} + φ, φ \in {\tilde{H}}^{\frac{1}{2}} (\partial Ω_{N}); T^{+} u = Ψ_{0} + ψ, ψ \in {\tilde{H}}_{* *}^{- \frac{1}{2}} (\partial Ω_{D})

and will regard the new unknown functions

φ

and

ψ

as formally segregated of

u \in H^{1, 0} (Ω; A)

. Thus we will look for the triplet

U = (u, ψ, φ)^{⊤} : = H^{1, 0} (Ω; A) \times {\tilde{H}}_{* *}^{- \frac{1}{2}} (\partial Ω_{D}) \times {\tilde{H}}^{\frac{1}{2}} (\partial Ω_{N})

BDIE system (M11). Obtained under conditions (4) and (18), using equation (13) in

Ω

, the restriction of equation (21) on

\partial Ω_{D}

, and the restriction of equation (22) on

\partial Ω_{N}

, we arrive at the BDIE system (M11) of three equations for the triplet of unknowns,

(u, ψ, φ)

u + R u - Vψ + Wφ = F_{0} in Ω

(27)

γ^{+} R u - V ψ + W φ = γ^{+} F_{0} - Φ_{0} on \partial Ω_{D}

(28)

T^{+} R u - W^{'} ψ + L^{+} φ = T^{+} F_{0} - Ψ_{0} on \partial Ω_{N}

(29)

Where

F_{0} : = P f + V Ψ_{0} - W Φ_{0} in Ω

(30)

We denote the matrix operator of the left hand side of the systems (M11) as

M^{11} ≔ [\begin{matrix} I + R & - V & W \\ r_{\partial Ω_{D}} γ^{+} R & - r_{\partial Ω_{D}} V & r_{\partial Ω_{D}} W \\ r_{\partial Ω_{N}} T^{+} R & - r_{\partial Ω_{N}} W^{'} & r_{\partial Ω_{N}} L^{+} \end{matrix}],

F^{11} : = [\begin{matrix} F_{0} \\ r_{\partial Ω_{D}} γ^{+} F_{0} - φ_{0} \\ r_{\partial Ω_{N}} T^{+} F_{0} - ψ_{0} \end{matrix}]

Remark 4. Due to the mapping properties of operators involved in

M^{11}

, The operator

M^{11} : H^{1, 0} (Ω; A) \times {\tilde{H}}_{* *}^{- \frac{1}{2}} (\partial Ω_{D}) \times {\tilde{H}}^{\frac{1}{2}} (\partial Ω_{N}) \to H^{1, 0} (Ω; A) \times H^{\frac{1}{2}} (\partial Ω_{D}) \times H^{- \frac{1}{2}} (\partial Ω_{N})

is bounded. And also

F^{11} = 0

if and only if

(f, Φ_{0}, Ψ_{0}) = 0

Proof.

(\Leftarrow)

evidently true.

(\Rightarrow)

from equation (30) we have that

F_{0} \in H^{1, 0} (Ω; A)

and by our assumption

0 = F_{0}

implies

F_{0} \in H^{1, 0 ⊥} (Ω; A)

, Lemma 3 with

F_{0} = 0

for u implies

f = 0

and

V (Ψ_{0}) - W (Φ_{0}) = 0

, in

Ω

and The equalities

γ^{+} F_{0} = Φ_{0}

\partial Ω_{D}

and

T^{+} F_{0} = Ψ_{0}

\partial Ω_{N}

, implies

Φ_{0} = φ_{0} = 0

\partial Ω_{D}

and

Ψ_{0} = ψ_{0} = 0

\partial Ω_{N}

that is,

Ψ_{0} \in {\tilde{H}}_{* *}^{- \frac{1}{2}} (\partial Ω_{D})

and

Φ_{0} \in {\tilde{H}}^{\frac{1}{2}} (\partial Ω_{N})

. Lemma 4 (iii) implies

Φ_{0} = Ψ_{0} =

BDIE system (M12). Obtained under conditions (4) and using equation (13) in

Ω

and equation (21) on the whole of

\partial Ω

, we arrive at the BDIE system (M12) of two equations for the triplet

(u, ψ, φ)

\begin{matrix} u + R u - Vψ + Wφ = F_{0} in Ω \\ \frac{1}{2} φ + γ^{+} R u - V ψ + W φ = γ^{+} F_{0} - Φ_{0} on \partial Ω \end{matrix}

The left hand side matrix operator of the system is

M^{12} : = [\begin{matrix} I + R & - V & W \\ γ^{+} R & - V & \frac{1}{2} I + W \end{matrix}], F^{12} : = [\begin{matrix} F_{0} \\ γ^{+} F_{0} - Φ_{0} \end{matrix}]

Remark 5. Due to the mapping properties of operators involved in

M^{12}

, The operator

M^{12} : H^{1, 0} (Ω; A) \times {\tilde{H}}_{* *}^{- \frac{1}{2}} (\partial Ω_{D}) \times {\tilde{H}}^{\frac{1}{2}} (\partial Ω_{N}) \to H^{1, 0} (Ω; A) \times H^{\frac{1}{2}} (\partial Ω)

is bounded. And also

F^{12} = 0

if and only if

(f, Φ_{0}, Ψ_{0}) = 0

Proof.

(\Leftarrow)

evidently true.

(\Rightarrow)

from equation (30) we have that

F_{0} \in H^{1, 0} (Ω; A)

and by our assumption

0 = F_{0}

implies

F_{0} \in H^{1, 0 ⊥} (Ω; A)

, Lemma 3 with

F_{0} = 0

for u implies

f = 0

and

V (Ψ_{0}) - W (Φ_{0}) = 0

, in

Ω

and The equalities

γ^{+} F_{0} = Φ_{0}

\partial Ω

, implies

Φ_{0} = 0

. Lemma 4 (i) implies

Ψ_{0} = 0

BDIE system (M21). Obtained under conditions (4) and (18) and Using equation (13) in

Ω

and equation (22) on the whole of

\partial Ω

, we arrive at the BDIE system (M21) of two equations for the triplet

(u, ψ, φ)

u + R u - Vψ + Wφ = F_{0} in Ω

(31)

\frac{1}{2} ψ + T^{+} R u - W^{'} ψ + L^{+} φ = T^{+} F_{0} - Ψ_{0} on \partial Ω

(32)

The left hand side matrix operator of the system is

M^{21} : = [\begin{matrix} I + R & - V & W \\ T^{+} R & \frac{1}{2} I - W^{'} & L^{+} \end{matrix}], F^{21} : = [\begin{matrix} F_{0} \\ T^{+} F_{0} - Ψ_{0} \end{matrix}]

Remark 6. Due to the mapping properties of operators involved in

M^{21}

, The operator

M^{21} : H^{1, 0} (Ω; A) \times {\tilde{H}}_{* *}^{- \frac{1}{2}} (\partial Ω_{D}) \times {\tilde{H}}^{\frac{1}{2}} (\partial Ω_{N}) \to H^{1, 0} (Ω; A) \times H^{- \frac{1}{2}} (\partial Ω)

is bounded.

BDIE system (M22). Obtained under conditions (4) and (18) and using equation (13) in

Ω

, the restriction of equation (22) on

\partial Ω_{D}

, and the restriction of equation (21) on

\partial Ω_{N}

, we arrive for the triplet

(u, ψ, φ)

at the BDIE system (M22) of three equations of "almost" the second kind (up to the spaces),

\begin{matrix} u + R u - Vψ + Wφ = F_{0} in Ω \\ \frac{1}{2} ψ + T^{+} R u - W^{'} ψ + L^{+} φ = T^{+} F_{0} - Ψ_{0} on \partial Ω_{D} \\ \frac{1}{2} φ + γ^{+} R u - V ψ + W φ = γ^{+} F_{0} - Φ_{0} on \partial Ω_{N} \end{matrix}

The matrix operator of the left hand side of the system (M22) takes form

M^{22} ≔ [\begin{matrix} I + R & - V & W \\ r_{\partial Ω_{D}} T^{+} R & r_{\partial Ω_{D}} (\frac{1}{2} I - W^{'}) & r_{\partial Ω_{D}} L^{+} \\ r_{\partial Ω_{N}} γ^{+} R & - r_{\partial Ω_{N}} V & r_{\partial Ω_{N}} (\frac{1}{2} I + W) \end{matrix}],

F^{22} : = [\begin{matrix} F_{0} \\ r_{\partial Ω_{D}} \{T^{+} F_{0} - Ψ_{0}\} \\ r_{\partial Ω_{N}} \{γ^{+} F_{0} - Φ_{0}\} \end{matrix}]

Remark 7. Due to the mapping properties of operators involved in

M^{22}

, The operator

M^{22} : H^{1, 0} (Ω; A) \times H_{* *}^{- \frac{1}{2}} (\partial Ω_{D}) \times H^{\frac{1}{2}} (\partial Ω_{N}) \to H^{1, 0} (Ω; A) \times H^{- \frac{1}{2}} (\partial Ω_{D}) \times H^{\frac{1}{2}} (\partial Ω_{N})

is bounded. And also

F^{22} = 0

if and only if

(f, Φ_{0}, Ψ_{0}) = 0

Proof. The proof follows in the similar way as in the Remark 4 proof.

7. Equivalence and Uniqueness Theorems

Theorem7. Let

φ_{0} \in H^{\frac{1}{2}} (\partial Ω_{D}), ψ_{0} H_{* *}^{- \frac{1}{2}} (\partial Ω_{N}), L_{2} (ρ; Ω) ⊥ R

and let

Φ_{0} \in H^{\frac{1}{2}} (\partial Ω)

and

Ψ_{0} \in H_{* *}^{- \frac{1}{2}} (\partial Ω)

be some extensions of

φ_{0}

and

ψ_{0}

, respectively, and conditions (4) and (18) hold.

(i) If a function

u \in H^{1, 0 ⊥} (Ω; A)

solves the

BVP

(7)-(9), then the triplet

(u, ψ, φ)

, where

T^{+} u - Ψ_{0} = ψ \in {\tilde{H}}_{* *}^{- \frac{1}{2}} (\partial Ω_{D}), γ^{+} u - Φ_{0} = φ \in {\tilde{H}}^{\frac{1}{2}} (\partial Ω_{N})

(33)

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

(ii) If a triplet

(u, ψ, φ) \in H^{1, 0 ⊥} (Ω; A) \times {\tilde{H}}_{* *}^{- \frac{1}{2}} (\partial Ω_{D}) \times {\tilde{H}}^{\frac{1}{2}} (\partial Ω_{N})

solves one of the BDIE systems (M11), (M12) or (M22), then this solution is unique and solves all the systems, including (M21), while

u

solves BVP (7)-(9) and relations (33) hold.

Proof. (i) immediately follows from the deduction of the BDIE systems (M11), (M12), (M21) and (M22).

(ii) Let a triplet

(u, ψ, φ)^{T}

solve BDIE system (M11), (M12) or (M22). The hypotheses of Lemma 3 are satisfied for the first equation in BDIE system, implying that

u

solves PDE (7) in

Ω

, while the following equation holds:

V Ψ^{*} - W Φ^{*} = 0 in Ω

(34)

where

Ψ^{*} = Ψ_{0} + ψ - T^{+} u

and

Φ^{*} = Φ_{0} + φ - γ^{+} u

Suppose first that the triplet

(u, ψ, φ)^{T}

solves BDIE system (M11). Taking trace of (27) on

\partial Ω_{D}

using the jump relations of Theorem 6, and subtracting (28) from it, we obtain

γ^{+} u = φ_{0} on \partial Ω_{D}

(35)

i.e.,

u

satisfies the Dirichlet condition (8). Taking the co-normal derivative of Eq. (27) on

\partial Ω_{N}

, using the jump relations on Theorem 6 and subtracting Eq. (29) from it, we obtain

T^{+} u = ψ_{0} on \partial Ω_{N}

(36)

i.e.,

u

satisfies the Neumann condition (9). Hence

u

solves the mixed BVP (7)-(9).

Taking into account

φ = 0, Φ_{0} = φ_{0}

\partial Ω_{D}

and

ψ = 0, Ψ_{0} = ψ_{0}

\partial Ω_{N}

, (35) and (36) imply that the first equation in (33) is satisfied on

\partial Ω_{N}

and the second equation in (33) is satisfied on

\partial Ω_{D}

. Thus we have

Ψ^{*} \in {\tilde{H}}_{* *}^{- \frac{1}{2}} (\partial Ω_{D})

and

Φ^{*} \in {\tilde{H}}^{\frac{1}{2}} (\partial Ω_{N})

in (34). Let

Γ_{1} = \partial Ω_{D}, Γ_{2} = \partial Ω_{N}

. Then Lemma 4 (iii) implies

Ψ^{*} = Φ^{*} = 0

, which completes the proof of conditions in (33). Uniqueness of the solution to BDIE systems (M11) follows from (33) along with remark 4 and Theorem 2.

Finally, item (i) implies that triplet

(u, ψ, φ)^{T}

solves also BDIE systems (M12), (M21) and (M22).

Similar arguments work if we suppose that instead of the BDIE systems (M11), the triplet

(u, ψ, φ)^{T}

solves BDIE systems (M12) or (M22).

The situation with uniqueness and equivalence for system (M21) differs from the one for other systems and from its counterpart BDIE system (M21) in

[20]

, particularly because item (ii) of Lemma 4 is different from its analog, Lemma 2.11 (ii) in

[20]

. This leads to the following assertion.

Theorem 8. Let

φ_{0} \in H^{\frac{1}{2}} (\partial Ω_{D}), ψ_{0} \in H_{* *}^{- \frac{1}{2}} (\partial Ω_{N}), f \in L_{2} (ρ; Ω) ⊥ R

and let

Φ_{0} \in H^{\frac{1}{2}} (\partial Ω)

and

Ψ_{0} \in H_{* *}^{- \frac{1}{2}} (\partial Ω)

be some extensions of

φ_{0}

and

ψ_{0}

, respectively, and conditions (4) and (18) hold.

(i) Homogeneous BDIE system (M21) admits only one linearly independent solution

(u^{0}, ψ^{0}, φ^{0}) \in H^{1, 0 ⊥} (Ω; A) \times {\tilde{H}}_{* *}^{- \frac{1}{2}} (\partial Ω_{D}) \times {\tilde{H}}^{\frac{1}{2}} (\partial Ω_{N})

, where

u^{0}

is the solution of the mixed

BVP

A u^{0} = 0 in Ω

(37)

r_{\partial Ω_{D}} γ^{+} u^{0} = \frac{1}{a (x)} on \partial Ω_{D}

(38)

r_{\partial Ω_{N}} T^{+} u^{0} = 0 on \partial Ω_{N}

(39)

While

ψ^{0} = T^{+} u^{0}, φ^{0} = γ^{+} u^{0} - 1 / a (x) on \partial Ω

(40)

(ii) The non-homogeneous BDIE systems (M21) is solvable, and any it’s Solution

(u, ψ, φ) \in H^{1, 0 ⊥} (Ω; A) \times {\tilde{H}}_{* *}^{- \frac{1}{2}} (\partial Ω_{D}) \times {\tilde{H}}^{\frac{1}{2}} (\partial Ω_{N})

can be represented as

u = \overset{̀}{u} + C u^{0} in Ω

(41)

where

\overset{̀}{u}

solves the BVP (7)-(8) and

C

is a constant, while

ψ = T^{+} \overset{̀}{u} - Ψ_{0} + C ψ^{0}, φ = γ^{+} \overset{̀}{u} - Φ_{0} + C φ^{0} on \partial Ω

(42)

Proof. Problem (37)-(39) is uniquely solvable in

H^{1, 0 ⊥} (Ω; A)

by Theorem 2. Consequently, the third Green identity (13) is applicable to

u^{0}

, leading to

u^{0} + R u^{0} - V ψ^{0} + W φ^{0} = 0 in Ω

(43)

with notations (40), if we take into account that

W (1 / a (x)) = 0

Ω

due to the second relation in (16) and the equality

W_{Δ} 1 = 0

Ω

(cf. the proof of Lemma 4(ii)). Taking the co-normal derivative of (43) and substituting the first equation of (40) again, we arrive at

\frac{1}{2} ψ^{0} + T^{+} R u^{0} - W^{'} ψ^{0} + L^{+} φ^{0} = 0 on \partial Ω

(44)

Equations (43)-(44) mean that the triplet

(u^{0}, ψ^{0}, φ^{0})

solves the homogeneous BDIE system (M21).

To prove item (ii) and check that there exists only one linearly independent solution of the homogeneous BDIE system (M21), we proceed as follows. First, we remark that the solvability of the non-homogeneous system (M21) follows from the solvability of the BVP (7)-(8) in

H^{1, 0 ⊥} (Ω; A)

and the deduction of system (M21).

Let now a triplet

(u, ψ, φ)^{⊤} \in H^{1, 0 ⊥} (Ω; A) \times {\tilde{H}}_{* *}^{- \frac{1}{2}} (\partial Ω_{D}) \times {\tilde{H}}^{\frac{1}{2}} (\partial Ω_{N})

solve (generally non-homogeneous) BDIE system (M21). Take the co-normal derivative of equation (31) on

\partial Ω

and subtract it from equation (32) to obtain

ψ + Ψ_{0} - T^{+} u = 0 on \partial Ω

(45)

Taking into account that

ψ = 0

and

Ψ_{0} = ψ_{0}

\partial Ω_{N}

, this implies that

u

satisfies condition (9).

Equations (31) and (30) and Lemma 3 with

Ψ = ψ + Ψ_{0}, Φ = φ + Φ_{0}

imply that

u

is a solution of equation (7) and

V (Ψ_{0} + ψ - T^{+} u) - W (Φ_{0} + φ - γ^{+} u) = 0 in Ω

(46)

Due to (45) the first term vanishes in (46), and by Lemma 4(ii) we obtain

Φ_{0} + φ - γ^{+} u = - C / a (x) on \partial Ω

(47)

where

C

is a constant. Taking into account that

φ = 0

\partial_{D} Ω

and

Φ_{0} = φ_{0}

\partial Ω_{D}

, we conclude that

u

satisfies the Dirichlet condition

γ^{+} u = φ_{0} + C / a (x) on \partial Ω_{D}

(48)

instead of (8). Introducing notation

\overset{̀}{u}

by (41) in (45), (47) and (48) and taking into account (37)-(39) prove the claim of item (ii). The case

φ_{0} = 0, Φ_{0} = 0, ψ_{0} = 0, Ψ_{0} = 0, f = 0

leading to the homogeneous BDIE system (M21) also implies that

\overset{̀}{u}

for this case satisfies homogeneous BVP (7)-(9) and thus

\overset{̀}{u} = 0

in (41) and (42) meaning that the triplet

(u^{0}, ψ^{0}, φ^{0})

is the only linearly independent solution of the homogeneous BDIE system (M21). This completes the proof of item (i) and of the whole theorem.

8. Future Work

Future work on the BDIEs for Variable Coefficient Mixed BVP in 2D Unbounded Domain, will consider the Fredholm properties and invertibility of the corresponding BDIOs in weighted Sobolev spaces. and we will also consider the Direct segregated systems of BDIEs for the Neumann BVPs for a scalar second order divergent elliptic PDEs with a variable coefficient in an exterior two-dimensional domain. and we will again consider the equivalence of BDIE system to the original boundary value problems and the Fredholm properties and invertibility of the corresponding BDIOs are to be analyzed in weighted Sobolev spaces for in the future.

9. Conclusion

In this paper, we have considered a second-order elliptic partial differential equation with a variable coefficient in a 2D Unbounded domain, in appropriate weighted Sobolev space. The right-hand side functions were from

L_{2} (ρ; Ω) ⊥ R

and the Mixed data from the space

H_{* *}^{- \frac{1}{2}} (\partial Ω_{N})

and

H^{\frac{1}{2}} (\partial Ω_{D})

. The BVP was reduced to four systems of Boundary Domain Integral Equations and their equivalence to the original BVP and Uniqueness property was shown.

The properties of a parametrix-based potential operator that contain logarithmic singularity were investigated. Unlike properties in 3D case, The single layer potential needs special consideration to be invertible, which is critical on this study.

Abbreviations

BDIE	Boundary Domain Integral Equation
MBVP	Mixed Boundary Value Problem
PDE	Partial Differential Equation
BVP	Boundary Value Problem
BDIO	Boundary Domain Integral Operator

Acknowledgments

First and foremost, I would like to express my deepest gratitude to Engineer Masresha Kuma, who has been the inspiration behind this work and my entire mathematical journey. I am also profoundly grateful to my advisor, Dr. Tsegaye Ayele, for his unwavering support, insightful guidance, and constructive feedback throughout this endeavour.

Additionally, I would like to extend my heartfelt thanks to Professor Sergay Mikailov for his invaluable contributions and mentorship. Lastly, I would like to acknowledge my student, Makriana Birhanu who has been the inspiration behind this work.

Author Contributions

Eshetu Seid Ahimed is the sole author. The author read and approved the final manuscript.

Conflicts of Interest

The author declares no conflicts of interest.

References

[1]	Amrouche, C., Girault, V. and Giroire, J.: Weighted Sobolev Spaces for Laplace's Equation in. Rn J. Math. Pures Appl., 73, 579-606 (1994).
[2]	Amrouche, C., Girault, V., and Schechter,: Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approche in weighted sobolev spaces. Journal de Mathématiques Pures et Appliquées, 76, 55-81 (1997).
[3]	Chkadua, O., Mikhailov, S. E. and Natroshvili, D.: Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient. I. Equivalence and invertibility. J. Integral Equations Appl., 21: 499-543 (2009), 499-543, https://doi.org/10.1216/JIE-2009-21-4-499
[4]	Chkadua, O., Mikhailov, S. E. and Natroshvili, D.: Analysis of direct segregated boundary domain integral equations for variable-coefficient mixed BVPs in exterior domains. Analysis and Applications, 11, No. 4, No 4, 1350006 (2013), https://doi.org/10.1007/978-0-8176-8238-5_11
[5]	Chkadua, O., Mikhailov, S. E. and Natroshvili, D.: Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks, Numerical Methods for Partial Differential Equations 27 (2011) (1) 121-140, https://doi.org/10.1002/num.20639
[6]	Costabel, M.: Boundary integral operators on Lipschiz domains: Elementary results. SIAM J. Math. Anal., 19, 613-626 (1988).
[7]	J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Appli cations, vol. 1 (Springer, Berlin Heidelberg New York, 1972), ISBN 3-540-05363-8.
[8]	Kufner, A. and Opic, B.: How to define reasonably weighted Sobolev spaces. Commentationes Mathematicae Universitatis Carolinae, 25, 537554 (1984).
[9]	McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambrige University Press, Cambrige, 2000.
[10]	Mikhailov, S. E.: Localized boundary-domain integral formulations for problems with variable coefficients. Eng. Anal. with Boundary El., 26, 681-690 (2002).
[11]	Mikhailov, S. E.: Traces, extensions and co-normal derivatives for elliptic systems on Lipschiz domains. J. Math. Anal. Appl., 378, 324-342 (2011), https://doi.org/10.1016/j.jmaa.2010.12.027
[12]	Miranda, C.: Partial differential equations of elliptic type. Springer, 1970.
[13]	Nédélec, J. C.: Acoustic and electromagnetic equations: integral representations for harmonic problems. Springer, 2001.
[14]	Sayas, F.-J., Selgas, V.: Variational views of stokeslets and stresslets. SeMA 63, 65-90(2014), https://doi.org/10.48550/arXiv.1303.4299
[15]	S. E. Mikhailov, Localized boundary-domain integral formulations for problems with variable coefficients, Engineering Analysis with Boundary Elements 26 (2002) 681-690.
[16]	S. E. Mikhailov, Analysis of extended boundary-domain integral and integrodifferential equations of some variable-coefficient BVP, in Advances in Boundary Integral Methods - Proceedings of the 5th UK Conference on Boundary Integral Methods, ed. K. Chen (University of Liverpool Publ., Liverpool, UK, 2005), ISBN 090637039 6, pp. 106-125.
[17]	S. E. Mikhailov, Analysis of united boundary-domain integro-differential and integral equations for a mixed BVP with variable coefficient, Math. Methods in Applied Sciences 29 (2006) 715-739, https://doi.org/10.1002/mma.706
[18]	T. T. Dufera, Mikhailov, S. E.: Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Drichlet BVP in 2D. In: Integral Methods in Science and Engineering: Computational and Analytic Aspects, Springer (Birkhäuser), Boston, 163-175, (2015), https://doi.org/10.1007/978-3-319-16727-5_15
[19]	T. T. Dufera, Mikhailov, S. E.: Boundary-Domain Integral Equations for Variable Coefficient Dirichlet BVP in 2D Unbounded Domain, In: Analysis, Probability, Applications, and Computations. Lindahl et al. (eds.) Springer Nature Switzerland AG, ISBN 978-3-030-04459-6, 481-492, (2019), https://doi.org/10.1007/978-3-030-04459-6_46
[20]	T. T. Dufera, S. E. Mikhailov, Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Mixed BVP in 2D, in Analysis, Probability, Applications, and Computation, 467-480, (2017), https://doi.org/10.1007/978-3-030-04459-6_45

Cite This Article

Plain Text BibTeX RIS

APA Style

Ahimed, E. S. (2024). Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain. International Journal of Theoretical and Applied Mathematics, 10(2), 23-32. https://doi.org/10.11648/j.ijtam.20241002.12

Copy | Download

ACS Style

Ahimed, E. S. Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain. Int. J. Theor. Appl. Math. 2024, 10(2), 23-32. doi: 10.11648/j.ijtam.20241002.12

Copy | Download

AMA Style

Ahimed ES. Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain. Int J Theor Appl Math. 2024;10(2):23-32. doi: 10.11648/j.ijtam.20241002.12

Copy | Download

@article{10.11648/j.ijtam.20241002.12,
  author = {Eshetu Seid Ahimed},
  title = {Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {10},
  number = {2},
  pages = {23-32},
  doi = {10.11648/j.ijtam.20241002.12},
  url = {https://doi.org/10.11648/j.ijtam.20241002.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20241002.12},
  abstract = {In this paper, the direct segregated Boundary Domain Integral Equations (BDIEs) for the Mixed Boundary Value Problems (MBVPs) for a scalar second order elliptic Partial Differential Equation (PDE) with variable coefficient in unbounded (exterior) 2D domain is considered. Otar Chkadua, Sergey Mikhailov and David Natroshvili formulated both the interior and exterior 3D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order elliptic PDE with a variable coefficients. On the other hand Sergey Mikhailov and Tamirat Temesgen formulated only the interior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. However, in this paper we formulated the exterior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. The aim of this work is to reduce the MBVPs to some direct segregated BDIEs with the use of an appropriate parametrix (Levi function). We examine the characteristics of corresponding parametrix-based integral volume and layer potentials in some weighted Sobolev spaces, as well as the unique solvability of BDIEs and their equivalence to the original MBVPs. This analysis is based on the corresponding properties of the MBVPs in weighted Sobolev spaces that are proved as well.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain

AU  - Eshetu Seid Ahimed
Y1  - 2024/08/30
PY  - 2024
N1  - https://doi.org/10.11648/j.ijtam.20241002.12
DO  - 10.11648/j.ijtam.20241002.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 23
EP  - 32
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20241002.12
AB  - In this paper, the direct segregated Boundary Domain Integral Equations (BDIEs) for the Mixed Boundary Value Problems (MBVPs) for a scalar second order elliptic Partial Differential Equation (PDE) with variable coefficient in unbounded (exterior) 2D domain is considered. Otar Chkadua, Sergey Mikhailov and David Natroshvili formulated both the interior and exterior 3D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order elliptic PDE with a variable coefficients. On the other hand Sergey Mikhailov and Tamirat Temesgen formulated only the interior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. However, in this paper we formulated the exterior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. The aim of this work is to reduce the MBVPs to some direct segregated BDIEs with the use of an appropriate parametrix (Levi function). We examine the characteristics of corresponding parametrix-based integral volume and layer potentials in some weighted Sobolev spaces, as well as the unique solvability of BDIEs and their equivalence to the original MBVPs. This analysis is based on the corresponding properties of the MBVPs in weighted Sobolev spaces that are proved as well.

VL  - 10
IS  - 2
ER  -

Copy | Download

Author Information

Eshetu Seid Ahimed

College of Natural & Computational Science, Addis Ababa University, Addis Ababa, Ethiopia

Contact Email

http://orcid.org/0009-0004-3714-374X

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Ahimed, E. S. (2024). Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain. International Journal of Theoretical and Applied Mathematics, 10(2), 23-32. https://doi.org/10.11648/j.ijtam.20241002.12

Copy | Download

ACS Style

Ahimed, E. S. Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain. Int. J. Theor. Appl. Math. 2024, 10(2), 23-32. doi: 10.11648/j.ijtam.20241002.12

Copy | Download

AMA Style

Ahimed ES. Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain. Int J Theor Appl Math. 2024;10(2):23-32. doi: 10.11648/j.ijtam.20241002.12

Copy | Download

@article{10.11648/j.ijtam.20241002.12,
  author = {Eshetu Seid Ahimed},
  title = {Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {10},
  number = {2},
  pages = {23-32},
  doi = {10.11648/j.ijtam.20241002.12},
  url = {https://doi.org/10.11648/j.ijtam.20241002.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20241002.12},
  abstract = {In this paper, the direct segregated Boundary Domain Integral Equations (BDIEs) for the Mixed Boundary Value Problems (MBVPs) for a scalar second order elliptic Partial Differential Equation (PDE) with variable coefficient in unbounded (exterior) 2D domain is considered. Otar Chkadua, Sergey Mikhailov and David Natroshvili formulated both the interior and exterior 3D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order elliptic PDE with a variable coefficients. On the other hand Sergey Mikhailov and Tamirat Temesgen formulated only the interior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. However, in this paper we formulated the exterior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. The aim of this work is to reduce the MBVPs to some direct segregated BDIEs with the use of an appropriate parametrix (Levi function). We examine the characteristics of corresponding parametrix-based integral volume and layer potentials in some weighted Sobolev spaces, as well as the unique solvability of BDIEs and their equivalence to the original MBVPs. This analysis is based on the corresponding properties of the MBVPs in weighted Sobolev spaces that are proved as well.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain

AU  - Eshetu Seid Ahimed
Y1  - 2024/08/30
PY  - 2024
N1  - https://doi.org/10.11648/j.ijtam.20241002.12
DO  - 10.11648/j.ijtam.20241002.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 23
EP  - 32
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20241002.12
AB  - In this paper, the direct segregated Boundary Domain Integral Equations (BDIEs) for the Mixed Boundary Value Problems (MBVPs) for a scalar second order elliptic Partial Differential Equation (PDE) with variable coefficient in unbounded (exterior) 2D domain is considered. Otar Chkadua, Sergey Mikhailov and David Natroshvili formulated both the interior and exterior 3D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order elliptic PDE with a variable coefficients. On the other hand Sergey Mikhailov and Tamirat Temesgen formulated only the interior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. However, in this paper we formulated the exterior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. The aim of this work is to reduce the MBVPs to some direct segregated BDIEs with the use of an appropriate parametrix (Levi function). We examine the characteristics of corresponding parametrix-based integral volume and layer potentials in some weighted Sobolev spaces, as well as the unique solvability of BDIEs and their equivalence to the original MBVPs. This analysis is based on the corresponding properties of the MBVPs in weighted Sobolev spaces that are proved as well.

VL  - 10
IS  - 2
ER  -

Copy | Download