mean value theorems for integral equations in 2d potential theory

Mean value theorems for integral equations in 2D potential theory

Subrata Mukherjee*, Salil S. Kulkarni

Department of Theoretical and Applied Mechanics, Cornell University, Kimball Hall, Ithaca, NY 14853, USA

Received 27 June 2002; received in revised form 16 September 2002; accepted 18 September 2002

Abstract

This paper presents a number of theorems, with proofs, related to mean values of certain integrals that arise in integral formulations for

boundary value problems in two-dimensional potential theory. These theorems can be useful, for example, for the understanding and

evaluation of new integral formulations and for simplifying existing ones.

q 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Potential theory; Integral equations; Boundary element method; Mean value theorems

1. Introduction

Integral formulations for boundary value problems in

two-dimensional potential theory are well known [1]. These

problems can involve simply or multiply connected

domains; can be either interior or exterior problems.

Integrals involving the logarithm of the Euclidean distance

rðP;QÞ (where P and Q are source and field points,

respectively), and those involving the normal derivative of

lnðrÞ (at a boundary point), commonly appear in these

formulations. This paper is concerned with certain theorems

that involve mean values of such integrals. A mean value is

defined here as the value of a double integral when the first

integral (which is a function of the source point P ), is

integrated once more over a closed contour, as P moves over

this contour. The second contour can be the same as, or

different from, the first one. Whenever necessary, these

integrals are defined in the Cauchy Principal Value (CPV)

or Finite Part (FP—as defined by Mukherjee [2]) sense.

These theorems can be useful, for example, for the

understanding and evaluation of new integral formulations

and for simplifying existing ones. They can also be useful

for checking computer codes.

Some of this work has been inspired by Khvisevich [3].

Unfortunately, however, this paper does not contain any

proofs and appears to have some wrong results in it.

Mr Khvisevich could not be contacted, nor could his reports

referenced in his paper be obtained by the authors of the

present paper.

2. General theorems on circles

Theorem 1. Let BI , R2 be a bounded domain with a

circular boundary ›B and let B0 ¼ R2 w ðBI < ›BÞ (see

Fig. 1). Also, let mðsÞ be any continuous function defined on

›B: One has:

ðaÞ

�KðIÞFP ;

1

2pL

ð›B

dsðPÞð¼

ðIÞ

›B

›ln rðP;QÞ

›nðQÞmðQÞdsðQÞ ¼ �m ð1Þ

ðbÞ

�KðOÞFP ;

1

2pL

ð›B

dsðPÞð¼

ðOÞ

›B

›ln rðP;QÞ

›nðQÞmðQÞdsðQÞ ¼ 0 ð2Þ

where

1.

�m¼1

L

ð›BmðsÞds ð3Þ

with L the length of ›B:

2. rðP;QÞ is the Euclidean distance between the points P

and Q [ ›B:

3. The unit outward normal to ›B at a point Q on it is

denoted by nðQÞ:

4. The symbolÐ¼

ðIÞ›B denotes the finite part of the appropriate

integral in the sense of Mukherjee [2] when the limiting

process involves the inside approach p [ BI !P [ ›B:

0955-7997/03/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.

PII: S0 95 5 -7 99 7 (0 2) 00 0 95 -4

Engineering Analysis with Boundary Elements 27 (2003) 183–191

www.elsevier.com/locate/enganabound

* Corresponding author. fax: 1-607-255-2011.

E-mail address: [email protected] (S. Mukherjee).

http://www.elsevier.com/locate/enganabound

In other words:

KðIÞFP ;

1

2p

ð¼

ðIÞ

›B

›ln rðP;QÞ

›nðQÞmðQÞdsðQÞ

¼ limp[BI!P[›B

1

2p

ð›B

›ln rðp;QÞ

›nðQÞmðQÞdsðQÞ ð4Þ

5. The symbolÐ¼

ðOÞ›B denotes the finite part of the appropriate

integral in the sense of Mukherjee [2] when the limiting

process involves the outside approach p [ B0 !P [ ›B:

In other words:

KðOÞFP ;

1

2p

ð¼

ðOÞ

›B

›ln rðP;QÞ


¼ limp[B0!P[›B

1

2p

ð›B

›ln rðp;QÞ

›nðQÞmðQÞdsðQÞ ð5Þ

Proof of Theorem 1(a). First some notation.

KFP: finite part of integral

KCPV: Cauchy principal value of integral. This is denoted

by the symbol O�KFP : mean value of finite part over appropriate contour

KðOÞFP : finite part with outside approach p [ B0 ! P [

›B

KðIÞFP : finite part with inside approach p [ BI ! P [ ›B

Next, an observation. A double integral (for an admissible

function f and with L0 a fixed number):

J0 ¼ðL0

0dy

ðL0

0f ðx; yÞdx ð6Þ

can be written as:

J0 ¼ðL0

0dx

ðL0

0f ðx; yÞdy ¼

ðL0

0dy

ðL0

0f ðy; xÞdx ð7Þ

where the first integral in Eq. (7) results from switching

the order of integration and the second from interchanging

x and y.

This idea can be applied to the mean value of the CPV

integral ð1=ð2pÞO›B ð›ln rðP;QÞ=›nðQÞÞmðQÞdsðQÞ since this

integral is, in fact, regular. One gets:

�KCPV ¼1

2pL

ð›B

dsðPÞ O›B

›ln rðP;QÞ


¼1

2pL

ð›B

dsðPÞ O›B

›ln rðP;QÞ

›nðPÞmðPÞdsðQÞ

¼1

2pL

ð›B

mðPÞdsðPÞ O›B

›ln rðP;QÞ

›nðPÞdsðQÞ ð8Þ

Now consider a harmonic function fðx1; x2Þ; x [ BI: The

well-known simple layer potential representation for this

problem is:

fðpÞ ¼1

2p

ð›B

ln rðp;QÞnðQÞdsðQÞ; p [ BI ð9Þ

Take the gradient of both sides of Eq. (9) at the source point

p, take the limit p [ BI ! P [ ›B; and take the dot product

with nðPÞ; the unit normal to ›B at P. This gives:

›f

›nðPÞ¼

1

2pO›B

›lnrðP;QÞ

›nðPÞnðQÞdsðQÞ2

nðPÞ

2; P[›B ð10Þ

With nðsÞ[›B¼1; Eq. (10) yields:

1

2pO›B

›lnrðP;QÞ

›nðPÞdsðQÞ¼

›f̂

›nðPÞþ

1

2ð11Þ

where f̂ðpÞ¼fðpÞln¼1:

However, for ›B a circle of radius b, and with

nðsÞ defined on ›B ¼ 1:

f̂ðpÞ ¼b

2p

ð2p

0ln rðp;QÞduðQÞ ¼ b lnðbÞ;

p[BI; Q[ ›B

ð12Þ

where use is made of the fact that with a$ 0; b$ 0; r2 ¼

a2 þb2 22ab cosðuÞ :

ð2p

0lnðrÞdu¼ 2p ln½maxða;bÞ� ð13Þ

But b lnðbÞ is a constant ;p[BI: Therefore:

›f̂

›nðPÞ¼ 0; P[ ›B ð14Þ

From Eqs. (8), (11) and (14):

�KCPV ¼�m

2ð15Þ

From Mukherjee [4]:

KFP ¼KCPV þ jump term ð16Þ

Fig. 1. A 2D infinite region B with circles ›B and C.

S. Mukherjee, S.S. Kulkarni / Engineering Analysis with Boundary Elements 27 (2003) 183–191184

For an inside approach, the jump term is mðPÞ=2: Therefore:

�KðIÞFP ¼

�KCPV þ�m

2¼ �m ð17Þ

which proves Theorem 1(a). A

Proof of Theorem 1(b). For an outside approach, the jump

term is 2mðPÞ=2: This time, application of Eq. (16) yields:

�KðOÞFP ¼ �KCPV 2

�m

2¼ 0 ð18Þ

which proves Theorem 1(b). A

Remark 1. The harmonic solution f of an interior problem

in potential theory in a simply connected domain, together

with a simple layer potential representation for this problem,

has been used to prove Theorem 1(a). A harmonic function

fðx1; x2Þ; x [ BI; however, can be generated with Eq. (9) by

any (admissible) source density on ›B: Also, the result of

Theorem 1(a) only involves m and no other function.

Therefore, Theorem 1(a) is true in general, i.e. for any

admissible function m on a circle ›B: Of course, Theorem

1(b) is also true in general.

Remark 2. It has been verified numerically that Theorem 1

is only true, in general, for circles and not for other closed

curves. On a general closed C2 curve ›B; one has, from Eqs.

(8), (11), (17) and (18):

�KðIÞFP ¼

1

L

ð›B

›f̂

›nðPÞmðPÞdsðPÞ þ �m ð19Þ

�KðOÞFP ¼

1

L

ð›B

›f̂

›nðPÞmðPÞdsðPÞ ð20Þ

with ›f̂=›nðPÞ given by Eq. (11).

Corollary 1. For a sufficiently smooth function n̂ðPÞ defined

on ›B with:

ð›Bn̂ðPÞdsðPÞ ¼ 0 ð21Þ

one has:

ð›B

dsðPÞð›B

ln1

rðP;QÞ

� �n̂ðQÞdsðQÞ ¼ 0 ð22Þ

Proof of Corollary 1. The well-known direct boundary

integral equation (BIE) formulation for a harmonic function

fðx1; x2Þ; x [ BI is (see Fig. 1):

fðpÞ ¼1

2p

ð›B

ln1

rðp;QÞ

� �›f

›nðQÞdsðQÞ

þ1

2p

ð›B

›ln rðp;QÞ

›nðQÞfðQÞdsðQÞ; p[BI ð23Þ

Taking the limit p[BI !P[ ›B in Eq. (23), integrating

once more around ›B and dividing by L gives:

�f¼1

L

ð›BfðPÞdsðPÞ

¼1

2pL

ð›B

dsðPÞð›B

ln1

rðP;QÞ

� �›f

›nðQÞdsðQÞ

þ1

2pL

ð›B

dsðPÞð¼

ðIÞ

›B

›ln rðP;QÞ

›nðQÞfðQÞdsðQÞ ð24Þ

Use of Theorem 1(a) with m replaced by f gives:

M ;1

2pL

ð›B

dsðPÞð›B

ln1

rðP;QÞ

� �›f

›nðQÞdsðQÞ ¼ 0 ð25Þ

For a harmonic function fðx1;x2Þ; x[BI;Ð›B ›f=›n¼ 0:

The function ›f=›n is otherwise unrestricted. Therefore,

one can replace ›f=›n in Eq. (25) with n̂: This completes the

proof of Corollary 1. A

Note that since lnðrÞ is log singular as r ! 0; Eq. (25) is

also true for an outside approach p [ B0 ! P [ ›B:

Remark 3. The direct BIE formulation for a simply

connected domain has been used to prove Corollary 1.

In view of Remark 1, however, Corollary 1 is true in

general.

Theorem 2. Let C be a circle of radius R in B0 (i.e. R is

large enough so that C encloses ›B see Fig. 1) and p [ C:

One has:

I ;ð

CdsðpÞ

ð›B

›ln rðp;QÞ

›nðQÞmðQÞdsðQÞ ¼ 0 ð26Þ

where mðQÞ is any continuous function defined on ›B:

Lemma 1.

I ¼ 0 , �fC ¼ �f ð27Þ

where:

�fC ¼1

LC

ðCfðsÞds ð28Þ

with LC ¼ 2pR: and f any harmonic function

fðx1; x2Þ [ B0:

Proof of Lemma 1. A standard indirect formulation for an

exterior problem for Laplace’s equation in 2D is [5]:

fðpÞ ¼1

2p

ð›B

›ln rðp;QÞ


þ1

2p

ð›BmðQÞdsðQÞ; p [ B0 ð29Þ

S. Mukherjee, S.S. Kulkarni / Engineering Analysis with Boundary Elements 27 (2003) 183–191 185

Taking the limit p [ B0 ! P [ ›B and integrating Eq. (29)

once more around ›B :

ð›BfðPÞdsðPÞ ¼ L �f

¼1

2p

ð›B

dsðPÞð¼

ðOÞ

›B

›ln rðP;QÞ


þL2

2p�m; P[ ›B ð30Þ

Using Theorem 1(b) in Eq. (30):

�f¼L

2p�m ð31Þ

Next, integrating Eq. (29) around the circle C:

ðCfðpÞdsðpÞ¼ LC

�fC

¼1

2p

ðC

dsðpÞð›B

›ln rðp;QÞ


þLCL

2p�m; p[C ð32Þ

Use of Eq. (31) and the definition of I results in:

�fC ¼I

2pLC

þ �f ð33Þ

which proves Lemma 1. A

Proof of Theorem 2. The direct boundary integral equation

(BIE) formulation for a harmonic function fðx1; x2Þ in the

exterior region x [ B0 is (see Ref. [6], also Fig. 1):

fðpÞ ¼ �fC þ1

2p

ð›B

ln1

rðp;QÞ

� �›f

›nðQÞdsðQÞ

þ1

2p

ð›B

›ln rðp;QÞ

›nðQÞfðQÞdsðQÞ; p [ B0 ð34Þ

ð›B

›f

›nðPÞdsðPÞ ¼ 0 ð35Þ

Take the limit p [ B0 !P [ ›B in Eq. (34), and integrate

once more around ›B: This gives:

L �f¼ L �fC þ1

2p

ð›B

dsðPÞð›B

ln1

rðP;QÞ

� �›f

›nðQÞdsðQÞ

þ1

2p

ð›B

dsðPÞð¼

ðOÞ

›B

›ln rðp;QÞ


Write Eq. (36) as:

�f¼ �fC þM þ �KðOÞFP ð37Þ

From Eq. (25) (with outside approach), M ¼ 0: Also, from

Theorem 1(b), �KðOÞFP ¼ 0: Therefore, �f¼ �fC and use of

Lemma 1 proves Theorem 2.

Remark 4. Again, in view of Remarks 1 and 3, Theorem 2

is true in general.

Theorem 3. Consider any continuous function mðsÞ defined

on a circle ›B of radius b and let C be another circle of

radius R that encloses ›B (see Fig. 1). Then

J ;1

2pLC

ðC

dsðpÞð›B

ln rðp;QÞmðQÞdsðQÞ

¼ �mIb lnðRÞ; p [ C ð38Þ

where LC ¼ 2pR; L ¼ 2pb and �mI ¼ ð1=LÞÐ›B mðsÞds:

Proof of Theorem 3. Apply the direct BIE formulation

for the doubly connected region B bounded by the circles

›B and C. For a harmonic function fðx1; x2Þ; x [ B; one

has:

fðPÞ ¼1

2p

ð›B

ln1

rðP;QÞ

� �›f

›nðQÞdsðQÞ

þ1

2p

ð›B

›ln rðP;QÞ

›nðQÞfðQÞdsðQÞ

þ1

2p

ðC

ln1

rðP;QÞ

� �›f

›nðQÞdsðQÞ

þ1

2p

ð¼

ðIÞ

C

›ln rðP;QÞ

›nðQÞfðQÞdsðQÞ; P [ C

ð39Þ

Now integrate this equation once more around C and

divide by LC to get:

�fC ¼1

2pLC

ðC

dsðPÞð›B

ln1

rðP;QÞ

� �›f

›nðQÞdsðQÞ

þ1

2pLC

ðC

dsðPÞð›B

›ln rðP;QÞ

›nðQÞfðQÞdsðQÞ

þ1

2pLC

ðC

dsðPÞð

Cln

1

rðP;QÞ

� �›f

›nðQÞdsðQÞ

þ1

2pLC

ðC

dsðPÞð¼

ðIÞ

C

›ln rðP;QÞ


Let q ; ›f=›n: The second integral on the right hand side

of Eq. (40) vanishes from Theorem 2 and the last one

equals �fC from Theorem 1. Applying the idea expressed in

Eqs. (6) and (7) to the third integral on the right hand side

of Eq. (40):

1

2pLC

ðC

dsðPÞð

Cln

1

rðP;QÞ

� �qðQÞdsðQÞ

¼ 21

2pLC

ðC

qðPÞdsðPÞð

Cln rðP;QÞdsðQÞ

¼ 2�qCR lnðRÞ ð41Þ

where q ; ›f=›n and use is made of Eq. (13).

Now, from Eq. (40):

the first integral on the right hand side of Eq:ð40Þ

¼ �qCR lnðRÞ ð42Þ


Eq. (42), however, is not a general result since J should

obviously involve the value of q on the inner, rather than on

the outer circle. Therefore, noting that:

0 ¼ð

CqðsÞds þ

ð›B

qðsÞds ¼ 2pR�qC þ 2pb�qI ð43Þ

and the fact that qðsÞ defined on ›B is arbitrary, one finally

gets:

J ¼ �mIb lnðRÞ ð44Þ

and Theorem 3 is proved. A

Remark 5. Again, in view of previous remarks, Theorem 3

is true in general.

3. A Theorem for the adjoint kernel

Theorem 4. Consider any continuous function nðsÞ defined

on a C2 curve ›B: The region inside ›B is BI; which is simply

connected, the region outside ›B is B0; and BI < B0 <›B ¼ B ¼ R2 (see Fig. 1). One has:

ðaÞ �NðIÞFP ;

1

2pL

ð›B

dsðPÞð¼

ðIÞ

›B

›ln rðP;QÞ

›nðPÞnðQÞdsðQÞ

¼ 0 ð45Þ

ðbÞ �NðOÞFP ;

1

2pL

ð›B

dsðPÞð¼

ðOÞ

›B

›ln rðP;QÞ


¼ �n ð46Þ

Proof of Theorem 4(a). A harmonic function fðx1; x2Þ;

x [ BI can be represented by a simple layer potential:

fðpÞ ¼ð›B

lnðrðp;QÞÞnðQÞdsðQÞ ð47Þ

Take the gradient of Eq. (47) with respect to the source

point x; the limit p [ BI ! P [ ›B; and, finally, the

inner product of both sides with the unit normal nðPÞ:

The result is:

›f

›nðPÞ ¼

ð¼

ðIÞ

›B

›ln rðP;QÞ

›nðPÞnðQÞdsðQÞ ð48Þ

Integrating once more around ›B; one gets the required

result:

0¼ð›B

›f

›nðPÞdsðPÞ¼

ð›B

dsðPÞð¼

ðIÞ

›B

›ln rðP;QÞ


ð49Þ

where the well-known fact thatÐ›B ð›f=›nÞds¼0 has

been used. A

Proof of Theorem 4(b). Applying the idea expressed in

Eqs. (6) and (7) to the CPV integral below, one has:

�NCPV ¼1

2pL

ð›B

dsðPÞ O›B

›ln rðP;QÞ


¼1

2pL

ð›B

dsðPÞ O›B

›ln rðP;QÞ

›nðQÞnðPÞdsðQÞ

¼1

2pL

ð›B

nðPÞdsðPÞ O›B

›ln rðP;QÞ

›nðQÞdsðQÞ

¼1

2L

ð›BnðPÞdsðPÞ ¼

�n

2ð50Þ

where the well-known fact:

O›B

›ln rðP;QÞ

›nðQÞdsðQÞ ¼ p ð51Þ

has been used.

Of course, Eq. (50) can also be proved directly from

Theorem 4(a) and the equation:

NðIÞFP ¼ NCPV þ jump term ð52Þ

with the fact that the jump term in this case is 2nðPÞ=2:

Now apply the equation ½NFP ¼ NCPV þ jump term� for an

outside approach p [ B0 ! P [ ›B to get

�NðOÞFP ¼

�n

2þ

�n

2¼ �n ð53Þ

to prove Theorem 4(b). A

Remark 6. Again, in view of previous remarks, Theorem 4

is true in general.

4. Applications of theorems

4.1. Direct determination of constant for potential theory

in doubly connected planar domains

A doubly connected planar domain B, with an outer

boundary ›B0 (which is a circle of radius a ) and an inner

boundary ›BI (which is a C2 curve), is shown in Fig. 2. Also,

S is any point inside BI and L0 is the length of ›B0:

Let fðx1; x2Þ be a harmonic function in B. It is known that

[1]:

fðpÞ ¼1

2p

ð›B0

›ln rðp;QÞ


þ1

2p

ð›BI

›ln rðp;QÞ

›nðQÞmðQÞdsðQÞþA lnðrðp;SÞÞ ð54Þ

ð›BI

mðQÞdsðQÞ ¼ 0 ð55Þ

where A is a constant, p[B and rðp;SÞ is the Euclidean

distance between p and S.


Take the limit p [ B ! P0 [ ›B0 and integrate Eq. (54)

once more around ›B0 to get:

ð›B0

fðP0ÞdsðP0Þ

¼1

2p

ð›B0

dsðP0Þð¼

ðIÞ

›B0

›ln rðP0;QÞ


þ1

2p

ð›B0

dsðP0Þð›BI

›ln rðP0;QÞ


þ Að›B0

ln½rðP0; SÞ�dsðP0Þ ð56Þ

It has been observed from computations, but not proved

yet, that Theorem 2 is also valid when ›BI is any C2 closed

curve (i.e. not necessarily a circle). Accordingly, the

second integral on the right hand side of Eq. (56) vanishes.

Now divide Eq. (56) by L0 and use Theorem 1(a). The

result is:

�f0 ¼ �m0 þ A lnðaÞ; P0 [ ›B0 ð57Þ

Eq. (57) yields a direct expression for the constant A

which is:

A ¼�f0 2 �m0

lnðaÞð58Þ

4.1.1. An example with two circles

This example concerns a harmonic function fðx1; x2Þ

defined in a doubly connected region between two circles—

the outer one, ›B0; of radius a and the inner one, ›BI; of

radius b, centered at Sðb; 0Þ (Fig. 3). Dirichlet boundary

conditions, f ¼ f0 and f ¼ fI are prescribed on ›B0 and

›BI; respectively.

Application of averaged BIE. Define:

EC½f � ¼1

LC

ðC

f ðsÞds ¼ �fC ð59Þ

where C is a closed curve with length LC:

Applying Eq. (57) to the situation in Fig. 3(a), one

has:

�f0 ¼ �m0 þ A lnðaÞ ð60Þ

Now take the limit p [ B ! PI [ ›BI and integrate

Eq. (54) once more around ›BI and divide by LI ¼ 2pb:

Fig. 2. A planar domain B bounded by closed curves ›B0 and ›BI:

Fig. 3. An example with two circles (a) physical domain (b) mapped domain.


One gets:

1

LI

ð›BI

fðPIÞdsðPIÞ

¼1

2pLI

ð›BI

dsðPIÞð›B0

›ln rðPI;QÞ


þ1

2pLI

ð›BI

dsðPIÞð¼

ðOÞ

›BI

›ln rðPI;QÞ


þA

LI

ð›BI

ln½rðPI; SÞ�dsðPIÞ ð61Þ

Using Theorem 1(b), one gets:

�fI ¼ II0 þ A lnðbÞ ð62Þ

where II0 is the first integral on the right hand side of

Eq. (61).

Unfortunately, Eqs. (60) and (62) contain three

unknowns ð �m0;A and II0Þ that cannot be solved directly.

The reason for this, of course, is that only the averaged BIEs

(rather than the actual ones) are being employed here.

Use of analytical solution. It is interesting, however, to

use the analytical solution of this problem in order to check

Eqs. (60) and (62) for consistency. This analytical solution

can be obtained by employing complex variable mapping

techniques (see Ref. [7]). The mapping function used here

is:

wðzÞ ¼ðz=aÞ2 a

az 2 að63Þ

This mapping function maps the physical region of interest

into a pair of concentric circles of radii 1=a and a=a;

respectively (Fig. 3(b)), where a satisfies the equation a2 2

ða=bÞaþ 1 ¼ 0 and has the value given below. (The

physical outer circle becomes the smaller one in the mapped

domain and vice versa):

a ¼a þ

ffiffiffiffiffiffiffiffiffiffiffia2 2 4b2

p

2bð64Þ

The solution of the problem is:

fðx1; x2Þ ¼1

ln af0 ln

a

ar

� �þ fI lnðarÞ

� ð65Þ

where z ¼ x1 þ ix2 and r ¼ lwl:Now consider the circles C1 (concentric with ›BI; radius

c1) and C2 (concentric with ›B0; radius c2), respectively, in

Fig. 3(a). The circle:

C1 : z ¼ b þ c1 eiu ð66Þ

maps into wðzÞ for which:

r ¼ lwl

¼½ðaa 2 bÞ2 2 2c1ðaa 2 bÞcosðuÞ þ ðc1Þ

2�1=2

a½ða 2 abÞ2 2 2ac1ða 2 abÞcosðuÞ þ ðac1Þ2�1=2

ð67Þ

which is written as:

lnðrÞ ¼ ln½f ðuÞ�2 lnðaÞ2 ln½gðuÞ� ð68Þ

Now, using Eq. (13):

EC1½ln½f ðuÞ�� ¼ lnðaa 2 bÞ; since aa 2 b $ c1 ð69Þ

EC1½ln½gðuÞ�� ¼ lnðac1Þ; since ac1 $ a 2 ab ð70Þ

Using Eqs. (68)–(70):

EC1½lnðrÞ� ¼ ln

ab

ac1

� �ð71Þ

Finally, from Eqs. (65) and (71):

EC1½f� ¼ �f1 ¼

1

ln a�f0 ln

c1

b

� �þ �fI ln

ab

c1

� �� ð72Þ

The circle C2 in Fig. 3(a) has the equation z ¼ c2 eiu:

Following exactly the same procedure as the above, one gets

the analytical solution:

�f2 ¼1

ln a�f0 ln

ac2

a

� �þ �fI ln

a

c2

� �� ð73Þ

Now:

�f1 2 �f2 ¼ð �f0 2 �fIÞ

ln aln

a

ba

� �þ ln

c1

c2

� �� ð74Þ

Let p [ C1: Integrate Eq. (54) over C1 and divide by 2pc1:

Now let p [ C2: Integrate Eq. (54) over C2 and divide by

2pc2: Subtract the second expression from the first and

compare with Eq. (74). The result is:

A ¼ð �f0 2 �fIÞ

ln að75Þ

As expected, if f0 ¼ fI (constants), A ¼ 0 and fðx1; x2Þ is

uniform in B. It is interesting to note that A is also zero if�f0 ¼ �fI:

Solving Eqs. (60) and (62), with Eq. (75):

�m0 ¼1

lnðaÞ½ �f0 lnða=aÞ þ �fI lnðaÞ� ð76Þ

II0 ¼1

lnðaÞ½ �f0 lnð1=bÞ þ �fI lnðabÞ� ð77Þ

4.2. Verification of some known failures in exterior domains

4.2.1. Indirect formulation

Consider the following equation (see Fig. 1):

fðpÞ ¼1

2p

ð›B

›ln rðP;QÞ

›nðQÞmðQÞdsðQÞ; p [ B0 ð78Þ

Take the limit p [ B0 ! P [ ›B; integrate again over ›B;

and use Theorem 1(b) to get:

�f ¼ 0 ð79Þ


which is too specific since one must be able to prescribe

f defined on ›B in arbitrary fashion. A correct formu-

lation appears in Eq. (29).

4.2.2. Direct formulation

Consider:

fðpÞ ¼1

2p

ð›B

ln1

rðp;QÞ

� �›f

›nðQÞdsðQÞ

þ1

2p

ð›B

›ln rðp;QÞ

›nðQÞfðQÞdsðQÞ; p [ B0 ð80Þ

Take the limit p [ B0 !P [ ›B; integrate again over ›B;

use Corollary 1 (hereÐ›B ð›f=›nÞðQÞdsðQÞ ¼ 0) and Theo-

rem 1(b), to again get:

�f¼ 0 ð81Þ

which is again too specific. A correct formulation appears in

Eq. (34).

4.3. Examination of a proposed new formulation

A new formulation is proposed here that always leads to

an integral equation of the second kind for Dirichlet

problems. Let:

fðpÞ ¼b �f1

þ1

2p

ð›B

ln rðp;QÞþ›ln rðp;QÞ

›nðQÞ

� hðQÞdsðQÞ ð82Þ

where b¼ 0 for interior and b¼ 1 for exterior problems.

4.3.1. Simply connected domain—interior problem

Consider Fig. 1 with p [ BI and ›B a circle of radius b.

Its perimeter L ¼ 2pb: Take the limit p [ BI ! P [ ›B and

integrate Eq. (82) once more around ›B: Using Eqs. (6), (7)

and (12), one has:

1

2pL

ð›B

dsðPÞð›B

ln rðP;QÞhðQÞdsðQÞ

¼1

2pL

ð›B

hðPÞdsðPÞð›B

ln rðP;QÞdsðQÞ

¼1

L

ð›BhðPÞdsðPÞb lnðbÞ ¼ b lnðbÞ �h ð83Þ

Also, from Theorem 1(a):

1

2pL

ð›B

dsðPÞð¼

ðIÞ

›B

›ln rðP;QÞ

›nðQÞhðQÞdsðQÞ ¼ �h ð84Þ

Now, using Eqs. (83) and (84), one gets:

�h ¼�f

1 þ b lnðbÞð85Þ

4.3.2. Region exterior to a simply connected domain

Consider Fig. 1 with p [ BI and ›B a circle of radius b.

Following the same procedure as in Section 4.3.1, one gets:

�h ¼�f2 �f1

b lnðbÞð86Þ

It has been proved before in Theorem 2 that the value of �f

remains the same on any other circle that encloses ›B:

Therefore, �f ¼ �f1 and �h ¼ 0:

4.3.3. Doubly connected domain—interior problem

Referring to Fig. 3(a), let the outer circle have radius a

and the inner one have radius b. For this problem, Eq. (82)

becomes:

fðpÞ ¼1

2p

ð›B0

›ln rðp;QÞ

›nðQÞ

� hðQÞdsðQÞ

þ1

2p

ð›B0

ln rðp;QÞhðQÞdsðQÞ

þ1

2p

ð›BI

›ln rðp;QÞ

›nðQÞ

� hðQÞdsðQÞ

þ1

2p

ð›BI

ln rðp;QÞhðQÞdsðQÞ; p [ B ð87Þ

Let p [ B ! P [ ›B0; integrate once more around ›B0 and

divide by L0 ¼ 2pa: Use, respectively, Theorem 1(a), Eqs.

(6), (7) and (13), Theorem 2 and Theorem 3 for the four

resulting integrals on the right hand side of this new

equation. One gets:

�f0 ¼ �h0 þ �h0a lnðaÞ þ �hIb lnðaÞ ð88Þ

Now let p [ B ! P [ ›BI; integrate once more around ›BI

and divide by LI ¼ 2pb: Use Theorem 1(b) and Eqs. (6), (7)

and (13) for the third and fourth terms on the right hand side

of this equation. This time one gets:

�fI ¼ II0ðh0Þ þ JI0ðh0Þ þ �hIb lnðbÞ ð89Þ

where:

II0¼1

2pLI

ð›BI

dsðPIÞð›B0

›lnrðPI;QÞ

›nðQÞhðQÞdsðQÞ; PI[›BI

ð90Þ

JI0¼1

2pLI

ð›BI

dsðPIÞð›B0

lnrðPI;QÞhðQÞdsðQÞ; PI[›BI

ð91Þ

The goal here is to prove that, given f0 and fI; Eqs. (88) and

(89) can be uniquely solved for h0 and hI: Unfortunately,

however, this has not been possible since simple expressions

could not be obtained for II0 and JI0: In fact, it has been

verified numerically that these integrals depend not only on

�h0 but also on higher moments of the function h0ðsÞ

defined on ›B0: For the special case of h0ðsÞ¼h0; a

constant, however, one gets the simple expressions II0¼

h0; JI0¼h0a lnðaÞ: In this special case, it is easy to show that


this new formulation, given �f0 and �fI; leads to a unique

solution for �h0¼h0 and �hI:

Acknowledgements

The research presented in this paper has been supported

by grant # 9912524 of the National Science Foundation to

Cornell University.

References

[1] Mikhlin S. Linear integral equations. Delhi, India: Hindustan

Publication Corporation; 1960.

[2] Mukherjee S. Finite parts of singular and hypersingular integrals with

irregular boundary source points. Engng Anal Bound Elem 2000;24:

767–76.

[3] Khvisevich V. Solution of the exterior and the interior Dirichlet

problem of potential theory in a multiply connected domain. J Engng

Phys Thermophys 1991;61:1435–8.

[4] Mukherjee S. CPV and HFP integrals and their applications in

the boundary element method. Int J Solids Struct 2000;37:

6623–34.

[5] Greenbaum A, Greengard L, McFadden G. Laplace’s equation and the

Dirichlet–Neumann map in multiply connected domains. J Comput

Phys 1993;105:267–78.

[6] Kress R. Linear integral equations. Applied mathematical series, vol.

82. New York: Springer; 1999.

[7] Greenberg MD. Foundations of applied mathematics. New Jersey:

Prentice-Hall; 1978.


mean value theorems for integral equations in 2d potential theory

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