Integral equations of first kind

+ P(t0)5(p +/f(p = /(t„),

t„ - 7,

(1-25)

where 11 rr

5

iri J t 7

cp-

i

(p(t)dt (p(t)dt (1.26)

tft 0

j /C(t Q , t)cp(t

(1.27)

7 7 is a closed or nonclosedLyapunov curve in the plane of complex variable z = x + iy, £Q and £ are ?affixes of points on 7. Functions a, p, cp, K(tr,, t) are given. Function cp is to be found. All these functions are, generally speaking, complex-valued Hoelder-continuous, i.e., they satisfy the Hoelder condition with constant L and the Hoelder exponent h, 0 < h < 1. In the theory of equation (1.25), the class C°'h(i) of Hoelder-continuous functions turns out to be natural not only, because the singular Cauchy integral S in formula (1.26) exists in the sense of the principal value for this class of functions. The fact is important that the class of functions C ' with the Hoelder exponent h < 1 is invariant for operator S and the operator K defined according to formula (1.27) is completely continuous even in class C ' (7) in the case of curves 7 of a finite length. n h We shall sometimes refer to the class of functions C ' as class K. In the preceding Section it has was already been noted, that the occurrence of a weak movable singularity of the form (1.23) in kernel K(x, t) does not prevent operator K from being completely continuous. In fact, the singularity disappears in the function

K{m)(tQ,

t) = I K(tQ,

t1)K(m-1\tv

t)dtv

16 * ( 0 ) ( t 0 , t)

- K(tQ,

t),

obtained by the integration repeated a required number of times which is based on the transposition of the succession of integration

| K(tQ9

^ K{<m~1\t1,

t1)dt1

t)y(t)dt

=

= [ tf(w)(t0, t)y(t)dt.

(1.28)

On the other hand, in the case of a singular Cauchy kernel \t - t n instead of expression (1.28), Poincare-Wrtrani formula40■55,56,67,68,74,75,84,90) 1

r

dt

the

r Cp(t, £ j )

~2 J ; — — J —

— d t i = ~ *(*()> *o>

cp(£,

t.)

+ -^ dt dt,

dt

Jti ll * 7

7 (* ~ *oH*l " O

takes place for any function cp(t, £-.) which is Hoelder-continuous with respect to both variables t and t-, . This formula also holds under more general assumptions (see, for example, Reference 50). It should be noted that the function kr,(tr), t) defined by the formula tx) ( t - tQHt1 ' (t
* 0n ( 0t n , O

=

J

- t)

<**il

is either continuous or has a weak singularity at £ = £Q. Moreover, this function is identically equal to zero in the

17 case of a closed contour 7, if 9 does not depend on L . The singular integral S is closely related to the Cauchy type integral 1

f

y(t)dt

Hz)

= — J • (1.29) iri t - z 7 The expression F(z) defined by formula (1.29) is an analytic function of complex variable z everywhere except for the curve of integration 7. When 7 is a simple closed smooth curve, let D~ and Z)+ respectively denote the simply connected regions of plane z inside and outside curve 7. Function F(z) can be continuously extendible onto curve 7 both from inside D+ and from outside D~. In other words, the limits F+{tQ)

F (tQ)

z e 2)+,

= lim F(z), z-t0

= lim

F(z),

z E D ,

tQ

e 7,

z-£0 exist which satisfy the Sokhotskii-Plemelj formulae F+(tQ)

= cp(tQ) + Sr

F (tQ)

'

'

'

= -
where S is the singular Cauchy integral (1.26). In space C ' (7), operator S is bounded, thereforeequality (1.25) can be naturally considered as a linear equation in this space. In the theory of singular integral equations of the form (1.25), the requirement that everywhere in 7a(£Q) + (t Q ) * 0, is

essential.

This

a(t Q ) - p(t Q ) * 0

requirement

is

referred

to

as

the

18 condition of normal solvability. If this requirement is fulfilled, there is no sense to divide equations of the form (1.25) into integral equations of the first, second, and third kinds any longer. In the theory of normally solvable singular integral equations (1.25), the giving below statements are of fundamental importance. In the scientific literature they are referred to as the Neother theorems ' ' ' . The first Neother Theorem. The numbers I and I' of linearly independent solutions of the homogeneous equation

7\p0 = 0

corresponding to equation homogeneous equation

(1.30)

(1.25)

1

r

T if - a(t 0 H(t 0 )

+ [ K(t,

and its assosiated

p(t)+(t)

+

tQ)ty(t)dt

=0

(1.31)

1 are both finite. The second Neother Theorem. The non-homogeneous equation (1.25) is solvable if and only if

| f{t)*?k{t)dt

k = 1, ..., l\

= 0,

(1.32)

7 where j^t(^)f a r e a ll "the linearly independent solutions of equation (1.31). When conditions (1.32) are satisfied, general solution of equation (1.25) has the form

the

I



=

**<*<))

+

£

c

*

k=\ where i(pL:(t)f a r e all "the linearly independent solutions of

19 equation (1:30), {cA

are arbitrary constants, and cp*(tQ) is

a particular solution of equation (1.25). The third

Neother

Theorem.

The number K = I - I'

to so

called index of integral equation (1.25) or operator T is determined by the formula 1 r a - 3-i K = — L arg , 2TT a + pJ7 where

• • •

(1.33)

denotes the increment of the function inside

the square brackets for a single-valued circuit of the contour 7 in the positive direction, i.e. in the direction, which leaves the finite region D+ on the left (7 is a Lyapunov closed curve or a set of such curves). The Noether Theorems remain valid also in the cases, when: a) formula (1.25) is a system of singular integral equations with the number of equations equal to the number of functions to be found. The condition of normal solvability is det(a + p) *-0,

det(a - p) * 0.

(1-34)

Index K is given by the formula 1 K = — 2ir

r L

d e t ( a + p) T arg det(a -

p)

J

,

(1.35)

7

instead of expression (1.33). The condition of normal solvability (1.34) and the formula for the index (1.35) have been established in Reference 40). b) functions a(t) and p(t) are continuous. In expression (1.26) for operator S , the integral is understood in the sense of Cauchy-Lebesgue. Operator K is completely continuous in space L~(7)i c) contour 7 is an Lyapunov curve (or a set of such curves). Index K is also given by the modified expression

20 (1.35) 3 7 ' 5 0 ). In the theory of one-dimensional integral equations of the form (1.25), the possibility of reducing them to the Fredholm integral equations of the second kind is important. In the case of many-dimensional singular integral equations this is not always possible. The analogues of the above Neother Theorems have not so far been established for them.

5. Normally Solvable Linear Equations Spaces. The Hausdorff Theorems.

in

Linear

Metric

In this Section, some basic statements from the theory of linear equations in linear metric spaces due to Hausdorff will be started 4 3 ' 5 3 ' 5 4 ' 7 3 . Let E and E be linear metric spaces and let T be a x y linear mapping of E x into Ev rep = /.

(1.36)

Let also Lr be the image of E under mapping (1.36). Like in the introduction to the present Chapter, E and L are called linearly homeomorphic if the correspondence established between them by mapping (1.36) is one-to-one and the inverse mapping

wf = wTy = v cp. The equality

21 v = wT

(1.37)

determines a linear mapping of E onto L = E T Q E . It is called conjugate to mapping (1.36). Along with mapping (1.36), we will consider the mapping cp' = T' f of space Er into E and perform them successively

,

/■ = 7y = r(r/) = (7T')/. This induces mappings T'T and 77" of each of the spaces £ and £r into itself. If T'T = I is the identical operator which maps space E onto itself, operator T' is said to be left-inverse to operator 7\ while T is the right-inverse to T' . Similarly, if 77" = I is the identical operator which maps space Er onto itself, operator 7" is called the right-inverse to operator and T is called the left-inverse to 7" . If operator T has both a right-inverse and a left-inverse this operator is referred to as invertible. An operator T is invertible if and only if mapping (1.36) establishes a homeomorphism between spaces E and Er. F If spaces E

I + X k=\ makes sense and its sum T* mapping

Tk

gives the inversion of the

22 9 - 7\p = /.

Let F

denote the closed set of elements cp e E , for which cp * cp' Tcp = e, where 0 is the null element of space Er. Each The class modulO F is the set elements cp' e E for which element / e Lr has 9a class modulo F as its inverse image.

of the factor space Et = E IF of elements s£ with the norm y 5 9 cp II $11 =

inf

Hep -

cp0II = i n f IIcpII

eF


^

onto L, and 117*11 = Hall. If the inverse of mapping / = d£ is linear, mapping (1.36) is said to be continuously invertible. It should be noted that the fact that mapping (1.36) is continuously invertible does not imply that operator T has an inverse in the usual sense. The statement, that the mapping (1.36) of space E onto Lr is continuously invertible, is equivalent to the statement that the image of every set open in E is a set open in Lr. In order that mapping (1.36) of space E onto Lr be one-to-one and continuously invertible, it is necessary and sufficient that

inf

IIr II > 0 .

Il9ll=l

73^ The following three Hausdorff Theorems ' are principal for the theory of linear equations in linear metric spaces. The first Hausdorff Theorem. The image Lr of the Banach

23 space E under the linear mapping (1.36) is either complete or is a set of the first category in itself, according as to whether this mapping is continuously invertible or not. It is evident that, for equation (1.36) to be solvable with respect to cp, it is necessary that the equality

V = o, hold for all the functionals H>Q, which are solutions of the equation

w0r= e where 0 is the null element of space E . For equation (1.37) to be solvable with respect to w, it is necessary that

vcp0 = 0,

for all cpQ, which are solutions of the equation

7\p0 = 6, Equation (1.36) is called normally solvable if the above necessary condition for its solvability is also the sufficient one. The normal solvability of equation (1.37) is defined similarly. The second Hausdorff Theorem. Equation (1.36) is normally solvable if and only if the set

Lr = TE

f


is closed in space Er. For equation (1.37) to be normally solvable, it is necessary that the set

24 L

= E T v

w

be closed in space E . In the case when the space conjugate to E coincides with space E , this condition is also sufficient for the normal solvability of equation (1.37). The

third

Hausdorff

Theorem.

For equation (1.37) to be

normally solvable, it necessary and sufficient that mapping (1.36) be continuously invertible. It is important to note that, if E is a Banach space and Lr is complete, mappings (1.36) and (1.37) are both solvable and continuously invertible. Let operator T be of the form T

= cp + K

(1.38)

where K is a completely continuous operator. If E and Er are Banach spaces, then all the three Fredholm Theorems apply to the equation % = ? + % = / •

(1-39)

Equation (1.39) and operator (1.38) are also referred to as those of the Fredholm type. In a Banach space, operator T is of the Fredholm type if and only if the representation T = B + K

takes place where operator B is one-to-one and continuously invertible and operator K is completely continuous }. Let us now consider equation (1.36) in a Hilbert space E under the assumption that T is a completely continuous operator, which converts E into itself. Operator T is referred to as finite-dimensional if it can be represented in the form

25

Ty =

P 2 i=l

(
u.,

v. e H,

where the parentheses denote the scalar product in space E . TogetWAlongher with equation (1.36), we will consider the following two equations: T*w = 0

(1-40)

i|/ - \TT*ty = 0

(1.41)

and

where operator T* is conjugate to operator T. Equation (1.36) is solvable if and only if (/, w) = 0 for any solution w of equation (1.40), provided the series 00

2

\fcl«*l2

converges where A., are the eigenvalues of equation (1.41) and QLI are the Fourier coefficients of / with respect to the orthonormal set {^h} of the eigenfunctions of equation (1.41). Moreover, the following statements are valid: 1) Equation (1.36) is normally solvable if and only if operator T is finite-dimensional. 2) The Neother Theorems do not apply to equation (1.36), if operator T is completely continuous. If operator T is of the form 7^ = $ K(x,

t)y(t)dt

26 discussed in section 3 and

K(x,

t)

E L2(G*G),

/, cp e L 2 (G),

equation (1.36) becomes (1.15). According to section 3 of the present Chapter, this equation, therefore, is a Fredholm integral equation of the first kind. For equation (1.15) to be Hausdorff-normally solvable, it is necessary and sufficient that kernel K(x, t) be of the form <1

K{x, t) = X V*)fi£(0, i=\ i.e., be degenerate ' . The above short review of the methods and results of the available theory of linear equations in linear metric spaces shows that this theory can be applied only to narrow classes of linear equations of the first kind. Nowadays, a number of rather wide classes of linear integral equations of the first kind have been nevertheless studied in detail. These classes are most often used in modern mathematics and related branches of science and technique. They will be discussed in the next Chapters of the present monograph.

27 CHAPTER 2

GENERAL REMARKS ON LINEAR INTEGRAL EQUATIONS OF THE FIRST KIND

Let A be a completely continuous operator mapping a Hilbert space into itself. The equation Ay = f

(2.1)

where cp is the element to be found and / is a given element of this space, is called an equation of the first kind, unlike the equation 9 + Aq> - f considered in Chapter 1, which is called equation of the second kind. This Chapter is devoted to the clarification of the role played by integral equations of the first kind in mathematics and its applications. Let V denote a domain of the Euclidean space E of points x with Cartesian orthogonal coordinates x-^, ..., x , n ^ 1 with (n -1)-dimensional boundary dT) of class C ' for n * 2. The equality Kcp ^ f K(x, s

t)y(t)dst

= f(x),

x e 5,

(2.2)

is called an integral equation of the first kind. As in Sect. 3 Chap. 1, here the integration takes place over domain D or over its boundary S = dV for n > 1, K(x, t) and

28 f(x) are given functions and cp(*) is the function to be found, K(x, t) being referred to as the kernel of integral operator /C9 or integral equation (2.2) and f(x) being referred to as its right-hand side. In many cases, function Ky is of a higher degree of smoothness than cp(jc). For this reason, equation (2.2) has a solution not for every its right-hand side f(x) at all. As we know from Sects. 3 and 4 of Ch.1, an integral equation of the second kind is also not necessarily solvable for every its right-hand side. However, this is due to some simpler reasons.

1. Integral transformations as integral equations of the first kind

The equality (2.2) can be interpreted as an integral transformation of functions from space E to space Er. For example, the transformation 00

J eTx\(t)dt

= f(x)

(2.3)

0 is known as the Laplace integral equation (see [86], pp 60 61). If /(a) is a finite number, the equality (2.3) can be written as J e^-^^cp^t) = (x - a) I e^-^cp^Orft = 0

f(x)

where x{t)

= J e" a T cp(T)^ 0

for x > a. It follows that, for x > a, the integral in the left-hand side of (2.3) converges and for any E > 0 we have

29

-—

=J e x

E

[e

9l (t)J

rft.

0

In the notations

f (X) =

f(a + s + x) 1 8

cp < t ) = e "9.(0

+ JC

the equality (2.3) can be written as 00

J eTtx92(t)dt

=

f2(x).

0 Since function ^ is boundedand for large values of x > 0 behaves as e~£X, we can assume that function cp(je) in (2.3) is square summable or it belongs to the class of boundedcontinuous functions. Under these assumptions, it can be proved that the transformation (2.3) is uniquely invertible. The Stieltjes equation

r
•k0 t + x

= fix),

0 <x

and the Plank equation GO

\ -jf 0e

Xt

f(t)dt ~

= f(x),

0 < x

1

are close to equation (2.3) by their nature [57]. Supposing that cp(jc) is piecewise smooth at x > 0 and that the integral J jccr_1cp(jc)^A ix, 0 0

a < a <

converges absolutely we can prove that the transformation

30 f xz 0

l

q(x)dx

= /(z),

z = a + it,

a < a < 0,

(2.4)

known as the Mellin equation is reversible. Assuming that /(z) is an analytical function of variable z = a + it in the strip a < Rez < (3 under the condition that the integral

T f(o + it)dt converges in the same strip and, moreover, in a narrower strip a + 5 ^ a ^ 0 - 8 where 8 is an arbitrary positive number the limit lim

/(z) = 0

exists uniformly with respect to a, the transformation inverse to (2.4) being given by the formula .«

cp(jc) =

(T+i°o

[

x~zf{z)dz,

x > 0,

a = const.

(2.5)

g-loo

Equations (2.4) and (2.5) are generalizations of the well known formulae

f jcz-1e"^jc = r ( z ) , i U

— 9ar1

f

x~z Y(z)dz

= e _A: , a > 0,

J . ff-loo

connecting functions T(z) and eTx (see [57]).

2. On the representation of the solution of the Dirichlet problem for harmonic functions as a potential of a simple layer

31 In the theory of the Dirichlet problem for harmonic functions (elliptic equations of the second order) the solution of this problem is usually searched for as a potential (generalized potential) of a double layer. This is considered to be natural because, for determining the density of the potential, the Fredholm equation of the second kind is obtained, which is globally (locally) solvable under rather general assumptions concerning the behaviour of the solution to be found in a closed domain 2) U a2) and the smoothness of its boundary a2). The function u(x) harmonic in domain 2) and even continuous in 2) U dV cannot always be represented as a potential of a simple layer. For example, a function which is harmonic in domain 2), continuous in 2) U dV and does not have normal derivative on a2), can not be a potential of a simple layer, because the simple-layer potential should necessarily have normal derivative on a 2) under rather general assumptions. Nevertheless, under the requirement that the boundary dV of domain 2) be sufficiently smooth, a wide class of functions harmonic in this domain can still be represented as a potential of a simple layer. So we search for the solution of the Dirichlet problem "+(*0)

=

/(*o>'

*0

6 8Z)

'

(2

'6)

for functions u{x) harmonic in domain 2) in the form of a potential of a simple layer u(x)

= 1%

[ E(x,

t)y(t)dst

(2.7)

3D

where E(x, t) is an elementary equation in space E

solution of the Laplace

32 - logl* - t |,

H = 2 2%n/2

I E(x,

t) = 1

? „ |* - t\2'n,

" n>

r(n/2)

2,

n - 2 Since the potential of a simple layer is continuous in the whole space E for determining the density cp(t) we obtain the following integral equation of the first kind ([40], [55], [88]) —

f E(xQ9

t)
= f(xQ),

xQ E 3D,

(2.8)

% 3D Though the kernel of the integral operator in the left-hand side of the equation (2.8) has a singularity at XQ = t, this operator is completely continuous ([33], [95]) for a wide class of densities cp(£) .

3. A tangential derivative of the potential of a simple layer

The behaviour of the potential of a simple layer and its normal derivative is well known. Below we shall need the knowledge of the behaviour of a tangential derivative of the potential of a simple layer on the boundary of a domain. Let T) be a two-dimensional domain on the plane of complex variable z = x* + ijc2 with the boundary s = 3T) of class

Clfh.

In this case the potential of a simple layer (2.7) and the integral equation (2.8) can be written as

33

u(z)

= 2

f l o g | t - z\y(t)d ,

*

(2.9)

and

— [ l o g | t - tQW(t)dst 2ir „

= - f(tQ),

t - $1 + ^ 2 '

*0 ~ *10

tQ

+

e

S,

(2.10)

ix

2Q'

respectively. Under the above assumptions the equation (2.10) easily reduces to a singular integral equation of the first kind with the Cauchy kernel whose theory has been well developed. To make sure, we note that the following statement is valid: If
t'n

r

(p(t)t' dt

i

— =— dsQ

2v J

r

d

*(t, tQ)ds

+ — t - tQ

(2.11)

2ir J d j Q

where 5" and s^ are the arc abscissa of points t and £Q on 5,
_

1

The integral in the first term in the right-hand side of (2.11) is understood in the sense of the Cauchy principal value. In the second term it is understood in the usual sense, since S is a curve of class C ' and the derivative of each branch of function ti(t, £ Q ) can be written in the form d

m

ft(t, tQ) dsQ

= ft (t,

tQ)*(\t

-hQ - tQ|) u ,

34 where

ft*(t,

£ Q ) is Hoelder continuous and 0 < h^ < 1.

Let t+ and t 2 abscissas

denote the points on S with the arc

S Q - E and s^

+ e with a sufficiently

positive e, respectively, and S arc ^ I ^ Q ^ 2 ' ^

small

denote the part S out of

^ s °t )v i ous "that the limit lim uE(tQ)

=

u(tQ)

exists uniformly with respect to t n , where

"E^O^

=

— f log|t log|i " toMt^dst-

(2-12)

2* J E

Differentiating we o b t a i n

the equality

d

1

T~ MV

=

r d X

"—J —

rfjQ

( 2 . 1 2 ) with r e s p e c t t o s*

2TT J d j E

°SU " tQ|cp(t)^

Q

1 + — < p ( t 2 ) l o g | t 2 - tQ\ 2ir

1 9(t1)log|t1 -

tQ\.

2ir

Since t h e r e l a t i o n s

9(t2)log|t

-

tQ\

-


[(t 1 )]log|£ 2 - tQ\ - yit^log

-

tQ \

—i t

t

—,

l 2 ~~ 0 I i i . E-0

| t

i -

t

o

|

- i

t

1^2 " o'

hold and function cp(t) is Hoelder continuous, we get

35

lim [ q > ( t 2 ) l o g | t 2 - tQ\ - y(t1)log\t1

- tQ\]

= 0.

£-0

On the other hand, according to the equality \t - tQ\

= (t - tQ) exp (- id(t, t0))

we conclude that, for t e S , d

t'r, y

l o g |t - t f l | =

d i

*(t,

tn).

We t h u s have 1

d

i^ = -*■

f

l o g | t - tQ\
t'
r

t

t

i + —

d

r

■»(£, ta)v(t)ds.

(2.13)

E

Taking into account that the function t'(p(t) is Hoelder continuous and determining the integral in the sense of Cauchy principal value we obtain

lim

t'y(t)dt

r

=

t'y(t)dt

r

Jj TT77 I ~ 0 ~s

E->0

L

L

t

f

£

0

The limit r

d

lim

fl(t,

r

tn)
d

G(t, t 0 )cp(t)^

E

is the usual improper integral. The both integrals converge

36 uniformly with respect to £Q. Hence, passing to the limit as 8 —> 0 in (2.13), we obtain the formula (2.11). Thus, the above statement is proved ([12], [14], [16], [39], [55], [87]).

4. On the representation of the solution of Neumann's problem in the form of a potential of a double layer

Let us assume the boundary S of domain D to be a sufficiently smooth (n - 1)-dimensional surface. We shall consider the Neumann's problem as follows: it is required to find a harmonic in domain V function belonging to class C ' (2) U S) and satisfying the boundary condition dU dv

x

=
x

o

e s

(2-14)

>

where vxv is the unit vector of the outer normal to S in 0 point XQ and (p(x^) is a given function on S. It is known that for Neumann' s problem to be solvable it is necessary and sufficient that

J V(x)d sx = 0.

(2.15)

If this condition is satisfied, the solutions of the problem are of the form uAx) + C, where C is an arbitrary constant and uAx) is the potential of a simple layer U^x)

= —

f E(x,

t)v(t)dst.

Its density v is a solution of a Fredholm integral equation of the second kind. We search for the solution of the problem (2.14) in the form of a potential of a double layer

37

U(x)

=—

E(x,

t)iL(t)dst.

(2.16)

d

% { *t Under the requirement that boundary S and density \L be sufficiently smooth, we can use the well-known property that function u(x) has a continuous normal derivative in V U S, and write the boundary condition (2.14) as d

1

r

d

E(x, w

*

dv

xQ

s

t)\i(t)ds

= cp(jcQ),

JCQ e S.

(2.17)

dv

t

The equality (2.17) is an integral equation of the first kind with respect to ji(t).

5. An integral equation of the contact problem elasticity theory

in the

A wide class of the so called contact problems in the elasticity theory in a linear formulation is modelled in terms of integral equations of the form (2.10). For example, in the measurements of rigidity of the elastic bodies the determination of stresses in two elastic bodies pressed into each other is important. This problem has been studied by Hertz in the three-dimensional case under the assumption that near the contact area before deformation, the surfaces of 2these bodies 2were given by the equations of the form z = ax + 2bxy + Cy in the Euclidean space E^ of variables x, y, z, which are the orthogonal Cartesian coordinates. When the elastic bodies are infinite cylinders interacting along their rulings under the assumption that planar deformation spreads only over narrow layers near these rulings, the Hertz problem becomes two-dimensional. So, let us consider two planar elastic bodies D-, and Z^ with profiles S, and 5 2 , which come into contact with each other in point 0. Let us consider a Cartesian system of orthogonal

38 coordinates x, y with the origin in point 0. Let axis Ox be the common tangential to profiles S+ and S 2 ^ n "this point and axis Oy be directed towards /)-, . Let also the equations of S-, and S 2 be y^ = f^(x) and y 2 = fi^*) respectively. We assume that the principal vectors of external forces having an effect on D^ and Z>2 are parallel to axis Oy and are directed to D^ and Z>2, respectively, and their principal moments with respect to point 0 are equal to zero. Under the effect of these external forces, the bodies D+ and Z>2 will come into contact along some line 7 which is symmetric with respect to axis Oy. Let the segment [- a, a] of axis Ox be a projection of the line 7. The area of contact is regarded to be small as compared to the dimensions of D. and Z>2. The number a is to be determined. After a deformation of the considered bodies the equations of 5, and 5 2 on the segment [-a, a) become y = fx(x)

+ vx(x),

y = f2(x)

+ v2(x),

where v^ and v 2 are the vertical components of shift, they are expressed in terms of pressure p(x), [- a < x < a], as 1

. K

Vj(*) = 4

v2(x)

a

l

r

^1

-a

1

a

l + K2

f

log|£ - x\p{t)dt

+ const,

=

log|£ - x\p(t)dt + const. ^2 -a Here K^, K 2 , jij, fi2 are the elasticity constants. The pressure p(x) can therefore be determined from the following integral equation of the first kind with logarithmic kernel ([2], [40], [51], [52], [63]) 4

1 IT

a

f log |t - x\p(t)dt J

= f(x)

(2.18)

39 where

1 + K^ K =

1 + K^ 4-

4ji 1

.

4p. 2

6. A stream problem for the flow around a finite straight segment

A stream problem for the flow around a segment - a ^ x+ ^ a of the real axis of the plane of complex variable z = x, + i*2 by the vertical plane-parallel flow of an ideal in compressible liquid also reduces to an integral equation of the first kind [32]. In fact, let

w(z)

= cp(z) +

ity(z)

denote complex potential of the velocity field q = (q^, q xj) a r e "the velocity potential and the stream function respectively, i.e., dcp 8X-|

di|/ ftXry

dcp

6 i|/

dXss

dX*

If functions cp and i|/ have exact differentials, these relations impliy that function w is analytical in the domain occupied by the flow. We assume that l i m w'(z) z-*»

= qloo

- iq2oo

= const.

(2.19)

40 Since ^(z) is the current function, it should satisfy the boundary condition lim

i|/(z) = V K J ^ ) = C>

a

-

- xi

- a>

(2.20)

Z-^X-t

where C is a constant. On the basis of (2.19) and (2.20), we conclude that the stream problem under consideration reduces to the determination of the function co(z) = w'(z) - q+ + i#2oo' which is analytical on the plane cut along the segment - a ^ x-, ^ a. This function should vanish at infinity, be continuous on the segment - a ^ x+ ^ a and satisfy the boundary condition Ima)^) = i|/' {xx)

+ q2oo = q^.

(2.21)

Keeping in mind that function w(z) may turn to infinity at z = - a and z = a we shall search for a solution of the problem (2.21) in the form of the Cauchy type integral a

1

o)(z) = _ ■

■

>

-

"

-

\L(t)dt

(2.22)

!

TTl

«i

z

a * ~

with an unknown real density \i which is Hoelder continuous. According to the Sokhotskii-Plemelj formula, we obtain from (2.22) a

1

l

+ 0 (X.)

= \l(x.)

H

\i(t)dt

r |JL -a

- a <

" ~

JC-J

< a.

x

l

iri i t The boundary condition (2.21) can therefore be written as 1

a

-

J

r

\L(t)dt

*x TTl 4J tt —X Y

= " *2oo

( 2 - 23 )

where the integral is understood as the Cauchy principal value. The equality (2.23) is a singular integral equation of

41 the first kind. The density \L of the Cauchy type integral (2.22) should guarantee the complete continuously extendiblety of function u)(z) on the segment - a ^ x, ^ a. In Sect. 5 of Chap. 4 it will be shown that under the additional requirement for w(z) to have a limit on one of the ends of the segment - a ^ x+ ^ a, the solution \L of the integral equation (2.23) can be uniquely determined. We can easily see the similarity between the stream problem and the Hertz contact problem considered in the previous section.

7. Integral equations of the first kind in the theory of boundary value problems for elliptic systems

Let us consider

a

n X Aij(x)Uxx ij=l

scalar

linear

n X i=l

B.(x)UXm

+ l

J

partial

equation

+ C(x)U

= F(x)

(2.24)

l

which satisfies the condition of uniform ellipticity n

n

kr, = const,

n

k* = const,

in a bounded domain 2) of space E . In this case the fact that linear boundary problems are correctly posed is equivalent to the possibility for these problems to reduce to the Fredholm integral equations of the second kind. When (2.24) is the system of equations, i.e. y*. •, B^, C are square m*m matrices (m > 1) and u and F are the m-dimensional vectors, the requirement of uniformly ellipticity of this system can not always guarantee that the

42 Dirichlet problem is correctly posed. For example, the system of equations S22U, U, S dx1 dXy

s\ a\

2

-

l

32U, -T±± - 2 dX2 6X2

s8%\ s—_ = 0, 8x^3x2 dx^dx

e\

s\

- + —fdXlXldx axf B dx22 dx\

J- = 0 dx$ dx*

(2.25)

is uniformly elliptic on the whole plane of complex variable z = xx + i*2- However, the domain can be shown, where the problem of finding a regular solution w = u-[ + iu2 °f this system satisfying the Dirichlet boundary condition ™(t Ht00)

=

= ^O^o)' 70('o>'

0 G *0

t

aZ) dD

''

((222. 2 6 6) )

is not correctly posed. We can easily verify this by considering an example when © is a circle \z\ < 1. In this case, the corresponding to (2.26) homogeneous problem

V V oV> = 0,

\t I'Qo\' = 1, 1,

(2.27) (2.27)

has as non-trivial solution - the function wQ(z)

(z~z - l)cp(z), = (zz

(2.28)

where cp(z) is an arbitrary analytic function in circle 2). In the considered case of an elliptic system, the incorrectness of the Dirichlet problem (2.26) consists in the fact that the homogeneous problem (2.27) has an infinite set of linearly independent solutions, and non-homogeneous problem (2.26) is not necessarily solvable for each right-hand side 7 [4, 14]. To understand the nature of this phenomenon we consider

43 a simply connected domain D of class C1 'h on plane z. If we write system (2.25) in the form W

zz-°

we easily see that all its solutions regular in are represented by the formula w(z) = zcfr^z) + ^ ( z ) ,

domain D

(2.29)

where §i(z) and *h(z) are arbitrary functions analytical in » [3], [4], [14]. We assume 4>j(z) and ty^(z) to be continously extendible on 3D and write the boundary condition (2.26) in the following way w(t)

= ti\>1(t)

+ ^ ( t ) = 70(£),

t

e dD.

(2.30)

The equality (2.30) is the boundary condition for functions §Az) and *h(z) analytical in domain 2). We shall show that this problem is equivalent to the integral equation of the first kind

1

zJU - zTT7

r

ty--

—= 11

IT

H

=

= / ( 0 ,

(2-31)

* - T

l«l = l,

1=^*,

where z = z(£) is an analytic function conformly mapping the domain D on the circle

': {U\ < l} of the plane of complex variable £ and /(£) is a given

44 function defined on dd = cr [14], [42]. In fact, let us assume the existence of functions <\>^(z) and v|/1(z) of class Clfh (D U dV) analytic in domain D which satisfy the boundary condition (2.30). The functions 4>(0 4>i z ( 0 anc* + ( 0 - + 1 z ( 0 analytic in circle d are Hoelder continuous on d U J. In view of (2.30) these functions satisfy the boundary condition

^(7)4>(«) + +(«) = 7(0.

« e a,

(2.32)

where

7(0 = 70[Z(S)]. In circle d we represent the analytic function <)>(£) in the form of the Cauchy type integral 1

9(T)^T

r

♦ (5) =

•

(2.33)

2iri J T - 5 By virtue of Sokhoskii-Plemelj formula, from yields 1

1

r

(2.33)

(P(T)^T

4> ( O = " 9 ( 0 + , 2 2iri J T - 5

S - cr.

(2.34)

For the expression

+ + (0 = - 7(0 + iTTH + (5)

(2.35)

determined from (2.32) to be the limit values of function i|/(0 analytic in d it is necessary and sufficient that the equality +

1

p ++(T)<*T

iri Ja

T

- fb

45 hold. If we take into account Eq. (2.35), we can write 1

r 7(T)dT

iriJ

1

T — £

f

?(T)
iriJ

T - £

5 e g.

(2.36)

We now substitute the expression (2.34) for $ (£) in (2.36), and use the Poincare-Bertrand formula. In view of the evident equation = 0, (T - 0(T

- TX)

we conclude that the density cp(-r) of the Cauchy type integral (2.33) should be a solution of the integral equation (2.31) where

/(O = 2||7(0 - — L

d

. J

T

- ^

J

Conversely, let <)>((;) be a solution of equation (2.31). Determining the function <|>(5) ^ n circle d according to formula (2.33), we find (|>+(0- Eq. (2.32) then gives the expression for v|/ (£), 5 eff• By means of the Cauchy integral formula 1

r

*+(T)^T

+ (0 = — 2iri J T - £ we find function i|/(£) analytic in d. If we substitute the expressions

^(z) = ♦[r1(*)] ^d

^(z) = ^[r 1 ^)]

in the right-hand side of formula solution w(z) of the problem (2.30).

(2.29) we obtain the

46 8.The tautochrone problem

We now consider the motion of a material point in a gravity field as one more problem modelled in terms of integral equations of the first kind. Let a material point A/(jt, y) move under gravity in the plane with Cartesian orthogonal coordinates x, y from the position ?(£,, T\) , T] > 0 to the position 2(£n> 0), £Q > £, so that time t is the given function t = t (-n) of coordinate T\ measuring height. The problem is to determine the trajectory of point M(x, y) which is referred to as the tautochrone problem. 2 It is known that the scalar square v of the velocity vector dx

dy ^

dt

dt '

of point M(x, y) satisfies the relation v 2 = 2*(T| - y),

0 < y < ^,

(2.37)

where g is the gravity acceleration. If a(jc, y) is the angle between the velocity vector and the positive direction of axis x in the counterclockwise direction, relation (2.37) yields dy — = vsina = V2g(T\ - y)sina dt Since the trajectory x = x(y)

(2.38)

is unknown the quantity

9 00 =

(2.39) sina[*(y). y]

is unknown either. Eqs. (2.38) and (2.39) imply that

47

}

v{y)dy

J

Mgi-n - y)

0

or = fW 0 v^

-

(2.40)

y

where /(T!) = - •Zgt(T|). Function cp(y) should therefore be a solution of the integral equation of the first kind (2.40) with a variable upper limit referred to as Abel integral equation with exponent 1/2. In view of (2.39) only those solutions cp(y) of the equation (2.40) are physically meaningful which satisfy the condition |cp(y) | > 1. If we manage to find the solution cp(y) of this equation which possesses the above property, the geometrical equality j

= tga = (csc2« - 1)1/2= [ 9 2 ( > ) _ ^ 1 / 2

immediately gives the trajectory of the moving point in the form of the integral x-][,

2

iz)-iy/2dz.

9. Some general remarks on integral equations the book is concerned with

The reader may ask whether the class of integral equations

48 of the first kind being under consideration in this chapter is natural. The answer is positive and quite clear when we speak of integral transformations or Abel integral equations are meant. But it is also positive in all other cases. We shall now explain why it is so. Let us start with a well known classical electrostatic problem. Let V be the domain of the Euclidean space E^ of points x with Cartesian orthogonal coordinates x-,, x~<> ^3 occupied by conducting medium (body). The problem is to determine the density \i(t) of charge distribution along the boundary S of the domain D if the potential u(x) of electrostatic field induced by these charges is known to be constant in D. Since it is the Coulomb potential that is under consideration, function u(x) is a potential of a simple layer with density jx(t)

u(x)

= — 4TT

\i(t)dsi

\ — „ \t s \*

J

-*\

which is constant everywhere in D and in particular on i.e. , 1

4TT

r

S,

\i(t)ds [L

I It

= const,

JC0 e S.

(2.41)

Equality (2.36), which should be used to determine function \L(t), is a special case of integral equation of the first kind (2.8). This problem is also meaningful in the case of a planar electrostatic field. In this case T) is the plane cross section of a long cylindric conductor. The appearance of an integral equation of the first kind in the theory of contact problems (for example, equation (2.18)), in the Wing theory (see equation (2.23)), in the theory of sea rising tides, etc. is also due to the occurrence of the potential of a simple layer in the corresponding boundary problems. The integral equations of the first kind, that will be considered below, are mainly related (except for the Abel

49 integral equation) with the boundary value for equations of elliptic type, which are undoubtelly correctly posed. The only exception is equation (2.31) connected with the Dirichlet boundary value (2.26) for elliptic system (2.25). We remind the reader that, according to Hadamard, a problem is referred to as correctly posed if it has a solution and it is a unique and stable one. The context of Hadamard's studies [69], [70] where this definition was given, shows that he meant classical problems for the main types of partial differential equations of the second kind, in particular, the Dirichlet problem for elliptic equations, the Cauchy problem with a data of a spatial type and the characteristic Cauchy problem for hyperbolic equations, the first boundary problem, and the Dirichlet-Cauchy problem for parabolic equations. The property of correctness is inherent in the very problems considered in physics and other natural sciences modelled in terms of these equations. If in equations there are terms with lower derivatives, these problems are regarded to be investigated locally (not necessarily globally) [14]. When the problem of finding a solution of an equation of the first kind (2.1) is said to be incorrectly posed, it means the following. If operator A is completely continuous, its inverse operator, if any, is not even continuous. (Almost all the integral equations of the first kind considered in this book possess this quality). When it is possible, additional requirements imposed on the right-hand sides of these equations are nevertheless pointed out, under which the existence of a solution is proved. And when it is impossible the reasons for this are analyzed. Though simple methods of differentiation and fractional differentiation, which enable explicit inversion formulae of the considered integral equations to be derived, may be taken for regularization, they play a specific role in the up-to-date general theory of the regularization of incorrectly posed problems [1], [29], [33]. In the past few decades, incorrectly posed problems have been intensively investigated by specialists. It is

50 linear integral equations that a special attention has been paid to. Since in incorrectly posed problems most or some of the Hadamard equirements (existence, uniqueness, stability of solution) imposed on correctly posed problems are not satisfied. Most of the researchers either change the way the problems are posed by regularizing them or generalize the concept of solution by introducing various approximate, asymptotic solutions, quasi-solutions, etc. to make it possible to determine a stable way of developing these solutions. A new rapidly developing field of modern mathematics called "Incorrectly posed problems" has appeared. Though almost all the equations studied below have arisen from applications the development of there approximate solutions goes beyond the scope of the present book. In this sence too the book differs from other known books devoted to incorrectly posed problems [1], [29], [33].

51 CHAPTER 3

THE PICARD THEOREM OF SOLVABILITY EQUATIONS OF THE FIRST KIND

OF

A

CLASS

OF

INTEGRAL

1. Preliminary remarks

In this section we shall assume that in equation (2.2) the integration is performed over a finite interval (a, b) of change of real variables x and t i.e.

b Kip - [ K(x, a

t)q>(t)dt

= /(*),

a ^ x ^ b

.

(3.1)

The problem of determining a solution of Equation (3.1) is incorrectly posed. We can easily see this on simple examples. Example 1. The kernel K(x, t) of Equation (3.1) is a polynomial of power N with respect to variable x

N *(*, t) = X

an{t)xm

with continuous or square summable coefficients a (t). With the requirement that the function cp to be found is continuous or square summable, the given function / must be a polynomial N

/<*) = X

bmxm

where

b b m

=\

am(t)
52 It means that in the considered case equation (3.1) cannot have a solution for a non-polynomial right-hand side. Example 2. The right-hand side and the kernel of equation (3.1) are

and

K(x,

t)

= 1

respectively. It is clear that the function 1


arbitrary

b |

yQ(t)dt

= 0

a

is a solution of the corresponding to (3.1) homogeneous equation. Example 3. The integration interval coincides with the interval (-a, a) and the kernel has the form 1 K(x,

t)

=

. t - x

The integral is understood as the Cauchy principal value. Below in Sect. 6 of Chap. 6, it will be shown that if / belongs to class Cfi' , equation (3.1) possesses a single-parameter set of solutions in the same class. This preliminary discussion shows that while studying the solvability of equation (3.1) it is important to choose appropriate spaces of given functions and functions to be found

53 according to a given kernel K(x, t). The first basic result in this direction has been obtained by Picard [88].

2. Spectral problem for one system of the associated homogeneous integral equations

Developing the spectral theory of integral operators Schmidt has introduced a system of associated homogeneous integral equations b cp(jc) - X f K(x, a

t)ty(t)dt

b I|;(JC) - X f K(t,

x)y(t)dt

= 0

(3.2)

a with a spectral parameter X [97]. The value X for which system (3.2) has a non-trivial solution cp, i|/ is usually called an eigenvalue and the pair of functions cp, i|i is called the associated pair of eigenfunctions of this system corresponding to X. Let the kernel K(x, t) be a x

continuous function or function square summable at a ^

^

b.

t Together with k(x,

t)

let us consider symmetrical kernels b

K*(x,

t) = J /C(JC, i)K(t,

T)<*T

(3.3)

a and b Km(x,

t)

= J*(T,

X)K(T,

t)di.

(3.4)

a At b > a these kernels are positive. Let us verify this fact for the kernel K*(x, t) as an example. In fact, by virtue of (3.3) for any square summable function i(x) we have

54 b

b

b

b

2

| dx J K*(x, j07(*bO0^ = J dy[ J *(*, 307(*)«**] * 0a

a

a

a

Excluding first i|# and then cp, from (3.2) we obtain b cp(x) - p. {**(*, t)vm(t)dt a

= 0

(3.5)

and b V|/(JC) - [JL [#,(*, x)ty(t)dt a

(3.6)

2 where \L = X . Consequently, if X. is an eigenvalue of system (3.2) and (cp, i|/) is the associated pair of 2 2 eigenfunctions corresponding to X, we get \i = X . Moreover, X , cp and i|/ are an eigenvalue and the corresponding eigenfunctions of the kernels K*(x, t) and K*(x, £) respectively. The inverse statement is also valid: If \L is an eigenvalue of the kernel AT (JC , t) and

**<*> " %* J **<*> 0 9 w ( 0 ^ = 0, a

than q> and A

(3.7)

determined by the formula

+*<x> - ^m

b j *<*• 0
(3.8)

form an associated pair of eigenfunctions of system (3.2) corresponding to eigenvalue X = \ZjT. In fact, substituting the expression (3.3) for k*(x, t) into the second term in the left-hand side of (3.7) and using the equality (3.8), we obtain

55

**<x) - Sn ] K(*> O + ^ O d T = 0.

(3.9)

The equalities (3.9) and (3.8) coincide with the system (3.2) where X = X m = /jT~, cp = (p . and ib = A . It is clear that I|ITO is the eigenfunction of kernel K*(x, t) corresponding to eigenvalue X . As it is known (see [28], [122 - 123]), the real positive symmetric kernel K*(x, t) has eigenvalues. All of them being positive and also their number is not more than countable. Let us assume that these eigenvalues are numerated in the ascending order \\

< \\

< ...

Let us denote the system of eigenfunctions of kernels t) and k*(x, t) corresponding to (3.10) as


(3.10) K*(x,

(3-n)

and +!(*), + 2 (*), ...

(3.12)

respectively. Since the quantities under consideration are real, without any loss of generality, we can assume that all the numbers X , m =1, 2, ... are positive. In fact, if we simultaneously replace X and cp (x) by - X and - q>m(x), respectively, the system (3.9) and (3.8) remains unaltered. Let us arrange X in the ascending order \1

^ X 2 ^ ...

(3.13)

We have thus built the spectrum (3.13) of the eigenvalues of

56 system (3.2) and the corresponding system of associated pairs of eigenfunctions

( *i)» (
= q.

(3.14)

where s£ = 1, sjf = 0, k * m. On the basis of equalities (3.8) and (3.9) we have b a b b = kkkm J dx | /C(t, x)
= x

JtXw \

dt

a

\ **
b | K(T, x)cp^(T)^T a

O^O^T^T

a

\. r =

—

^ ( O ^ T ^ T

(3.15)

X

fc a In view of (3.15) we conclude that the system (3.12) is orthonormal if only the system (3.11) is orthonormal. Let f(x) be a real, square summable function on the segment a ^ x ^ b. Let also systems (3.11) and (3.12) be orthonormal. We denote the Fourier coefficients of function / with respect to systems (3.11) and (3.12) as a and B , i.e., a

b m = J f(*)*m(x)dx, a

The series

b Pm = J f(x)*(x)dx. a

(3.16)

57

*kVk(*)

X

(3.17)

Jc=l

is called the Fourier series of function f(x). The fact that / and cp, belong to space L~ does not necessarily mean that the series (3.17) converges or its sum gives function f(x) if the series converges. However, the obvious relation b

n

(

\2

/ - 2 *kvk a

dx £ 0

v

valid for any natural number n yields the Bessel inequality n

b

f2dx

I ^ f k=\

a

which implies that the series of the squares of Fourier coefficients a» converges and the following estimation is valid oo

b

z 4 - ] f2dx-

(3-18>

k=\ a The orthonormal system of functions (3.11) is called complete if the equality sign is realized in the estimation (3.18). The completeness of the system (3.11) means that there is no non-vanishing function cp e L~ orthogonal to each function cp7 of this system (see [28], p. 114). Below we shall speak about the representation of a class of functions in the form of a series in the functions of system (3.11) or (3.12).

3. Source-like functions. The Hilbert-Schmidt theorem

A given function f(x), a ^ x ^ b, is said to be source-like in terms of kernel K(x, t) if there exists a square summable function g{x) such that

58 b f(x) = J\ K(x, a

We assume that K(x, K(x, K(x,

t) t)

t)g(t)dt,

(3.19) (3.19)

a ^ x ^ b. b.

t) has the form \x t \ = \x - t I

K^yx, K^(x,

t) t)

(3.2U) (3.20)

where a is a constant, 0 * a * 1/2, and the function ^(x, t)is continuous with respect to the set of variables x, t at a ^ ^ t

-

c

m

-i

*-» i t

4-Vin-l-

t-Vn=»

-P" -n/^»4--I /-k-no

ofi/ !

I

-in

c v c f p m c

( "X 1 1 \

and (3.12) form associated pairs generated by equasions (3.2) in which the kernel coincides with the kernel of the integral operator in the right-hand side of equation (3.19). With these assumptions, the following Hilbert-Schmidl therrem holds: The source-liko function is the sum of the abeorutely and uniformly converging series 00

/(*) =- XZ
(3.21) (3-21)

m=l m=l

where am are the Fourier coefficients determined from the first formula (3.16). First of all we note that in the case of kernel of the form (3.20) a source-like function is continuous. Let ym denote the Fourier coefficients of the function g determined from the second formula (3.16), i.e. b

71mm = g(tHmm(t)dt, (t)dt, = J \ g(t)+

m = 1, 2, ... m=l,2, ...

(3.22)

a

Since am and ym are the Fourier coefficients of square summable functions / and g with respect to the orthonormal systems (3.11) and (3.12), the numerical series 00

v m=\ m=\

00

.Z

v^ m=\ m=\

Z

59 converge. We now write formula (3.8) in the form 1 - ♦„<*) ♦„<*) ==rJJ k(t, *(*, x)x)9m9(t)dt. - I (t)dt. m

(3.23)

a

By vertue of formulae (3.16), (3.19) and (3.23) we have b

b

0Lm = J cpm(*) f K(*, K(x, a b

t)g(t)dt

a

b

b

= r g(t)^t *<*, t)*(x)dx g(t)dt r k(x, t)*(X)dx = — r g(ouo^ x

=^.

It is clear that for each value of *, a * x * b, the series 1

00 oo

2 T<J(X)

(3.24) (3.24)

m=l ro=l ^m *m

converges. According x k

to

(3.23),

each term

of this series

c li

*nS )/ m is the s l are of the Fourier coefficient of function K(x, t) with respect to system (3.12). Due to (3.18), we have (3.25) where M is the upper bound of the continuous function b

J |t(x, Ol 2 *. It is easy to see that the series 00

1

X — VP»(*) VP*(*) x — /n=l /n /n=l m

(3-26) (3-26)

converges absolutely and uniformly. In fact, since the series X 2 converges, for any given E > 0 there exists a natural number 7 N such that

60 N+p

(3.27) m=N

for every natural /? On the basis of the classic inequality '

N+p

L

1

i

-

VM(x)\

N+p

N+p

, m=N

in view of (3.25) and (3.27), we obtain the following estimation f M+p

x I m=N

9»<*) X

i(E

m

which implies the absolute and uniform convergence of the series (3.26). Let R(x) = f{x)

-

S(x),

(3.28)

where S(x) is the sum of the series (3.26) We shall now show that R = 0, i.e. f(x) = S(x). We can easily verify this statement when kernel K(x, t) is continuous and symmetric. In fact, in this case the systems (3.11) and (3.12) coincide so that

**(*>

= k

m | *<*' 0 < P O T ( 0 ^

and

7 W (*) = | *(* )?„(*)<**. Since the Fourier coefficients of functions f(x)

and

S(x)

61 with respect to system (3.11) coincide, all the Fourier coefficients of R(x) with respect to the same system are equal to zero, i.e., b J R(x)vm(x)dx a

m = 1, 2, ... .

= 0,

The continuous and symmetric kernel possesses the followproperty: If function R(x) is orthogonal to all of the functions
t)dx = 0.

Multiplying the obtained equality by y(t) it, we find b b J dt J R(x)k(x, a a

t)g(t)dx

b = J R(x)f(x)dx a

and integrating

= 0.

Besides, it is obvious that b f R(x)S(x)dx a

= 0.

Hence, we have b | R2(x)dx a and therefore R(x)

b = f R(x)\f(x) a

- S(x)]dx

= 0

= 0, i.e., f{x)

- S(x).

We now return to the general case. Since the series (3.26) is uniformly convergent, we can integrate this series term by term. Its sum is equal to K(x, t)S(x). As a result we get

62 J*(x, t)S(x)dx

= X T ♦*(*).

(3.29)

/B=1

On the other hand, for the Fourier coefficients 8 with respect to the functions of system (3.12) of the source-like function b

b

h(x) =]"*(*, t)f(t)dt

=JK,(*,

a

t)g(t)dt,

(3.30)

a

we have that 8

m

=

6 b , b J /(x)<*x J *<*• 0**(0<*t = — j /(x)9j(I(x)dx a a ma

m

Since the kernel K*(JC, t) is continuous and symmetric, function h(x) is the sum of the uniformly converging series 00

h(x) = X ^ ^ ( J ) . TB=1

X

(3.31)

m

According to (3.28), (3.29), (3.30) and (3.31) we conclude that b f K(x, t)R(x)dx

= 0,

and therefore

6

fc

f dx f /C(x, t)R(x)g(t)dx a

= \ R(x)f(x)dx

= 0.

a

Further, we have 6 | K(x)9j||(*)
| R(x)(x)dx

= 0,

63 u

u

a

a

I R2{x)dx = I *(*)[/(*) - £(*)] dx = 0, i . e . , R(x) = 0, or 7

/ ( * ) = S(*) = S -5
X

(3.32)

/n

The Hilbert-Schmidt theorem is thus proved ([28], [114-117]). As a result of similar reasoning, we can conclude that the function b /*(*) = \ K(t,

x)h(t)dt

(3.33)

a i s t h e sum of t h e uniformly converging s e r i e s \)

00

/*(*)= X —♦„(*). 772=1

1„ = J MjO^OOd*.

771

(3-34)

a

4. The Fischer-Riesz theorem

Let an orthonormal system of square summable functions Xi(*)>

Xi(*), x2(*)> ••• be given number K The summable

(3-35>

on a segment J: a ^ x ^ b of finite length, and a real being put into correspondence to each Xm(x)• following Fischer-Riesz theorem holds: The square function (O(JC), x e I with the Fourier coefficients K.|

,

K ^ ,

• ..

(3 .36)

with respect to the system (3.35) exists if and only if the

64 series 00

X

K?

(K)

converges. The validity of the first part of the theorem is obvious, because ( K ) is the sum of the squares of the Fourier coefficients K. of the square summable function CO(JC) with respect to the orthonormal system (3.35). The proof of the second part requires some additional reasoning [28], [44], [62], [91]. With the help of the functions of sequence (3.35) we form a new sequence u)^x), B.2(-X), o)2(x), ..., (^(x), ....

(3.(3.

where m m

(3.38)

fc=i We assume that the series ( K ) converges. It means that for any positive number e there exists a natural number N such that P

v ^ 2

2 N+m ■&.<

K

(3.39) 3 39

* e£

< - >

m==

for every natural p. By virtue of (3.39) the obvious equality

| (*N+p ~ ^ I

d x

= 2 *£+„ m=l

yields the necessary and sufficient condition convergence of the sequence (3.37) in average

for

the

65

J (®N+p ~ ^2dx

< £

*

(3.40)

I The sequence (3.37), which satisfies the condition (3.40) for the convergence in average, possesses the following important property: In the interval I, there exists a square summable function CO(JC), which is the limit of this sequence in average. We shall now prove this statement. First of all, we note that if the condition (3.40) of the convergence of the sequence (3.37) in average is satisfied, for any given positive number 8, there exists a natural number N(8) such that the measure \i of the set of those points x in the interval 7, for which the inequality

!%<*) " wn<*)l > 8'

m

~ N(>>

n

~ N<8)'

holds is smaller than 8. In fact, since the sequence (3.37) converges in average, for any positive number 8 there exists a natural number N(b) such that the inequality

I holds for any m ^ N, n ^ N. We can therefore write M 2 * ! ( « „ - <»m)2dx < 83, I i.e. , |i < 8 . Let 81, 82, ••• denote a decreasing sequence of positive numbers with the converging series

k=l For every value 8r there exists a number M & p

such that the

66 measure of the set of points x of the interval I, inequality

where the

l« B (*) - »„(*) I > Sfc holds, is smaller than 8, at any n > N(h^) and m > N(&fc)- We denote the function of the sequence (3.37) satisfying the above condition with the smallest index as ^f(jc) and form the following sequence:

(d[(x),

(*l(x),

....

(3.41)

We shall now show that for any E > 0 and 8 > 0 it is possible to find a set I of points x of the interval I with measure greater than b - a - E, where the inequality Wp(x) - w*(*)| < 8 holds for all p and q beginning with a certain N( 8). Indeed, for a given E > 0, beginning with a certain N we have 00

1

bk < E.

We remove from interval I the set of such points x, where

l«J(*) - »J(*) I > s n for n ^ N and for any p > n and q > n. The measure of the set of these points is smaller than 8 . The remaining part of the set of points I just constitutes the set I we are interested in. In fact, at no point x e I the estimations

67 \<*p(x) ~ <»>*(*) I > & can hold. Otherwise, for any number 8 > 0, there would always tha 8 „ < 8, and therefore in the exist a number n > N such that n specified points the inequality

!>(*) ~ %(*) I

> 8

n'

^> n'

*> "'

would hold. But these points cannot belong to I . The measure of I is obviously greater than b - a - t . Thus, in all the points x e I we have

W*p(x) ~ ©*(*) I < 8, i.e., the sequence (3.41) converges uniformly on I . Let O)(JC) denote the uniform limit of sequence (3.41) on I . For any e > 0 and 8 > 0 there exists a real number N Q such that

[sndx - sm)2dx

| „gdx*21 8

E

+

2 J „;2,** E

for all n ^ Nr, and m ^ NQ. If we fix /w here, for any « ^ Af we obtain an estimation a*n2dx ^ 2T\ + 2M I £

where

Hence we have

68 (o* dx =

lim

a) dx.

E

6

In view of the obvious fact that the limit

lim to n-*» I e

dx = \ to dx

exits, we can consider function ca(jc) to be determined everywhere on I and square summable. We shall now show that function a) is the limit of the whole sequence (3.37) in average. This results from the estimations 2 J (u ~ <*p) dx ^ 2 J (to - to^) d*

+ 2

J

v(to*

m

-

to)/ p

dx

I

283 < 2 U\2(b - a - e) + JK IE ] + which hold for any E > 0 and T\ > 0 at arbitrary natural /?, beginning with a certain number N* and fixed m > N* . The limit to(x) of the sequence (3.37) in average is unique. If there were two functions to(x) and toWr.(x) with the specified property, the following estimations

(to - O)Q) dx

J

^ 2

(to - to ) rfjc

J

+ J (w0 " ®p)2dx>

P > N*

69 would hold. Then, passing to the limit as p tends to infinity, we obtain

f (a) - aQ)2dx I

= 0,

i.e., (D(X) = O)Q(JC) almost everywhere on I. We now proceed to proving the second part of the Fischer-Riesz theorem. Since the convergence of the series (K) implies the convergence of the sequence (3.37) in average to the square summable function (D(JC) , we have

lim J mn(x)Xm(x)dx

= J *(x)xM(x)dx

n-*» I

(3.42)

I

But for n > m, according to (3.38), we obtain

I %(x)xm(x)dx

= Km

(3.43)

I for any m ^ 1. By virtue of equality (3.43) equation (3.42) K

m

=

<*(x)xm{x)dxy

m = 1, 2, ... .

5. The Picard theorem

We shall assume that the kernel K(x, t) and the right-hand side f(x) of the integral equation of the first kind (3.1) are square summable, and also that the system of functions (3.11) is complete. The main statement of this Section is the Picard theorem: Under the assumptions made, equation (3.1) is solvable if and only if the series

70

( 3 - 44 )

i *£«£ ro=l

converges. Here X are the eigenvalues (3.13) of the problem (3.2) and a are the Fourier coefficients of function f(x) determined from the first formula (3.16) with respect to the system (3.11) [81], [88]. The first part of this theorem can be proved rather easily. Indeed, let cp (x) a square summable solution of equation (3.1). According to the first formula (3.16) and equation (3.1) we have b a

=

m

b

J *(*)<** | K(*> a

t)ym(x)dxtx.

a

Then, using the equality b a and the second formula (3.16) we find that b a

m=~] k

f(t)*m(t)dt m a

*m k

m

i.e. ,

K = V « = ]v(y)*m(y)dy

■

0.45)

a Since the function

71 guarantees the existence of a square summable function cp (x), for which A. or are the Fourier coefficients with respect to the r mm system (3.12)

K = kmam

=

J *(y)*M(y)dya

Let us introduce the notation ft /*(*) - \ K(x, a

y)y(y)dy

(3.46)

According to the Hilbert-Schmidt theorem a source-like function / is a sum of the absolutely and uniformly converging series 00

/•<*) -

2

■;?,<*)

/w=l where ft

«; = J/'W^')^-

(3-47>

a Substituting expression (3.46) for function /* in right-hand side of equation (3.47) and using the equality

ft +»<*)

=

k

m\

K( x

<>

y)v(y)*y*

a we obtain a

m

=

J V^dy J *(*' >0
ft

r

m a By virtue of (3.45) we then find that

the

72 , a m

^m =

m

m m =

—

=

a

A

A.

m

™

m

,

v

m

m = 1, 2, ... . Since the Fourier coefficients of functions / and /* with respect to the complete system of functions (3.11) coincide we obtain that f(x) = /*(*)• Thus, the right-hand side of equality (3.46) coincides with function f(x), and therefore function cp(jc) is a solution of equation (3.1).

6. The case of a symmetric kernel

When the kernel K(x, t) is symmetric the proof of the Picard theorem becomes more simple because the system of equations (3.2) reduces to a single equation b cp(jc) - X f K(x,

t)q>(t)dt

= 0

and the systems of functions (3.11) and (3.12) coincide. For symmetric kernels the criteria of the completeness of the system of eigenfunctions (3.11) are known. It is easy to see that the system (3.11) is complete if and only if the kernel k{x, t) is closed. Indeed, by definition, a kernel K(x, t) is called closed if the identity b [ K{x, a

t)a(t)dt

=0

implies that the square summable function w(x) vanishes Due to the obvious identity b b J
b $ K(x,

t)u{x)dx

(O(JC)

= 0

73 and the completeness of the system (3.11), we find that the kernel k(x,t) is closed. Conversely, if the kernel k(x, t) is closed, the system (3.11) is complete [28], [43], [44].

7. Some remarks concerning to the Picard theorem

The main reasons, used in the proof of the Picard theorem enables some important remarks to be made. Note 1. If we enrich the conditions of the Picard theorem with the requirement that the system (3.12) is complete, the theorem itself will have the additional statement that equation (3.11) is uniquely solvable. To be specific, in this case the trivial function cpn(t) = 0 is the only square summable solution of the homogeneous equation b | K(x, a

t)q>Q(t)dt

= 0

corresponding to (3.1). Note 2. If the system (3.12) is not complete, for any square summable function (pn(t) orthogonal to all the functions of this system, we have b $ K(x, a

t)[cp(0 + vQ(t)]dt

=

f(x).

Thus, together with the function cp constructed in the preceding Section the function cp(jc) + 9 Q ( * ) is also a solution of equation (3.1). Note 3. When completeness of system (3.11) is violated the fulliment of the condition of the Picard's theorem on the convergence of series (3.44) by virtue of Fisher-Risz's theorem, which in this case in also valid, certainly garantees the existence of square summable function cp(jc) whose Fourier coefficients with respect to system (3.11) are numbers am. But in this case we can only state the fact that the equality

74 b $ K(x, a

t)y(t)dt

= f(x)

+

f*(x)

holds, where /* is an arbitrary satisfying the conditions b | f\*)*m{x)dx a

= 0,

square summable function

m = 1, 2, ... .

In the considered case equation (3.1) may have no solution. This case takes place, for example, when the kernel is degenerate, i.e. , K(x,

n t) = X i=l

Pt(x)qi(t)

where p- and q- are square summable functions and n is a natural number. In fact, in this case the kernels K*(x, t) and K*{x, t) introduced in Section 2 of this chapter are also degenerate. The spectrum (3.10) and the system of eigenfunctions (3.11) are therefore finite. Consequently system (3.11) cannot be complete, (see examples 1 and 2 in Section 1 of this chapter). Note 4. Since function /*(*) is orthogonal to all the functions of system (3.11) regardless of whether it is complete or not, the Fourier coefficients of functions f(x) and f(x) + / (x) with respect to this system coincide. Hence, if system (3.11) is non complete the Fourier coefficients of function 0 there exists a natural number N ( E ) such that the inequality

75

J [/<*) " /„<*)] dx < £ a

holds for all n £ N, where n

/„(*) = 2

<W>

and a^ denotes the Fourier coefficients of function / with respect to system (3.11). Using formula (3.9) we find that b

a where n


*«*»+.(*)■

m=l

Thus, in the considered case we can point out a function
J*(x, t)**n(t)dt = f(x) + / V )

(3.48)

a

infinitesimally differs from / in average.

8. The spread of the Picard theorem to the equations of the first kind in the Hilbert space

Let us consider equation (2.1) when A is a completely continuous linear operator, mapping a separable Hilbert space into itself. Similarly to Section 2, we introduce a system of two equations

76 q> - kA*$ = 0,

i|r - kAy> = 0,

(3.49)

where A* is the operator adjoint to A, cp and ty are elements of space E , and X is a spectrum parameter. Let X be an eigenvalue of system (3.49), cp and i|> be the corresponding associated pair of eigenelements. Under these assumptions equation (3.49) yields cp - \2A*Ay

= 0,

(3.50)

i|r - k2AA*ty = 0 .

(3.51)

2 We thus conclude that X is an eigenvalue and 9 and <|r are eigenelements of the homogeneous equations of the second kind (3.50) and (3.51), respectively. 2 The converse statement is also valid: If X is an m e i g e n v a l u e of e q u a t i o n ( 3 . 5 0 ) , i . e . , q> - xV,4cp

= 0,

the element cp equality

(3.52)

together with the element
determined by the

form the associated pair of eigenfunctions of system (3.49), corresponding to the eigenvalue X . In fact, according to equation (3.52), equation (3.53) becomes

A

**m = v -^ = r 9

(3

-

54)

The equalities (3.53) and (3.54) are nothing else than the system (3.49) with X = X , cp = m , and i|/ = d; . Eliminating functions cp from (3.53) and (3.54) we arrive at the equality

77

K ~^*+„ = °

(3.55)

It means that x* and A are the eigenvalue and the corresponding to it eigenelement of equation (3.55). By means of straightforward verification, we make sure that operators AA* and A*A are completely continuous, self-adjoint, and positive. The set of eigenvalues X is therefore not more than countable. We will assume that the y are numerated in the ascending order \11

& < ..... . .. * \X2 ^

(3.56)

Let Let *l» 1>2'
((33..5577))

*!, -t-2, *2, ••• ... +!•

(3.58)

and and

be the corresponding sequences of eigenelements. An element f e E is referred to as source-like if / == i4q>, y*q>, /

(3.59) (3.59)

cpp ee ZZ yy c

The Hilbert-Schmidt theorem discussed in Section 3 can be generalized in the following way: An element / represented in the form (3.59) is the sum of converging series 00 00

oo oo

1

/ = I )vmM = Z )fmm X ((/, / , 9vmm)9 X -- (OP, 9, v
m=l km

where (/, cpw) and (
78 respectively [38]. We shall now confine ourselves to the case when operator A is a self-ad joint, A = A*. The following generalized Fischer-Riesz theorem holds (see Section 4 ) : Let

XV Xp

be a given sequence of elements of space E Kjp, K

(3.60)

X, ... and

K, ... , K , . .. ,

(3.61)

be a sequence of numbers. The convergence of the series 00 00

is the necessary and sufficient condition for the existence of the element / e E with the Fourier coefficients with respect to system (3.60) equal to numbers (3.61). These two statements generalizing the Hilbert-Schmidt and the Fischer-Riesz theorems make it possible to formulate the Picard theorem in the following way: If the system (3.58) is complete, equation (2.1) is uniquely solvable if and only if the series 00

2 25>*(/> xjj;
(3.62) (3.62)

7K=1

converges ([29], [87-88], [38], [42]) It has been shown above that in the case of a symmetric kernel K(xf t ) , the completeness of system (3.11) is equivalent to the closeness of this kernel. In the considered case when the requirement that operator A be self-adjoint is imposed, the closeness of the kernel K(x, t) is equivalent to the requirement that the corresponding to (2.1) homogeneous equation ^cp0 = 0

(3.63)

79 have no non-trivial solutions. The proof of the Picard theorem discussed in Section 5 shows that the convergence of the series (3.62) makes it possible to determine an element (p e E so that its Fourier coefficients be equal to numbers ^ (/,
/ - Ay = h holds where h is the following equalities

(h,

element

cpj = 0,

of

space

m = 1, 2, ... .

E

satisfying

the

(3.64)

Among elements h which satisfy the conditions (3.64) there is the only element h* for which the equation

f - Ay* = h* has a solution. This fact is one of the reasons for the aspiration to developing the method which would permit to find "approximate" solutions of equation (2.1).

81 CHAPTER 4

Integral Equations of the First Kind with Kernels Generated by the Schwarz Kernel

1. The concept of the Schwarz expressions for some domains

kernel

and

its

explicit

Let D be a domain of the plain of complex variable z = x + ±y with the Lyapunov boundary s = 3D. An analytic function F(z) defined in D is called continuously extendible over s if there exists the limit F(z)

lim

= F+(t0),

z

e

D

for any point tQ e S. The function K{t,z), t e s,z e D, is called the Schwarz kernel for the domain D if the formula

F(z)

= $ K(t,

z)f(t)dt

+ C

(4.1)

S determines a function analytic in Z), which is continuously extendible over s from its real or imaginary part defined on S where C is an arbitrary imaginary or real constant depending on weather f(t) or

= ReF + (t)

82 = ImF+(t),

f(t)

t e S.

The formula (4.1) is used to be called the Schwarz formula. When D is a simply connected domain (with the Lyapunov boundary) the existence of the Schwarz kernel is a direct consequence of the basic principle of the theory of conformal mappings of plane domains on the circle. In the case of multiply connected domains, a similar statement is proved under certain inessential restrictions on D. It is well known that, for a unit circle Z):{|z| < 1} and for a half-plane Z):{Im z > 0}, the Schwarz kernels are given by the formulae 1 z) = —

K(t,

1

r

TTl ^ t

1 .

-

It

Z

(4.2)

}

and K(t,

1

1

iri t

-

z) =

,

(4.3)

z

respectively [16,22]. When D is a strip {- oo < Re z < «>, 0
k(t,

z)

=

1

IT

c t h — (t 2±q 2q

- z) =

1

iri t - z

83

+

2

°°

— *i

X n=1 (t

t - z

2 2-j - z)z + 4 * V

(4.4)

is a result of its conformal mapping on the unit circle with the center in the origin of the frame of reference. The Schwarz kernel can also be explicitly written for a circle ring D:{q < \z\

< 1},

namely, K(t,

z) = — «fi I i log log

(4.5)

where £(T) is a the Weierstrass £-function

5(0 = ir

T + - Ctg 2 2 <72nsinT

+ 2

2 n=1

?H 1 + q

zn ~ 2^

ZW

>

(4.6)

COST

and log t/z means the principal multiple-valued function [22,101].

branch

of

this

2. Integral Equations of the First Kind with the Hilbert Kernel

For a function F(z) = u(z) + iv(z) of the class C°'h(D U S) analytic in the circle D:{\z\ < 1}, due to Equations (4.1) and (4.2), we have the representations

84

F(z) = — J f lf

)u{t)dt + i q

iri

t - z

(4.7)

It >

S and l F(z) = TT

,

f

1

J

l

£ -

Z

1

x

It

>

v(t)rft + C2,

(4.8)

t e 5,

(4.9)

L

s where F+(t)

= a ( t ) + iv(t),

2ir

2ir

J

2ir

2ir

0 u(t)

= u(Q),

v(t)

0 £ = e1^,

= v(d),

0 ^ ft * 2ir.

The problem of the definition of the function F(z) analytic in D from the boundary condition (4.9), i.e., from the boundary values of its real and imaginary parts on s, is the Cauchy problem. It is known that this problem is incorrectly posed. From Equations (4.7) and (4.8), the Sokhotskii-Plemelj formula yields F+(t0)

1 + — iri

f J

S

and

= u(tQ)

,

+ iv(t Q ) =

1

^ t - tn

1 , \u(t)dt }

U

It

u(tQ)

+ iC, l

(4.10)

85 F+(t())

+ -

= n(tn) + iv(tn) = iv(tft)

f (

}v(t)dt

+ C2,

t 0 e S.

(4.11)

IT

s Since (t

+ t*)dt Ol_ 2t(t - tQ )

=

1 fl _ ctg 2

^ ,

fln

. t = e1*,

2

tn = e u

ifln °,

from Equations (4.10) and (4.11), we have 2ir v(ft Q ) =

2ir

[ctg 2ir ^ 0

^ 2

w

(^)^ + _ f v(d)dG. 2ir ^ 0

(4.12)

and 2ir w(^0) = — f c t g 2ir ^ 0

2ir 2

^ w(fl)dfl + — f u(ti)d$. 2ir ^ 0

(4.13)

The singular integrals in the right-hand sides of the formulae (4.10), (4.11), (4.12), and (4.13) are regarded in the sense of the Cauchy principal value. The validity of each of the equalities (4.12) and (4.13) is the necessary and sufficient condition for the existence of the (unique) solution of the Cauchy problem (4.9) for the function F(z) analytic in D. This means that the equalities (4.12) and (4,13) are equivalent to each other. Under the additional requirement

86 2ir

J F+($)d($) = 0

(4.14)

0 we write the equalities (4.12) and (4.13) in the form 2ir

1 —

r

ft ctg

-

ftn ^ w(ft)dft = -

(4.15)

2lT

0 and 2
— [ ctg 2ir J 0

^ v(ft)dft = u(*Q),

0 ^ ftQ ^ 2ir.

(4.16)

2

It is natural to treat the equalities (4.15) and (4.16) as integral equations with respect to «(ft) and v(ft), respectively. It is used to call them integral equations of the first kind with the Hilbert kernel 1 —

*-ftp ctg

2TT

The equivalence between (4.16) means that they are literature [17,18,20,94], these Hilbert inversion formulae (see

. 2 the equalities (4.15) and mutually inverse. In the equalities are known as the Ref.[40], pp.75-76).

3. The Hardy Inversion Formulae

0 h For a function F(z) of the class C ' (D U s) analytic in the upper half-plane, which vanishes for | z \ -* » and has an absolutely integrable boundary value

87 F+(t)

= u(t)

+ iv(t),

-oo < t < oo,

the equality 00

I F+(t)dt = 0.

(4.17)

—00

holds, provided the Cauchy theorem is valid. F(z)

Then due to Equations (4.1) and (4.3), the function can be represented by the formulae 00

1 = — TTl irr

F(z)

u(t)dt

r

(4.18) JJ

tt -— z7

—00

and

F(z)

11 = Tl

r J

V\L

v(t)dt IUL

t

-

,

z

e

(4.19)

D.

Z

These formulae can easily be obtained from the formulae (4.7) and (4.8) as a result of the conformal mapping z - r w =

z + i of the half-plane D on the circle \w\ < 1. If for sufficiently large | t \ the evaluations

u(t)

= 0(\t\~h),

v(t)

= 0(\t\~h),

h = const > 0,

take place, the integrals in the right-hand sides of Equations (4.18) and (4.19) are regarded as the limits of the expressions

88 N"

N"

n U(t)dt

r

-* t - z -N'

V(t)dt

J t - z -N'

when the positive constants N' and N" tend to infinity independently. According to the Sokhotskii-Plemelj formula the formulae (4.18) and (4.19) yield F+(x)

= u(x)

• + iv(jc) = u(x)

1 P u(t)dt + — , iri ^ t - x —00

and 00

F+(x)

= u(x)

+ iv(x) = iv(jc) + IT

f— J

t

, -

X

—oo

Hence, as well as in the previous section, we come to the conclusion that the equalities 1 V(JC) = - TT

u(t)dt

r J

t ~ X

(4.20)

—00 '

and U(X)

1 = _ IT

°°f J

v{t)dt t

-

(4.21)

X

—oo'

are equivalent. For the functions u and v defined according to equations (4.20) and (4.21), the equality (4.7) obviously takes place. The formulae (4.20) and (4.21) are usually called the Hardy inversion formulae and their right-hand sides are known as Hilbert transformations [33]. These formulae remain valid under certain restrictions

89 with respect to u(x) and V(JC) which are weaker than the above ones, provided the equality (4.17) holds [74,75].

4. The Inversion of the System of the Singular Integral Equations of the First Kind with the Kernels generated by the Schwarz Kernel in a Strip

Let F(z) be a function of the class C°' the strip

(D U s) analytic in

/):{-<» < Re z < oo, 0 < Imz < q} which vanishes for | z \ -* °° and has absolutely boundary values F+(t) on

integrable

.sij-oo < £ < « > , l m z = 0} and s:

l {~°°


<00

> Imz

=

#}•

The Cauchy theorem ensures the validity of the equality 00

00

| (u + iv)dx + —00

J ("l

+

ivj)*/* = 0,

—00

where

u(x) + iv(x)

= F+(x)

(4.22)

90 w1(x) + iv1(a:) = F+(x + ±q) ,

-oo < x

< °° .

Since in the case under consideration, the Schwarz kernel is given by the formula (4.4), the function F(z) can be represented in D with the help of its real and imaginary parts on dD = s U s« according to the formulae 00

F(z)

= J

2iq

i#+°° r

1

ir

cth — 2q

J

2±q

z)u(t)dt

[ cth — (t 2q

(T -

i

z)u,(i)di

iq-co

1

r

+

J

2iq

1

p J

2i£

TT

cth — 2q

(t

-

z)u(t)dt

IT

th —

2#

(t

- z)uAt)dt A

(4.23)

and F(z) = — 2q

1

2q

r J

J

[ cth — (t 2q

-

z)v(t)dt

IT

th — (t 2q

- z)v.{t)dt. l

(4.24)

—00

Since D stays to the right as the point T = £ + iq (-00 < ^ < oo) traverses 5 1 , the Sokhotskii-Plemelj formula gives us from (4.23) that

91

F + ( t n ) = u(x)

+ J

2iir

t h — (t 2q

J

2iq

f c t h — (t 2q

- x)u.(t)dt L

-

= u(x)

x)u(t)dt

+ iv(jc),

x = tQ G S,

and

^('n)

=

w

l<*)

+

2i^r

29.1/7 i#

J

f th — (t J 2q

c t h — (f - x)u,(t)dt l 2q

x)ii(t)dt

= u,(x) l

+ i v l^ j c ) ,

x = iq = tQ e 5 1 ?

i.e. , V(JC) = 2q

— 9/7 2#

c t h — (t 2q

J

f t h — (t 9/7 2q

JJ

-

x)u(t)dt

- x)uAt)dt

(4.25)

X

and 1

^(JC) = 1

2q

r

IT

t h — (t * 2q

-

x)u(t)dt

92

+ — 2q

J

(t - x)uAt)dt.

f cth — Iq

(4.26)

l

—00

Using the representation (4.24), we similarly obtain 00

1

u(x)

— 2q

r

=— 2q

J

IT

cth — ( t 2q

J

-

x)v(t)dt

f th — (t - x)vAt)dt 1 2q

(4.27)

and

1

u.(x) 1

=— 2q

r J

IT

th — (t 2q

x)v(t)dt

IT

— 2q ?/7

f cth — (t - x)v.(t)dt. J 2 2^ J 1~

(4.28)

By direct calculations we see that the functions u(x), V(JC), uAx), and V-,(JC) connected by the relations (4.25), (4.26), (4.27), and (4.28) satisfy the condition (4.22). Each pair of the equalities (4.25), (4.26) and (4.27), (4.28) represents the system of singular integral equations of the first kind with respect to the pairs of functions M(JC), UAX) and V(JC), vAx) respectively. These systems are equivalent, i.e., mutually inverses. This assersion is an obvious consequence of the fact that the ???satisfiction of statement of each pair of equalities (4.25), (4.26) and (4.27), (4.28) is the necessary and sufficient for the

93 solvability of the Cauchy problem F+(tQ)

= ii(tQ) + iv(tQ),

tQ E S,

for an analytic function F(x) on the strip D. When q -> », by condition we have that lim

F(JC

+ iq) = 0

uniformly with respect to x, -» < * < <». Therefore, when q ^ oo, in the limit, the equality (4.22) passes into the equality (4.17) and formulae (4.25), (4.26) and (4.27), (4.28) transform into the Hardy inversion formulae (4.20), (4.21) [40].

5. Integral Equations of the First Kind with the Kernels Generated by the Schwarz Kernel for an annulus

An analytic function F(z) extending to dD = s U slt

on the annulus D : {q < |z| < 1},

s:{\t\

= 1},

^ { M

= q},

under the requirement that F+(t)

e C°'h(D U dD)

due to equations (4.1), (4.5), is given by the formulae

94 2TT

F(z)

= -

\ j ( i log J

IT IT

)ii(*)dd

7

0 2ir t

-

{ 3 ( i log - )« 1 (*)d* + i q

IT

J

(4.29)

Z

0 and 2TT

1 F(z) = - -

£ J ( i l o g - )v(*)
r

IT J

7

0 2TT

i1 + -

r

{,(i

t log - )v1(«)d* + C2,

IT J IT

(4.30)

7

0 where K(«) + iv(d) = F+(t),

Ml(d)

t = eid,

+ iv^O) = F+(t),

£ = ^e1*,

£ 3 ( T ) = U T + co') + £ ( « ' ) ,

(4.31)

a)' = i l o g q,

(4.32)

and C^ and C~ are arbitrary real constants ([22], 100, 315, 101). Since in D the function {^(T) is given by the formula

95 q2n~ls^

£00 T

£3< )

=

T + 2 X= 1 n=\

1 + q

An

-

2q

(4.33)

Jn=T^

COST

and £ ( T ) the functions are odd. Owing to equations (4.29), (4.30), (4.4), and (4.32), we conclude that for the function F(z) to be a single-valued one in the annulus D it is necessary and sufficient that the conditions

J [«(*) - «!(*)]«» = 0, 2ir

(4.34)

J [v(*) - v^*)]*** = 0. 0 be fulfilled. Formula (4.29) ca be written as F(z)

= Fx(z)

+ F2(z)

(4.35)

+ iC

where, due to Equations (4.6) and (4.33), we have i

V2> = i J

£(ir) i

t

i t + z

z

2 £ -

log z

0 <72n(z/t -

+ 1

X 72=1

t/z)

1 + * 4 n " * 2 n ( * A + t/z)

w(fl)dfl, J

96 t = ei{>,

(4.36)

and 2TT

2)

* = -J +

1

q2n(z/t

|(ir)

i t +z log t/z 2 t - z

IT

-

t/z)

I £ !+,*-><,/*'♦W"0 t =

tl(*)d«,

qeU.

(4.37)

From (4.29), (4.35). (4.36), and (4.37) yield F+(tn) = «(dn) + iv(* n )

2ir

= - [ j(* 0 - *)«(*)<» + «(*0) IT J

2TT

- J h(\

^)w1(^)^) + i C p

i f l ur

tQ = e

and

^ + (t 0 ) = ^ ( V + i v ^ ^ )

6

5,

(4.38)

97 97 2TT

++

== «l(* M 0V> "- JJ *3<*o *3<*o -*)«(*)*> -*)«(*)*> IT IT

J

2
ii rr - IT{(ft0 IT

ft)^*)^

J J

J

0 0

+ iC 2 ,

iflnn ifl t Q = ?e u e S

(4.39) (4.39)

n 0

Since the functions { and ^ are odd equations (4.38) and (4.39) yield 2ir

v ( V - - - r K. - v « ( . ) * > IT IT

0

2ir w

+ - f s3(* - V IT

*^ ^

i ( ^ + ccii

(4.40) (4-4°)

0

and 2TT

v1(V--;Js3(*-V"W* o 0

2TT

cc + - f S«> S«> - *VM*)** 0 )"i(*)^ + rr IT

(4.41) (4-41>

J

0

0 (4.37)

In a similar way we obtain from equation (4.37) 2TT 2ir

1 _f

«(« 0 ) = -ir

J

* 3 (* - V v <*> d *

0

2TT

- " J 53(» (* " V M * ^ * IT IT

0

+ CC

2

<( 44.'4422))

98 and 2TT

MV

=

- r M * - ^0)v(^)^ IT

J

0 2ir r

- -1

5 ( * - * 0 ) v 1 ( d ) ^ + C2.

(4.43)

IT

0 As i t i s known [ 1 0 0 ] , 2
S(* -

*0)
= -

2vg(ir) + 25(Tr)*f

(4.44)

0 2ir

J g 3 (d - d 0 ) ^ Q = - 2*g(ir) + 25(ir)*,

(4.45)

0 As a result of the integration of equations (4.40), (4.41), (4.42) and (4.43) by (4.44) and (4.45), we get 2ir

2ir

| V(ft0)dft0 = J 0

2ir V^^Q)^

=

0

— I

(TT

- <>)[ll(<>) -

^(fl)]^

0

+ 2TTC1

(4.46)

and 2ir

2ir

2ir

r u($Q)d$0 = r n ^ ) ^ = —

r <* - *HVO>) - vjooi**

0

0

0

+

2irC2.

(4.47)

99 We now require that the equalities 2ir

2ir

[ w(fl)rf# = f v(fl)rffl = 0. 0

(4.48)

0

hold. Then, due to (4.43) the equalities 2ir

2ir

[ u^^d* = J v1(«)d« = 0. 0

(4.49)

0

also hold. On the basis of the equalities (4.48) and (4.49), we find from (4.46) and (4.47) that 2ir

e(*) r IT

J

0 and 2TT

$(*) r 2*[v(*) -

C2 =

IT

v^fl)]^.

J

0 Consequently, the equalities (4.40), (4.41) and (4.42), (4.43) can be written in the form 2ir

2ir

1

r

l

11

o

* o

r

2ir

+ i-J- f *[n(d) IT

J

0

K^*)]***,

(4.50)

100 2ir

2ir

'!(* 0 = - - [ £3(* - *0)uWd$ + - f |(d - d0)Ml(d)dd IT

J ■*

IT

J

0

0 2ir

«(') r + — j - *["(*) " " x C * ) ] ^ IT

(4.51)

J

0

and 2ir

2ir

«(*0) = - [ « ( * - *0)v(«)dd - - [ 53(* - d0)v1(d)d* IIT T

•* **

IT IT

0

•*

0 2ir

5(') r —2" *[v(*) - v x ( d ) ] ^ , IT

(4.52)

J

0 2ir B

1
= IT

2TT

d

f M * ~ *nM*) * - - f 5(* - * 0 )v 1 (*)d*

J

IT

0

J

0 2TT

6(») r 2~ *fv(«) - v ^ * ) ] ^ * . IT

(4.53)

J

0

Since the validity of each pair of the equalities (4.50), (4.51) and (4.52), (4.53) is the necessary and sufficient condition for the solvability of the Cauchy problem F+(tn) = ii(*n) + iv(fln),

tn = e

u

e 5,

101

F (t Q ) = ^(ftg) +

iv

i(V'

*0 =

qe

e5

1'

for the function F(z,) analytic in the annulus Z), we can conclude that, in the presence of equalities (4.48) and (4.49) these pairs are equivalent. Therefore, considering the equalities (4.50), (4.51) and (4.52), (4.53) as systems of integral equations with respect to w, u+ , , and v, v2 respectively, we can state that these systems are mutual inverses, provided the conditions (4.48) and (4.49) are fulfilled. [22]. When q —► 0, the annulus D transforms into the circle \z\ < 1 and z = 0 is no longer as an isolated sign????point of the analytic function F(z), so that lim [w1(^) + iv^fl)] = F(0) = 0 q — 0 by virtue of (4.48). Hence, taking into account that according to equations (4.6) and (4.33) lim q —

£(T)

0

=

«00 *

1 ^ T + - ctg 2

2

and 6 00 lim 63(f) =

T,

we see that equalities (4.51) and (4.53) become the identity 0 = 0 for q — 0, and the equalities (4.50) and (4.52) pass into the Hilbert inversion formulae (4.15) and (4.16).

103 CHAPTER 5

INTEGRAL EQUATIONS OF THE FIRST KIND WITH THE KERNELS GENERATED BY THE POISSON AND NEUMANN KERNELS

1. The Concept of the Poisson Kernel

We consider D to be a domain of the space E of the class Al>h with the boundary S ([15,22-24]). The function P(x,y) is called the Poisson kernel for the domain D if the formula U(x) = | P(x, y)f{y)dsy

(5.1)

S determines the harmonic function U(x)

e C°>°(D U s),

on Z), which satisfies the boundary condition U(y)

= f(y),

y - s,

(5.2)

whereThe f(y) existence is a continuous definedis on well S. of the function Poisson kernel known at 1 h

least for the domains of the class A ' . In the case n = 2, the Poisson kernel is expressed in terms of the Schwarz kernel by the formula

104 y) = Re K(t,

P(x,

z)

dt —, ds1 ds

where t = yx+ iy2, z = xx + £* 2 , is the complex notation of the points y = (yvy2), x = <* lf *2), and ds is an element of the arc s at the point t. When D is the unit sphere |*| < 1, the Poisson kernel is explicitly written as

i1 i1 -- \x\ M 22 !»(*, y) = -a, % \y \y -- x\ *\ % where \y - *\

n = I (yt i=l

2

2 *xt)t)2-.

-

In this case, Equations (5.1) is the Poisson formula itself [15, 75-76] II ff 11 "" N 1*1 U(x) U(x) = =— — % s \y - *i

2

(5.3) (5.3)

f(y)ds f(y)ds

In the case of the half-space D : {*n > 0}, we can write

*> ==

PP(*'

(*'

xn

2 l , ,

S

2-^72-

X |

+X ]

«"B » I l > "- x |

+ *" „]

22 |y \y -- *\ A22 -- zz V * -- Xi Xi) 1=1

and _ 2_ UyXj

—

Q) Q)

, X

n "

I J J

Vi

/(^)dy M % II M yj ; — _ %

^

;y 4. £Z + £^

(5.4)

105 where En_^

2.

is the hyperplane yn = 0 and dy = dy^...dy

_+.

The Concept of the Neumann Kernel

If a harmonic function U(x) by the formula

of the class C 1 ' h (D U s) defined

U(x) = — f N(x, y)
+ C,

(5.5)

"s where C is an arbitrary constant and cp is a function of the n h class C ' defined on 5, which satisfies the condition

J 9(y)dsy

= 0,

(5.6)

S is a solution of the Neumann problem d —

I/O) = q>(y),

y « 5,

(5.7)

dv

y

then the function N(x, y) is called the Neumann kernel for the domain D. 1 h The existence of the Neumann kernel for the domains of the class A ' is well known. Below we shall need the explicit expressions for the Neumann kernels for the unit sphere and for a half-space in the space En [19,24]. In the case of the ball D : {| x \ < 1}, the outward normal v to S : {\y\ = 1} coincides with the radius-vector of the point y e S. The boundary condition (5.7) can therefore be written in the form

y gradU(y)

Since,

together

=
with

U(x),

(5.8)

the

function

106 x grand U(x) = v(x) is also a harmonic one in the ball Z), the equality (5.8) represents the boundary condition of the Dirichlet problem V{y)

= V(y),

y - S

(5.9)

for v(x). The solution of this problem is given by the formula (5.3) in which the function cp(y) should be written instead of /(v), i.e., 1

V{x) = — (on

1 -

r J

\x\2

y(y)ds y

\y - x\

(5.10)

S

In view of equality (5.6) the formula (5.10) yields V(0) = 0.

(5.11)

The solution of the Neumann problem (5.7) is contained among the solutions of the linear partial differential equation of the first order x gradU(x) = V(x)

(5.12)

with a known right-hand side of (5.10), regular in the ball D.

It is easy to see that the required solution u(x) the equation (5.12) is given by the formula [19, 24,58]

of

1 U(x) =

r J

dt

V{tx)

—

+ C.

(5.13)

t

0 Really, due to Equwation (5.11) the integral in the left-hand side of equality (5.13) converges uniformly. It is obvious that

107 1

1

0

0

Consequently, we have 1 gradU(x)

r

= J

dV — dt = V{x) - K(0)

(5.14)

At

and 1 At/ = J AKtdt.

(5.15)

0 The equalities (5.14) and (5.15) imply the validity of the above statement.

3. The Neumann Kernel for the Unit Ball

We shall use the following designations R(t)

m (y - tx)2

\y\ = i,

= 1 - Ixyt

+

x2t2,

n *y = 2 ^ ^ £ , i=l (5.16)

fi (t) - [P(tx,

1 y) - —

1 ]-

108 ,

tX

1

2

1 1 xA 1

and additionally define the function Q„(t) for t = 0 lim gn(t) = 2(0). t-*0 Since the formulae (5.5) and (5.13) are equivalent, the Neumann kernel N(x,y) becomes 1

N(x, y) = J Qn(t)dt.

(5.17)

0 The integral in the right-hand side of (5.17) is easy to calculate. In fact for t > 0 equality (5.16) yields p

p

*>„ J Qn(t)dt

~ p

dt

tdt

= J - ^ - x z J ^ T J - log t.

(5.18)

We first assume that rc is even, i.e., n = 2m. If we Dduce the notation 8 = x 2 - (xy) 2 and, for t > 0, use introduce the well known formulae dt

ttf"

1

p dt 1 1 +

2(m - l)*" "

2 p tdt

J if

1

** J ^

+

p dt

J t/^" 1,

1

7(m - i ^ "

p ^ 1 + Xy

J ^

for /w > 1, and p dt 1 t — = - log J tR 2 R(t)

jcy tx2 - xy + -yTarctg r75—, 81/2 81/Z

109 2

r

x

tdt LUL

1i

=

J T" 2

tx LX

xy

l o g R(t) +

1 arctg

—

xy

1

W

~W

for IB = 1 we find from (5.18) that

°lm

(m -

l)lf-

m-2

dt \)F^~k~l

2{m - k -

xy

1

- - log R(t)

+ -jjz

+ xy

tx

arctg

j

- xy

(5.19)

+ C

1/2

for w > 1, and (o2 | Q2(t)dt

= - log K(t) + C

(5.20)

for m = 1. Let now « = 2m + 1 where dt

P

/w ^ 1. Since

1

J 7 ^ ^

=

r

1 2+

(2. - i ) * - /

<*£

"J ^^

r

+

dt

=T

J ^ 72'

tdt

f— ^ ^

(2m - l)/?m-1/2

+ xy

T72'

for t > 0, we have P 2

*

fl

2 x

' "•*■"-! "' "

(2BI - i)fl*

Hence, taking into account that

r

(it

110

2 - 2txy + 2K 1 / 2

dt

J

-^72

= - log

f o r t > 0, we find 2 = -JJ2 - log (2 - 2tjcy + 2K 1 / 2 ) + C

w3 f Q3(t)dt

(5.21)

for m - 1 and 2

r •2f 1 J 22w+l(0^ « -lr A:=l

_

(2w

ftk-m+1/2

1 ) j p .l/2

dt

r

T72

2(m - k)

- log (2 - 2tjr>' + 2/? 1 / 2 ) + C

(5.22)

for m > 1. In view of t h e e q u a l i t i e s r

dt /^

+1

tx2 - xy 2/w + 1

m 1

~ (2m - 1 ) . . . ( 2 m - 2k + 1) ( * 2 ) *

£Q

m...(m

-

(2m - l ) ! ! ^ 2 ) * " arctg

mlz o

J

dt

tx

k)2

tx

- xy

J7T - xy

1

*k+\ nM-k

k+l

+ c,

111 ro-1

2K(m

-

1)...(m -

Rk

k)

(x2)k

i +x

(2m -

k=1

3 ) . . . ( 2 m - 2k -

1)

1

+ C 8*

valid for m > 1, we come to the conclusion that the Neumann kernel N(x,y) is expressed in terms of elementary functions. In particular, due to Equations (5.20) and (5.21), equality (5.17) gives the known expressions for N(x,y), namely, 2vN(x,

v) = - 2 log|v - JC|

(5.23)

for n = 2 and 4*N(x,

y)

(5.24)

+ log \y -

1 - xy + \y -

x\

x\

for n = 3 ([80],247). Here we also give the expression for N(x,y) when n = 4 which is a consequence of (5.17) and (5.19) [17]. 1 2
=

\y - x

+

-^ - log \y - x\

,1/2 xy - ^ arctg -

(5.25)

xy

4. A Many-Dimensional Analogue of the Hilbert Inversion Formulae

Further we shall suppose that / e C ' (s), cp e C ' (s) . The problem of the determination of the function U(x) of the class C1}h(D U s) harmonic in the ball D:{\x\ < 1}

112 from the boundary conditions d

U(y)=f(y),

— 0 0 0 = 900,

y 6 s,

(5.26)

dv

y is the Cauchy problem for harmonic functions. In order that the solution of the problem (5.26) should exist it is necessary and sufficient that the functions / and cp be connected with each other by the identity J f(y)dsy

/(x0) w

n

= J N(xQ,

y)
(5.27)

S x

0

under the condition (5.6). Actually, the fulfillment of the condition (5.6) guarantees the existence of the solution U(x) of the Neumann problem (5.7) which is given by the formula (5.5). The validity of the identity (5.27) implies that U(x) satisfies not only the second but also the first of the conditions (5.26). The necessity of the conditions (5.6) and (5.27) is obvious. Due to (5.3) under the condition (5.6) the identity =

1


n

v

x„

g

rad

o ™ x-*xQ

r

1-

JJ —

\y -

\x\2

^

x\n

f(y)dsr

(5.28)

s

\* 0' is also the necessary and sufficient condition for the solvability of the problem (5.26). That is, if the condition (5.6) is fulfilled, the identities (5.27) and (5.28) are 1 h

equivalent. It should be noted that, since / e C ' (s), the limit in the right-hand side of Equation (5.28) exists (the

113 Lyapunov-Tauber Theorem). Thus, considering the equalities (5.27) and (5.28) as integral equations of the first kind with respect to the functions cp and /, we come to the conclusion that if the condition (5.6) is fulfilled, each of the equalities (5.27) and (5.28) is the inversion of the other one. For n = 2, the formulae (5.3) and (5.5) become 2ir 2TT

1 U(X)

=

r l -

~ \x\ \x\Zz fWdfWd

((55 . 2 2 99))

- *|cp(ft)dft + C

((55..3300))

^ JJ \y - x\2

* *

0 0

*

and 2TT 2TT

U(x) = [/(JC) =

J log|y IT

0 0

where y, y1

= cos

ft,

y2 = s i n

ft,

2TT . 0 ^ ft ^ 2TT.

Integrating (5.29) by parts, we find

U(x)

2ir o 2 2 1 r (1 + |\x\ JC | + 2jc1)tg(ft/2) l = - - \ arctg —* IT

J

1 "

0

x /'(ft)rfft. /'(ft)rfft.

U|

-

2x2

2

(5.31)

For | JC | = 1, we have |v - x\ \y

= 2sin * " * °- ,

(5.32) (5.32)

114 d dvYX 1 cos(V 2 ) 2 cos(^0/2)

(1 + x2 + 2jc1)tg(d/2) - 2x2 * 1 - \x\2

arctg

±

1

s *

s ^0

1 * - An 1 A = - ctg 5i - - tg - (5.33) 2 2 2 2

jc-, = COS#Q,

x2 = sin^Q.

Since 2ir

r fwd$ = o 0 in view of (5.6) and due to (5.29), (5.30), (5.32), (5.33), the equalities (5.27) and (5.28) respectively become 2ir

/(V - — \ /(*)<** 2ir 0 2TT

- Jf l o g I s i n IT

^ |
(5.34)

2

0

and 2ir

1
2

^ / ' (*)*>.

(5.35)

0

Since / e C ' (5") we differentiate (5.34) to obtain 2% [ct§ "

/'(V = — 2ir

J

cp(ft)^.

(5.36)

2

0 The formulae (5.35) and (5.36) are just the Hilbert

115 inversion formulae (4.15) and (4.16), in which U(v) = cp(v) and v(v) = /'(v). For n > 2 it is therefore natural to refer to the formulae (5.27) and (5.28) as many-dimensional analogues of these formulae [17, 19].

5. Many-Dimensional Formulae

Analogues

of

the Hardy

Inversion

The Hilbert one-dimensional transformation

1 " r U(y)dy J IT J V V — X Y

= V(x)

—oo

can be written in the form I

r y - X

— TT

y U(y)dy = V(x),

J \y - x\Z

-» < x < ».

—00

In this section, the transformation

r(n/2) c y - x —frrr\

7LV{y)dy

= V(x),

(5.37)

Vl in which U is a scalar, v = (v^, ..., v ) is a vector, and ^ __- is the (n - 1)-dimensional Euclidean space of the points x = (jtj, ..., Jt-j) with Cartesian orthogonal coordinates x« , . . . ,x « , n > 2, is meant to be a many-dimensional analogue of the Hilbert one-dimensional transformation. Let E be the Euclidean space of the points x* = (jtp ..., x i> xn) and D be the upper half-space xn> 0. Let us consider the Cauchy problem for harmonic functions posed in the following way: The problem is to find

116 a function U(x*) harmonic in D, which is continuous in DUE*, vanishes as |JC*| ~n when |**| —* » and satisfies the boundary conditions u(x)

= f(x),

dU(x*) lim

= - cp(jc),

(5.38)

x e E i.

(5.39)

a

0<xn—*0 *n In addition we suppose that

/ eC^V^),

cp e C 0 ^(£ n _ 1 ) !

grad /,cp e L ^ E ^ ) . The problem of the determination of the function U(x*) harmonic in D from the boundary condition (5.38) is the Dirichlet problem the solution of which is given by the formula (5.4). The problem of the determination of the function U(x) harmonic in D from the boundary condition (5.39) is the Neumann problem. By a direct check it is easy to see that the function U x

(

)=

Y(n/2) TT77

(p(y)rfy 7 T—T, ?•

r

E

(5-4°)

i

is its solution. We discuss a simple way of obtaining the formula (5.40). First of all note that in the half-space D the function du(x*)/dx is harmonic together with U(x*). Owing to the formula (5.4) we therefore have

11.7 dU(x")

r(«/2) f WTxn\

d*

9(y)dy (ly

_

x |

2

2 } n/2

+

n-1

After the integration we obtain the formula (5.40). In order that the harmonic function U(x*) determined by the formula (5.40) be the solution of the Cauchy problem (5.38), (5.39), it is necessary and sufficient that the equality T(n/2)

(.

y(y)dy

r

-»).-/»I iT^r*"'**'-

"

V l

<5 41)

'

Vi be valid. On the other hand equality (5.4) yields W(x')

a,„

_ _ =

,

/(^)dy

Z

" (« - 2),»/ l?n J ( | . - . | n_1

r(n/2) =

a2

r(»/2)

f

a2

2 +

^ ) ^

i

(» - 2)//^ J ' < » £ 7% ( | „ - „ *

+

x*)<^2)/2 ^

Vi r(«/2)

(y - x)grad / ( y )

f

IT

n-\ whence

0

lim < x ^

ai/(^) >*n

\n/t.) r r(n/2) r n / * %J

^rr\

\y -- x)grad x)&±<xu /(>>) jyy} (y n \y-*\

—nrm^

dy

-

It then follows that the harmonic function U(x*) defined by the formula (5.4) is the solution of the Cauchy problem

118 (5.38), (5.39) if and only if the equality r(n/2)

(y - -*)grad x)grad . f(y)

f

^TTTn

*(*)---572" J E

dy

(5 42)

-

n-l

holds for all x e £n—t1 . Thus, under the above assumption regarding the behaviour of the functions f(x) and cp(jc) the equalities (5.41) and (5.42) are equivalent. That is the identical validity of any of them implies the identical validity of the other one. In other words equations (5.41) and (5.42) are the inversion formulae. Now note that, for / e C 2 ' 0 ^ ^ ) the equality (5.42) is the same as

T(n/2)

(x)

'

,

--{n-2) ^

Lf(y)

IiTT^*'

(5 43)

-

Vi where A is the Laplace operator with respect to the variables v^, ..., v _. . Besides the equality (5.41) is equivalent to the equality T(n/2)

v- x

r

TTI ^y)dy

—prrr J

IT ' E

|V -

= grad /(*)•

(5-44)

X\

n-X

The applied significance of the equation (5.41) for n = 3 is discussed in References 27,34,47. For n = 2, we require that the functions f'(x) and cp(jc) fl h

belong to C u '" (- oo < x < oo) and that the estimates A

1*1

A

1*1*

take place for large |JC| . Since in this case the solutions of the Dirichlet and

119 Neumann problems (5.38) and (5.39) are given half-plane D:{JC2 > 0} by the formulae

in

the

u(x) = — ~ 2 * M y - *j) + * 2 and

«<*> = — r logics - *!>2 + ^] 9 (y)^ 2ir 9 IT

J

J

-1 respectively, we: h a v e 00

1 IT

I

l o g | v -• * l < p ( :

(5.45)

—00

and 00

(y)dy

®( x) J

IT

V

-

- oo < x < oo.

(5.46)

X

—oo

instead of (5.41) and (5.42). After differentiating, equation (5.45) yields 00

1

IT

r

y(y)dy

= /'(*), J

- . < x < oo.

(5.47)

V - X

—oo

Since equations (5.46) and (5.47) give the Hardy inversion formulae (4.20) and (4.21), it is natural to consider equations (5.42) and (5.43) to be their many-dimensional analogues. The expression y - x \y - *\n

in the right-hand side of equation (5.42) is called the vector singular Riesz kernel [21,37,46].

120 The formula (5.42) and (5.44) remain valid even under weaker assumptions regarding f(x) and cp(jc). They are wittingly valid if

grad / ( * ) ,
L2(El).

6. The Neumann Generalized Problem

In this section, the Neumann generalized problem is meant to be the problem of the determination of a function U(x) from the class C"1' (D U s) harmonic in the domain D c E which satisfies the boundary condition dmu(y)

-ps--gm(y).

ys,

(5.48)

where v is the outward normal to S at the point y, gm(y) is a function from the class C(S) defined on S, and m is a non-negative integer. When m = 0 and m = 1 the equality (5.48) is the boundary condition for the Dirichlet and Neumann problems, respectively. The function N (x,y) is called the Neumann generalized kernel if the formula

u\x)

= J NM(x, y)gm{y)dsy

(5.49)

gives the solution of the Neumann generalized problem (5.48). If D is the unit ball {| x \ < 1} in the space En the solution of the problem (5.48) is written in quadratures. Actually since S is the unit sphere {|JC| = 1} for the

121 normal derivative of the order m of the function U(x) on S, we have the expression m-l

n

(xv - k)u(x)

= gm(x),

x e S,

(5.50)

fc=0 where V is the gradient operator with respect to the variables x*i . ..., x n. It is obvious that the function m-l

n (*V - k)u(x) fc=0

- v(x).

is harmonic in the sphere D

(5.51)

together with the function

U(x).

On the basis of equality (5.51), taking into account (5.50) we conclude that (4.48) is the boundary condition for the Dirichlet problem for the harmonic function V(JC) the solution of which is given by the formula 1

r 1 -

v(x) = - JJ a)n

\x\2

\y - x\

7^Sm{y)ds y

(5.52)

s

Consequently, the solutions of the problem (5.48) to be found are the solutions of the linear partial differential equation (5.15) of the order m harmonic in the ball D, the right-hand side V(JC) of which is defined in D by the formula (5.52). Since the harmonic solutions of the homogeneous equation m-l

n (*v - k)u(x) = o fc=0

corresponding to (5.51) are reduced to the harmonic polynomials of the variables x+ , .., x of the orders ranging from the zeroth to the (m - l)-th, then the linearly

122 independent harmonic polynomials of all these orders constitute the set of all the linearly independent solutions of the homogeneous problem dmU()(x) 0

dv

=0,

m

A

x E S.

(5.53)

x

corresponding to (5.48). It is known that all linearly independent homogeneous harmonic polynomials of the order k of the variables x. , ..., x are given by the formulae [20]

V - - V - 1 0 0 "j-0? ( " 1 ) J TiTTT ^i1---*-!1) < 2 J>'

(5 54)

-

and

%---K-i(x)

- Z ( - l ) i _ ^ A J ( j=o <^ + ^ !

x

; i . . . , ^ l )

where A is the Laplace operator variables xv . . . , xn_1, and av are non-negative integers such that

(5.55)

with respect to the . . . , a ^ , px , . . . , P n - 1

n-1 Xct-=A:, fc = 0, . . . , /w - 1, 1=1

n-1 X 3- = A: - 1, £=1

k = 1, ..., ro - 1.

(5.56)

In the case n = 2, the value Z of all the linearly independent homogeneous harmonic polynomials of the order k, k > 0, is equal to two, and, for n > 2, it is equal to

123 n + 2k

-

2

it - 2

f

I

^ j n

~

N

a

3

J

Consequently, the homogeneous problem (5.53) corresponding to (5.48) has N = 2m - 1 linearly independent solutions for n = 2 and

^=0

rt

~

2

linearly independent solutions for « > 2. For the equation (5.51), the point x = 0 is a singular one. For an arbitrary right-hand side v(jc),it may therefore have no solutions regular in the ball D. If we take into account that this equation is reduced to the equation with constant coefficient by the change of independent variables JC •

= e

,

i = l, . . . , n

in view of the expressions (5.52) for V(JC) we can easily find that the solutions of the equation (5.51) regular and harmonic in the sphere D exist only if the conditions

J V.«l_1<>>*«<'>*)'-0' s Je P r .. P n i (v)^(v)^ = 0,

(5.57)

S are fulfilled, where P a

Qa are homogeneous a l " -V-l' Pr-'P/i-l harmonic polynomials defined by the formulae (5.54) and (5.55). If the conditions (5.57) are fulfilled, the particular solution u*(x) of the problem (5.48) can be taken in the

124 form 1 1 u*(x)

= (m - 1)!

r J

, 1 A/B-1 v(t>x) - - 1j dt. M S

(5.58)

0 Thus, we come to the conclusion that the problem (5.48) is solvable if and only if the function gm(x) satisfies the conditions (5.57). When these conditions are fulfilled, the solutions of the problem (5.48) are given by the formula u(x) = II* (x) + X ^ a

P

(x)

+ Z*B * Q* a (*) Pi-•-Pn_l Pi-•-Pn_i

(5.59)

where >1

and # Q are arbitrary real constants Q p r * 'an-l> l- * 'Pn-l with integral non-negative indices a-,,... a n i and Pl>---Pw_l subjected to the conditions (5.56). For m = 1 the formula (5.59) transforms into the formula (5.13). On the basis of (5.52) and (5.58) we find that 1 1 x/w-1 1 dt, r ( 1 Nix, v) = - - 1 -. (5.60) 71 0 a

The formula (5.60) permits one to find the inverse of the integral equation of the first kind

lNm(*>

y)gm(y)*'y-fml*)>

X e

S,

the kernel of which Nm(x,y) is defined by the formula (5.60). In the similar way, we can study the problem (5.48) when D is the half-space {x > 0} of the space E [24].

125 CHAPTER 6

SOME OTHER CLASSES OF INTEGRAL EQUATIONS OF THE FIRST KIND

1. The Integral In The Sense Of A Cauchy Principle Value

In the present chapter we shall restrict ourselves to the study of the integral a r

1

I(t ) =

o

\L(t)dt

~a < *0
~ J ;—r>

-a representing, by definition, the limit of the expression tr\-t

a

dt

J

J

-a

t

irJL ^

'

Q

tQ-s

+

(t

rr ^ o> ( J iri v

J

-a

t

-

tn

0

+E

a

+

^

\

}

J

}

t

-

tr,

^n+e

for £ —► 0. It obviously exists for all £Q, - a < tr, < a, if the density \i(t) is Hoelder continuous and

I(tQ) where

=

I ^ Q )

+ i^O*'

126 1

r

|L(t) - |i(tQ)

a - t0 a + £Q

1

J

iri t - £Q iri -a T h e expression -f(tn) a s a function of the parameter £Q has a very important property: if the density M-(£Q) is Hoelder continuous on the segment 7 : - a ^ t ^ a, the expression I(tn) is also Hoelder continuous on the segment 7 f i : - a + 8 ^ t Q ^ a - 8 , where 8 is an arbitrarily small positive number. a -

Since the expression log

tQ

is analytic on 7. as a

a + tQ

function of £Q, the function ^9^0^ ^ s Hoelder continuous with Hoelder index v the same as \i(tr,). We have only to show that IJ^tQ + h) - Ix{tQ)\

± Lx\h\\

II^tQ + h) - X 1 (t 0 )| < L2\h\

0 < v < 1,

log — ,

when the points £Q and tQ + h are both on 7.. Without loss of generality we can 0 < h < a. The difference II(£Q + h) - IAtr.) represented in the form Ix(t0

+ h) - Ix{t^)

v = 1,

assume that can then be

= A + B,

where 1

irx

t(.+2h

r J

tQ-2h

r

|i(t) - n(tQ t — tQ — H

+ A)

(i(0

- |L(£0>

t — t rk

dt,

127 1

B(tn)

tn-2h

a

r 1 T v-(t) - n(t n + h)

f r

= —

wi I

+J

J

tQ+2h

-a

I I

t - tn - h °

M-(0 - C-(tn) 1 —

^

0

Since ji is Hoelder continuous, we have tr.+lk 1

M M - i

J (It - t 0 " ^l""1 + I* - f o l v _ 1 ) ^ tQ-2h ( 1 + 2 - 2 v + 3v)fcv,

< _ VTT

where L i s the Hoelder coefficient of the function ^(t). Let us represent the function p(t Q ) as a sum C + D where

c « 0 , -'- ( J ♦ J )„<.> -a

tQ+2/i

1

- |i(tn + A)] 0

dt (t - tQ - h)(t

-

tQ)

and

D(tQ)

1

=-

[n(i0) - ^(tQ + h)] log

It is obvious that |D(t0)| < IT

lMh\

a - tQ

5i.

128 where M = max I log

a

~ t0

a +

'

\

,

t Q e -y .

tpv '

Taking into account that JJL is also Hoelder continuous and the inequalities 2h < t* -t and 2h < t -t hold in the intervals - a < t* < t -2h tnr-2h a Lk r r r |c| +

* r I J tr,-2h f

(it U

t Q +2/*

U

a

r

r

+

"7 ' J

\

we obtain

J J it - 1 0 - A i ^ i t - t 0 i

-a Lk

tr, + 2h
and

-a

\

dt

J J |i - v ( t - t0)i1-vit - t0i2-v U

tQ+2h i_ 2

Lh r

trr-2h r

a r

-a

tQ+2/i

—T\

ir(l -

\

U

^t u

7 h 9 0 < V < 1 v) '

_?*logJ,

-1,

for sufficiently small h. Thus we have proved the above statement. We should also note that the Hoelder condition for I(tr\) is of the form |J(t0 + h) - I(tQ)\

*

L*hv

for 0 < v < 1 and |I(tQ + h) - I(tQ)\ for v = 1 [40].

* l"Alog

—

129 The discovered property of the continuity of the Cauchy principal value integral I(trx) also holds in the cases when the integration line is the Lyapunov closed or non-closed curve or a set of such curves. When 7 is closed the Hoelder continuity I(tn) takes place on the whole 7. In the case of a non-closed 7, this property takes place in any interval of 7* of the curve 7, the ends of which are at any different from zero distance from the ends of 7 [50].

2. The Behaviour of the Derivative of the Cauchy Integral Near the Line of Integration

Let us consider the Cauchy type integral integration path coincides with the segment 7

when

Type

the

a 1 1

F(z)

\i(t)dt )UL

rr

=—

iri ir1

LLl L

J

(6.1)

tt -— z7

J

-a

with the density JUL(t) which is Hoelder continuous. At any point z of the complex plane out of the segment 7, we have a 1 iri

\i(t)dt

r J

(t

-

z)L

-a When the point £Q belongs to the interval 7^, i.e., to - a + 5 < t Q < a - 5 , we are interested in the points z out of the interval 7 and satisfying the conditions IT

\z - tQ\

< T|,

a Q ^ arg(z - t Q ) = a * -,

where T\ is an arbitrary positive number, T\ < 6, an is an arbitrarily small positive fixed number, and arg(z - £ Q ) means the principal branch of this multiple-valued function.

130 We now show that under these assumptions take place the evaluations v = 1,

|F'(z)| < C|log(z - t Q )|,

\F'(z)\

< C\z

t0\v-X9

-

0 < v < 1,

where C is a fixed positive number and v is the Hoelder exponent of the function \i(t). Indeed, we represent F' (z) in the form F'(z)

= F^z)

+ F 2 (z),

where -a+8 1

iri ^

a

J

J j

-a

a-h

(t -

z)A

and a-b Z

iri J (t - z ) Z -a+8

Since \z

~ *§)

<

T\>

it is obvious that the function FAz) is bounded. We write the expression ^oC2) a s F2(z) where

= ^(z) + * 2 (z),

131 2(a - 5)

^(tn)

iri

(z-a+8)(z+a-8)

and a-8 1

*2(z)

r

=-

\L(t) M-(t) -- »i(\i(tQ)

T1—

I — J

TTl

dt.

(t - Z)

-a+8 The function ^ ( z ) is bounded for \z - tA the function $9(z) w e have a-b

< T\ and for

l*2l * ~ f

2 2 2 i* (I* ~ ^o^ ~ \z ~ £n|cosa) + \z - tg| sin a -a+8 IT

*J

L = - (o(z,t0) IT

where a is the smallest actue angle equal to arg(z - £Q) and L is the Hoelder constant of the function \i. We rewrite the expression (o(z, £Q) in the form <0

(t -

Z

<»H > * r J =

2 J

-a+8

((tr,*n-

a-8

+

I J

tQ)vdt

tt --

\z \z --

(t ( t - tQ

2—

|cc £t n0|cosa) + \z - £Q | sin a

tnydt

2 2 1—' \z - t0|cosa) + |z - £Q| sin a

For v = 1 these integrals are expressed in terms of elementary functions, and, as a result of simple calculations, we obtain

132
-

tQ\l,

we obtain the equality t'

o)(z, tQU ) = \z - t U0 r

_1

t"

rr»

~2

f f+f)

J J (T _ cosa)Z + s i n a 0

I J 0

where t'

=

a - 8 + tn —,

a - 5 - t t" =

n

—.

It is obvious that 00

VJ

(o(z, t Q ) < 2 | z - tQ\v

X

| 0

Cx|z

-

-r, ai cosa)

-

T

+ sin^a

t Q , v-1

which accomplishes the proof of the above statement. We draw the ray X : arg(z - t Q ) = a through the point tQ, - a + 8 < tn a - 5, where a is the number introduced above and consider two points Zi and z~ on X. Without loss of generality, we assume that | z~ ~tr\\ > \z\ ~ tc\\The evident equality

133 z z

2 2

F(z F(z22)) - F( F(zj) | F'(z)dz F'(z)
yields F(xZl)\)\ \F(z2) --F(z

-- \ztx -- tQQ\vv)

l

v <<-- (|z(|z 2 2 -- ttQ\0 r

s* c'\z C*\z 2 2 - z\

((6.2) 6.2)

for 0 < v < 1 and |F(z22)) - F(z F(Zlr)| \F(z )\

< C"\z C**|z z1r|log|z |v|log|z22 -- zxz\\x\\ 2 2 - Zi

(6.2^ (6-2x)

for v = 1, due to the evaluation for F' (z) proved above. In the formulae (6.2) and (6.2^, C* and C** are quite definite positive constants [40].

3. The Behaviour of the Cauchy Type Integral (6.1) Near the Segment 7§

Under the requirement of the Hoelder continuity for the function ^(t), the Sokhotskii-Plemelj (S-P) formulae remain valid and, if at any point tQ, - a < tQ < a, F+(tQ) and F~(tQ) are understood as the expressions F+(tQQ)

= lim F{z), F(z),

Im z > 0,

z—£Q z—tQ

fF~(t ~ ( tQ0)) = lim F(z),

z^tQ

Im z < 0,

134 we have

^(t0)

1

r

= ^(tQ) + - J

|i(t)dt

-,

-a

a F (tQ) = - |L(tQ) + —

f J

TTl

\i(t)dt

£ - £Q

-a In view of Section 1 these formulae permit us to conclude that F + ( £ Q ) and F~(£Q) are Hoelder continuous on 7, for an arbitrarily small 8. When the point £Q is on the upper boundary of the cut 7, we additionally define the function F(z) at this point as F + ( £ Q ) . The function obtained in this way -is Hoelder continuous in the upper half-plane Im z > 0 up to the upper boundary of the cut 7& . To prove this statement we close 7. by the Jordan smooth arc 7' situated in the upper half-plane, which constitutes, together with 7., the Jordan smooth closed curve T. Let D+ be a finite domain of the plane of complex variable z bounded by the curve r. The function F (£ ) exists everywhere on r and is Hoelder continuous. Let us consider the function z

♦< >

F(z) - F+(tQ) = (7U - M t«Q) '

*oe r >

where a positive number a is smaller than the Hoelder index p of the function F+(t) and (z - £n) a is any branch of this function. The boundary values ¥ (t) of the function ^(z) exist on T and the expression +

♦ (o

F+(t) (t -

-

F+(tQ) tn)a

135 is a Hoelder continuous function, since a < p. Let us remove the point £Q together with a circle of a sufficiently small radius E smaller than the standard radius 8Q of the curve T from r ([40], 16-17) and represent the function ^(z) in the rest part D of the domain D in terms of the Cauchy formula 1

ty+(t)dt

r

+ (z) =

,

(6.3)

3D £

where 3D is the Jordan piecewise smooth curve. Since in the neighbourhood of the point z = t^, evaluation

the

uz) -o(\z - t 0 r a ) , holds for the function ^(z) in the formula (6.3), we can pass to the limit e —► 0 to obtain 1

r

2iri J

*?+(t)dt

t - z

r for all the points z of the domain D+. After the additional definition of V(z) in D+ U r as V(t) on r, due to the maximum-modulus principle for analytic functions, we conclude that |+(z)| ^ max |i|/+(t)| ^ C,

C = const,

uniformly with respect to t^ and a, which, in its turn, yields the validity of this evaluation also for a = 3. Therefore, the evaluation \F(z)

- F+(tn)\

* C\z - tn|<\

136 holds, which proves the validity of the formulated statement when one of the two points in the Hoelder condition belongs to r. Now let these points z. and z 2 be in D+. For the function F(z), the Hoelder condition is to be checked only in the case when at least one of these points, for example ZQ, is at the distance smaller than 8 Q from r. We denote the point, nearest to Z Q on r, as £Q and cut the domain D along the straight segment 7^ connecting the points £ Q and z (this segment is orthogonal to T). In the domain D. , obtained after cutting the domain D , any branch of the function

=

FU) - ?(»„)

*•<" (, - ,„>»

is analytic with respect to z and has continuous boundary values on r. Therefore repeating the above reasoning, we get the estimate

|+Q(*)| * C,

t

e

r.

According to (6.2) and (6.2^), the evaluation we have just obtained is preserved at each boundary of the cut and, therefore,

|I|/Q(Z) | ^ const

in the domain D , i.e.,

\F(z)

- F(z Q )| ^ C\z

- zQ|P.

4. The Behaviour of the Cauchy Type Integral (6.1) Near the Ends of 7

137 In this section we shall study the behaviour of the Cauchy type integral (6.1) near the points z = a and z = - a when the density pi is of the form cp(t) n(t) = , Va - t where cp(t) is Hoelder continuous for - a ^ t ^ a. We shall show that, for the function 4>(z), represented by the formula a 1 r y(t)dt

Hz) = —

-,

t

(6.4)

iri J Va - t(t - z) near the point z = a, the evaluation const

l°(z)l

<

■;

rs>

<6-5>

p - a\ takes place, where p is any \z real number from the interval

1/2 < p < 1.

(6.6)

To prove this statement we rewrite the formula (6.4) in the form a 11 -- |\t - a\ rr \z ft \Z ~ a\V U\ cp(£)d£ \z - a\*Hz) =- J -™ z) iri (t a)i/Z(t -a a 1 r \t - a\.P-l/2 + — cp(£)rf£. (6.7) iri J £- z -a Due to (6.6), the second term in the right-hand side of this formula is bounded near z = a. Since

138 \z - a|P -

- a|p ^ \z -

\t

t\{

we have 1 * |z - a|P - |t - a|P TT

TF)

J

|t - a| 1/2 (t - z)

(p(t)dt

-a a

-I 1

<

|
-a for the first term in the right-hand side of the formula (6.7). The last integral converges because 3 > 1/2 in view of (6.6). We have thus proved the validity of the evaluation (6.5). We shall need the evaluation (6.5) to obtain a more exact evaluation. In the case
dt

r

*i(*) =-J

7=7 (t

iri -aJ Va -

-

z)

t(t

near the point z = a is easy to see. Actually according to the Sokhotskii-Plemelj formulae we have = 2 Va

*i(*o> " W Let

d>2(z) =

—

~ 'o-

139 and

when to belongs to upper boundary of cut 7 Q . It is obvious that

1

i.e. ,

*2<*0> " *2<'o>

= 2(a

" t 0>" 1 / 2 -

For the function

H(z) = ^ ( z ) -
we therefore have

a+(tQ)

- n~(£0) = 0,

- a < t0 < a,

i.e., (l(z) can be analytically continued over the interval 7 and, besides, the evaluation (6.5) holds for it near the

point z = a. Moving the point z = a from the plane of complex variable z together with a circle of a sufficiently small radius E and applying the Cauchy integral formula in the rest part, we conclude that O(z) is analytic near the point

z = a, i.e.,

140

^(z) = d>2(z) + o(z). We can therefore write the function <|>(z) in the form *(z) =
1

r

+— J


_

Yrr-p 7


Z

)~

1 / 2

_ '

(a - t)

z

dt + n(z),

(6.8)

iri (a - t) (t - z) -a where E' is a positive number so small that the function
'

remains Hoelder continuous. The evaluation (6.5) also applies to the second term in the right-hand side of the formula (6.8), in which we can assume that p = 1/2. Consequently, the evaluations const 7 \z - a\ YJ !^

|$*(z)| < C,

<*>*(z) = (z - a) 1/2 (D(z).

holds for the function <|>(z) near the point z = a. We note that a stronger statement is also valid: if (|)*(z) is bounded near the point z = a, then $*(a) = 0 [40].

5. The Construction of Solutions of a Class of Integral Equations with the Cauchy Principle Value Integral in a Finite Interval

141 In the present section the solutions of a class of the integral equations of the type 1 1 \U(x)

+ ir

u(t)dt

r

= V(JC), J

t -

0 < x < 1,

(6.9)

x

0 are constructed, where X is a given real constant, v(x) is a given real function, u(x) is a real function to be found, and the integral is treated in the sense of a Cauchy principle value [11,12,15]. The equations (6.9) are referred to as normally solvable singular integral equations for \ * ± i (see Chapter I, Section 4). Further we shall suppose that this condition is fulfilled. Under the assumption that v and u belong to the class h co,/i s C°> (h

< JC < 1 - 5)

the solutions of equation quadratures. The Cauchy type integral

(6.1)

are

constructed

in

1 1

F(z)

with the density u(t)

=

r

u(t)dt

(6.10)

2iri J t - x 0

e C?' represents an analytic function

on the plane E cut along the segment 0 < x ^ 1 of

SL complex

variable z = x + iy having the powers F(z) = 0[ - ) and

= 0(|z|P),

F(z)

Q < ^ <

I.

F(z)

= 0{ \ 1 - z|p)

(6.11) for

certain

p,

142 When 0 < x < 1, due to the Sokhotskii-Plemelj formulae (6.10) yields 1 +

F (x)

=

l1

l i m F ( z ) = - u(x) + ?

1 u(t)dt

rr

9.ir1 JJ t - JC 2iri

1 1 1 F (x) = l i m F(z) = - - W(JC) + 2 2iri

r J

u(t)dt , t - x

0

K(X)

= F+(JC)

- F~(JC),

(6.12)

1 1

r

u(t)dt = F+(x)

—

^

Q* -

X

+ F-(JC),

(6.13)

0<*<1

Assuming that the density u of the Cauchy type integral (6.10) is a solution of the integral equation (6.9) and due to (6.12) and (6.13) this equation can be written in the form F+{x)

= GF~(x)

+

V(JC),

(6.14)

X + i where X - i G =

.

X + i The equality (6.14) represents a boundary condition of the so called linear conjugation problem in the following form: to find the function F(z) of the class C°> analytic in E which satisfies the conditions (6.11) and (6.14).

143 The solution of this task is easy to create. Indeed, we introduce the function *(z) = X(z)F(z),

(6.15)

where Z

_1

X(z) = (

z

fz JZ

(6.16)

1

arg G,

29 =

-IT

< a r g G ^ ir,

(6.17)

IT

into consideration, where (z-l/z)6 is a branch of this function which becomes unity for z —► «. Since * + (*) = ( ^ —

X"(x) = X+(*)e-2ilve,

) x,

the boundary condition (6.14) for F(z) condition . $+(x)

1 - aT(x)

=

becomes a boundary

. X+(JC)V(JC),

(6.18)

X + i for the function $(z) of the class C?' . Due to (6.11), (6.15), and (6.16) the function $(z) should be bounded on the infinity. It is restored from the jump (6.18) by the formula 1 0(z) =

1 r

.

T{t)v{t)dt

2iri J (\ + i)(t - z) 0

where C is an arbitrary real constant. (6.15) and (6.19) yield

+ iC,

(6.19)

144 1 F(z)

=

2iri

r

X + (£)

v(t)dt

1

±C +

J

0

X(z)

X(t) X + i t - z

Calculating F+(x) and F~(x), 0 < x < 1 and using the Sokhotskii-Plemelj formula and the equality (6.12) we obtain the solution u(x) of equation (6.9) to be found 1

\ u(x)

=

i

v (

j-

j v(*)

t a-e, i - t ^e -

v{t)dt

(6.20)

(i - ,)V- 0 *

The last term in the right-hand side of equation (6.20) represents a general solution of the homogeneous equation corresponding to equation (6.9). Further we shall need the solution u{x) of equation (6.9) which can be obtained from equation (6.20) for 1 1 C =

f

, 1 - t .0 v(t)dt,

27

i.e.

v(x)

«(*) = 1 + k'

*(1 ~ t) Z

n(l + X )

J

t(l - x)

e v(t)dt (6.21) t - X

0 Such a solution is unique. Equation (6.9) becomes an equation of the first kind

145 1 1 1

rr

U

u(t)dt

(6.22)

V(*),

IT J t

0 for X. = 0 and the formula (6.20) yields its solutions 1 1/2 v(£)dt

u(x) = - - r IT

*0 - *)

J

[x(l

t - x

(6.23)

x)} TJ1-

-

In applications the solutions of equation (6.22), bounded at one of the ends of the segment 0 < x < 1, are of great importance. This solution always exists and is unique. It can be obtained if we assume that in the formula (6.23), 1 1 f , 1 - t ,1/2 0 or t I f f \1'2 C = J IT ^ 1 " t >

v(t)dt.

In the first case 1

r

X(l

-

t)

1/2

£(1 - x) J

v(t)dt

(6.24)

t - x

and in the second

u(X) = - - r IT

J

t(l - X)

x(l

- t)

1/2

v(t)dt

(6.25) J

t- *

146 The solution of Equation (6.22) bounded at both ends of the segment 0 < z < 1 exists if and only if the function v satisfies the requirement 1 v(t)dt J [t(l

(6.26)

= 0.

A/2

- t

0 It is obtained from (6.24) or, which is the same, from (6.25) and has the form 1 *(1 ~ x)

u(x) = - - [ IT

t(\

J

-

v(t)dt

1/2

(6.27)

t)

t - x

0 Since as a result of the change of variables y + a

2a

u\ ^

0 <

JC

I = u*(y), >

2a

< 1,

0 < t < 1,

i + a

t =

v[ I

2a

y + a 2a

j= >

- a < y < a,

v*(y),

- a <

T

the equalities (6.22) and (6.23) become the equalities 1

f

-n J•* -a

and

u*(y)dT T

- y

= v (y)

(6.28)

147 a

T

«*00 = -- J ( -2 J

it

K

a

^1/2 v (y)di

2 T - y

- v

-a 2aC (6.29) 2

- y2)1/2'

{a

respectively, we can conclude that all the solutions of the class C°h'h{ - a < y < a) of equation (6.28) are given by the formula (6.29). Besides, the formulae

u\y)

(a - y)(a + T )

1 r

1/2 v*(y)rfT

=-- I

(6.30) (a - x)(a + y)

-a and a

u\y)

(a - i)(a

1 r

= - - \ IT

+ y)

1/2 v*(y)
(a - y)(a + T ) J

J

T - y

-a give the solutions of equation (6.28) bounded near y = a and y = a, respectively. The solution of this equation bounded at both ends of the segment - a < y < a holds only if the condition a V(T)^T

J 7~2 (a J

Z7

- T )172

(6.32)

= 0

-a is valid and has the form a

1

, a2 - y1

r

"(30---J i r

According

to

^ J

the

V

,1/2

v*(>)dT

(6.33)

2

a - T

y

note

made

T - y

in

Section 4

of

this

148 chapter, this function becomes equal to zero at the ends of the segment - a < y < a. The condition (6.32) holds in particular when the function v*(y) is odd. The formulae (6.28) and (6.29) become the Hardy formulae (4.20) and (4.21) respectively for a —> °°. The transformation (6.28) and its inverse transformation (6.30) are usually called the Hilbert transformations for the finite segment - a < y < a.

149 6. The Integral Equation of the First Kind with a Logarithmic Kernel

Here we shall study the equation a - I log \t IT J

-

x\u{t)dt

= V(JC),

-a

^ x ^ a,

(6.34)

-a

where V(JC) is a given real function and u(x) is a real function to be found. This equation often arises in the applications (see Equation (2.18)). In particular, the Dirichlet problem reduces to it in the following statement: to find the function harmonic on the plane of a complex variable z = x + iy cut along the segment - a £ x ^ a, which has a logarithmic singularity on the infinity and which belongs to the class C?' under the boundary condition u(x)

= f(x),

- a * x * a.

(6.35)

If we seek a solution u of this problem in the form of the potential of a simple layer a 1

u(z)

= -

r

log \t - z\u(t)dt,

(6.36)

IT ^

-a

we get the equation (6.34) itself. Under the assumption u{x)

h

e C°b> (- a < x < a),

the

that equation

v'(x), (6.34)

transforms into the equation 1

a f

u(t)dt = - V'(JC),

ir JJ tt -_ xY -a

-a < x < a.

(6.37)

150 studied in the previous section, as the result of differentiation. According to (6.29) the general solution of equation (6.37) is given by the formula [2, 59, 60, 61] 2 2 1 r , a - t

TT

J

^ a

- *

xl/2 v'(t v'(t)dt >

t

-

-a C +

(fl2 _

<6-38)

1^/V

where C is an arbitrary real constant. The expression uAx)

1 = -

1

TT

f J

, a2 - t2 xl/2 v'{t)dt -2 2 l a - xL > t - x

-a

is a particular solution of equation (6.37) and C

V * ) " ,1 _ 2x1/2

(a* ~^y

is a general solution of the homogeneous equation a 1 r un(t)dt UQ(t)dt

0. IT J

t

-

(6.39)

X

-a

corresponding to equation (6.37). Obviously the function uAx) equation a - T log \t

- x\ul(t)dt

satisfies

= v(x)

+ CQ

(6.40)

IT

-a

where C Q is a constant. Since a dt at

r

J log \t - x\

2

_

a

*x/1

=

TT

the

log 2

151 the function equation

WQ(JC)

is a solution of the homogeneous

a - I log \t - x\uQ(t)dt

= 0.

IT J IT

-a corresponding to Equation (6.34) only for a = 2. When a * 2, we choose the constant C in the representation (6.8) of the solutions of equation (6.37) in such a way that the equality a

C log - + Cun = 0. 2 holds. It is obvious that the function CQ u(x)

= uAx) 1

^ log a/2 {a1 -

1 TTT1 xZ)l/2

(6-41)

is a required solution (the only one) of equation (6.34). In this case the solution u(z) of the Dirichlet problem is given by the formula (6.36) in which the density u is defined by the formula (6.41). When a = 2, the function C 1

(a

-

xLy,L

is a solution of the equation a 1 r IT

|t - x\u(t)dt

= v(x)

+ Cn

J

-a for an arbitrary constant C, and this time the solution (the only one) of the Dirichlet problem (6.35) is given by the formula

152 2 1

u(z)

r

=-

log \t - z\ux(t)dt

IT

- CQ.

J

-2

7. The Dirichlet Problem for a Plane with a Gap

The plane with a crack means an extended plane of complex variable z = x + iy cut along a smooth arc 7. It, of course is a simply connected domain. We shall 6 denote it as E7 . We shall assume that 2 coincides with the interval - a < x < a of the real axis Im z = 0. In this section the solution of the Dirichlet problem is constructed in quadratures posed in the following way: to find a function u{x,y) = u(z) harmonic in E , which vanishes on infinity and takes given values

u{t)

= f(t),

-a < t < a,

(6.42)

on 7 under the assumption that the evaluations const Z

l"( )l

<

~,

a>

0 < « < 1.

(6-43>

\z - c\ hold in the neighbourhood of the points z = c which coincide with the ends of 7. The function f(x) is Hoelder continuous for - a ^ x ^ a. Let F(z)

= u(z)

+ iv(z),

(6.44)

be analytic in E with the real part u{z) and vanish on the infinity, for which the evaluations

153 const \F(z)\

<

-,

0 < a < 1.

(6.45)

\z - c\

hold near c = ± a. The function, defined by the formula F(z)

= u(x,

-y) - iv(jc, -y)

(6.46)

is also analytic in E . It vanishes in the infinity and satisfies the condition (6.45) near the ends of 7. We know nothing about the behaviour of the functions F(z) and F(z) when the point z goes over the segment 7 from the upper half-plane Imz > 0 to the lower one Imz < 0 and back until these functions are constructed. Besides the functions F(z) and F(z), we introduce the functions
+ F(z)]

(6.47)

and *(z) = F(z) - F(z),

(6.48)

where (a 2 - z2) 1/2 denotes the branch of this function which behaves in infinity as iz. The function $(z) is analytic in E , bounded in infinity, and the evaluations const (Z)I <

'*

|z-c|-
(6

- 49)

take place for it near the points z = c = ± a. Due to (6.42), (6.44), (6.46), and (6.47), we

154 have -2(a 2 - t 2 ) 1 / 2 / ( 0 > ~a
lim ReO(z) = Re4> (t) = \ z—^t

-»< t < -a,

0,

Im z > 0

a .

(6.50)

To define the function $(z) in the upper half-plane Im z > 0 according to the boundary conditions (6.50), we apply the Schwarz formula 00

1

r

d>(z) = — iri J

dt

+ Re
+ C . t - z

—00

where C* is an arbitrary complex constant. As a result we obtain a

*(Z)=- \ ( a irl irr

2

- t V

J

/ 2 f /• —

^

+

C*.

(6.51)

7

-a

Due to (6.42), (6.44). (6.46), and (6.48), we obtain that Re¥ + (£) = 0 for -oo < t < «> everywhere, probably, except for the points t = ± a. Consequently, ^(z) = 0 everywhere on the plane of complex variable z and, thus, according to (6.48), we find T(z)

= F(z).

(6.52)

The formulae (6.47), (6.51), and (6.52) yield

155 a

F

1

r(

a1

~t

?

^

^ -iri- J a ^- z 2 J v

y

£- z

-a

(a2 - z2)V2-

(6

- 53)

It is obvious that the evaluation (6.49) holds for the function F(z) defined by the formula (6.53), and its real part u(z)

= Re F(z)

(6.54)

satisfies all the requirements of the Dirichlet problem formulated above. The constant C* is directly determined from the requirement of the boundednes of u(z) at one of the ends of the segment 7. For example when a 1 f r a + t >l/2 C* = — J v f(t)dt (6.55) iri a - t } -a the formulae (6.53) and (6.54) yield the (unique) solution u(z)

= Re —

a f

(a + t)(a

- z)

1/2 m w *

(a - t)(a + z) J -a bounded for z = a. We now consider the function a 1 f u(z) = log |t - z|cp(t)Jt,

.

(6.56)

(6.57)

or J IT

-a which is a potential of a simple layer of masses, distributed over the segment 7 with the density cp(t). Since

156 a 1 y(t)dt 1 rr = Re — J iri t - z -a

0

— u(z) dy we have du >,

—

^ ay

= q>w,

J

/ aw

— "= " *<*)'

l

ay

- a <

JC

(6 58)

-

;

< a.

under the assumption that