Integral equations with fixed singularities

172 Pages·1979·8.448 MB·English

by  Duduchava, Roland

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Roland Duduchava TEUBNER- TEXTE
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Id ooQClusioD I wish to express my deep gratitude to prof.
A* Kalandiya who kindly drew my attention to the basic problems
considered here.
I shall be very much obliged to the readers for their comments
on the book*
Author
Tbilisi, Georgian SSR
July, 1978
CONTENTS
Introduction . 7
Chapter I. ' Integral Equations in Convolution with 11
Discontinuous Presymbols
Sect® 1. Definitions and Auxiliary Propositions 11
Sect* 2* Basic Properties of the Integral Operators 25
in Convolution
Sect* 3* Equations in Convolution with Continuous 42
Presymbols
Sect* 4* Equations in Convolution with Discontinuous 47
Presymbols
Sect* 5* Operators in Convolution in the Spaces of 61
Bessel Potentials
Sect* 6* Operators of Multiplicative Convolution in 64
Holder Spaces of Functions
Sect* 7* Convolution-Type Equations with 71
Discontinuous Coefficients
Chapter II* Singular Integral Equations with Fixed 83
Singularities in Kernels
Sect* 8* Singular Equations with Fixed Singularities 83
Sect* 9* Equations (8*1) in the Space Lp(J,a£*(l-x)T ) 94
Sect*10* Singular Integral Equations with Two Fixed 99
Singularities
Sect.11* Smoothness and Asymptotical Behaviour of 104
Solution of eqs* (8*1) and (10*1)
Sect*12* Some Further Results 120
5
Chapter III. Some Applications 127
Sect.13. H. Bueckner* s Integral Equation 127
Sect.14. A Notched Half-Plane 136
Sect.15* A Notched Half-Plane with a Rigid Core 140
3ect.16# Influence of a Stringer on the Stress 144
Distribution Near a Circular Hole
Sect.17. Transfer of Load from a Stiffener 146
to a Sheet
Sect.18. Bending of a Semi-Circular Plate 149
Sect.19* A System of Integral Equations 152
Sect.20# F. Tricomi1 s Integral Equation 155
Sect.21. On the Integral Equations Appearing in 162
the Dislocation Theory
References 165
List of Notations 170
Subject Index 171
INTRODUCTION
The book presents some new results for singular integral equa­
tions of the type
C0 u(x) + ;1 Hkisz + l -°-**2 * k /1 /~ °k 3 f(x),
ui o y - x k=o Hi o (y + x) 1
x € J = [0,1], 0 < Re nfc < k , (0.1)
and also of the type
c u(r) + s +
0 Hi -1 y - x
“ 1 1 ck(*»y)(l-x) k(l+x) k u(y)dy
+ Zj J ' — y~Tr -■ ■ ■1 u. = f )»
k=° -1 (1_y) * (l+y)“*-k(l-*y)k+1
x € J° = [-1,1], 0 < Re m^ < k (0.2)
The specific feature of eqs. (0.1) and (0.2) is that along
with a movable singularity on the line x = y their kernels possess
fixed singularities at the points x = y = 0 and x = y = + 1,
respectively; these singularities influence essentially the Noethe-
rian properties and the index of equations*
Eqs. (0.1) and (0.2) have found many applications to problems
of mechanics and mathematical physics (cf* [3,4, 6 a-b, 9, 10, 20,
21 a-c, 25, 32, 33, 38, 39, 41, 45, 46, 47] and Chapter III below).
By substituting x = exp(-t) and y = exp(-nr) eq* (0.1) may
be reduced to the equations in convolution on the semi-axis
v(t) + / k(t-x) v(t) dr = g(t), t € R+ = [0,oo), (0.3)
o
where the Fourier transform K(x) = Fk(x) (X € R = ( - 00,00)) of the
kernel k(t) is discontinuous at infinity.
Chapter I will therefore deal with eq. (0*3) in the case when
k(t) is the distribution with discontinuous Fourier transform K(\)
(the latter may be, for example, a function of bounded variation
on R). Such a kernel may be illustrated by the function
7
fc(t) = k0(t) + E — e3j)(itO , k e l1(r), a1 € e j
m=1 t °
the integral in eq. (0.3) should be now understood in the sense of
the Cauchy principal value*
Eqs. (0*3) are obviously interesting in themselves; they have
applications in mechanics and mathematical physics and are investi­
gated thoroughly in the case of the continuous presymbol a(X) =
= 1 + F k(A) (cf. [14, 27, 34])• A complete theory of eqs. (0.3)
is stated in Chapter I (cf. Sects. 2-3); in particular, the neces­
sary and sufficient conditions for being Noetherian, the index for­
mula and formulas for solving these equations in the space Lp(R+)
(1 < p < oo) and in Bessel potential spaces are obtained. Chapter I
is a revised and enlarged presentation of the author1 s works [I3a-e]
the investigation is conducted by means of the local principle from
[13a], Chapter XII, Sect. 1.
In Sect. 6 are stated some results of A. P. Soldatov [36a] re­
lating to equations of multiplicative convolution with continuous
presymbols in Holder spaces with weights.
Sect. 7 is concerned with some new results for equations of the
convolution type
s
£ [^ (t) u(t) + b (t) u(y) dy] = g(t), t € S. (0.4)
m=1 -oo m
These results simplify essentially and supplement the results
of the author®s works [I3b,f], where the sufficient conditions for
being Noetherian and the index formula of eq. (0.4) were obtained.
In the case k(t) € L.,(R) eq. (0.3) was investigated in [21 ]
by M. G. Krein, while under more rigid restrictions on k(t) it
was studied in the earlier works by N. Wiener and E. Hopf (cf. [14,
27, 34] and the referencescited therein); the case k(t) e L1(R),
however, can be handled quite easily (see Sect. 3 below), since the
presymbol a(t) = 1+5* k(t) is a continuous function. For some
particular cases eqs. Co.3) and (o.4) are investigated in [22a-b,
28, 37].
Chapter II is entirely devoted to eqs. (0.1) and (0.2). Con­
ditions to be Noetherian, the index formula and formulas for sol­
ving eq. (0.1) in spaces Lp(J, aP^I-x)^) and in differentiable
function spaces are obtained in Sects. 8 - 9. A simple geometrical
procedure of finding conditions of being Noetherian and calculating
8
the index of eq. Co.1) is also indicated! this procedure will prove
helpful in further applications (see Sects. 13-20).
It is of interest to note that solutions of the homogeneous
equations (0.1) and (0.3) can be chosen to form the so-called
“d-chains* (see formulas (4-.15) and (8.18)).
Sect. 11 deals with solutions of eq. (0.1). It is proved, in
particular, that the solution has the same smoothness as f(x) at
every interior point of the interval J = [0,1]| at the point x = 1
the solution is bounded (for cQ = 0 it vanishes) or has a singu­
larity of order v = (2iti)"1 ln(cQ + c1)^ , 0 < Re v < 1 (v = 1/2
for c = 0)j at x = 0 the solution is either bounded or has a
^ —r f \
singularity of type x (0 < Re r < 1), where r is defined by
the presymbol of eq. (0.1) and all the coefficients cQ,...,ca+2
influence the value of r. The results of this section were formerly
accepted as hypotheses for establishing order of singularities of
solutions of eq. (o.l) at the points x = 0 and x = 1 (cf. [32]
and Sect. 1? below). A relatively simple formula for finding the
coefficient of the principal member of the asymptotic expansions
at x = 0 is derived (cf. (11.3))*
In Sect. 10 eq. (o.2) is treated; the conditions for being
Noetherian and the index formula are obtained for the case when
ck(x,y) has limits c£ = ck(l,l) and c£ = ck(-1,-1) (k = 2,3,
...5n+2). Next, in Sect* 11 the asymptotic behaviour at the points
x = + 1 and the smoothness at the interior points of the interval
JQ = [-1,1] of solutions of eq. (0.2) are investigated.
Eq. (0.1) is a particular case of an integral equation with
the homogeneous kernel of order -1, k(tx,ty) = t”"1 k(x,y)
(0 < t < oo); we have considered the case because it plays an impor­
tant part in applications. We have also given a great attention
to the scalar case but have avoided a detailed treatment of systems
of equations and Banach algebras generated by the operators under
consideration! we have also avoided a detailed investigation of
the case when cQ and are functions in (0.1) and (0.2). The
proofs of these and some other generalizations are outlined in
Sect. 12.
Equations having homogeneous kernels of order -1 and compli­
cated with displacement operators are studied by A. P. Soldatav in
Holder spaces with an infinite weight (cf. [42a].
For c* = 0 eq. (o.l) was investigated by L. G. Mikhailov who
had recourse to M. G. Rrein*s results [27].
For ci = 0, c0 = 0, Ci = 1, 3m c2 = 0 eq. (0.1) is completely
9
solved la a different way by H* Bueckner (the case n * 0, cQ = 1,
c1 = 0 was considered earlier in [27]). In Sect*. 13 eq* (o*1) will
be investigated for n = 0$ in particular, the solutions of the
homogeneous equation f = 0 will be written out (in contrast to
the case cQ = 0. or c< = 0 this equation may have two linearly
independent solutions)*
Eq* (0*1) is investigated by J* Bierman [3], but his investiga­
tion is somewhat incomplete; eq* (0*1) is solved in the class
U L1(J,sP:), which is not invariant with respect to the ope-
0 < a < 1
rators, occurring in eq* (0*1); by the substitution indicated above
eq* (0*1) is reduced to eq* (0*3) and then solved by means of the
method of WieneivHopf [34] without investigating the integrability
of the solutions obtained* However the Wiener-Hopf method in its
classical form is suitable only for the solution of the equation
in convolution (0*3) with the continuous presymbol a s 1 + Fk
and is not applicable to eq* (0*1) with c* = 0 unless some additi-
nal investigation of the solutions is carried out*
Eq. (0*2) is solved for n = 0, c2 * - c< by F* G. Tricomi [45]
and S* G* Mikhlin [31] (see also [4,41])f while in a more general
case (even when Q = 0, cQ and c< are functions) it is treated in
[46]| the investigation is conducted by means of singular integral
equations with displacement* Ve shall go into a detailed treatment
of eq* (0*2) for n = 0 in Sect. 20 of this book*
Chapter III is a concluding one; it deals with the application
of the author1 s results to various problems of mechanics (mostly
to problems of the elasticity theory) and mathematical physics*
The formulations of the problems are given in brief and the integral
equations of type (0*1) - (0*3), appearing while solving the pro­
blems, are written out; next, the equations are proved to be Noetbe-
rian and their indexes are calculated; the asymptotical behaviour
and smoothness of the solutions are also investigated*
Despite the fact that the numerical solutions of the problems
considered in Chapter III are treated in quite a number of works
(cf* [15a] and the references cited therein) the conditions for
being Noetherian and index values for most of them remained at
issue*
10
C h a p t e r I
Integral Equations ia Convolution with
Discontinuous Presymbols
Sect* 1» Definitions and Auxiliary Propositions
In this section are presented the definitions and auxilary pro­
positions from the theory of operators, the theory of singular in­
tegral equations, Banach algebras and Fourier transforms*
1°* The proofs of the propositions fonaulated in this subsec­
tion can be found in [1] and [13a]*
Let A be a linear bounded operator in the Banach space X*
We denote by Eer A (c X) a linear space of solutions of the homo­
geneous equation Ax = 0 and by Coker A the algebraic cofactor
of the image Im A of the operator A* Let A* be the conjugate
operator to A*
DEFINITION 1*1« The operator A is said to be normally solvable
if the equation Ax = y has a solution x € X iff g(y) = 0 for
any g € Eer A* , i.e* Im A = 0 Ker g, where g ranges over Eer A**
It appears that A is normally solvable iff it has a closed
image Im A = Im A*
DEFINITION 1*2« The operator A is called Noetherian iff the
sets Eer A and Coker A have finite dimensions dim Eer A < oo
and dim Coker A < oo . The integer
Ind A = dim Eer A - dim Coker A
will then be called the index of the operator A*
It is proved in [1] that Noetherian operators are always nor­
mally solvable and dim Coker A = dim Eer A**
The operator A is Noetherian iff its conjugate A* is Noethe­
rian and then, obviously, Ind A* = - Ind A*
THEOREM 1*3* If A is Noetherian, then there exists a number
e > 0 such that the operator A + B + T is Noetherian for any B
with ||B || < s and any compact T; also, Ind(A + B + T) = Ind A*
COROLLARY 1*4* The index Ind A of the Noetherian operator A
is a homotopic invariant; in other words, if A^. depends conti­
nuously on t € [0,1 ] and A^ is Noetherian for all values of t,
11
then lad Aj. is independent of t*
THEOREM 1*5* If "the operators A aad B are Noetherian, the a
AB is also Noetherian aad lad AB = lad A + lad B*
THEOREM 1*6* The operator A is Noetheriaa iff there exists
a liaear bouaded operator M (called the regularizer) such that
AM = I + T.| aad MA = I + T2, where T* aad T2 are compact
operators*
let L(X) be aa algebra of all liaear bounded operators ia the
Baaach space X aad K(X) be an ideal of all compact operators
in this space | we denote by [l](X) the quotient algebra L(X)/K(X)
with norm S|j A |jj = |i [A] || = ihfjjA + T l| (T 6 K(Z)), where [A]e[X.](Z)
is the class, containing the operator A € L(X)»
THEOREM 1*7* The operator A € l(X) is Noetheriaa iff the class
[A] is invertible ia the quotient algebra [l](X}*
DEFINITION 1*3* The operator A is called the $+-or § -opera­
tor if A is normally solvable and dim Ker A < oo or dim Coker A
< co , respectively*
THEOREM 1*9* If A, B € l(X) and AB is the $+-operator
(§_-operator), then so will be the operator 3 (the operator A)«
GORQUARX 1*10* Let A, B € l(X)$ if the operators AB and BA
are Noetherian, then so. will be the operators A and B*
THEOREM 1*11* If A € l(X) is $+- or $_-operator, there exists
a number e > 0 such that the operator A + B + T will also be
§+- or $_-operator for any || B || < s and any compact T € £(X)$
moreover, if dim Coker A = oo (dim Her A = oo), then dim Coker(A+B+T)
= oo (dim £er(A+B+T) = oo).
2°* The definitions that follow will be helpful in the sequel
(cf* [14])*
DEFINITION 1*12* The operator A e l(X), where X is a Banach
space, is said to be left (right) invertible * if there exists an
operator B € L(X) such that BA = I (AB = I), where I stands
for the identity operator? if A is left and right invertible it
—*1 —*1
will be called invertible* The notations Ai , A^, and A will be
used to denote the left, right and two-sided inverse of A, respec­
tively*
DEFINITION 1*13* Invertibility of the operator A will be said
to be compatible with the integer k(A), if this operator is inver­
tible, left invertible or right invertible only, depending on
12