Multidimensional inverse problems for differential equations

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z0rich

167 Lavrentiev. Romanov Vasiliev Computing Center of the Academy of Sciences Novosibirsk / UdSSR

Multidimensional Inverse Problems for Differential Equations

$ Springer-Verla9 Berlin-Heidelberg. New York 1970

I S B N 3-540-05282-8 S p r i n g e r V e d a g B e r l i n • H e i d e l b e r g • N e w Y o r k I S B N 0-387-05282-8 S p r i n g e r Verlag N e w Y o r k • H e i d e l b e r g • Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopymg machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin - Heidelberg 1970. Library of Congress Catalog Card Number 70-140559. Printed in Germany. Off~druck: Julim Bdtz, Weinhdm/Bergr,.r.

CONTENTS

INTRODUCTION CHAPTER

I

................................................... Some

-

1. P r o b l e m

Problems

Ellipsoids

Problem

Integrals

along Center

the

4. On the

Problem

Values

CHAPTER

of D e t e r m i n i n g

2

over

Equation I. S t a t e m e n t

CHAPTER

Formulations

Problem

3 - Linearized

1. F o r m u l a t i o n

CHAPTER

4

-

of the

I.

Inverse

Heat

2. n - D i m e n s i o n a l 3. A p p l i c a t i o n

CHAPTER

5 - Inverse

Problem

and

Inverse

Linearlzed

Space

from

13

Its M e a n

for the

19

Telegraph

Its L i n e a r i z a t i o n

Problem

in

22 ..

Inverse

Problem

Equation

Inverse

Kinematic

Problem

Problem

of the

Heat

Its

Linearlzed

to G e o p h y s i c s

Conduction

Sources

for the

Linearization

Version

.......

of the

with

Inverse

Problems

Heat

Problems

Problems

for

Conduction

for S e c o n d - O r d e r

33 33

Inverse 36

Continuously

a Half-Plane

59

.......

59

..........

45

...............

49

Problems

to G e o p h y s i c s

31

Wave

.........................

Problems

28

for the

.....................................

Conduction

of the

and

24

in

................................. Differential

22

Two-

.........................................

Problem

Inverse

to R o t a t i o n

.......................................

of the

Active

from

...........................................

2. A n A p p l i c a t i o n Kinematic

Invariant

a Function

Problem

of a N o n l i n e a r

Equation

a Circle

..........................

Dynamic

Inverse

Three-Dimenslonal

Inverse

2 10

.....................................

Inverse

Space

4. D e r i v a t i o n

inside

1

over

.......................

of C u r v e s Circle

0ne-Dimenslonal

Dimensional

Integrals

...........................................

of the

2. L i n e a r i z e d

5. T w o

of the

of D e t e r m i n i n g

Linearized

-

Curves

a Function

a Family

Circles

.................

from

................................

to A n a l y t i c

about

Geometry

a Function

of R e v o l u t i o n

2. G e n e r a l i z a t i o n 3.

of I n t e g r a l

of D e t e r m i n i n g

IV

Elliptic

Equations

51

i.

Inverse

Problem

for E q u a t i o n

(i)

in a H a l f - P l a n e

........

52

2.

Inverse

Problem

for E q u a t i o n

(1)

in a H a l f - S p a c e

........

54

BIBLIOGRAPHY

.........................................................

57

INTRODUCTION An inverse problem for a differential

equation

the determination

or right-hand

tial equation

of the coefficients

from certain

Two significant

advances

inverse problems

functionals

[19],

[23],

operator

function of the operator. ving the determination equation

for STURM-LIOUVILLE

equations

the coefficient

(Ill,

in a second-

is required to be found from the spectral

In [19] and

[2], a number of problems

of the coefficients

are shown to be reducible

equations.

in the study of

equations.

[36]). In these problems,

order differential

side of a differen-

of its solution.

have been made previously

for differential

The first is in inverse problems [13],

is any problem involving

of a partial

to inverse problems

It is assumed there that the coefficients

invol-

differential for STURM-LIOUVILLE

are functions

of

a single variable. The second is in problems

of potential

[33]). In the inverse problems elliptic partial differential restrictive Thus,

additional

in [17] and

equation

theory

([17],

conditions

restrictions

is a star-shaped

dealing with the inverse problem of potential Until now, multidimensional ly little consideration. coefficients

rally arbitrary function space. Multidimensional M. BERZANSKII.

functions

inverse problems

some multidimensional

In [8] and

i

only,

side in other papers

problem,

the required

equations

are gene-

belonging to a certain

problems were first investigated

functions.

and

The same sort of

have been given comparative-

of several variables

in the papers of Ju.

theorem was proved for the solution

to the inverse problem for SCHR~DINGER'S wise analytic

0

sides of the differential

In [4] a uniqueness

side.

theory.

In a multidimensional

or right-hand

Highly

side in POISSON'S

domain.

are also imposed on the right-hand

[29],

side of an

are imposed on the right-hand

is assumed to be a function having values 1

[25],

equation has to be determined.

[20], the required right-hand

and the set where it is

[20],

of this type, the right-hand

equation in the class of piece-

[9] functions were constructed

inverse problems

of quantum scattering theory

that are similar to the GEL'FAND-LEVITAN

functions

occurring in the

for

-

V -

inverse problem for the STURM-LIOUVILLE This monograph

investigates

whose formulations

equation.

a number of multidimensional

inverse problems

differ from those of the papers mentioned above. A

portion of the results have been published

as short notes

([21S,

[22],

[30], Di]). A characteristic property

aspect of multidimensional

of not being well-posed

advantageous

to make use of the general notions

posed problems theoretical HADAMARD

developed

in [16],

[20],

of proofs of uniqueness

theorem.

theorems

is their

Thus,

and approaches

[34], and

it is

to ill-

[55]. Central to the

study of a problem that is not well-posed

is the proof of a uniqueness

primarily

inverse problems

in the sense of HADAMARD.

in the sense of

The monograph

consists

for the formulations

in

question. The inverse problems

for which uniqueness

theorems will be proved are

linear and their solution is reduced to the solution of first-order linear operator equations. them numerically [20] and

Thus algorithms

by application

of the general methods

the problems.

developed

the stability

compact

are uniformly

sets

of the various

(for example,

in

differential

formulations

the set of functions

on certain whose gradients

here to the study of inverse problems equations

over to higher-order

equations

The inverse problems

considered

to problems

for

of estimates

bounded).

We shall confine ourselves second-order

lead to special algorithms

They also make possible the derivation

characterizing

aspects

to solve

[35] for linear equations.

The methods used to prove uniqueness

specific

may be constructed

of integral

geometry

although some of the methods

for carry

~8]. for hyperbolic

equations

are reducible

and so Chapter i is devoted to some

of it. Chapter 2 establishes

uniqueness

to the inverse problem for the telegraph

theorems

for the solution

equation with the help of the

results of Chapter i while Chapter 3 does the same thing for the wave equation. Chapters 4 and 5 deal with inverse problems

for the heat equation and

for elliptic equations. They are reduced to the solution of certain integral equations of first kind. In Chapters

3 and 4, some applied problems

to corresponding

versions

are discussed that lead

of the inverse problem.

CHAPTER I Some Problems of Integral Geometry In accordance with the terminology used in

~2],

an Integral-geometrlc

problem is any problem involving the determination of a function defined in a domain through its integrals along a family of curves in the domain. One of the earliest and most familiar versions of such problems determination of a continuous

function of

is the

n-variables through its mean

values over spheres of arbitrary radius with centers lying on an (n-l)dimensional hyperplane.

The uniqueness

proved in R. COURANT'S book

of a solution to the problem is

[6]. At present,

the determination of a

function from its integrals over hyperplanes has been the problem dealt with most fully In monographs

[12]. The authors of the book, I.M. GEL'FAND,

M.I. GRAEV and N.Ja. VILENKIN were led to its consideration while working on problems In representation theory. The appearance of this book furthered to a considerable extent the development of Integral-geometrlc

problems.

and systematic elaboration

Some differently formulated problems of

integral geometry are contained in F. JOHN'S book As already indicated in the introduction,

we were led to problems

integral geometry while considering llnearized problems for the simplest

[18]. in

inverse

equations of mathematical physics.

Some of

them lead to the familiar problem of determining a function from its mean ~alues over spheres and others to new problems which it is the purpose of this chapter to consider.

The problem of finding a function

from its integrals is not well-posed in the sense of HADAMARD.

We shall

therefore proceed from A.N. TIHONOV'S notion and we shall prescribe the function space to which the solution of a problem is to belong. With applications

of integral geometry to the study of linearlzed problems

mind, the most natural one for our purposes is the space uous functions.

Throughout

C

in

of contin-

the following we shall assume the solutions

of corresponding problems in integral geometry to belong to the space C

or to some subset of it. We shall be interested in uniqueness

for theseproblems,

the determination of inversion formulas

theorems

for them

and the set of functions for which these formulas are valid. The latter is related to the question of existence of a solution. Section i considers the problem of reconstructing a two-dlmensional function from its integrals over a family of ellipses with one focus

-

2

-

fixed and the other running over the points

of a llne.

At the end of

the section it is shown how these results may be rephrased to encompass the p r o b l e m of r e c o n s t r u c t i n g grals over ellipsoids I are generalized

a function of

of revolution.

is also g e n e r a l i z e d

The formulation

by the introduction

grals along the curves.

Section

invariant to rotation

already,

uniqueness

for c o n s t r u c t i n g

of a weight

case

the d e t e r m i n a t i o n

of a funct-

over a family of In Section

for the p r o b l e m of d e t e r m i n i n g Although,

has been proved before function,

problem

factor in the inte-

from its integrals

over circles.

the required

of Section

a special

about the center of the circle.

formula is obtained

from its mean values

from its inte-

2 the results

of the integral-geometric

3 considers

ion inside a circle of unit radius an inversion

variables

to curves of a more general nature

being that of ellipses.

curves

n

In Section

a function

as we have indicated

and a basic a l g o r i t h m

the inversion

exists

formula has been

lacking for this problem. 1. Problem of D e t e r m i n i n 6

a Function

from Inte6rals

over E l l i p s o i d s

of Revolution Consider the following p r o b l e m

In

(n+l)-dimensional

grals of a function

u(x,s)

family of ellipsoids

with one focus

running over all points tained by r e v o l v i n g to be considered.

= u(xl,x2,...,Xn,S)

The inte-

are p r e s c r i b e d

on a

fixed at the origin and the other

of the hyperplane

an ellipse

space.

s = O. Only ellipsoids

ob-

around the llne Joining the two foci are

It is required to determine

u(x,s)

from the given

integrals. Denote the coordinates and by

S

of the second

the ellipsoid

focus by

of revolution

(x°,O)

= (x~,x~ .... ,x°,O).

defined by

x°,t (I) where

r(x,s,O,O) r(x,s,O,O)

and the foci Thus,

(0,0)

and

+ r(x,s,x°,O)

r(x,s,x°,O)

and

(x°,O),

= t

are the distances

between

(x,s)

respectively.

let the function

(2) be known.

v(x°,t) =/u(x,s)d~ Here

the origin.

~

Sxo,t is the solid angle

In accordance

in

(x,s)-space

with the above discussion,

4

with vertex at we shall assume

-3-

that

u(x,s)

apparently

belongs to the space of C-functions.

only m e a n i n g f u l

From

solution to this p r o b l e m in the class of even functions we shall do in the following.

In addition,

of generality

in a s s u m i n g that

computations,

we shall take

be played by ellipses At the c o n c l u s i o n

u(O,O)

n = I

of the section,

of a

s

is o b v i o u s l y

and this no loss

so that the role of ellipsoids w

will

will be the polar angle ~ .

we show how the c o r r e s p o n d i n g

results

n.

n = I , the following u n i q u e n e s s

T h e o r e m 1: If equation

there

of

: 0. To simplify the subsequent

and the solid angle

carry over to the case of a r b i t r a r y For

(2) it is

to pose the question of uniqueness

t h e o r e m holds.

(2) has a solution belonging to

C

satisfying

a H~Ider condition in a neighborhood of the origin,

then it

is unique. By solution here, we mean an even function of

s

v a n i s h i n g at the origin.

The idea of the proof of the t h e o r e m is to find all moments To this end,

it is convenient

related to the cartesian (3)

of an ellipse

(4)

and

eccentricity and

~os~

t

~

(x,s)

,

Sin ~

s : r

.

is

cos~)-1

are p a r a m e t e r s

of the ellipse;

u(x,s). (r,#)

by the formulas

in polar coordinates

r = p(1-~

p

where

coordinates

X : r

The equation

of

to go over to p o l a r coordinates

characterizing

the polar distance

they are e x p r e s s i b l e

in terms of

and

x°

by x° z : K- '

(4a) Formula

p :

t

(I - c 2)

.

(2) then becomes 2~

(5)

J u(r cosg,

r ~in~)d~

: v(p,~) ,

0 with

r

g i v e n by ( 4 ) .

We a p p l y t o b o t h s i d e s

of

defined by

(6)

Lv = p ~

v(z,¢)

dz

-

Z

0

•

(5) t h e o p e r a t o r

L

-4-

When equation (5) has a solution in C , both its legitimacy and the result of applying it are substantiated by the following sequence of equations : p 2~ ~c / LV : P T~

f

=PT[ (6a)

dz o ~ U ( r z C O S ~ , ~--

d~

0

: Py

rz s i n ~ ) d ~

u(r z e o s ~ ,

r z sin~)

d,.

~--

0 P

d~ ~-~ S

0 2~

u(r c o s ~

,

r sin~)

dr ~--

0

=/

U(rp c o s ~ ,

rp s i n ~ )

0

rp c o s ~ d f : / u ( x , s ) x d ~ p,c

The subscripts p and z on r indicate which of these two parameters is to be substituted in formula (4) when calculating r. Sp,¢ denotes an ellipse with parameters p and ¢ . Applying the operator

(7)

L

repeatedly to the resultant equation

Y

U(X,S)Xd~=

Lk

the k-fold iteration of

/

u(x,s)xkd~=

LV

S

and denoting by way

(8)

Sp,¢

L, we obtain in a similar

Lkv (k = 1,2,3,...)

If we define L°v m v(p,¢), formula (8) is also valid for k = 0 . Thus we have constructed in unique fashion a system of moments on each ellipse. Since u(x,s) is an even function of s, it is uniquely determined by these moments. exists, it is unique.

In other words,

We next consider how relations u(x,s)

explicitly in terms of

if a solution to equation

(4)

(8) may be used to express the function v(p,¢). At the same time, we shall study

-5

the properties to equation

needed by

v(p,c)

-

to assure the existence

(5). At this Juncture,

we shall slightly

of functions

for which we have proved the uniqueness

ted function

from its integrals

sider functions I °. each

u(r,~)

u(r,~)

disc

over ellipses.

is a continuous

positive

2 ° . In a n e i g h b o r h o o d

contract

the class

of the r e c o n s t r u c -

Namely,

we shall con-

satisfying the following conditions

r ~ r o , it is even in

arbitrary

of a solution

function of its arguments ~

and

u(O,~)

= O. Here

: in the r°

Is an

number. of the polar origin,

u(r,~)

satisfies

a HOLDER

condition,

(9)

lu(r,~)

where 3 ° . Each

A

and

,

u(r,~)

I ~ Ar~,

(~ > O)

are constants.

satisfies

(Io) k:O

the inequality

max l uk(r) r

]< ®

wherein 2~ ~(r)

: ~/u(r,~)

(Ii)

(k : 1,2,3,...)

cosk~d~,

o 2~

0 The functions class

for which conditions

I°-3 ° hold will be designated

as

U.

We shall also make a slight Consider

a circle of radius

of eccentricity to determine

0 ~ c < 1

a function

change in the statement r°

in the

of the problem.

(x,s)-plane

and all ellipses

falling inside the circle.

u(r,~)

~ U

It is required

from its integrals

over this

family of ellipses. Let

v(p,¢)

be a function

for which a solution to

parameter

c

radius

and the following relation

(12)

p

A

(P,~ ) c°sk~df=

0

(5) exists.

tend to zero in (8). Each ellipse becomes results

P-k-Lkv-z=O[] .

Let the

a circle of

in the limit

:

(k = 0,1,2,...)

-6-

For each fixed value of construct u G U

by virtue of condition

ellipses

having eccentricity

arbitrarily disc

r, these relations

a Fourier series for

u(r,~)

3 ° . Thus knowing the integrals ranging in the interval

small positive number,

r ~ ro

alone can now be used to

which is convergent

in an arbitrarily

small strip

u(r,~)

the function

0 ~ e i 6 , we determine

~ ~ e < i. This implies

~ is an

in the

along all ellipses

lying in this disc. This means that if we prescribe v(p,e)

along

0 i e ~ 6where

we can determine

and hence we can find its integrals

for any

it

completely

for

in turn that to any arbitrary

continuous

function there exists no solution to the stated problem.

This result relates to the fact that the given problem is not well-posed in the sense of HADAMARD.

Actually

belonging to the function

spaces

property

obtained above for

perties the functions sufficient

v(p,e).

u(r,~ )

for the existence

no closed linear manifold C k, H, Lp

of a

Mk v : I

of functions

Wp . (z) possesses the

Below we shall indicate what pro-

must possess u(r,~)

Consider a family of linear operators

(13)

or

Mk

that are necessary satisfying

and

equation

(5).

defined by the relations

[~] [~-J k ) (J~2))p2J-kLk-2J v ' ~ (-l)J( ~ (2(J+g) J=O ~=0 (k=i,2,...)

and I MoV : ~

(13a)

Using the system of relations to (5) results

(14)

1

It should

be

5u(~{[t+/t2_l]k+[t_

t2~_l]k }

cos~d~: t-

p,e

noted that for

Nevertheless,

e = O, equation

(8), we find that the application

of

Mk

in the formula / S

complex.

i v(p,~) L°v --~-~

Itl < i

(14) can be written

l-ecos~

the expressions

the entire expression

Mkv

in brackets

in braces

are

is real. For

as

2~ ~i f O

(14,)

Hence,

u(r,~)

cosk~d~

:

[MkV]~: o

if a solution to (5) exists which belongs to

in terms of

(i5)

v(p,e)

through the formula

u(p,~) =kiO[Mkv]e=O cosk~' .

U, it is expressible

-7-

The convergence of this series for any

u~U

follows from (14') and

condition 3 ° on the function u ( r , ~ ) . We now examine what properties are possessed by V, the image of the set U under the correspondence definable by (5). Theorem 2: The image

V

of

U

under (5) has the following properties:

i °. The functions MkV (k = 0,i,2,...) corresponding to a by (13) exist and are contlnuous t and [MkV]p=o = O. 2° . For any

v(p,a)

v(p,~)~V

the series

(16) k=O

maxlMkVlc:O p

: ~v

is convergent. 3 ° . The function u ( r , ~ ) constructed for satisfies the HGLDER condition, (17) 4 °. Each

lu(r,9')l v(p,¢)

v(p,¢)

from formula (i5)

~ Ar ~, (~ • 0) .

satisfies the inequality

(18)

Iv(p,~)i

~ 2~ ~v "

Properties 1°-3 ° follow in a trivial way from the corresponding properties of the functions u(r,~) and equations (14), (14') and (15). Only inequality (18) remains to be proved. To this end, note that the function u ( r , ~ ) given by (15) has to satisfy equation (5), i.e., the following identity must hold: (18a)

f 0

~ [MkV] e=0 c o s k ~ d ~ k=O p÷r

s v(p,¢)

.

[MkV] c :0 should be interpreted to mean that MkV Is first to be p÷r evaluated at ¢ = 0 and then p replaced by r as given by (4). From this Identlty we deduce that (18b)

Iv(p,c)l ~ 2~

and the theorem Is proved.

~ maxlMkVl¢:0 k=O p

= 2~ ~v

-8-

Theorem 3: A necessary and sufficient condition for equation

have a solution belonging to The necessity v(p,e)

to

follows

V

is that

v(p,¢)

belong to V.

from Theorem 2. Let us show that the belonging

is sufficient

consider for such a

U

(5) to

v(p,¢)

(i9)

for the existence

of a solution.

of

Indeed,

the series

[Mkv] :ocosk : k=O

By virtue of property continuous

function

coefficients

for

By properties even in

~

equation

2 ° , it is uniformly u(p, ~ )

u(p,~)

I°-3 ° of

and

of

Moreover, [MkV]~=O

the function u(p,~)

S

$ ~ V. Applying

Or by the linearity

of

(20b)

Mk

is in

is

the function

.

to equation

(20), we obtain

=

(k : 0,I,2,...)

V-V

V. Using inequality

(20c)

w(p,a)

or equivalently,

u

Mk ,

W

W

(clearly,

p,e

[MkW]e=O = O,

The function

the FOURIER

(k = 0,1,2,...).

u~U

construct

~(p,~) : f u ( r , ~ ) d ~

Evidently,

and defines a

= 0). It remains to show that it satisfies

(5). On the basis of

(2o)

produces

andS.

coincide with v(p,c),

u(O, ~ )

p

convergent

$(p,¢)

= v(p,e).

a solution to equation

(18), we find from this that

~ 0

This means that each function

(5) through

formula

v(p,e)

(i9). By Theorem I

it is unique. We return now to the case of arbitrary obtained Thus,

let

for e l l l p s ~ c a r r y Sxo t

n

and we outline how the results

over to the case of ellipsoids

be a family of ellipsoids

of revolution

of revolution. one of whose

foci is at the origin and the other at any point of hyperplane Knowing the integrals

of a function

u(x,s)

s = O.

over these ellipsoids,

one

-9

is required equation

to determine

-

it. In other words,

(2). The uniqueness

of a solution

strated Just as in the case of ellipses function.

To this end, we perform

formation

in

the origin

(x,s)-space

formation

the point

goes into

(y°,O),

(Yl,O,.. . o

,0,0).

parameters

(x,s) where

goes into

coordinate

amounting

(y,s)

and

transabout

(x°,O)

(y°,O)

depends

to be, in particular,

Yl : r~i

of the

to a rotation

and the point

= (yl,Y2,...Yn,S)

cosines ql,q2, .... qn of the radius Introduce spherical coordinates for

is demon-

that under this trans-

of the transformation

which may be taken

(21)

Q

s = 0 . Suppose

(y,s)

The matrix

to this equation

to solve

by finding the moments

an orthogonal

with matrix

in the hyperplane

it is necessary

=

on several

the direction

vector to the point (x°,O). y and s by the formulas

' s : r~n+ i ,

(i : 1,2,...,n) where

~i(i

vector

r

= 1,2,...,n+l) in

are the direction

(y,s)-space.

can then be written

The equation

of revolution

r : p(l - ~ i )-i

p

and

c

are given by o Yl ~ c2 E : K- ' p : (1 )

(22a) Equation

of the radius

in the form

(22) where

cosines

of an ellipsoid

(2) can now be rewritten

(23)

/

as

u(rQ.~)d~

= v(q,p,¢)

,

S q,p,a where ~ = (~1,~2,...,~n+I), q = (ql,q2,...,qn) and Sq,p, E is the surface of the ellipsoid of revolution with parameters q,p and c. Just as in the preceding,

by applying

keeping

L

q

fixed,

(24)

where

/

the operator

is the operator

(24a)

to equation

(k = 0,1,2,...)

of the orthogonality Yl Yl0 = x l

of the transformation,

Xl0 + X 2 X ~ + "°" + X nXn0

(23)

by (6), we obtain

u(rQ. )y d : Lkv

Sp,q,~ Note that by virtue

Lk

defined

"

-

Since

(x°,O)

is an arbitrary

taking the invarlance (24) for

c = 0

of

dm

10

point of the hyperplane into consideration,

SO, t

to Analytic

symmetric

section,

dimensional

uniquely.

an inversion

a function

These

Using the resultant

formula in a similar

from its integrals

section can easily be g e n e r a l i z e d

n = 1, in other words,

preceding

s

Curves

more general than ellipsoids. sider curves

(i = 1,2,...,n)

with center at the origin.

of

one can construct

The m e t h o d of determining

case

:

n = I.

2. G e n e r a l i z a t i o n

the p r e c e d i n g

t

our even function

system of moments, way to the case

u(x,s)

(Ai = 0,1,2,...),

is a sphere of radius

determine

for

AI ~2 ~n x I .x 2 ... x n dm,

u(x,s) So't

where

s = O, by

one can use equation

to derive the following moments

(24b)

moments

-

However,we plane

about the

the entire discussion

surfaces

Moreover,

to the

we shall only con-

(s = 0). However, easily carries

resulting by way of rotating

in

of r e v o l u t i o n

shall confine ourselves

curves. x-axls

discussed

to surfaces

as in the

over to

a plane

(n+l)-

curve around

an axis of symmetry.

Consider

a two-parameter

coordinates

family of curves

given in polar

r = pf(E,E COS~), f(E,,)

is an analytic

small n e i g h b o r h o o d ~--~n f(O,O)

/~(~,~

and

f(O,O)

,

in an arbitrary

~ 0

and

cosy) u(r,~)d~: v(p,c),

p,e

be given along these curves, of

e

and

n

~ O. From formula

sider d e t e r m i n i n g Therefore

¢

of the origin such that

S

function

function of

~ O. Let the integrals

(2)

~(0,0)

pjc

by

(I) where

S

~(E,,)

from

v(p,¢)

is a known analytic

of the origin such that

(2), it is apparently

uniquely

throughout

wherein

in a n e i g h b o r h o o d

only m e a n i n g f u l

a function

u(r,~)

to coneven in ~.

the following when we speak of a solution to

we shall mean an even function of ~.

(2)

-

il

-

The following theorem holds. Theorem 4: If equation (2) has a solution that is bounded everywhere, belongs to C and satisfies a HULDER condition at the origin, then it is unique.

The method of proving this theorem is similar to that of Theorem I and so we shall find the moments of Expanding

@(~.n)

u(r. ~ )

on each circle

in a series with respect to

,

Sp. o.

and substituting It

in formula (2). we obtain V(p,¢) :

(3)

~ ckak(¢) Vk(P,¢), k=O

where

f (4)

vk(P,¢) : 9 u ( r , ~ ) Sp.~

(5)

ak(e)

cosk~d ~,

1

~k$

kl

~n k In=O

=

I

If we are able to determine the functions Vk(P.O) uniquely in terms of v(p.e), thls wlll prove that the reconstruction of u ( r . ~ ) is unique. From (3). we find that

(5a)

Vo(p,O) :

Introduce the operator

L

aotU)

defined by

P dz Lv : ~-~ .r * V(Z,¢) ~-0 and apply it to (47. Its validity is Justified by the H~LDER condition (6)

for the function

u(r,~).

Indeed,

P LV k

~.~

Z-" O

(6a)

Sp,,

S

SE. c

L f(''' oos ) j

CO sk~d~ Sp. a

pf(~,c COS~) U (r,~) ~-O

-

12

-

Expanding the expression in brackets g c o s ~ , we obtain

in the integrand in powers of

LVk = b°(g) vk(P'E) +n i i [Cn(g ) + Ebn(E) ] cn-ivk+n(p,a),

(7)

wherein bn(g) derivatives of

and Cn(g) are expressible in terms of the partial f(¢,n). By virtue of the conditions imposed on the

latter function,

ci(O) ~ O. The last relation allows us to now derive

an equation for This results in

vl(P,O). Apply

L

to (3) and use formula (7) for Lv k.

Lv = do(z) Vo(P,g) + ~ ck-ldk(g) Vk(P,e) k:i

(8)

+ ~o(g )

Vo(Z,g ) z-- + O

gk-

dz

k=i

O

in which dl(O) = ao(O)Cl(O)M 0. Letting ~ ÷ O in equation (8) and dividing by di(O) , we arrive at the following VOLTERRA equation for vl(P,O)

: P

(9) X

vl(P,O) + X

d~ Vl(~,O) ~-- = fl(p)

0 being some numerical parameter. The equation is meaningful

in that

vi(P,O) satisfies a HOLDER condition at p=O . Equation (9) does not have a unique solution in general since it has an elgenfunction of the form Cp -x if X < O, C being an arbitrary constant. The equation has no other elgenfunctions.

Note however that the boundedness

of

u(r,~) together with (4) implies the boundedness of all vk(P,a) and so the solution we seek for (9) has to be bounded. But since for X < O, Cp -x tends to infinity as p ~ = if C ~ O, we can assert that any bounded solution to (9) is unique. If we apply L to equation (3) k times in succession, we let g tend to zero and we divide by the non-zero coefficient ao(0)c~(O) , _ obtain for the function equation of the form (9a)

z

Wk(Z) +

J[ko(Z

Wk(Z) ~ vk(ex p z, O)

we

a VOLTERRA integral

- ~)k-l+Xl(z - ~)k-2+...+Xk_2(z

- ~)+Xk_i] Wk(~) : fk(z),

where ~o' XI' ..., Xk_ 2 and ~k-1 are certain numerical coefficients and fk(z) is a contlnuous function expressible in terms of the functions [LJv]g=O (J : O,l,2,...,k). It is easy to show by reducing

-

the V O L T E R R A efficients

13

-

equation to a differential

that there are at most

k

equation with constant

linearly

independent

co-

elgenfunctions

each of w h i c h tends to infinity as

z ÷ ~. Hence any b o u n d e d

of the integral

equation is unique.

Thus all of the functions

(k = O,1,2,...)

can be found in a unique way. This implies the unique-

ness of a solution to equation in the class of functions

(2). An inversion

having convergent

solution Vk(P,O)

formula can be derived

FOURIER

series by p r o c e e d i n g

in the same way as in Sec.i. The following

example

shows that n o t h i n g

significant

if the hypothesis

of the t h e o r e m

could be relaxed.

Even for our earlier considered

a solution

is unbounded,

certain situations uniquely Indeed,

it is possible

boundedness

determined

of

u(r,~)

case of ellipses,

to choose a weight

in such a way that the required

function

if in

function will not be

by its integrals.

consider the function

(i0) where

concerning

could be gained

u(r, ~ )

T

is a positive

are arbitrary numerical

number,

= rY N

N X Ak cosk~, k=l

is a positive

coefficients.

integer and the

Choose as weight

Ak

function

¢ ( E , n ) ~ (1 -n) Y. Then if

Sp,a

is the ellipse with parameters

equation

(4) of Sec.1,

and

E

given by

we have

f

(lOa)

p

_~

(1-s c o s ~ P ) T u ( r , ~ ) d ~

= 0

Sp,s for any

p

and

0 ~ ¢ < 1. But this means that equation

ponding to the function also the nontrivial

v(p,E)

solution

3. Problem of D e t e r m l n i n ~ a Family

of Curves

~ 0 not only the trivial

given by (iO).

a Function

Invariant

inside a Circle

to Rotation

from Integrals on

around the Center of the CIEcle

In this section we shall consider the following problem: function

is defined

two-parameter of interest.

inside the unit circle

family of curves The first

(2) has corressolution but

are given.

A continuous

and its integrals As before

is w h e t h e r the function

along a

two questions

determined

are

through these

-

integrals

is unique

and second

14

-

if the function

is unique how to con-

struct it. Introduce polar coordinates

(r,~)

with the pole situated at the

center of the circle. Consider a t w o - p a r a m e t e r

family of curves having the following proper-

ties: 1 ° . The family

is invariant

2 ° . Each curve begins

to rotation about the center of the circle.

and ends on the unit circle.

3 ° , Each curve consists

of two branches whose

equations

are expressible

in the form ~j

(I)

= ~ - (-l)Jgj

(r,p)

r~-p , J = 1,2

Here

p

is the distance

(O O)

the solution being known at a single point: (4) The point

Ul(Xl,O,t) Mo

: ~(t)

.

will be regarded as the origin. The solution

uo(x,y,t)

of the two-dlmensional analogue of equation (7) of Sec. I in the halfplane y ~ 0 is given by [32]

2 _×2_y2' 0

,

t 2


0 ; 2. f(x,y) ~ LI(D) , D = {-- < x < ®; O < y < ®}. Taking FOURIER transforms in equation (6) with respect to

x

and

using the relation (see [7]) (7)

/ K o LFb ( c 2 + t 2 )]I ~cos

e-lcl

ut dt

0 we obtain

(Ta)

/F(¢, n)(e- I r'-Yll ~ e 0

= ~

I "+Yl I ~1"1

/g(x,X)

e -i~x dx ,

with

(Tb)

-

F(~,n)

Taking into consideration that

I

/f(~,n)

f(x,y) = 0

e -i~

for

d~ .

Y < Yi' we can write

this last equation in the form

/a2~2+2 ' F(~,n) e -n

dn :

Q(m,~)

,

Yl

(8)

Q(~,~) =

/a2X2+~2'

G(~,X) ,

2w sinh yl ~ G(~,k) -

1 /

2~

g(x,k)

e-i~x

dx .

Let us show that we have the right to take FOURIER transforms in equation (6) with respect to

x

under the above assumptions on

f(x,y).

From (7) it follows that the kernel of the integral equation (6) is an absolutely integrable function. Then by the theorem convolution for two absolutely integrable functions, we infer that absolutely integrable in the argument is not hard to derive for

G(~,k)

2~sinhy1~-~

x

and hence

the estimate 22 2' ~ e -Yl/a ~ +~ / Yl

g(x,k) is also G(~,X)

exists. It

-

The fact that

f(~,n)g LI(D)

42

-

implies @o

(9a)

lF(~,n)l

~

-

~

/.F(~.n)[dn

i [lf(¢,n)Id¢


O. Under these conditions, equation

(5) Sfb(~,~)[Ko(arl)-Ko(ar2)]~Ko(ar3)-Ke(ar4)]d~do 0

-~

= f(xl,x 2)

-

where

Ko(ar)

(5a)

53

-

is the HANKEL function of imaginary argument and I I r I : [((-xl)2+(n-yl)2]~ , r 2 = [(6-xl)2+(n+yl)2] ~, I 1 r 3 = [(~-Xm)2+(n-yl)2]~ , r 4 = [(~-x2)2+(n+yl)2]~ , Yl : const • 0 .

We impose on the rlght-hand side f(xl,x2) of integral equation (5) the single requirement that the solution b(~,n) of the equation belong to LI(D). We take FOURIER transforms In (5) wlth respect to x I In thls connection equation (7) of Sec.1, Chapt.4.

and

x2

using

Equation (5) then assumes the form f ;

b(~,n) e -I(~1+~2)~ e - n ( ~ +

~~dn=F

i(ml'm2 )'

Yl -~

=

F(~I,~2) , 2 n slnhy I / ~ i s Inhy I / ~ 2

:

~

f(xl,x 2) e -cm

dXldX 2 .

--c@

It Is not hard to show that (6a)

IF(~1,~2) [
O;

Let the function f(xl,x2,yl,y2) be such that the solution of equation (I) belongs to LI(D). As before, on taking FOURIER transforms wlth respect to Y2 In (I), we obtain fi/b(~'n'~)e-i(~l+~2)~

b(~,n,~)

Xl,X2,Y I

and

- I(~2+~4)n

z1-®-®

•e

(2)

/2 21 2 '2 2 2 -~(va +~IT~2 + /a +~3+~4)

d~dnd~

a'2+ 2~ 2 /'2+ 2. 2 ~ 1 ~ 2 ~a ~3T~4 F(~I '~2' ~3 '~4 ) F(~I,~2,~3,~ 4)

/ 2 + 2+ 2 / 2 + 2+ 2 ~slnh zl~a '"I ~2 slnh zlfa '~3 ~4 ~e

=

I

f

oo

f(xl,x2,yl,y2) e

-I(~1xl+~2Yl+,.,3x2+~4Y2 )

( 2~ ) 2_~_®

dXldX2dYldY 2

It is not hard to show that the FOURIER transform of f(xl,x2,Yl,Y 2) exists and therefore one may consider equation (2) instead of (1). Let

w I > O, ~3 < O, m4 = 0

(3)

u

-- ~ I

+

v

=

'

~2

and

w 2 6 (-®,®)

~

'

2

2'+ /a2+ 2'

and introduce the notation

u6(-®,')

,

v~(-®,-)

,

I

w

/a 2 :

+wl+~

2

w~(2a,®)

)

The JACOBIAN (3a)

~ (u,v,w)

=

~3

.

~i

$ 0

,

-

56

-

is continuous for the considered values of (~I,m2,~3) + (u,v,w) Substituting

the mapping

(3) into (2), we have ;

(4)

~i " Therefore,

is one-to-one and has an inverse.

/;b(~,n,~)

e "lu~-iVne -~Wd~dnd~

= F2(u,v,w)

,

zI .... F2(u,v,w)

: Fl[~l(u,v,w) , ~2(u,v,w),

~3(u,v,w)]

Let

j

/e-i(uE+vn).b(E,n,;)dEdn

= h(u,v,E),

(5)

/h(u,v,~) zI

e-~Wd~ = F 2 ( u , v , w )

Repeating the reasoning of the first section, we again arrive at the conclusion that the inverse problem for equation (I) in a half-space has at most one solution for which the following representation is valid:

b(~,n,~)

i / jh(u

= (2~) 2

~) el(U~+Vn)dudv

,v,

,

(_:t~ntn~n÷l an F 3 ( u , v , ~ ) (5a)

h(u,v,t+z i) = lim

i t~i

n÷~

aw n

n!

zlw F3(u,v,w)

= e

F2(u,v,w)

.

Thus the only difference between the inverse problem for a half-space and the corresponding problem for the half-plane is that we have made use of excess information. Namely, to determine a function of three variables a function of four variables

b(~,n,~), we employ

f(xi,x2,Yl,Y2).

The case where the given function is gated further.

f(xl,x2,Y i)

has to be investi-

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-

58

-

[12] GEL'FAND, I.M., GRAEV, M.I. and VILENKIN, N.Ja., Generalized Functions, Vol.5: Integral Geometry and Representation Theory, Academic Press, New York 1965. ~13]

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[141 GUTENBERG, B., Physics of the Earth's Interior, Academic Press, New York 1959. [15] HERGLOTZ, G., 0bet die Elastizit~t der Erde bei BerGcksichtigung ihrer varlablen Dichte , Z. fGr Math. Phys., 52, 3, 1905, pp. 275-299. ~[16] IVANOV, V.K., On ill-posed problems, Mat. Sb., 61, 103, 1963. ~[17] IVANOV, V.K., Integral equations of first kind and approximate solution of the inverse potential problem, Dokl. Akad. Nauk SSSR, 142, 5, 1962. [18~ JOHN, F., Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, New York 1955. [19J KREIN, M.G., On the transition function for a one-dimensional boundary value problem of second order, Dokl. Akad. Nauk SSSR, 88, 1953, pp.405-408. [20] LAVRENT'EV, M.M., Some Improperly Posed Problems of Mathematical Physics, Springer-Verlag, New York 1967. [21]~ LAVRENT'EV, M.M., On a class of inverse problems for differential equations, Dokl. Akad. Nauk SSSR, 160, 1, 1965, pp.52-35. * [22] LAVRENT'EV, M.M. and ROMANOV, V.G., On three linearlzed inverse problems for hyperbolic equations, Dokl. Akad. Nauk SSSR, 171, 6, 1966. [23] MAR~ENKO, V.A., Some questions in the theory of second order linear differential operators for one independent variable, Trudy Moskov. Mat. O b ~ . , 1, 1952. [24] MJUNTC, G., Integral Equations, Part I, GTTL, Moscow-Leningrad 1934.

-

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~[25] NOVIKOV, P.S., On uniqueness for the inverse problem of potential theory, Dokl. Akad. Nauk SSSR, 19, 1938. [26] PALEY, R.E.A.C. and WIENER, N., Fourier Transforms in the Complex Domain, Amer. Math. Soc. Coll. Publ., 19, New York 1934. ~[27] PRILENK0, A.I., On uniqueness of a solution to the exterior inverse problem for a Newtonian potential, D.U., 2,i,1966. ~[28] PRILENKO, A.I., On inverse probrems in potential theory, D.U. 5,i, i967. ~9]

RAPPOPORT,I.M.,On a two-dlmenslonal inverse problem in potential theory, Dokl. Akad. Nauk SSSR, 28, 1940.

~0]

ROMANOV, V.G., On the determination of a function from its integrals over ellipsoids of revolution with one fixed focus Dokl. Akad. Nauk SSSR, 175, 4, i967.

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