On the uniqueness of the solution of the converse problem in potential theory

ON THE UNIQUENESS OF THE SOLUTION OF THE CONVERSE PROBLEM IN POTENTIAL THEORY * A. KH. OSTROMOGIL’SKII Moscow (Received

23 January

1969)

WE shall investigate the uniqueness of the solution of the converse problem in ~tential theory, i.e. the problem of finding the dis~ibution of at~ct~g masses, given their external potential. This problem is well known not to have a unique solution unless additional assumptions are made regarding the mass distribution. The problem has therefore been considered by many authors in the following form: given the Newtonian potential outside some sphere, generated by a body with a given mass dis~ibution density p is a function such that its total variation over any straight line is not greater than the minimum of ,u(f, n, 0. Uniqueness theorems were later obtained for certain classes of body that are not star-shaped. Uniqueness was proved in 121 in the class of bodies possessing central planes parallel to the (’ = 0 plane, i.e. it was assumed as regards the region T, occupied by the body that

where

*Zh.

vjichisl.

Mat. mat. Fiz.,

10, 2, 352-361,

91

1970.

92

A. Kh.

Here, I,!J/,~and tiza are positive bounded

set of the < = 0 plane,

known and independent The restrictions

Ostromogil’skii

continuous

functions

and Ha is a constant.

(0.2 on the class

of bodies

in [31 are proved for the n-dimensional

these

that d,a and ~5~~ are continuous

bounded

in an open was assumed

of the depth.

the theorems papers

defined The density

were removed case.)

functions,

in [3, 41. (Ail

It was assumed

defined

in

in an open

set of the < = 0 plane.

In the present paper we examine the uniqueness of the solution of the converse problem in potential theory when both the region and the density arc unknown, but certain restrictions are imposed on the class of bodies and the mass distribution density inside them. We assume regarding the class Rs of bodies body can be written in the form (0.11, where cpP($,rl)

=min

that the region

occupied

by a

{x(E, 11); f”(Elll)I1 (0.3)

and the surface

S =

{(E, Y, C.), 5 = x(& rl))

is assumed

known Cs($, 7) is a

continuous function throughout the C = 0 plane). Under these assumptions, we (1) the function f a (,‘, v), defining the shape of the bod,v, and (2) the mass seek: distribution additional

inside it. The uniqueness assumption that the density

of the solution is independent

is proved subject of the depth, i.e.,

to the

(0.4) These

results

arc stated

and proved

Section 1 being of an auxiliary nature. that the restriction (0.3) is essential.

in Section 2 of this paper (Theorem l), We mentioned in regard to Theorem 1 It cau in fact be shown that, when no

restrictions are imposed on the density, the problem solution. The density restriction (0.4) may prohably worth trying

to establish

density

restrictions

does not have a unique be weakened, so that it is

such that a unique

solution

is

assured. We assume

in Theorem

2 of Section

2 that the depth-independent

distribution

density is known, and we consider the external potentials Vv generated by the on the mutual bodies 7’” of the form (0.1) (9 = 0, 1, 2). F-or certain restrict,ious disposition of the bodies, we can prove that the identity Vo + VI = VZ is

On the uniqueness

of the solutions

IT impossible for points not belonging to Jp

93

V.

In Section 3 the results of Section 2 are transferred to the class of bodies that can be written in the form P = { (4, rl, 5) : $1 Mu -=z P -=z 92(wfu 19 where (p, (ri, 8) are the spherical coordinates of the point (5, n, .Q_ All our results are also stated and proved for the ~~imensional case. 1.

Definitions

and fundamental

lemma

1. The function

is called the potential of the mass m, distributed in an open bounded set 7’ with distribution density p(y), where z = = (a~,. . . , ST*), y = (~1,. . . , YTZ) are points of ~dimensional5uclide~ space En, and p(y) is a bounded measurable function in En, and K (L, y) is the fundamental solution of Laplace’s equation

We shall also requne the projections of the point x, y on the x, = 0 plane, which we shall denote by x’ , y’ respectively. 2. We shall consider open funds the form

sets Tarc En, which can be written in

Ta={x~E*,x’~Ta.,~=(;e))


(1.3)

where T, * is sn open bounded set in the xn = 0 plane, consisting of a finite number of (n - lkbmensional regions. If n > 2, we shall assume that their boundaries are Lyapunov surfaces. The fuuctions +cla.(x’)< qa(x’) satisfy a Holder condition in the set T,*.

Ld

ra*Ya be respectively l-m =

(x

Ez

the “upper” and “lower* bounds of the get T,:

EE”: xn =(pa(x’),

ya = (5 es En: xn =qk&‘),

xfE

Ta‘},

5’ E Tar}.

(1.4)

A. Kh. Ostromogil’skii

94

If U(x) = U(x’, xm), we put u (2) jr, = u (6 Let R be the class of sets satisfying 3. Let &(y) be a bided, condition

~eas~able,

fpa (2’)).

(1.13). positive fiction,

satisfying

the

(1.5)

Let 2 be the class of functions satisfying

this condition.

Lemma 1 Let

ya(4 =

lJ(Y>K k l

Pa

(1.s)

Y)dY*

a

where (4.7) and let b(s)

=

7-2(s)*

ZEE”\(TIlJT2).

Let U ty) be a function continuous in Tt U T.J and satisfying the Laplace equation at interior points of Q, (5, y) = X(x, y) - U(y) , if

U.8)

TI U Tz.

Proof. Let D be a region such that D II) Tt u Tz. By Novikov’s lemma [l, 41, whatever the function H(y), harmonic in D,

On the uniqueness

of the solution

(1.11)

We put H (y) = awl+, on using (1.7),

where W(y) is a function harmonic in D; we then obtain,

s (Y’)[W IF,- w

I,,1 dY’-

Pl

T**

j*P2 (Y’)[W IF,- w Iv*1dy'

= 0. (1.12)

Using the concept of a generalized solution of the Dirichlet problem, we can extend equation (1.12) in the same way as in [51, to the function LI(y) satisfying the conditions of the Lemma. Differentiation

av, as, =-

of (1.6) with respect to x, gives

s+qx, a

spa

(Y’)

y)dy=-

n

[K Ir, - K Iral +I’*

Ta*

Hence,

a(V1- v,> ax,, On adding the last equation and (1.9). we get (1.10). Section

2

Suppose we are given a surface S in the form

Js=

{Y;

yn = x(y')},

(2.1)

where x (y’ ) is a continuous function, defined on the y, = 0 plane and satisfying a Holder condition there. If T, E R, every straight line passing through SC’E T,* and parallel to the y, axis will cut the boundary of T, at exactly two points. We shall assume that one of these points lies on S. The class of such sets will be denoted by R,. If T, E R,, we put

A. Kh. Ostromogil’skii

r2=r,,s*

ycP=yans.

Let T, and T, be open sets. Let [ be the boundaryof the set TI U F2. Theorem 1 Let the masses m, be distributed in the sets T, E R, with density h Z (a = 1, 2). Let V, (3~) be the potential of the mass m, and let

VI (xl = Vz(x),

x E E” \ (T1u T2).

(2.2)

Then T, = T, and ~1,= ~1~. Proof. 1. We introduce the sets (Yl”UY29

nx

u r29 n 2 = G,

(JL”

=g,

g* is the projection of the set g on the 3cn= 0 plane (the projection of the part of T, U T, which lies Yaboven the surface S), G* the projection of the set G on the xn = 0 plane (the projection of the part of T, U T, which lies ‘below” the surface S), go’ = (2’1’ U 10’) \ (g’ U G’) (the projection of the part of T, U T, which lies on both sides of S). Let &

(2))

x’ E T;;

=

x’ cj5 T;.

(2.3)

Let

c = (TI”U T2-)\ B, rC= {x:xd-lur2,x%=c),

B = {x’: p.z(x’)a PI), rB = (5: x E r, u r2,5’ E B}, ys =

yc=

{5:x~ydy2,~‘~B),

(x:zEy,Uy2*5’EC}.

Obviously, TI’ U T2*

=

so’

Uf

U G”.

Let us define on the surface ZZthe function

(2.4)

On

f(Y)=

the

of the solution

uniqueness

Y~Y,U(rln~)~ YE:GAJ(Ydw*

I i I

97

Iv=

YardYam% ifor other

0

Y' E 8,

Y’EG’, Y’ E go*;

(2.5)

YEZ.

Approximating f(y) by continuous functions, we can extend (1.9) in the same way as in [51, to a function F, harmonic at interior points of the set TI U Tz, which takes the value f(y) everywhere on E except for points of surface measure zero. I

Then,

[Fl == 1 Fi (4 v I& - F I,,) cw - TOTAL_. (4 IF I&- F I,> ds’ =o. TtUTP

(2.6) L&t Io[Fl=

~~(.t’)IFl~,--lhldz’IX*

$ ~a - (2’) [F Ir, g.’

J,[Fl== SILl(r’)tFlr,-FI,ldz’8.

Let us

2. The set g,,* is the projection tion of y2s n J?ls then,

of

yl* n I’*s

of the set

(vlB n I’s6) U (y#n

r12).

Let

on to the xn = 0 plane, and A,, the projec-

on t.o the x, = 0 plane.

$ & (x’) IF IF,;l

- $$a (x’) [Fr, - F I,1 h’. GO

find bounds for these functionals.

A,, be the projection

1. [F] =

(-Pa(~‘1[F Ir, - p I,,1dx’, g*

jk (2’) IF I,,-- P I& d

I, [Fl=

F I,1 dz’,

F &,I dx’ -

obviously,

5 k (5’) (+--

me8

(AIs n Azl) =

0;

F Id ax’ +

~(~')(F~~,--P~n)dz'~~G;(~')tF~~,-Fl~l~~'= At1

At1 = ,,” s

(x’) t 1 -

F I,,] ax’ +

+&Gd s

(x’)

!1;1(x’) At1 [I -

F l&x

F Ir,l h’+

+

$i&‘) Am

F1,. >o.

Thus, zo[q 3. Consider

l,[Fl. Let A, be the

3

0.

projection

(2.7) of rcr U E on to the x, = o

98

A. Kh. Ostro~og~Z’ski~

plane (a = 1, 2). Consider

*a Consider 1, PI- Let a, be the projection of the set ya U I: on to the xn = 0 plane (a = 1, 2). Consider 121 [PI = $1, =

(3’) P Jr, 1

$1

(x’)

-

El-a (t’)

IL2

F

(2’)l

Ir,l

dz’

dx’

t Fl

=

j*rF2

(4

1

>

tii

(;t”)

-

G2

Gv~nc

o+nc I22

=

p

I,

-

jil

(z’)

F

jy,dx’

‘j

=

[F2P

I,

-

EF

WH w I,,,1

&I+

+

G*nat

3

G ,j

’

nc [cb2 -

6’8

I,,1 dx’.

P

Thus, Z#J

=Z21[F]

+

122[FJ

a

(2.9)

0.

5. From (2.71, (2.8) and (2.9, we have WI

=

JoCFl

f

11 [PI

+

12 [PI

3

0.

(2.10)

In view of (2.6), the equality sign must hold in (2.10) (this will only be the case when F = const). The necessary condition for this is that a connected component D, of one of the sets T, exist (e.g. of T,), such that the following is satisfied for any

99

On the urpiqueness of the solution

connected component D, of I’,;

D,czDo

or

Let ?$(x, y) be Green’s function for the Dirichlet problem on the set Putt~g dtt (5, y) = k?Z(r, g), in fl.lO), we get

a(V1-

v!J f= _

%I

iJ pl(y’)

[ij

(2.11)

9.

&flDo=

r,-q,Jdy’-+

71 U Tz.

$P*(Y')[~I,-~l~~l~Y'.

1

T*’

I’

(2.12)

If y E X, then G G, rf > 0 for y ~longing to the connected com~nent !?i U T2, containing X, and a(,, y) = 0 at remaining points of Ti IJ Tz.

of

Let D, be a connected component of Ti IJ Tz, containing D,, and D,* the projection of D, on to the x, = 0 plane. Using the properties of Green’s function C&12), we can rewrite

if x E D,. When conditions (2.11) are satisfied, Pr and ya must necessarily iie inX When Di flDo= (bt the necessary condition for F I co&t is that D, and D, lie on different sides of the surface S and the union of their projections is not empty. Using this, we get

Wl-+)= ka

1$(y’)

lA*

qpy

+

$&Y’) q$Y’>O*

(2.43)

Da*

This means that V, - V, is monotonically increasing with xl fixed as a function of xn inside the region D,. This is impossible in view of condition (2.2). The contradiction proves the theorem. Note. The condition T, E R, is essential. The example of two homogeneous concentric spheres (see E81,p. 97) shows that it cannot be dispensed with. 6. Given the sets T, (a = 0, 1, 21, in which the masses m, are distributed with density it (Y). Let V, (x1 he the potential of the mass m Q. Theorem 2 Let

100

A. Kh. Ostromogil’skii

(2.14)

and is the boundaryof Tr IJ!i’z).

(2.15)

Then the identity

Vo(z) is impossible,

if

+ VI(S)

=

SEE*

vi@),

\

(Tl

U~2UUo)

(2A6)

T1 U To # T2.

Proof. Let T12 = Tr \ T2, T21 = 2’2 \ TI; Ur2 is the potential of the mass mr2, distributed over the set ‘I’,,, and U,, the potential of mass m,,, distributed over ‘I’,, with the same distribution density ,uCy). Let (2.16) be satisfied; Ul2 +

vo

then

=

x=E”-

&I,

(T12 u TPI u To).

(2.17)

Arguing as in Lemma 1, we can obtain an equation similar to (1.91, i.e. whatever the function H, continuous in T12 U T2, U To points of this set, we have

sPWIro-

and harmonic at interior

- II /,,I dx’ = 0. - H ly,*l c.b’-- j p [ff l1‘21 T, I*

Tti*

7-00

(2.18) Let S, be the boundary of the set

To U TSI U T12.

We define on the surface S, the function

f(Y) = [

;’ 9

YE(T‘oU~l2UY2l)n

Y E

&

\

Sl,

(2.19)

Q-0 u r12 u Yad.

We extend (2.18) to a function F(x), harmonic inside the set To U T12U T2, and taking the value f(y) everywhere on S, except for a set of points of surface measure zero.

Then,

On the uniprreness

101

of the solution

If f(y) f const, we can easily show by using the properties of the function F that (2.20) is impossible. Let f(y) = 1.; this is only possible when l-0 11 r12

u Y21~

(2.21)

St.

It follows from (2.21) that T, = (b. Let 5 (x, y) be keen’s function for the Dirichlet problem on the set T, U ‘I’,. Putting @((5, y) = F(s, y), in 11.10). we get

Using the properties

of Green’s function and (2.21). we can rewrite (2.22)

as

g-

n

[VO-- v21=

jtq*w +$PCI,,,W>O. -l-Z,*

T,’

(2.23)

This means that V, - V, is a montonically increasing function of x, inside the set T, U T, when x’ is fixed. This is impossible in view of condition (2.16). The ~~t~di~tion proves the theorem. Section

3

All the Theorems of Section 2 can be extended to the class of regions described below. In view of the strong similarities, we shall only quote the statements of the theorems. 1. Let Q be the spherical coordinates sphere at the point y which can be written

unit sphere in space En, a-= (prox) fo, ez Q are the of the point x), and do, an elementary area of the unit E Sk We shail consider open bounded sets P, c En, in the form

P, = (5 = (~~0,) E En,

ax E P,“,

oa (0) C pr C &(a)},

(3.1)

A. Kh. Ostromogil’skii

102

where P,* is the open set on the sphere $I consisting regions on the sphere.

of a finite number of

If n > 2, we shall assume that their boundaries are Lyapunov surfaces; are functions satisfying a Holder condition in the set 0 e %x(o) < k(o) Pa*. Let R” be the class of sets satisfying condition (3.1). Let cl(y) be a bounded measurable positive function, satisfyily

Let 2’ be the class of functions satisfying

(3.2).

Lemma 2 Let

va(x)

-

5Pa(Y)K (xvY)d/t

Pa

wherePa~~o,~a(y)~Zo(a=1,2) Vi(S)

andlet =

I

x=En\

Vz(x),

(&il&).

Let U (y) be a function continuous in P, U p 1 and harmonic inside PI U I$, Then, W? Y) = K(x, Y) - U(y). 5 PI (0) If “17 IQ,- PU I,,1 da% Pl’

5 Pa (W) [P”u IQ, - PnU I,1 d% = 0, PI*

(3.3) (3.4)

The proof of (3.4) is based on the obvious equation

2. Suppose we are given the surface

On the uniqueness

103

of the solution

where xO(oY) is a continuous function defined on the sphere Cl and satisfying a Holder condition on it. If P, E R’, every ray from the origin cutting P, cuts it in precisely two points. We shall assume that one of these must necessarily lie on S,.

Let RsO be the class

of such sets.

Theorem 3 Let masses m, be distributed on the sets P, E RsO with distribution density pa E Z’J (a = 1,2). Let V, Cc) be the potential of mass m, and let

V,(x)

=

5 E

V2(x),

E’” \

(T1 u

T2).

Then T, = T, and p, = pz.

Theorem 4 Given the sets tion density

p(y)

Pn E

R“, on which masses m, are distributed with distribuLet V, (x) be the potential of the mass

E 2” (,o = 0,1,2).

m, (a = 0, 1, 2) and let

(C is the boundary V,(r)

is impossible

if

of +

PI U A). V,(r)

=

Then, Vz(x),

the identity

XEE”

\

(P,U&tJUo)

PI U PO #Pp. Translated

by D. E. Brown

REFERENCES

1.

NOVIKOV, P. S. On the uniqueness of the solution of the converse problem, Akad. Nauk SSSR, 13, 3, 165-168, 1938.

potential

2.

SRETENSKII, L. N. On the uniqueness of determining the shape of an attracting body, given the values of its external potential, Dokl. Akad. Nauk SSSR, 99, 1, 21-22. 1954.

104

A. Kh. Ostromogil’skii

3,

PRILETSKO, A. I. On the uniqueness of the solution of converse metaharmonic potentials, Differ. Umiya, 2, 2. 194-204, 1966.

problems

4.

TODOROV, I. T. and ZIDAROV, D. On the uniqueness of determining the shape of an attracting body, given the values of its external potential, Dokl. Akad. Nauk SSSR, 120. 2, 262-264. 1958.

5.

PRILETSKO, A. I. On the uniqueness of the solution of the external converse problem of Newtonian potential, Differ. urniya, 2. 1, 107-123, 1966.

6.

SRETENSKII, L. N. Theory of Newtonian potential (Teoriya potentsiala), Gostekhizdat, Moscow-Leningrad, 1946.

n’yutonovskogo

on