数理解析研究所講究録
第 890 巻 1994 年 19-30

19Inverse source problems
in Poisson’s equations
SABUROU SAITOH

(fi

rk

$\underline{=}$

$\Re$

)

Abstract –In the Poisson’s equations$\Delta u=-\rho(r)$for $L_{2}(dr)$ functions
values of$u$$\rho$

onand$R^{1},$ $R^{2}$$R^{3}$, the natural and fundamental inverse formulas representing

in the outsides of supports of$\rho$$\rho$

in terms of theare established.

1. INTRODUCTION

We shall consider the Poisson’s equationson$\triangle u=-\rho(r)$$\mathbb{R}^{1},$and$\mathbb{R}^{2}$(1.1)$\mathbb{R}^{3}$for real-valued $L_{2}(dr)$ source functions whose supports are contained in a sphere $r<a$or in the outside of the sphere; denotes the usual distance $r=|r|$ from the origin.
By using the general method ([3] and [4]) for integral transforms using the theory of
reproducing kernels, we first give the characterizations and natural representations of
the potentials on the outsides of supports of . As an application, we shall givein terms of of the outsides of supports of ,surprisingly simple expressions ofwhich have the minimum $L_{2}(dr)$ norms among the source functions satisfying (1.1)
on the outsides of supports of . These representations give practical applications toby the potentials . These inverse problems were
determine the source functions
proposed by Laplace 200 years ago and many mathematicians have attacked to these
problems. In this paper, we shall give the reasonable solutions for these problems in
the framework of $L_{2}(dr)$ spaces. See Foreword and References of Isakov [1]. For a basic
and general reference for the inverse source problems, see [1].
$\rho$

$r$

$u$

$p$

$\rho^{*}$$u$$\rho$$p$$\rho$$\rho^{*}$

2. CASE $mTH$ COMPACT SUPPORT ON$u$$\mathbb{R}^{3}$

We aaeume that the supports of are contained in the sphere $r<a$ with radius . So,
forand $|r|=r$ , we shall examine the integral representation of the solutionsof the Poisson equation (1.1)$\rho$$r\in \mathbb{R}^{3}$

Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu 376, JAPANE-mail address: ssaitoh@eg.gunma-u.ac.jp

$a$

$u$

20

(2.1)

$u( r’)=\frac{1}{4\pi}\int_{<a}\frac{1}{|r’-r|}\rho(r)dr$on$\mathbb{R}^{3}$for the source functions$\rho$

satisfying$\int_{<a}\rho(r)^{2}dr<\infty$

.

(2.2)

In order to determine the characteristic property of the potentials on $r>a$ using thegeneral method ([3] and [4]) for integra} transforms, we consider the kemel form$u$

$K_{i,a}( r’, r’’)=\frac{1}{4\pi}\int_{r<a}\frac{1}{|r’-r||r’’-r|}dr$

for $r’,$ $r”>a$ . In order to calculate $K_{i,a}(r’,$ $r”)$ we shall use the expansion$\frac{1}{|r’-r|}=\sum_{n=0}^{\infty}\sum_{m=0}^{n}\frac{\epsilon_{m}(n-m)!}{(n+m)!}P_{n}^{m}(\cos\theta’)P_{n}^{m}(\cos\theta)\cos m(\varphi’-\varphi)\frac{r^{n}}{r^{\prime n+1}}$

for $r’>r$(2.3)(cf. [2, p. 1274]). Here,andin terms of spherical coordinates
is the Neumann factor. By using the two orthogonality$(r’, \theta’, \varphi’)$$(r, \theta, \varphi)$$\epsilon_{m}=$$2-\delta_{mo}$$\int_{0}^{2\pi}\cos m(\varphi’-\varphi)\cos m’(\varphi’’-\varphi)d\varphi=2\pi\delta_{mm’}(\epsilon_{m})^{-1}\cos m(\varphi’-\varphi’’)$

(2.4)

and
$\int_{0}^{\pi}P_{n}^{m}(\cos\theta)P_{n}^{m}(\cos\theta)\sin\theta d\theta$$= \int_{-1}^{1}P_{n}^{m}(x)P_{n}^{m}(x)dx$,$= \delta_{nn’}\frac{2}{2n+1}\frac{(n+m)}{(n-m)}!$(2.5)

we have the expansion$K_{i,a}( r’, r’’)=\sum_{n=0}^{\infty}\frac{a^{2n+3}1}{(2n+1)(2n+3)r^{\prime n+1}}\frac{1}{r^{\prime/n+1}}$$x\sum_{m=0}^{n}\frac{\epsilon_{m}(n-m)!}{(n+m)!}P_{n}^{m}(\cos\theta’)P_{n}^{m}(\cos\theta’’)$$x(\cos m\varphi’\cos m\varphi’’+\sin m\varphi’\sin m\varphi’’)$

for

$r’,$

$r”>a$ .

(2.6)

21

This series converges absolutely on $r’,$ $r”>a$ . The kernel $K_{i,a}(r’, r’’)$ is a positive matrix
on $r>a$ and so, there exists a uniquely determined IIilbert space
admitting the
reproducing kemel $K_{i,a}(r’, r’’)$ . Furthermore, the images $u(r’)$ of (2.1) belongju to the. The expansion (2.6) impliae that the imagae $u(r’)$ are expressible
IIilbert spacein the form
$H_{K_{i.a}}$$st$$H_{K:.a}$$u( r’)=\sum_{n=0}^{\infty}\frac{a^{2n+3}}{(2n+1)(2n+3)}\frac{1}{r^{\prime n+1}}$$x\sum_{m=0}^{n}\frac{\epsilon_{m}(n-m)!}{(n+m)!}P_{n}^{m}(cos\theta’)(A_{n}^{m}\cos m\varphi’+B_{n}^{m}\sin m\varphi’)$$(2^{\cdot}.7)$

for some constants$\{A_{n}^{m}, B_{n}^{m}\}_{n,m=0}^{\infty}$

satisfying

.

(2.8)

$\sum_{n=0}^{\infty}\frac{a^{2n+3}}{(2n+1)(2n+3)}\sum_{m=0}^{n}\frac{\epsilon_{m}(n-m)!}{(n+m)!}\{(A_{n}^{m})^{2}+(B_{n}^{m})^{2}\}<\infty$

Conversely, any $u(r’)$ defined by (2.7) with (2.8) belongs to$H_{K}:,a$. Since the family$\{P_{n}^{m}(\cos\theta)\cos m\varphi, P_{n}^{m}(\cos\theta)\sin m\varphi\}_{n,m=0}^{\infty}$

is complete in the Hilbert space composing of the functions$f(\theta, \varphi)$$\{\int_{0}^{\pi}\int_{0}^{2\pi}f(\theta, \varphi)^{2}\sin\theta d\theta d\varphi\}^{i}<\infty$

we have the representation of the norm$||u||_{H_{K_{i.a}}}$with finite norms,

in the form

.

(2.9)

$||u||_{H_{K_{i,a}}}^{2}= \sum_{n=0}^{\infty}\frac{a^{2n+3}}{(2n+1)(2n+3)}\sum_{m=0}^{n}\frac{\epsilon_{m}(n-m)!}{(n+m)!}\{(A_{n}^{m})^{2}+(B_{n}^{m})^{2}\}$Furthermore, we have the isometrical identity$||u||_{H\kappa_{:}}^{2},= \min a\int_{<a}\varphi(r)^{2}dr$$= \int_{<a}\varphi^{*}(r)^{2}$ $dr$.

(2.10).
Here, the minimum is taken over all
satisfying (2.1) and (2.2) for $r’>a$ andthe uniquely determined function with the minimum norm ([3] and [4]).
$\rho(r)$$\varphi^{*}$

is

22

In (2.7), by using the orthogonality$\int_{0}^{\pi}\int_{0}^{2\pi}P_{n}^{m}$(cos )$\theta$$\cos m\varphi P_{n}^{m’}(\cos\theta)\{\begin{array}{l}cosm^{/}\varphiorsinm^{/}\varphi\end{array}\}\sin\theta d\theta d\varphi$$= \delta_{mm’}\delta_{nn’}\frac{4\pi(\epsilon_{m})^{-1}}{2n+1}\frac{(n+m)!}{(n-m)!}$,we have the expressions, for any fixed $b>a$$A_{n}^{m}= \frac{(2n+1)^{2}(2n+3)b^{n+1}}{4\pi a^{2n+3}}\int_{0}^{\pi}\int_{0}^{2\pi}u(b, \theta, \varphi)$(2.11)$xP_{n}^{m}(\cos\theta)\cos m\varphi\sin\theta d\theta d\varphi$

and$B_{n}^{m}= \frac{(2n+1)^{2}(2n+3)b^{n+1}}{4\pi a^{2n+3}}\int_{0}^{\pi}\int_{0}^{2\pi}u(b, \theta, \varphi)$$xP_{n}^{m}(\cos\theta)\sin m\varphi\sin\theta d\theta d\varphi$.

(2.12)

These expressions with (2.7) imply that for any point $r’(r’>a)$ , the potentials $u(r’)$
are expressible in terms of$u(b, \theta, \varphi)$

for any fixed$b(b>a)$ .

(2.13)

in terms of (2.13).
We shall derive the inverse formula representing$K_{i,a}(r,r’)$$H_{K_{jq’t}},we$ havein
Using the reproducing property of$\rho^{*}$$u(r’)=(u(r), K_{i,a}(r, r’))_{H_{K}}.,a$$=(u(r),$$\frac{1}{4\pi}\int_{1}<a\frac{1}{|r-r_{1}||r’-r_{1}|}dr_{1})_{H_{K}}\cdot\cdot a$$= \frac{1}{4\pi}\int_{r<a}1\frac{dr_{1}}{|r’-r_{1}|}(u(r), \frac{1}{|r-r_{1}|})_{H_{K_{i,a}}}$(2.14)$= \frac{1}{4\pi}\int_{1}<a\frac{dr_{1}}{|r’-r_{1}|}\rho^{*}(r_{1})$and so, we have$\rho^{*}(r_{1})=(u(r), \frac{1}{|r-r_{1}|})_{H_{K}}.,n$Here, note that$\frac{1}{|r-r_{1}|}$belongs to the Hilbert space.

$H_{K_{i,a}}$(2.15).

23

Indeed, from (2.7) and (2.3) the corresponding coefficientsthe representation (2.7) are$\overline{A_{n}^{m}}$and$\overline{B_{n}^{m}}$of$\frac{1}{|r-r_{1}|}$in$\overline{A_{n}^{m}}=\frac{(2n+1)(2n+3)}{a^{2n+3}}P_{n}^{m}(\cos\theta_{1})r_{1}^{n}\cos m\varphi_{1}$and$\overline{B_{n}^{m}}=\frac{(2n+1)(2n+3)}{a^{2n+3}}P_{n}^{m}(\cos\theta_{1})r_{1}^{n}\sin m\varphi_{1}$

.

Then,
$\sum_{n=0}^{\infty}\frac{a^{2n+3}}{(2n+1)(2n+3)}\sum_{m=0}^{n}\frac{\epsilon_{m}(n-m)!}{(n+m)!}\{(\overline{A_{n}^{m}})^{2}+(\overline{B_{n}^{m}})^{2}\}$$= \sum_{n=0}^{\infty}\frac{(2n+1)(2n+3)}{a^{2n+3}}\sum_{m=0}^{n}\frac{\epsilon_{m}(n-m)!}{(n+m)!}P_{n}^{m}(cos\theta_{1})^{2}r_{1}^{2n}$$= \sum_{n=0}^{\infty}\frac{(2n+1)(2n+3)r_{1}^{2n}}{a^{2n+3}}P_{n}(1)$$= \sum_{n=0}^{\infty}\frac{(2n+1)(2n+3)r_{1}^{2n}}{a^{2n+3}}<\infty$for$r_{1}<a$.

(2.16)

See [2, p. 1274].
Hence, the formal arguments in (2.14) and (2.15) arejustified. See also the following(2.17) and [4] for these arguments. We thus have
Theorem 2.1. The source functions
terms of (2.13) in the form$\rho^{*}$

in the sense of (2.10) in (1.1) are expressible in$\rho^{*}(r_{1})=\frac{1}{4\pi}\sum_{n=0}^{\infty}\frac{(2n+1)^{2}(2n+3)}{a^{2n+3}}r_{1}^{n}b^{n+1}$$x\sum_{m=0}^{n}\frac{\epsilon_{m}(n-m)!}{(n+m)!}P_{n}^{m}(\cos\theta_{1})$$x\int_{0}^{\pi}\int_{0}^{2\pi}u(b, \theta, \varphi)P_{n}^{m}(\cos\theta)\cos m(\varphi-\varphi_{1})\sin\theta d\theta d\varphi$

As we see from Theorem 2.1, the source functions
tions on $r<a$ .
Indeed,$P_{n}^{m}(\cos\theta_{1})r_{1}^{n}\cos m\varphi_{1}$,$\rho^{*}$.

obtained are harmonic func-$P_{n}^{m}(\cos\theta_{1})r_{1}^{n}\sin m\varphi_{1}$24are harmonic functions, and from (2.15) and (2.16)$|\varphi^{*}(r_{1})\leqq||u||_{H_{K_{i,a}}}$$\frac{1}{|r-r_{1}|}ff_{K}:.a$(2.17)$=||u||_{H_{K}}:,a \{\sum_{n=0}^{\infty}\frac{(2n+1)(2n+3)r_{1}^{2n}}{a^{2n+3}}\}^{\neq}$in the right hand side in Theorem 2.1 converges absoHence, the expansion oflutely and uniformly on $r\leqq a’<a$ for any fixed .
The fact that are harmonic means that the source functions obtained are smooth$\rho^{*}(r_{1})$$a’$$\rho^{*}$

and uniform on the ball $r<a$ , in a sense.
In order to get a more general source, we shall consider the integral transform$u( r’)=\frac{1}{4\pi}\int_{<a}\frac{1}{|r’-r|}\rho 1(r)D(r)dr$

for a nonnegative measurable function $D(r)$ depending only on
form

$r$

such that the kernel.

$K_{i,a}( r’, r’’;D(r))=\frac{1}{4\pi}\int_{<a}\frac{1}{|r^{l}-r||r’’-r|}D(r)dr$$(=K:,a(D(r)))$exists for $r’,$ $r”>a$ . Then, we can obtain the inversion formula representing
satisfyingof$\rho_{1}^{*}$

in terms$u(b, \theta, \varphi)$,$||u||_{H_{K_{i,a}\langle D(r))}}^{2}= \frac{1}{4\pi}\int_{<a}\rho_{1}^{*}(r)^{2}D(r)dr$

as in Theorem 2.1. Hence,$\rho(r)=\rho_{1}^{*}(r)D(r)$.

For this argument, see Section 3 for $D(r)=r^{-2}$ By taking a suitable $D(r)$ , we will be. This technique is valid in theable to obtain a more general and practical source
following situations, similarly.
$\rho(r)$3. CASE $KTH$ UNBOUNDED SUPPORT ON$\mathbb{R}^{3}$

Next, we shall consider the caae such that the supports of are contained in the sphere$r>a$ . In this case, in order to request the existence of the corresponding reproducing
kemel we assume that$\int_{>a}\rho(r)^{2}r^{2}dr<\infty$ .
(3.1)$\rho$

25
In order to examine the integral representation(3.2)$u( r’)=\frac{1}{4\pi}\int_{>a}\frac{1}{|r’-r|}p(r)dr$of , we shall consider the integral transform in the form
$u$

(3.3)

$u( r’)=\frac{1}{4\pi}\int_{>a}\frac{1}{|r’-r|}\rho_{1}(r)\frac{1}{r^{2}}dr$

for the functions$\rho_{1}=r^{2}\rho$satisfying$\int_{>a}p_{1}(r)^{2}\frac{1}{r^{2}}dr=\int_{>a}\rho(r)^{2}r^{2}dr<\infty$

.

(3.4)

We form the reproducing kernel$K_{o,a}( r’, r’’)=\frac{1}{4\pi}\int_{>a}\frac{1}{|r’-r||r’-r|}\frac{1}{r^{2}}$ $dr$

.

(3.5)

By using (2.3) and the orthogonality of (2.5) and (2.6), we have$K_{o,a}( r’, r’’)=\sum_{n=0}^{\infty}\frac{r^{\prime n}r^{\prime\prime n}}{(2n+1)^{2}a^{2n+1}}\sum_{m=0}^{n}\frac{\epsilon_{m}(n-m)!}{(n+m)!}$$xP_{n}^{m}(\cos\theta’)P_{n}^{m}(\cos\theta’’)(\cos m\varphi’\cos m\varphi’’+\sin m\varphi’\sin m\varphi’’)$

for

$r’,$

$r”<a$ .

,(3.6)

Hence, by the parallel arguments to Theorem 2.1, we have

Theorem 3.1. For the source functions
satisfying (3.2) and (3.1) with the minimum
norms in (3.1), we have the inverse formula, for any fixed $\hat{b}(0<\hat{b}<a)$ ,$p^{*}$$p^{*}( r_{1})=\frac{1}{4\pi r_{1}^{2}}\sum_{n=0}^{\infty}(2n+1)^{3}a^{2n+1}r_{1}^{-(n+1)}\hat{b}^{-n}$$x\sum_{m=0}^{n}\frac{\epsilon_{m}(n-m)!}{(n+m)!}P_{n}^{m}(\cos\theta_{1})\int_{0}^{\pi}\int_{0}^{2\pi}u(\hat{b}, \theta, \varphi)$$xP_{n}^{m}(\cos\theta)\cos m(\varphi-\varphi_{1})\sin\theta d\theta d\varphi$

4.

$\mathbb{R}^{2}$

CASE

.

26

We shall consider the 2-dimensional potential(4.1)$u( r’)=\frac{1}{2\pi}\int_{r<a}\log\frac{1}{|r’-r|}p(r)dr$

for the source functions$\rho$

satisfying(4.2)$\int_{<a}\rho(r)^{2}dr<\infty$

whose support are contained in the disc $r<a$ .
In order to examine the integral transform (4.1) with (4.2), we form the reproducing
kemel
$s$

$K_{i_{r}a}^{(2)}( r’, r’’)=\frac{1}{2\pi}\int_{<a}\log\frac{1}{|r’-r|}\log\frac{1}{|r’’-r|}dr$

for$r’,$ $r^{u}>a$

.

(4.3)

By using the expansion,$\log\frac{1}{|r’-r|}=\log\frac{1}{r}+\sum_{n=1}^{\infty}\frac{1}{n}(\frac{r}{r})^{n}cosn(\varphi’-\varphi)$

for $r’>r$in the polar coordinates$(r, \varphi)$and$(r’, \varphi’)$(4.4)(cf. [6, p. 1188]), we have$K^{(2)}:,a( r’, r’’)=\frac{a^{2}}{2}\log\frac{1}{r}\log\frac{1}{r’}+\frac{1}{4}$$x\sum_{n=1}^{\infty}\frac{a^{2n+2}}{n^{2}(n+1)}\frac{1}{r^{n}r^{n}}(\cos n\varphi’\cos n\varphi’’+\sin n\varphi_{\lambda}’\sin n\varphi’’)$for$r’,$ $r^{n}>a$

,

(4.5)

which converges absolutely. Hence) as in Theorem 3.1 we have
Theo $rem4\cdot 1$ . The source functions
with the minimum norms (4.2) satisfying (4.1)
for $r’>a$ are expressible in the form, for any fixed $b(b>a)$$\rho^{*}$$\rho^{*}(r_{1})=(u(r);\log\frac{1}{|r-r_{1}|})_{H_{K_{i,a}^{\langle 2)}}}$$= \frac{1}{\pi a^{2}\log\frac{1}{b}}\int_{0}^{2\pi}u(b, \varphi)d\varphi$$+ \frac{4}{\pi}\sum_{n=1}^{\infty}\frac{n(n+1)b^{n}r_{1}^{n}}{a^{2n+2}}\int_{0}^{2\pi}u(b, \varphi)cosn(\varphi-\varphi_{1})d\varphi$.

27

Next, we shall examine the case such thatconsider the potential$\rho$

have unbounded supports. In order to$u( r’)=\frac{1}{2\pi}\int_{>a}\log\frac{1}{|r’-r|}p(r)dr$

,

(4.6)

we shall examine the integral transform(4.7)$u( r’)=\frac{1}{2\pi}\int_{>a}\log\frac{1}{|r’-r|}p_{1}(r)\frac{1}{r^{3}}dr$

for the functions$\rho_{1}=r^{3}p$satisfying

.

(4.8)

$\int_{>a}\rho 1(r)^{2}\frac{1}{r^{3}}dr=\int_{>a}\rho(r)^{2}r^{3}dr<\infty$

Then, the corresponding reproducing kemelas follows:$K_{o,a}^{(2)}(r’, r’’)$can be calculated, using (4.4),$K_{o,a}^{(2)}( r’, r’’)=\frac{1}{2\pi}\int_{>a}\log\frac{1}{|r’-r|}\log\frac{11}{|r’’-r|r^{3}}dr$$= \frac{1}{a}\{2+(\log\frac{1}{a})^{2}-2\log\frac{1}{a}\}$$+ \frac{1}{2}\sum_{n=1}^{\infty}\frac{r^{\prime n}r^{\prime\prime n}}{(2n+1)n^{2}a^{2n+1}}(\cos n\varphi’\cos n\varphi’’+sinn\varphi’\sin n\varphi’’)$

.
(4.9)

Hence, we have

Theorem 4.2. The source functions
satisfying (4.6) and (4.8) with the minimum
norms are expressible in the form, for any fixed $\hat{b}(0<\hat{b}<a)$$\rho^{*}$$\rho^{*}(r_{1})=\frac{1}{r_{1}^{3}}(u(r),$$\log\frac{1}{|r-r_{1}|})_{H_{\kappa_{:}^{(2)}}},a$$= \frac{1}{r_{1}^{3}}[\{\frac{1}{a}(2+(\log\frac{1}{a})^{2}-2\log\frac{1}{a})\}^{-1}$$x(\frac{1}{2\pi}\int_{0}^{2\pi}u(\hat{b}, \varphi)d\varphi)\log\frac{1}{r_{1}}$$+ \frac{2}{\pi}\sum_{n=0}^{\infty}\frac{n(2n+1)a^{2n+1}}{\hat{b}^{n}r_{1}^{n}}\int_{0}^{2\pi}u(\hat{b}, \varphi)\cos n(\varphi-\varphi_{1})d\varphi$

28

5.

$\mathbb{R}^{1}$CASE

We shall consider the one dimensional potential in the form$u(x)= \int_{-a}^{a}-\frac{1}{2}|x-t|\rho(t)dt,$

for the source functions$x>a>0$(5.1)

satisfying$\rho$$\int_{-a}^{a}\rho(t)^{2}dt<\infty$

.

(5.2)

In order to examine the integral transform (5.1) with (5.2), we form the reproducing

kemel$K_{f}(x, y)= \frac{1}{4}\int_{-a}^{a}|x-t||y-t|dt=\frac{a}{2}(xy+\frac{1}{3}a^{2})$

for

$x,$

$y>a$ .

(5.3)

admitting the reproducing kernelThis expression means that the Hilbert space$K_{f}(x, y)$ is two dimensional, and any members $f(x)$ ofare expressible in the form
$H_{K_{r}}$$H_{K_{r}}$$f(x)= \frac{a}{2}(c_{1}x+c_{2}\frac{1}{3}a^{2})$for some constants$c_{1}$

and$c_{2}$

for $x>a$

(5.4)

.

(5.5)

, and furthermore$||f||_{H_{K_{\Gamma}}}^{2}= \frac{a}{2}(c_{1}^{2}+c_{2}^{2}\frac{1}{3}a^{2})$andare determined by any two point values, say $f(b_{1})$ and $f(b_{2})$The constants$b_{2}>a$ as follows:for$c_{1}$$c_{2}$$b_{1},$(5.6)$c_{1}= \frac{2}{a}\frac{f(b_{1})-f(b_{2})}{b_{1}-b_{2}}$and$c_{2}= \frac{6}{a^{3}}\frac{b_{1}f(b_{2})-b_{2}f(b_{1})}{b_{1}-b_{2}}$

.

(5.7)

We thus have
Theorem 5.1. In the potential (5.1) satisfying (5.2), the source functions
with the$u(b_{2})(b_{1},b_{2}>a,b_{1}\neqb_{2})$
andminimum norms in (5.2) are expressible in terms ofin the form$\rho^{*}$$u(b_{1})$$\rho^{*}(t)=(u(x), -\frac|x-t|)_{H_{K,}}\overline{2}1$$= \frac{3(b_{1}u(b_{2})-b_{2}u(b_{1}))}{a^{3}(b_{1}-b_{2})}t-\frac{u(b_{1})-u(b_{2})}{a(b_{1}-b_{2})}$.

29

In order to determine source functions with unbounded supports we shall examine
the integral transform, for $a>0,$ $x<a$$u(x)= \int_{a}^{\infty}-\frac{1}{2}|x-t|\rho(t)dt$(5.8)$= \int_{a}^{\infty}-\frac{1}{2}|x-t|\rho_{1}(t)\frac{1}{t^{4}}dt$

for the source functions$\rho=\rho_{1}t^{-4}$satisfying$\int_{a}^{\infty}p_{1}(t)^{2}\frac{dt}{t^{4}}=\int_{a}^{\infty}p(t)^{2}t^{4}dt<\infty$

.

(5.9)

We form the corresponding reproducing kemel$K_{1}(x, y)= \int_{a}^{\infty}-\frac{1}{2}|x-t|\cdot-\frac{1}{2}|y-t|\frac{dt}{t^{4}}$$= \frac{1}{12a^{3}}(x-\frac{3}{2}a)(y-\frac{3}{2}a)+\frac{1}{16a}$.

(5.10)

Hence, by the parallel arguments to Theorem 5.1, we have

Theorem 5,2. In the potential in (5.8) satisfying (5.9), the source functions
withthe minimum norms in (5.9) are expressible in terms ofandfor any fixed$b_{2}<a$ as follows:
two points$u$$\rho^{*}$$u(b_{1})$$u(b_{2})$$b_{1},$$\rho^{*}(t)=\frac{1}{t^{4}}(u(x), -\frac{1}{2}|x-t|)_{H_{K_{l}}}$$= \frac{1}{t^{4}}[\frac{8a}{b_{1}-b_{2}}\{(b_{2}u(b_{1})-b_{1}u(b_{2}))-\frac{3}{2}a(u(b_{1})-u(b_{2}))\}t$$+ \frac{12a^{2}}{b_{1}-b_{2}}\{(b_{1}u(b_{2})-b_{2}u(b_{1}))+2a(u(b_{1})-u(b_{2}))\}]$.

ACKNOWLEDGMENTS
The author wishes to express his sincere thanks to Professor Victor Isakov for his very
valuable comments for the harmonicity ofin Theorem 2.1 at the 2nd Intemational(Inverse
Symposium
Problems in Engineering Sciences” (Osaka, July 27-30, 1994) to theold version. For the harmonicity of , Professor Fumi-Yuki Maeda kindly taught theauthor a more general proof using Weyl’s lemma at the Meeting of “Integral Transforms
and their Applications”at Kyoto University (August 25-26, 1994). The research was$p^{*}$$p^{*}$

30

partially supported by the Japanese Ministry of Education, Science and Culture; Grantin-Aid General Research (C) 06640201.
REFERENCES
1. V.Isakov, “Inverse Source Problems,” Mathematical Surveys and Monographs Number 34, American Mathematical Society, 1990.
2. P.M.Morse and H.Feshbach, (Methods of Theoretical Physics,” McGraw-Hill Book
Company Inc., New York Toronto London Tokyo, 1953.
3. S. Saitoh, Hilbert spaces induced by Hilbert space valued functions, Proc. Amer.
Math. Soc. 89 (1983), 74-78.
4. S. Saitoh, “Theory of Reproducing Kemels and its Applications,” Longman Scientific &Technical, Harlow, Essex, 1988.