Construction of the fundamental solution of the equation Ари(Х) + Ы( Х) = 0

A N N A L E S SO C IE TA T IS M A TH EM ATICAE P O LO N AE Series I : COMMENTA T IO N ES M A TH EM A T IC AE

M. Filar (Kraków)

Construction of the fundamental solution of the equation Ари(Х) + Ы( Х) = 0

1. The purpose of this paper is a generalization of certain results of the paper [2] concerning the construction of a fundamental solution of the equation A2u(X)-\-ku{X) — 0. We shall now deal with the equa­

tions

(1) Apu ( X ) - c 2pu{X) = 0,

(Г ) Apu(X)-\-c2pu (X ) = 0,

where X = {xl f . . . , xn), n > 2, p is an integer greater than 1, and c is a positive constant. In Section 2 we consider solutions of (1) which depend only on the distance r = [(aq — 2/

i

)2+ ••• + (# » —yn)2]1/2 between X and Y = (yi , yn). In Section 3 we give the definition of a fundamental solution, and the following sections concern the fundamental solutions grouped according to' the cases we single out.

2. We shall first be concerned with the solutions of the equation (1) which are of the form u( X) = U{r). Equation (1) may be written in the form

( A — k0)(A — k\) ... (A — Tcp_i)u(X) = 0,

where icj (j — 0,1, . . . , p — 1) are the roots of the equation lcp—c2p = 0, / 2

tt

j \

i.e., kj =

c

2€

j

, where % = e x p (--- i\ are the roots of order p of the

\ P I unity.

Lemma 1.

Each solution of class C2p of the equation (2) { А - Ц ) и { Х ) = 0 (j = 0 , l , . . . , p - l ) is a solution of (1). ,

P r o o f. We shall prove this by induction with respect to p. It follows from (2) that

АЫ A{Au) === 4{kjU) — kj-Au = k2 Au .

1 3 2 M . F i l a r

Let us suppose that Aku — kku for 2 < b < p. Then A k+1u = A(Aku) = A {t fu) = к) Au = t f +1u.

Thus, if k —p, we have Apu = kf и = c2psfu = c2vu in virtue of kj = c2^ , and this concludes the proof.

It is known that each function u( X) = V(r) satisfying equation (

2

) must also fulfil the equation

лi 1

V'\r) + --- 7 '( r ) - b F ( r ) =

0

. r

Substituting a new independent variable

(3) ^e = ^{- t / Ę r} - c r e x v ^ t z i h . ij

we obtain an equation of the form

w"(e)+ 1~~2((!r ”- )/.2I тг'(е)+Ж(е) = о,

Q

which is a particular case of the equation (see [

1

])

u"(z)- 1 —2 a

z * '( * ) + W

y

# - x)' u(z) —

0

whose integrals are of the form u(z) = zaZv(ftzv), Zv(z) being an arbitrary Bessel function of order v. In our case a — ( 2—n)f2, z' =

q

, у — (3 — 1, v = — a — ( n —

2

)/

2

; if n is odd, i.e., if v — ( n —

2)/2

is not an integer, then

(4) «,<*) = </-.(*) = ^ fc= о

( - 1

)k(z/2)-p+2k Г { к + 1 ) Г ( - v + b + 1 ) У

0

< 1^1 < oo, |arg

0

| < те. If n is even, then v = (n—2)J2 is an integer and therefore

Zv(z) = Y,(z) =

*=o ' ’

1

у

( - 1

)\z!2)’ +tk

+ % Z

j

h\(r+le)\

²

In— — y>{k ~f"l) ~~~ip{k ~f~ v-j-

1

)^,

where 0 < \z\ < 1, |arga| < те. In the case v = 0 we can admit freely

that the first sum in the above formula equal to zero.

In view of (3), (4), and Lemma 1, solutions of (1) for n odd are equal to

{ —W ( —i/itfr) = ( - 2 e , ) - V

and also to the functions

w

,<r) = 2 k=0

e^ crft)2^

—p -|-7s -|-1)

_ Г ( к + 1 ) Г ( - у + к + 1 )

(j = 0 ,1, 1)

(j = 0 ,1 , . . . , p —l)

which are equal to the above solutions up to a constant factor. Each linear combination of those solutions is a solution too, so p - parameter solution depending on constants a0, % , ap_ x is of the form

-v) P -1

where bk — У це*.

(5) V(r) = V ---,

Z j r(lc+l)r(-v+lc+l)' ^

Similarly to the case where n is even, the solutions are of the form

< - * / v r r , < - i ^ ) = P +

0 * '

P 00

2^

+ — У , П ГТТ^\2Ы^Г -V > ( * + l) -? '( H -» + l) + 2 « a r g (- iI ^ ')],

•л:

Kl(v-{-k)l L

2 J

(j == 0 ,1 , . . . , p — 1), and also of the form

+ V £i-+ł’^ / 2)2 k i n — —yi(fc-bl)—y(fe-hvH-l)},

^ 7C Z ; fc!(r+T;)! \ 2 ^ r T

(j = 0,1, 1). The functions Vj(r) are solutions of (1) because they are linear combinations of the solutions

Щ(г) = У and { - i V Ę r ) Tv{-i\/Ę r) k\{v-\-k)\

0 = o, l ,

13 4 M . F i l a r

Thus, a p-parameter solution depending on the constants a0, ap_ x is of the form

1 v- j j. (v—k^{— 1 ) !} Ic Ą W -')

(

6

) Y(r) = - ~ 2

j

(

t

- I ) V ----

77—

^ - l - l +

fc

=0

Jc\

oo —

( —

1

)” X'ibkicr 12)2к { cr

1

|21n¥ - T ( * + D - r » + » + 1)).

P - l

where

i=o

v-i

Ł V " 1 к T V I f c + V

О/ę — у t B j Clj

,

O k — у

f

^{B j Щ}

.

1 = 0

In the case v — 0, the first sum in (

6

) is equal to zero.

3. We assume the following definition of a fundamental solution:

Definition 1.

A

function

U

(r) is called a

fundamental solution

of the equation (1) [(1'), respectively] in a hounded region D if (i) U(r) =

TJ{\X— Y|)

is defined and of class

C2p

in the set {(

X , Y): X , YeD, X Ф Y},

(ii) if

X Ф Y,

then the function

U(r),

as a function of

X

or

Y

alone, satisfies the equation (1) [(1')] in D, (iii) for each function

W( X)

of class

C2p

in

D

and of class

C2P~

1 in D, satisfying the equation

(1) [(l')]

in

D

and for each

X

in

D,

we have

p - i

(7) W ( X ) =

Q,

n г

iss

= 0 F(D)

Al U

d A ^ i - i w

dn AlW dAv^ ~ l V

dn I dS

_{Y >}

Qn being the surface of the n - dimensional unit sphere. -

4. We shall first construct the fundamental solution of equation (1) in the case where n is odd. This solution will be of the form (5), with aj properly chosen. The constants af are determined so that bk = 0 {1c =

0

,

1

, . . . , p —

2

), bp_ x = b, where

=

2

- « + ' с - ™ ^ Г ( Р) Г ( - у + р ) Ip- , - i r

1

( 2 - « ) [ ( p - l ) ! ? I P - l I Then, because of (5),

P - I P - l

(9) ^ а 3-вк = 0 (к = 0 ,1 , JT ayef

*1

= b.

1=0 1=~

0

This is a system of p linear equations with p unknows, its deter­

minant (being a Vandermonde determinant) is different from zero, since

Bj are different roots of the unity. Thus, if we choose the solution of (9)

as ay, then in view of the identities bsp+i = Ьг, 1 = 0,1 , . . . , p ~ 1; s = 0 ,1, 2 , the solution (5) of (1) takes on the form

( 10 ) ^ (cr/2)2(pfc-,,-1)

U(r) = b > — — ---,

where 6 is given by (8).

T

heorem

1. In the ease where n is odd, n > 3, and

2

) is an integer satisfying p > 2, the fundamental solution of (1) & Йе function given by formula (10).

P r o o f. We have U(r) =

(?)

2

(p—

^{v -}1

)

where

H{r) = V — — Г ( р ) Г ( - г + р )

&(cr/2)2(pfc- ”- I) Г(р1е)Г(—г-\-рк)

+ Я (г),

O ^ 225- ' - ! ) )

(the symbol 0(f{r)) refers to the case r -> 0). It can be verified that

(11) A*U(r) = b(cl2)W— " / p _ v_ i \ / p _ i \

2zl J 1j ^ i 1J ( i !) V (p-*’-1 -i)-f-

4 -0(r2(2p~”-1_i)) (i = 0 ,1, . . . , p - l )

(12) — (

Al U

{

r

) ) =» - ---— 22i+1p . 1) ( р 1 V)(i!)2( ^ - v - l - i ) . d r 1 M . Г < р ) Г ( - г + ;р ) ~ \ M l * /

x r 2(p- ,' - 1- i)~1+ 0 ( r 2(2p" ,'~1_<)“ 1) (t = 0 ,1 , . . . , 2 > - l ) .

X

Let W (Y ), where Г = (?/2, . . . , y w), be a function of class C2P in D and of class C2p_1 in D and let it satisfy equation (1) in D. Let K R be the ball with the centre X and radius R, K R c D. Applying, in D —E R, the fundamental formula of [3] to the function U(r) given by (10) and to the function W { Y ) , we obtain

SSS {UApW - W A pU)dV

d- Kr

p - i

— 2 SS{ 1 A1 TJ —— =— — - A * W - — r— — I dSr.

d ^ - t - x j j

<=* о F ( D - K R )

dn dn

13 6 M . F i l a r

Since the function U(r) and W ( Y) satisfy equation (

1

), the left-hand side of the above equation is equal to zero. Thus

p- 1

г п л р / ,

d A p-*~1W

,

dAp-i-1U \

(13) 2

j o

S \ A U --- - --- A W --- ---

{ « 0 F(D)

dn dn

P-i

- S S S { AlW

d A V -^ jj

dr Al V d A ^ - 'W ' dr / ' >dSv.

i=0 F(KR)

Applying the mean value theorem to the surface integrals we obtain

(14)

Jn = S S d<V

F(KR)

<^ = S S 4 < r

F{KR)

dAv- l- lW _ _ , ____

dr dSY = QnR^-1A%U{R) dAp~i~1U

dr dSY = Q ^ ' J W i Q x )

dr

d A ^ - ' U i R ) dr

where Qn€F(KR), Q2ie F ( K R) (i = 0 ,1 , p — 1). Taking into account (

8

), (

1 1

), (

1 2

), we obtain

j u = QnRn- 1— P~ - = o ( ^ p- ^ -i)_ _ > o,

0

< % < 29—1,

■hi = Й . Г , Л ( Ы О | ] !* <- , н ) = 0(,R2i) — t 0, iv— ► O 1 < * < p - 1 , J20 = QnRn- 1W{Q2,) l R - n+1^ 0 { R 2v- n+3)1

= O n W iQ n n i+ O iR 2™ ) ] - ^ QnW{X), hence, by (13), we obtain (7). In virtue of Definition 1 and of regularity of the function TJ (r), we infer that the fundamental solution is the func­

tion JJ(r) defined by (

1 0

) and (

8

).

5. We now pass to the construction of the fundamental solution of equation (

1

') in the case where n is odd.

Theoeem 2.

The function

V{r) = - b 2

к- 1

( - l ) fc(cr/

2

)s(pfc- ,- 1>

Г{рк)Г{ — v-j-pk) ’

where Ъ is given by (

8

), is a fundamental solution of (

1

') in the case where

n is odd.

P r o o f. Since replacing of c by cv i converts equation (1) into (1')

P Г

a solution of (1') is given by

00

U{P Vir) = (Vi)~nb J j?

fc-i

( - l ) fc(cr/2)a(pfc- ,’“ 1>

Г(рк)Г(~г-\-рк) p / 7zi \

where Vi = exp I— ) and U{r) is given by (10). A solution of (1') will

\2PI also be equal to

F(r) i--- ( V i ) " V ( ń r ) , and the proof is analogous to that of Theorem. 1.

6 .

Next, we shall deal with equation (1) in the case where n is even.

As we already know, the function defined by (6) is a solution of (1). We shall obtain the fundamental solution by an appropriate choice of the constants aj (j = 0,1 , . . . , p — 1). We single out two cases: v v < p, where v

=

(n —

2 ) / 2 .

Theorem

3. In the case where n is even and v = (n —2)/2 the fundamental solution of (1) is given by

(15) U(r) = V * * 4

n Z j ( i p - 1)! \ 2 /

⁺

fc=l

( —1)” \ i b k(crl2)zk -1 У y h i c r p T I TC k \( v+k) l{

fc—0

cr

k i ( v + m \ 2^ T ~ v { k + 1 ) - v ( k + v + 1 ) !•

where

5 = 0 or b — d,

d = ( —l ) v+1nc~2p+n2n~2^{n—2)(v—p)l(p —1)! l. •

P r o o f . We choose u;-in formula (6) so as bk = О (Тс = 0 ,1 , . . . , p — 2), bp-l = d. Then

p-i

^p

-

i

(16) ejdj = 0 (к = 0 ,1 ,.

*-0 /-0

An argument analogous to that in Section 4 leads to the conclusion that

there exists precisely one solution of the system (16). When щ are chosen

so as to satisfy (16), then the function defined by (6) takes on the form

138 M . F i l a r

(15). It can shown by induction that

_|_

0

(r2(2p- ’’“ 1- <)),

X (г!)2(p —1 —v —i)r

2(p_1~’’- i ) ~ 1

-|-0(r2(2p_’'_1_i)_1) Ц = 0 ,1, 1).

Since the function U(r) is regular, it is enough to check the condition (iii) of Definition 1. By an argument similar to that in the proof of The­

orem 1, we obtain (13) and (14). Then, by the above formulas,

J u QnRn~l dAP 1 0 ( R 2p- 2i~n) = 0 {R 2p~2i- 1)

dr

⁰

,

0

< i < p —

1

, J 2i = QMs r lAiW (Q « )0 {R r '* x+«) = 0 ( R 2i) ^ 0, 1 < i < p - 1 , J

20

= ^ E n-

1

F (Q

2

i) [ r B+4 0 ( ^ - n+1)] i ^ OnW (X y,

hence, by (13), we obtain the formula (7). t "

Theorem 4.

I f n is even and v = ( n—2)f2 < p, then the fundamental solution of equation (

1

) is given by formula

(17) U(r) = ( - D 7 v («•/

2

)2(te- ’ - ,) 71

— v—1)! (ftp

—

1)!

2 1

n --- ip(kp—v) —y)(kp) cr ]•

where

<17')- f = j u i » —2)П Г 2’

P r o o f. We choose u,- in formula (

6

) so as bk — 0 (к = 0 , 1 , . . . , v—1),

_ _

p

-

i

bk = 0 (к = 0,1, p —v —2), bv_ v_i — f. Then we obtain ^fejaj = 0

- #=o

(fc =

5 £#+4

=

0

(& = o, i , . . . , p —v—2), 2 4 ~ ' « / = f -

? = 0v 3=0

This is again a system of p linear equations with p unknowns % with the

determinant different from zero. If % are chosen so as to satisfy this system

of equations, then-because of the identities Ъ8Р+г — Ъг (I —

0

,

1

, . . . . , p —

1

,

0 , 1 , 2 ,. . .) the solution (

6

) of (1) takes on the form (17). If U{r)

is given by (17), then we can prove by induction that (-1 )У с 2(р- ' - 1>

A'U(r) =

tz (p—v^{— 1 ) !} (p ^{— 1 ) !}

2 ~ 2 ( p — l ) + l + 2 f

X (i!)2r2(p-,' - l- <)ln — + A ir2<p- v- 1- H O ( r 2<2p- v- 1- <>-1), cr 2

Ap- v+iU(r) — 1

- f f i (--- — (

P 1 \ [ ( p _ v _ i ) ! ]

2

X

v ; Tz(p—v—l ) \ { p —l ) \ \ p —v—l J lKF ^{' J}

1+1

fc-i

where i — 0 ,1 , . . . , p — v—1, is constant and j = 0 , 1 , v— 1. If j = v— 1, then

Av~l V{r) —2/c2(p" *'—1) [(тг—2) !!]2 Ц ( п - 2 ) /2)1 ( n~2)

r-n+2 _(_(}(r2(p-^-l^

Differentiating both sides of this formula with respect to r and taking (17') into account, we get

d

dr [АЯ-'Щг)] = r~n+l + 0 ( r 2(p~'-1)) and the proof follows the lines of that of Theorem 3.

7. In this section we shall consider equation (!') in the case where n is even.

Theorem

5. I f n is even and v

=

( n—2)/2

^

p , then the function (18) V(r) = T L e [-{V tfv+2U { \ T ir )],...

where U(r) is defined by (15), is a fundamental solution of (1').

P r o o f . The substitution of c v i instead of c changes equation (1) into p,- (1'); thus, in the case v = ( n - 2)/2 ^ p , the function JJ{rVi) is a solu­

tion of (I'), where U{r) is given by formula (15), and

и ( h r ) =

fc=l

C[pfp]

( - l ) klp+1)( v - hp ) l {kp —1) i

' cr\ 2(P*— 1)

j) +

| ( - 1 Г y i М У У

—ip (&-(-!) —y (fc + r + l)+ 2 7 a r

140 M . F i l a r

where Vi = exp V Г . A solution of (1') will also be equal to (18). The proof is similar to that of Theorem 3.

Th e o r e m 6 .

I f n is even and v — (n —

2 ) /2

< p, then the fundamental solution of equation ( ! ') is given by formula

( 19 ) 7 (f)

S S

( - 1 Г 1/

TC 1 ( — l ) k(crl2)4kp- v-V Г or

(kp — v—l)\(1cp—

1

)! L П

2

—y>(Jcp~v)—y>(kp) ]•

P r o o f . In the case where v < p the function u f y ir ) is a solution of (1'), where U(r) is defined by (17). Thus,

= Ы > 1 flV fl-ч н -ч V ( - i ) t ^ /2 ) ł(tg- ,- 1)

тс

j

L

i

(ftp—» —l)!(ftp —

1

)!

[

er

P/-1

2 In —---y > ( k p — v) — y > { k p ) - \ - 2 i & r g v i \ .

The function (19) is a solution of (1') as being a linear combination of the solutions

U(Vir)

F(r) у ( ~ i ) (kp—v

l ) k(crl2)4kp~v~1)

g j ( l c p - v - l ) l ( J c p - l ) l and u {V i).

An argument analogous to that used in the proof of Theorem 4 gives Theorem

6

.

References

[1] N . N. L e b i e d i e w , Funkcje specjalne i ich zastosowania, Warszawa 1957.

[2] J. M u s ia łe k , Construction o f the fundamental solution fo r the equation A * u { X ) + k u ( X ) = 0, Prace Mat. 9 (1965), pp. 2 13-23 6.

[3] M. N i c o l e s c o , L es fonctions polyharmoniques, Paris 1936.