A generalization of BoÃcher's theorem for polyharmonic functions

31 (2001), 59±70

A generalization of BoÃcher's theorem for polyharmonic functions

Dedicated to Professor Maretsugu Yamasaki on the occasion of his sixtieth birthday

Toshihide Futamura, Kyoko Kishi and Yoshihiro Mizuta

(Received May 9, 2000)

Abstract. In this paper we generalize BoÃcher's theorem for polyharmonic functions u. In fact, if u is polyharmonic outside the origin and satis®es a certain integral condition, then it is shown that u is written as the sum of partial derivatives of the fundamental solution and a polyharmonic function near the origin.

1. Introduction

Let Rⁿ be the n-dimensional Euclidean space with points xˆ …x₁;x₂;. . .;x_n†. For a multi-index lˆ …l₁;l₂;. . .;l_n†, we set

jlj ˆl₁‡l₂‡ ‡l_n; x^lˆx₁^l1x₂^l2. . .x_n^ln and

D^lˆ q qx1

_l1 q qx2

_l2 q

qxn

_ln :

We denote by B…x;r† the open ball centered at x with radius r>0, whose boundary is written as S…x;r† ˆqB…x;r†. We also denote by B the unit ball B…0;1† and by B0 the punctured unit ball B f0g.

A real valued function u on an open set GHRⁿ is called polyharmonic of ordermonGifuAC^2m…G†andD^muˆ0 onG, wheremis a positive integer,D denotes the Laplacian and D^muˆD^{m 1}…Du†. We denote by H^m…G† the space of polyharmonic functions of ordermon G. In particular,u is harmonic onG if uAH¹…G†. The fundamental solution of D^m is written as Km, that is,

2000 Mathematics Subject Classi®cation. Primary 31B30

Key words and phrases. polyharmonic functions, BoÃcher's theorem, the fundamental solution of polyharmonic operator, Green's formula, mean value property

Km…x† ˆam jxj^{2m n}log…1=jxj† if 2m nis an even nonnegative integer, jxj^{2m n} otherwise,

(

where the constant am is chosen such that D^mKm is the Dirac measure dat the origin.

Our aim of this paper is to prove the following theorem.

Theorem. If uAH^m…B₀† satis®es …

B0

u…x†^‡jxj^sdx<y …1†

for some integer sV0, then u is of the form

uˆ X

jmjUs‡2m 1

c…m†D^mK_m‡h

for some hAH^m…B†, where c…m† are constants and u^‡…x† ˆmaxfu…x†;0g.

We shall show that u in the theorem satis®es …

B0

ju…x†jjxj^sdx<y: …2†

Hence in case sU0, u is integrable on B. In cases>0, we shall show thatu de®nes a distribution T_u such that

hT_u;viˆlim

r!0

…

B B…0;r†u…x†v…x†dx for vAC₀^y…B†:

Armitage [1] treated the case where uAH^m…B₀† satis®es 1

o_nr^{n 1} …

S…0;r†ju…x†jdS…x† ˆo…r ^s0† as r!0; …3†

where o_n denotes the surface area of S…0;1†. If s⁰<s‡n, then (3) clearly implies (1).

As an easy consequence of the theorem we have the following result due to Ishikawa-Nakai-Tada [5]:

Corollary 1. If u is a harmonic function on B₀ such that lim sup

x!0 u…x†jxj^{n 1}U0;

then u is of the form

uˆcK1‡h for some hAH¹…B† and a constant c.

As another application of our theorem, we obtain the following result.

Corollary 2. If uAH^m…B₀† satis®es (2) for some integer s, then u is of the form

uˆ X

jmjUs‡2m 1

c…m†D^mKm‡h for some hAH^m…B†, where c…m† are constants.

As to the behavior at in®nity, we refer the reader to the recent papers Kishi-Futamura-Mizuta [4] and Nakai-Tada [7].

2. Lemmas

In this section we prepare some lemmas, which will be used in the proof of the theorem.

Lemma 1. If uAH^m…B0†, then 1

o_nr^{n 1} …

S…0;r†u…x†dSˆX^m

kˆ1

fakr^{2…1 k†}Km…r† ‡bkr^{2…m‡1 k† n}‡ckr^{2…m k†}g …4†

for all rA…0;1†, where fa_kg, fb_kg, fc_kg are constants and a_k ˆ0 when k>

m‡1 n 2.

Proof. We prove this lemma by induction onm. In the case that mˆ1, this is known; see e.g. [3, Lemma 3.10]. So we suppose that (4) holds formˆl, and takeuAH^l‡1…B₀†. ThenDuAH^l…B₀†becauseD^l…Du† ˆD^l‡1uˆ0, so that

…

S…0;t†Du dSˆo_nX^l

kˆ1

fa_kt^{2…1 k†‡n 1}K_l…t† ‡b_kt^{2…l‡1 k† 1}‡c_kt^{2…l k†‡n 1}g for 0<t<1, where a_kˆ0 when k>l‡1 n

2. We integrate this equality with respect to t from r1 to r2, where 0<r1 <r2<1, and obtain

…

fx:r1<jxj<r2gDu dxˆ

…_r2

r1

…

S…0;t†Du dS

! dt

ˆon

X^l

kˆ1

…_r2

r1

fakt^{2…1 k†‡n 1}Kl…t† ‡bkt^{2…l‡1 k† 1}‡ckt^{2…l k†‡n 1}gdt

ˆX^l

kˆ1

fa_k⁰r₂^{2…1 k†‡n}K_l…r₂† ‡b_k⁰r₂^{2…l‡1 k†}‡c_k⁰r₂^{2…l k†‡n}g X^l

kˆ1

fa_k⁰r₁^{2…1 k†‡n}K_l…r₁† ‡b_k⁰r₁^{2…l‡1 k†}‡c_k⁰r₁^{2…l k†‡n}g …5†

where a_k⁰ ˆ0 when k>l‡1 n

2. On the other hands, we have by Green's formula

…

fx:r<jxj<r2gDu dxˆc…r₂† …

S…0;r†

x

r`u…x†dS…x†

ˆc…r₂† r^{n 1} …

S…0;1†z`u…rz†dS…z†

ˆc…r2† r^{n 1} d dr

…

S…0;1†u…rz†dS…z†

!

ˆc…r2† r^{n 1} d dr

1 r^{n 1}

…

S…0;r†u dS

!

; where 0<r<r₂ and c…r₂† ˆ„

S…0;r2†

x

r2`u…x†dS…x†. Hence (5) leads to X^l

kˆ1

fa_k⁰r^{2…1 k†‡1}K_l…r† ‡b_k⁰r^{2…l‡1 k†‡1 n}‡c_k⁰r^{2…l k†‡1}g ‡d⁰r^{1 n}

ˆd dr

1 r^{n 1}

…

S…0;r†u dS

!

for 0<r<r2. We integrate both sides with respect to r from r1 to r2 to obtain

1 r₁^{n 1}

…

S…0;r1†u dSˆX^l

kˆ1

fa_k⁰⁰r₁^{2…1 k†‡2}Kl…r† ‡b_k⁰⁰r₁^{2…l‡1 k† n‡2}‡c_k⁰⁰r₁^{2…l k†‡2}g

‡b⁰⁰logr1 d⁰⁰r₁^{2 n}‡d⁰⁰⁰logr1‡e

ˆX^l‡1

kˆ1

fa_k⁰⁰r₁^{2…1 k†}Kl‡1…r1† ‡b_k⁰⁰r^2…l‡2₁ ^{k† n}‡c_k⁰⁰r₁^{2…l‡1 k†}g;

where 0<r1<r2<1, a_k⁰⁰, b_k⁰⁰, c_k⁰⁰ are determined such that a_k⁰⁰ˆ0 when k>l‡2 n

2, b⁰⁰ˆb_k⁰ when kˆl‡2 n

2 and d⁰⁰⁰ ˆd⁰ when nˆ2. But we see that the constantsa_k⁰⁰, b_k⁰⁰, c_k⁰⁰ are determined independently of r2. Now the induction is completed.

With the aid of Lemma 1, we prove

Lemma 2. If uAH^m…B0† and l is a multi-index, then

1 o_nr^{n 1}

…

S…0;r†ux^ldS

ˆ^m‡jljX

kˆ1

fAkr^{2…1‡jlj k†}Km…r† ‡Bkr^{2…m‡1‡jlj k† n}‡Ckr^{2…m‡jlj k†}g …6†

for 0<r<1, where A_k, B_k, C_k are constants and A_k ˆ0 when k>m‡1 n 2. Proof. We prove this by induction on the lengthjlj. First we note from Lemma 1 that the conclusion is true for jlj ˆ0.

Now let jlj ˆl‡1 and write lˆl⁰‡e_j, where jl⁰j ˆl and je_jj ˆ1. By the Gauss-Green formula we have

…

S…0;r1†ux^l0…x_j=r₁†dSˆ …

fx:r1<jxj<r2g

q

qxju…x†x^l0 dx

‡ …

S…0;r2†ux^l0…x_j=r₂†dS: …7†

Using the assumption on induction, we have …

S…0;r†

qu

qxj…x†x^l0dS

ˆ^m‡jlX^0j

kˆ1

fAkr^{2…1‡jl0j k†‡n 1}Km…r† ‡Bkr^{2…m‡1‡jl0j k† 1}‡Ckr^{2…m‡jl0j k†‡n 1}g and

…

S…0;r†u…x† q qxjx^l0dS

ˆ^m‡jlX^{0j 1}

kˆ1

fA_k⁰r^{2…jl0j k†‡n 1}K_m…r† ‡B_k⁰r^{2…m‡jl0j k† 1}‡C_k⁰r^{2…m‡jl0j 1 k†‡n 1}g;

where AkˆA_k⁰ ˆ0 when k>m‡1 n

2. As in the proof of Lemma 1, using polar coordinates in (7), we have the required conclusion for l withjlj ˆl‡1.

Lemma 3. If uAH^m…B₀† satis®es (2), then the limit limr!0

…

fx:r<jxj<1guv dx exists for every vAC₀^y…B†. Further, the mapping

Tu:v7!lim

r!0

…

fx:r<jxj<1guv dx

de®nes a distribution on B. In what follows we identify u with Tu. Proof. We write

…

fx:r<jxj<1guv dxˆ …

B B…0;r†u v X

jmjUL

x^m m!D^mv…0†

0

@

1 Adx

‡ X

jmjUL

D^mv…0†

m!

…

B B…0;r†ux^mdxˆI…r† ‡J…r†;

where L is an integer such that LVs 1. Since v P

jmjULx^m

m!D^mv…0† ˆ O…jxj^L‡1†; we have

jI…r†jU …

B B…0;r†juj v X

jmjUL

x^m

m!D^mv…0†

dxUC …

B B…0;r†jujjxj^L‡1 dx;

so that limr!0I…r† exists and is ®nite by (2).

In view of Lemma 2, we see that limr!0

…

fx:r<jxj<1gu…x†x^mdxˆlim

r!0

…₁

r

…

S…0;t†u…x†x^mdS…x†

! dt

exists and is ®nite; this limit is denoted by C…m†. Hence J…r† converges to X

jmjUL

D^mv…0†

m! C…m†

as r!0. Therefore hu;vi 1lim

r!0

…

fx:r<jxj<1guv dx

ˆ …

B0

u v X

jmjUL

x^m m!D^mv…0†

0

@

1

Adx‡ X

jmjUL

C…m†D^mv…0†

m!

is de®ned to be ®nite. Clearly, u is a distribution on B.

3. The proof of the main theorem

In this section we give a proof of the theorem.

(I) We ®rst prove the theorem under the strong condition (2). We recall Green's formula for D^m (see e.g. [2], [8]):

…

B B…0;r†…uD^mv vD^mu†dx

ˆ X^m

iˆ1

…

S…0;r† …D^{i 1}u†q…D^{m i}v†

qn …D^{m i}v†q…D^{i 1}u†

qn

dS …8†

foruAC^y…B₀†, vAC₀^y…B†and 0<r<1, where q=qn denotes the inner normal derivative. If uAH^m…B0† and vAC₀^y…B†, then we obtain

…

B B…0;r†uD^mv dxˆXⁿ

jˆ1

X

jlj‡jmjˆ2m 1

C…l;m;j†

…

S…0;r†D^luD^mvxj

r dS 8<

:

9=

; with constants C…l;m;j†. For simplicity, we put c…x† ˆD^mv…x†, and use its Taylor expansion

c…x† ˆ X

jnjUl

xⁿ

n!Dⁿc…0† ‡Rl‡1…x†;

where lˆs‡ jlj. Then we have …

S…0;r†D^luD^mvx_j r dS

ˆ …

S…0;r†D^lu…x†Rl‡1…x†x_j r dS‡1

r X

jnjUl

1

n!Dⁿc…0†

…

S…0;r†D^lu…x†xⁿxjdS

ˆI…c† ‡J…c†:

To evaluate I…c†, we need the following lemma, which is an easy con- sequence of [6, Lemma 8.4.5].

Lemma 4. If uAH^m…B₀† and 0<r<2=3, then …

S…0;r†jD^lujdSUCr ^{jlj 1} …

fx:r=2<jxj<3r=2gjujdx with a positive constant C.

From Lemma 4 it follows that

jI…c†j ˆ …

S…0;r†D^lu…x†Rl‡1…x†xj

r dS UCr^l‡1

…

S…0;r†jD^lujdS UC⁰r^{l‡1 …jlj‡1† s}

…

fx:0<jxj<2rgju…x†jjxj^sdx;

so that I…c† tends to zero as r!0, since lˆs‡ jlj.

On the other hand, we see from Lemma 2 that J…c† ˆ1

r X

jnjUl

1

n!Dⁿc…0†

…

S…0;r†D^lu…x†xⁿxjdS

ˆ1 r

X

jnjUl

1

n!Dⁿc…0† X^m‡jnj‡1

kˆ1

…Ak…l;n;j†r^{2…2‡jnj k†‡n 1}Km…r†

‡Bk…l;n;j†r^{2…m‡2‡jnj k† 1}‡Ck…l;n;j†r^{2…m‡1‡jnj k†‡n 1}†

! X

jnjUl

C⁰…l;n;j†Dⁿc…0† as r!0;

since Ak…l;n;j† ˆ0 when k>m‡1 n

2. Hence it follows that hu;D^mviˆlim

r!0

…

fx:r<jxj<1guD^mv dx

ˆ X

jl‡mjˆ2m 1;jnjUs‡jlj

C⁰⁰…l;n†D^m‡nv…0†

ˆ X

jljUs‡2m 1

C⁰⁰⁰…l†D^lv…0†: …9†

Finally let us ®nd constants c…m† for which u P

jmjUs‡2m 1c…m†D^mK_m is polyharmonic of order m. For this purpose, we have by (9)

D^m u X

jmjUs‡2m 1

c…m†D^mK_m 0

@

1 A;v

* +

ˆ u X

jmjUs‡2m 1

c…m†D^mK_m;D^mv

* +

ˆhu;D^mvi X

jmjUs‡2m 1

c…m†hD^mK_m;D^mvi

ˆ X

jljUs‡2m 1

C⁰⁰⁰…l†D^lv…0† X

jmjUs‡2m 1

c…m†hD^m…D^mK_m†;vi

ˆ X

jljUs‡2m 1

C⁰⁰⁰…l†D^lv…0† X

jmjUs‡2m 1

c…m†hD^md;vi

ˆ X

jljUs‡2m 1

C⁰⁰⁰…l†D^lv…0† X

jmjUs‡2m 1

c…m†… 1†^jmjhd;D^mvi

ˆ X

jmjUs‡2m 1

C⁰⁰⁰…m†D^mv…0† X

jmjUs‡2m 1

c…m†… 1†^jmjD^mv…0†:

Hence if we take c…m† ˆ … 1†^jmjC⁰⁰⁰…m†, then

D^m u X

jmjUs‡2m 1

c…m†D^mKm

0

@

1 Aˆ0 as required.

(II) Now we assume that (1) holds for uAH^m…B₀†. Using polar coordinates and Lemma 1, we have

…

B B…0;r†u…x†jxj^sdx

ˆ …₁

r t^s …

S…0;t†u…x†dS

! dt

ˆ …₁

r ont^{s‡n 1}X^m

kˆ1

fakt^{2…1 k†}Km…t† ‡bkt^{2…m‡1 k† n}‡ckt^{2…m k†}gdt

ˆX^m

kˆ1

fa_k⁰r^{2…1 k†‡s‡n}Km…r† ‡b_k⁰r^{2…m‡1 k†‡s}‡c_k⁰r^{2…m k†‡s‡n}g ‡d;

where a_k⁰ ˆ0 for k>m‡1 n

2. Hence „

B B…0;r†u…x†jxj^sdx is bounded for 0<r<1. Therefore (1) implies (2). In view of the ®rst half of the proof,uis of the form

uˆ X

jmjUs‡2m 1

c…m†D^mK_m‡h

for some hAH^m…B†. The proof of the theorem is now completed.

4. Proofs of Corollaries 1 and 2

In this section we give proofs of Corollaries 1 and 2.

Proof of Corollary 1. Assume that u is a harmonic function on B₀ such that

u…x†Uo…jxj ^n‡1† as x!0: …10†

Then we have

…

B0

u…x†^‡dx<y;

so that (1) holds for sˆ0. Hence our therem shows that u is of the form

uˆ X

jmjU1

c…m†D^mK₁‡h

for some hAH¹…B† and some constants c…m†. In view of (10), we see that c…m† ˆ0 when jmj ˆ1, which proves the corollary.

Proof of Corollary 2. Assume that uAH^m…B0† satis®es (2) for an integer s. Then

…

B0

ju…x†jjxj^s0dx<y

for a nonnegative integer s⁰Vs. Hence it follows from our theorem that u is the form

uˆ X

jmjUs⁰‡2m 1

c…m†D^mKm‡h

for some hAH^m…B†. According to (2), P

s‡2m 1<jmjUs⁰‡2m 1c…m†D^mKm should disappear, which completes the proof of Corollary 2.

5. Remark

Remark 1. If uAH^m…B0†, then u is expressed as Laurent series ex- pansion:

u…x† ˆX

m

c…m†D^mKm…x† ‡h…x† …hAH^m…B††

(cf. [3, Chapter 10]).

To show this, ®x xAB0 and ®nd from (8) (Green's formula for D^m)

0ˆ …

B…0;r2† B…0;r1† B…x;r†…u…y†D^mKm…x y† Km…x y†D^mu…y††dy

ˆX^m

iˆ1

…

S…0;r2†

…D^{i 1}u…y††q…D^{m i}K_m…x y††

qny

…D^{m i}Km…x y††q…D^{i 1}u…y††

qny

dS…y†

X^m

iˆ1

…

S…0;r1†

…D^{i 1}u…y††q…D^{m i}Km…x y††

qn_y

…D^{m i}K_m…x y††q…D^{i 1}u…y††

qny

dS…y†

X^m

iˆ1

…

S…x;r†

…D^{i 1}u…y††q…D^{m i}Km…x y††

qny

…D^{m i}Km…x y††q…D^{i 1}u…y††

qn_y

dS…y†

ˆa…r2;x† b…r1;x† g…r;x†

for 0<r₁<jxj<r₂ and r>0 with B…x;r†HB…0;r₂† B…0;r₁†. Note that limr!0g…r;x† ˆcu…x† for some constant c and limr2!1a…r2;x†AH^m…B†. Fur- ther, using the Taylor expansion for Km and Lemma 2, we have

rlim1!0b…r1;x† ˆX

m

c…m†D^mKm…x†;

where c…m† are constans. Hence we see that u…x† ˆX

m

c⁰…m†D^mK_m…x† ‡h…x† …hAH^m…B††:

Our aim in this paper has been to ®nd conditions which assure that the series in the above expression contains only ®nite terms.

Open problem. Under the weaker condition that lim inf

r!0 r ^{s n‡1} …

S…0;r†u…x†^‡dS<y

instead of (1), we do not know whether uAH^m…B0† is a polynomial or not.

References

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[ 2 ] N. Aronszajn, T. M. Creese and L. J. Lipkin, Polyharmonic functions, Clarendon Press, 1983.

[ 3 ] S. Axler, P. Bourdon and W. Ramey, Harmonic function theory, Springer-Verlag, 1992.

[ 4 ] T. Futamura, K. Kishi and Y. Mizuta, A generalization of the Liouville theorem to polyharmonic functions, to appear in J. Math. Soc. Japan 53 (2001).

[ 5 ] Y. Ishikawa, M. Nakai and T. Tada, A form of classical Picard principle, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 6±7.

[ 6 ] Y. Mizuta, Potential theory in Euclidean spaces, GakkoÅtosho, Tokyo, 1996.

[ 7 ] M. Nakai and T. Tada, A form of classical Liouville theorem for polyharmonic func- tions, Hiroshima Math. J. 30 (2000), 205±213.

[ 8 ] M. Nicolesco, Recherches sur les fonctions polyharmoniques, Ann. Sci. EÂcole Norm Sup.

52 (1935), 183±220.

Toshihide Futamura and Kyoko Kishi Department of Mathematics Graduate Schoolof Science

Hiroshima University Higashi-Hiroshima 739-8526, Japan T. Futamura: toshi@mis.hiroshima-u.ac.jp

K. Kishi: kyo@mis.hiroshima-u.ac.jp Yoshihiro Mizuta

The Division of Mathematicaland Information Sciences Faculty of Integrated Arts and Sciences

Hiroshima University Higashi-Hiroshima 739-8521, Japan

mizuta@mis.hiroshima-u.ac.jp