数理解析研究所講究録1116 巻 1999 年 45-50

45

A generalization of the Liouville theorem to polyharmonicfunctions
外村俊英 (Toshihide Futamura)
岸恭子 (Kyoko Kishi)
水田義弘 (Yoshihiro Mizuta)

広島大学理学研究科
広島大学理学研究科
広島大学総合科学部

1

Introdu.ctionLetbe the -dimensional Euclidean space with a point $x=(X_{1}, X_{2}, \ldots, X)n$ . Fora multi-index, we set$\mathrm{R}^{n}$$n$$\lambda=$$(\lambda_{1}, \lambda_{2}, \ldots , \lambda_{n})$$|\lambda|=\lambda_{1}+\lambda_{2}+\cdots+\lambda_{n}$,$x^{\lambda}=x1^{\lambda_{1}}x2^{\lambda_{2}}\cdots xn^{\lambda_{n}}$and$D^{\lambda}=( \frac{\partial}{\partial x_{1}})^{\lambda_{1}}(\frac{\partial}{\partial x_{2}})^{\lambda_{2}}\cdots(\frac{\partial}{\partial x_{n}})^{\lambda_{n}}$We denote by
the open ball centered at the origin with radius $r>0$ , whoseboundary is denoted by $rS^{n-1}$ .
A real valued function is called polyharmonic of order $m$ on
if $u\in C^{2m}$and $\Delta^{m}u=0$ , where $m$ is a positive integer,denotes the Laplacian and. We denote by
the space of polyharmonic functions of order $m$
on . In particular, is harmonic onif.
The Liouville theorem for polyharmonic functions is known in several forms(cf. [1, 3, 4]).
$rB^{n}$$u$$\mathrm{R}^{n}$$\triangle$$\Delta^{m-1}(\triangle u)$$\mathrm{R}^{n}$$u$$\mathrm{R}^{n}$$u\in H^{1}(\mathrm{R}^{n})$THEOREM A. Let
and $s>2(m-1)$ . Thendegree less than if one of the following conditions holds:$u\in H^{m}(\mathrm{R}^{n})$

$s$

(i)

$\lim_{rarrow\infty}\frac{1}{r^{s+n-1}}\int_{rS^{n-1}}u^{+}dS=0$(see [1]);(ii)$\lim_{rarrow\infty}\frac{1}{r^{s+n}}\int_{rB^{n}}u^{+}dx=0$(see [3]);$\lim_{rarrow}\sup_{\infty}(\max_{x\in rS^{n-1}}\frac{u(x)}{|x|^{s}})\leq 0$(see [4]).

(iii)$\triangle^{m}u=$$H^{m}(\mathrm{R}^{n})$Now we propose the following theorem.

$u$is a polynomial of46THEOREM. Let
degree at most if and only if$u\in H^{m}(\mathrm{R}^{n})$and $s>2(m-1)$ . Then$u$

is a polynomial of$s$(1).

$\lim_{rarrow}\inf_{\infty}\frac{1}{r^{s+n-1}}\int_{rS^{n-1}}u^{+}dS<\infty$

We here note that each condition of Theorem A implies (1), so that our theorem
gives an improvement of Theorem A.

The main lemmas

2Let us begin with the following lemma, which expands polyharmonic functions toharmonic functions (cf. [2], [5]).
LEMMA 1 (THE FINITE ALMANSI EXPANSION). A real valued function onsuch, then there exists a uniquebelongs toth at(2)$u(x)= \sum_{i=1}^{m}|x|^{2(}i-1)h_{i}(X)$$u$$f\mathrm{a}\mathrm{m}ily\{h_{i}\}_{i=}^{m_{1}}\subset H^{1}(R^{n})$$H^{m}(\mathrm{R}^{n})$$\mathrm{R}^{n}$for very$e$$x\in \mathrm{R}^{n}$.

We prove this lemma by induction on $m$ . For $m=1$ the conclusion. Thenis trivial. Suppose the conclusion is true for $m=k$ , and letsuch thatthere exists a family
PROOF.

$u\in H^{k+1}(\mathrm{R}^{n})$$\{g_{i}\}_{i=1}^{k}\subset H^{1}(R^{n})$$\triangle u=\sum_{i=1}k|x|^{2()}i-1gi(x)$If a family$\{h_{i}\}_{i=1}^{k1}+\subset H^{1}(R^{n})$.

(3)satisfies (2), then we should have$\triangle u=\sum_{i=1}^{k+1}\Delta(|x|2(i-1)h_{i}(x))$$=$$\sum_{1i=}’\Delta(|X|2ih_{i}+1(x))k$.

If we write $r=|x|$ , then$\triangle(|X|^{2i}h_{i}+1(x))$$= \sum_{j=1}^{n}(\frac{\partial}{\partial x_{j}}\mathrm{I}^{2}(r^{2i}h_{i+}1(x))$$= \sum_{j=1}^{n}\{\frac{\partial^{2}r^{2i}}{\partial x_{j}^{2}}h_{i+1}(X)+2\frac{\partial r^{2i}}{\partial x_{j}}\frac{\partial h_{i+1}(_{X)}}{\partial x_{j}}+r2i_{\frac{\partial^{2}h_{i+1}(_{X)}}{\partial x_{j}^{2}}\}}$

47

$=$

$2ir^{2(i-1)} \{(2i-2+n)h_{i+}1(x)+2r\frac{\partial h_{i+1}}{\partial r}(X)\}$$=$$|x|^{2(i}-1) \{2i(2i-2+n)h_{i+}1(x)+4ir\frac{\partial h_{i+1}}{\partial r}(x)\}$

From the uniqueness of Almansi expansion forfind a unique solution
for the equation$\triangle u$.

, it is necessary and sufficient to$h_{i+1}$(4)$g_{i}(x)=2i(2i-2+n)h_{i+1}.(x)+4ir \frac{\partial h_{i+1}}{\partial r}(x)$for each $i=1,$$\cdots,$$k$. We see that the unique solution for (4) is given by$h_{i+1}(x)= \frac{1}{4ir^{i-1/2}+n}\int_{0}^{r}t^{i2n}-+/_{gi}2(tX/r)dt$

is harmonic on
Here we have only to show that
where $r=|x|$ and $\zeta=x/|x|=x/r$ , we have$h_{i+1}$$h_{i+1}(_{X)}$Since$g_{i}$$\mathrm{R}^{n}$.

. Actually, putt.ing $x=r\zeta$ ,$=$$h_{i+1}(r\zeta)$$=$$\frac{r^{1-i-n/2}}{4i}I_{0}^{r}t^{i-}n/2g_{i}(2+t\zeta)dt$$=$$\frac{r^{1-i-n/2}}{4i}\int_{0}^{1}(rS)^{i2+/}-n2g_{i}(rs\zeta)rdS$$=$$\frac{1}{4i}\int_{0}^{1}s^{i-2+/2}g_{i}(Xs)dns$.

is harmonic, we see that $\triangle h_{i+1}(X)=0$ for $i=1,$$h_{1}(x)=u(x)- \sum_{i=2}^{k+}1|x|^{2()}i-1hi(x)$$\cdots,$$k$. Now put.

Then $\triangle h_{1}(x)=0$ by (3), and the induction is completed.

Next we prepare the following lemma, which gives a relation between sphericalmeans and derivatives for harmonic functions.

LEMMA 2. Supposecons tant $C=C(\lambda)$ such that$u\in H^{1}(\mathrm{R}^{n})$. For each$m$ulti-index , there exists a positive$\lambda$(5)$\int_{rS^{n-1}}ux^{\lambda}dS=Cr^{2|\lambda|1\lambda}+n-Du(0)+P_{2|\lambda|+}-3(n)r$for every $r>0$ , where$P_{k}(r)$is a polynomial of degree at most$k$

depends on$u$.

PROOF. We prove this lemma by induction on the length of . Assume firstthat
and $\lambda_{i}=0(i=1, \ldots, n-1)$ . Using Green’s formula and the meanvalue property for harmonic functions, we have$\lambda$$\lambda_{n}=1$48$\int_{rS^{n-1}}ux^{\lambda}dS$$=$$\int_{rS^{n-1}}ux_{n}dS$$=$$r \int_{rS^{n-1}}u\frac{x_{n}}{r}dS$$=$$r \int_{rB^{n}}\frac{\partial u}{\partial x_{n}}dx$$=$$\sigma_{n}r^{n+1}\frac{\partial u}{\partial x_{n}}(0)$,.
where
is the -dimensional volume of the unit ball. Hence (5) holds for, where
Next suppose that (5) holds foris a positive integer. Let
such that $|\mu|=k+1$ . We may assume without loss of generality, and set $\mu’=(\mu_{1}, \ldots, \mu_{n-}1, \mu_{n}-1)$ . Then we writethat$|\lambda|=1$$n$$\sigma_{n}$$k$$|\lambda|\leq k$$\mu=(\mu_{1}, \ldots, \mu_{n})$$\mu_{n}\geq 2$$\int_{rS^{n-1}}ux^{\mu}dS=r\int_{rS^{n-1}}ux^{\mu_{\frac{x_{n}}{r}}’}dS$.

From Green’s formula we obtain$\int_{rS^{n-1}}ux^{\mu}dS$$=$$r \int_{rB^{n}}\frac{\partial(ux^{\mu})\prime}{\partial x_{n}}dx$/$=$

Set$r \int_{rB^{n}}(x^{\mu_{\frac{\partial u}{\partial x_{n}}}’}+(\mu_{n}-1)uX_{1}\mu_{1}$$\mu^{\prime/}=(\mu_{1}, \ldots, \mu_{n-1}, \mu_{n}-2)$

$(*)$

,..

$x_{n}^{\mu_{n^{-2}})}dx=(*)$.

. Since $|\mu’|=k$ and $|\mu^{\prime/}|=k-1$ , we find$=$$r \int_{0}^{r}(\int_{tS^{n-1}}(x^{\mu’}\frac{\partial u}{\partial x_{n}}+(\mu_{n}-1)uX^{\mu}\mathrm{I}\prime\prime dS)dt$$=$$r \int_{0}^{r}(\int_{tS^{n-1}}\frac{\partial u}{\partial x_{n}}xd\mu\prime s)dt+(\mu_{n}-1)r\int^{r}0(\int_{tS^{n-1}}ux^{\mu}dS)\prime\prime dt$$=$$r \int_{0}^{r}(C(\mu)/t+n-1D2|\mu|’\mu’(\frac{\partial u}{\partial x_{n}})(0)+P_{2|\mu’|n-}+3(t))dt$$+( \mu_{n}-1)r\int_{0}^{r}(C(\mu^{\prime/})t2|\mu\prime\prime|+n-1D\mu’\prime u(0)+P_{2|\mu’’}|+n-3(t))dt$$=$$C(\mu)r^{2}Dk+n+1\mu u(0)+P_{2k+n-}1(r)$,where $C( \mu)=\frac{C(\mu’)}{2k+n}>0$ anddenotes various polynomials of degree at mostwhich may change fron one occurrence to the next; throughout this note, we use
this convention. Hence (5) also holds for $|\mu|=k+1$ . The induction is completed.
$P_{\ell}$$\ell$

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3

Proof of the theorem

First we show that our theorem is valid under the two sided condition on sphericalmeans for polyharmonic functions.

LEMMA 3. Let
degree at most if$u\in H^{m}(\mathrm{R}^{n})$and $s>2(m-1)$ . Then$u$

is a polynomial of$s$$\lim \mathrm{i}\mathrm{n}\mathrm{f}rarrow\infty\frac{1}{r^{s+n-1}}\int_{rS^{n-1}}|u|dS<\infty$

PROOF. By (6) we can find a sequence$\{r_{j}\}_{j=1}^{\infty}$.

(6)such that$\sup_{j}(r_{j}^{-s-n}\int_{rS}+1jn-1|u|dS)<\infty$$r_{j}arrow\infty$and.

(7)Using (2) and Lemma 2, we have$\int_{rS^{n-1}}ux^{\lambda}dS$$=$$\int_{rS^{n-1}}(\sum_{i=1}^{m}|x|^{2(i}-1)h_{i}(x))X^{\lambda}dS$$=$$\sum_{i=1}^{m}r^{2(1)}-\int irS^{n}-1h_{i}(X)x^{\lambda}ds$$=$$\sum_{i=1}^{m}r-1)(2(ic_{i^{\Gamma^{2}}}|\lambda|+n-1D\lambda h_{i}(0)+P_{i},2|\lambda|+n-3(r))$,

where $Ci=C_{i}(\lambda)$ is a positive constant anddenotes various polynomials ofdegree at most depends on . Hence it follows that
$P_{i,k}$$k$$h_{i}$$r^{|\lambda|} \int_{r}S^{n-1}|u|dS\geq|_{i=1}\sum^{m}r-1)(c_{i}r-1D^{\lambda}2|\lambda|+nhi(0)+P_{i},2|\lambda|+n-3(r))2(i|$,so that we obtain$r_{j}^{-\mathit{8}-n}+1 \int_{r_{j}S^{n-1}}|u|dS$as$r_{j}arrow\infty$$\geq$$r_{j}-s+2(m-1)||\lambda|c_{m}D^{\lambda}h_{m}(0)+o(rj)-2|$. By (7), we find$D^{\lambda}h_{m}(0)=0$

for all $|\lambda|>s-2(m-1)$ . By analyticity of harmonic functions, we see thata polynomial ofat most $s-2(m-1)$ . Hence we note that$h_{m}$$\mathrm{d}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{e}\mathrm{e}$$r^{2(m-1)} \int_{rS^{n-1}}h_{m}(X)_{Xd}\lambda s=O(r^{s+})|\lambda|+n-1$

as$rarrow\infty$.

Consequently,$r_{j}^{-S-n}+1 \int_{r_{j}S^{n-1}}|u|ds\geq rj||\lambda|-s+2(m-2)C_{m}-1D^{\lambda}hm-1(0)+o(r_{j}^{-})2|+o(1)$is50. This implies thatfor $|\lambda|>s-2(m-2)$ , so thatis a polynomial of degree at most $s-2(m-2)$ . By repeating this arguments,
we see that each is a polynomial of degree at most $s-2(i-1)(i=1, \ldots, m)$ .
Thus it follows that is a polynomial. In view of (2), the degree of is at most2 $(i-1)+s-2(i-1)=s$ .

as$r_{j}arrow\infty$$D^{\lambda}h_{m-1}(0)=0$$h_{m-1}$$h_{i}$$u$$u$PROOF OF THE THEOREM. If$u\in H^{m}(\mathrm{R}^{n})$, then we see from (2) that$\frac{1}{\omega_{n}r^{n-1}}\int_{rS^{n-1}}udS=\sum^{m}i=1r^{2()}i-1h_{i}(\mathrm{o})$denotes the surface measure ofSince $|u|=2u^{+}-u$ , we havewhere$\omega_{n}$$S^{n-1}$,.

$\lim_{rarrow}\inf_{\infty}r^{-}s-n+1\int_{rS^{n-1}}|u|dS$$=$$\lim \mathrm{i}\mathrm{n}\mathrm{f}rarrow\infty(2r^{-s-n+1}\int_{rS^{n-1}}u^{+}dS-r^{-s-}n+1\int_{rS^{n-1}}udS)$$=$$\lim \mathrm{i}\mathrm{n}\mathrm{f}rarrow\infty(2r^{-s-}n+1\int_{rS^{n-1}}u^{+}dS-r^{-s_{P_{2}}}(m-1)(r))$.

Hence (1) implies (6) since $s>2(m-1)$ , so that the present theorem follows fromLemma 3.

References[1] D. H. Armitage, A polyharmonic generalization of a theorem on harmonic functions, J. London Math. Soc. (2) 7 (1993), 251-258.

[2] N. Aronszajn, T. M. Creese, and L. J. Lipkin, Polyharmonic functions, Clarendon Press, 1983.
[3] Y. Mizuta, An integral representation and fine limits at infinity for functions
whose Laplacians iterated m times are measures, Hiroshima Math. J. 27 (1997),415-427.
[4] M. Nakai and T. Tada, A form of classical Liouville theorem for polyharmonic
functions, to appear in Hiroshima Math J.
[5] M. Nicolesco, Recherches sur les fonctions polyharmoniques, Ann. Sci. EcoleNorm Sup. 52 (1935), 183-220.