数理解析研究所講究録 1211 巻 2001 年 112-119

112
Existence of singular solutions with bounds of linear partial
differential equations in the complex domain(上智大学)

忠大内Sunao \={O}UCHI (Sophia Univ.)
Introduction\S 0

In this paper we consider alinear partial differential equation in the comis an $m$-th linear partial, $L(z, \partial)u(z)=f(z)$ .
plex domain in
differential operator with coefficients are holomorphic in aneighborhood
The inhomogeneous term $f(z)$ has singularities on acomof $z=0$ in
plex hypersurface $K$ . The author reported the results concerning the growthproperties and the asymptotic behaviors of solutions, and those concerning the existence of solutions with asymptotic expansion, when $f(z)$ has an
asymptotic expansion, at the conference held here, RIMS of Kyoto Univ.(see$[5],[6],[7]$ and [8] . In the present paper our concern is the existence of s0lutions, when $f(z)$ has not necessary asymptotic expansion, but the singularities are tempered, that is, singularities are of the fractional order. The
details will be given elsewhere.
$\mathbb{C}^{d+1}$$L(z, \partial_{z})$$U$$\mathbb{C}^{d+1}$$)$

Notations and Definitions\S 1

In order to state our problem and results more precisely, let us introduce
notations, function spaces and characteristic polygon.
. $|z|=$$\max\{|z_{i}|;0\leq i\leq d\}$ and $|z’|= \max\{|z_{i}|;1\leq i\leq d\}$ . Its dual variables are
.
, and
.
is the set of all integers and is the set of all nonnegative integers. For a. For apolydisk,
multi-index a$U=U_{0}\cross U’$ in, where $U_{0}=\{z_{0}\in \mathbb{C};|z_{0}|<R_{0}\}$ and $U’=\{z\in \mathbb{C};|z’|<$, set Uo(6) $=\{z_{0}\in U_{0}-\{0\};|\arg z_{0}|<\theta\}$ and $U(\theta)=U_{0}(\theta)\cross U’$ .
is asectorial region with respect to . $K$ is acomplex hypersurface through$z=0$ in . We choose the coordinate so that $\{z\in U;z_{0}=0\}$

1.1.

Notations.

$z=$$\xi=(\xi_{0}, \xi’)=(\xi_{0},\xi_{1}, \cdots, \xi_{d})$$(z_{0}, z_{1}, \cdots, z_{d})=(z_{0}, z’)\in \mathbb{C}\cross \mathbb{C}^{d}$$\partial_{i}=\partial/\partial z_{i}$$\partial=(\partial_{0}, \partial_{1}, \cdots, \partial_{d})=(\partial_{0}, \partial’)$$\mathrm{N}$$\mathbb{Z}$$=(\alpha_{0}, \alpha’)\in \mathrm{N}\cross \mathrm{N}^{d}$$| \alpha|=\alpha_{0}+|\alpha’|=\sum_{i=0}^{d}\alpha_{i}$$\mathbb{C}^{d+1}$$U(\theta)$$R\}$$z_{0}$$U$-mail s-0huti@hoffman.cc.sophia.ac.jp
KEY WORDS: complex partial differential equations, asymptotic expansion, existence of;Sec1991 Mathematical Subject Classification(s): Primary
singular solutions.
ondary$\mathrm{e}$$35\mathrm{A}20$$35\mathrm{A}07,35\mathrm{G}20$113
1.2. Function spaces.
holomorphic functions on U.

For aregion U in C, $O(U$ is the set of all

Definition 1.1. (i).
that for any 0’ with $0<\theta’<\theta$$\mathcal{O}_{temp,c}(U(\theta))$(1.1)

is the set of all $u(z)\in \mathcal{O}(U(\theta))$ such$|u(z)|\leq C|z_{0}|^{c}$holds for a constant $C=C(\theta’)$ .
(ii).

$\mathcal{O}_{temp}(U(\theta))=\bigcup_{c\in \mathbb{R}}\mathcal{O}_{temp,c}(U(\theta))$$z\in\Omega(\theta’)$. We say thattempered singular, or regular singular, on $K$ in$U(\theta)$$u(z)\in \mathcal{O}_{temp}(U(\theta))$.

Definition 1.2. $A_{\{\kappa\}}(U(\theta))(0<\kappa \leq+\infty)$ is the set of all
such that for any 0’ with $0<\theta’<\theta$(1.2)$| \partial_{0}^{N}u(z)|\underline{<}AB^{N}\Gamma(N(1+\frac{1}{\kappa})+1)$

holds for all$n\in \mathrm{N}$

is

for

$u(z)\in \mathcal{O}(U(\theta))$$z\in U(\theta’)$and for some constants $A=A(\theta’)$ and $B$ $=B(\theta’)$ .

$u(z)\in A_{\{+\infty\}}(U(\theta))$means that $u(z)$ is holomorphic at $z=0$ .
in the preceding papers $[5],[7]$ and [8], whichwith asymptotic expansion of Gevrey type,$A_{\{\kappa\}}(U(\theta))$

is coincident with $Asy\{\kappa\}(U(\theta))$consists of all $u(z)\in \mathcal{O}(U(\theta))$that is, for any 0’ with $0<\theta’<\theta$

(1.3)

for

$|u(z)- \sum_{n=0}^{N-1}u_{n}(z’)z_{0}^{n}|\leq AB^{N}|z_{0}|^{N}\Gamma(\frac{N}{\kappa}+1)$$z\in U(\theta’)$,$=B(\theta’)$ . The notation
where $u_{n}(z’)\in \mathcal{O}(U’)(n\in \mathrm{N})$ , $A=A$ (?’) and$u(z)\sim 0$ inmeans that $u_{n}(z’)\equiv 0$ for all in (1.3).
$B$$A\{\kappa\}(U(\theta))$$n$1.3. Characteristic polygon. Let
be an $m$-th order linear
partial differential operator with holomorphic coefficients in aneighborhoodof $z=0$ ,$L(z, \partial)$(1.4)$L(z, \partial)=\sum_{|\alpha|\leq m}a_{\alpha}(z)\partial^{\alpha}$.

We introduce the characteristic polygon ofwith respect to hypersurface $K=\{z_{0}=0\}$ , which is indispensable for our purpose, to study theexistence of solutions with bounds. Let us introduce anotation $\lrcorner(a, b):=$$\{(x, y)\in \mathbb{R}^{2} ; x\leq a, y\geq b\}$ , which means an infinite
rectangle. Let
be the$L(z, \partial)$$j_{\alpha}$

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valuation ofwith 6.(0,with respect to zO) that is, if $a.(z)$ S 0,
for
0. Definef 0 and set$\ovalbox{\tt\small REJECT}.(\ovalbox{\tt\small REJECT})$$\mathrm{z}’)$where$e_{L,\alpha}=j_{\alpha}-\alpha_{0}$$e_{L,\alpha}=+\infty$if$a_{\alpha}(z)\equiv 0$,.

The characteristic polygon of$\sum:=the$$z\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}’ b.(z)$$a.(z)\ovalbox{\tt\small REJECT}$$\circ \mathrm{p}$$\ovalbox{\tt\small REJECT} 7_{\mathrm{D}}\ovalbox{\tt\small REJECT}$(1.5)$\mathrm{a}\mathrm{a}(\mathrm{z})\ovalbox{\tt\small REJECT}$$\Sigma$is defined byconvex hull of$\cup\lrcorner(|\alpha|, e_{L,\alpha})\alpha$.

The boundary of
consists of avertical half line
and ahorizontal$p^{*}-1$$\Sigma(i)(1\leqi\leqp^{*}-1)$
half lineandsegments
with slope ,$\Sigma$$\Sigma(0)$$\Sigma(p^{*})$$\gamma_{i}$$0=\gamma_{p^{\mathrm{r}}}<\gamma_{p^{*}-1}<\cdots<\gamma_{1}<\gamma_{0}=+\infty$.

Let $\{(m_{i}, e(i))\in \mathbb{R}^{2} ; 0\leq i\leq p^{*}-1\}$ be vertices of , where $0\leq m_{p^{*}-1}<$$\ldots<m_{i}<m:-1<\cdots<m_{0}=m$ . So the endpoints of $\Sigma(i)(1\leq i\leq p^{*}-1)$
are $(m_{i-1}, e(i-1))$ and $(m_{i}, e(i))$ . We call the slope of
the i-thcharacteristic index ofwith respect to $K=\{z_{0}=0\}$ .
$\Sigma$$\gamma_{i}$$\Sigma(i)$$L(z, \partial)$$\rfloor(m,e(0))\Sigma(0)$$\{$$1$$(m,e(0))\Sigma(0)$$|(|1)$$\Sigma$,‘$’\supset(1))$$(|\alpha|, e_{L,\alpha})\nearrow_{\Sigma(2)}^{(m}1$$\mathrm{o}$$\circ$$(m_{2}, e(2))$$\nearrow(/m_{i}, e(i))$$\Sigma(p^{*}-1)\nearrow_{/_{m}}$$\Sigma(p^{*})$$(m_{p^{*}-2},e(p^{*}-2))-\Delta/-^{*}-\rceil\backslash \backslash$.
$\backslash \prime\prime bp^{\mathrm{s}}-1,$$\circ\backslash P-[perp]//$Figure 1: Characteristic polygon

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Let

$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}(\mathrm{i})$be asubset of multi-indices and$l_{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}$

E N(0

i

$\ovalbox{\tt\small REJECT}$p’–l) defined$\ovalbox{\tt\small REJECT}$

by

(1.6)

$\{\begin{array}{l}\Delta(i)..=\{\alpha\in \mathrm{N}^{d+1}\cdot.|\alpha|=m_{i},e_{L,\alpha}=e(i)\}l_{i}\cdot.=\max\{|\alpha’|..\alpha\in\Delta(i)\}\end{array}$Define asubset$p^{*}-1)$ by

(1.7)

of

$\triangle(i)$and apolynomial$\chi_{L,i}(z’, \xi’)$

in$\xi’$$(0\leq i\leq$$\{\begin{array}{l}\triangle_{0}(i)=\{\alpha\in\triangle(i)\cdot,|\alpha’|=l_{i}\}\chi_{L,i}(z’,\xi,)=\sum_{\alpha\in\Delta_{0}(i)}b_{\alpha}(0,z’)\xi^{\alpha’}\end{array}$$\chi_{L,i}(Z_{)}’\xi’)$\S 2$\triangle_{0}(i)$is homogeneous in 4’ with degree$l_{i}$.

Existence of singular solutions

Let us return to the equation(Eq)$L(z, \partial)u(z)=f(z)\in \mathcal{O}(U(\theta))$.

The existence of singular solutions are studied by [2], [4], [10] and otherpapers referred in these papers. More generally we have

Theorem 2.1. Suppose that(?) ) of (Eq) for some V$\mathcal{O}(V$$\chi_{L,0}(0, \xi’)\not\equiv 0$. Then there is a solution $u(z)\in$.

$\subset U$

In this paper we consider the case $f(z)$ has tempered singularities on $K$ .
We have asolution $u(z)$ by Theorem 2.1, but $u(z)$ has not always tempered
singularities. We can generally show $|u(z)|\leq A\exp(c|z_{0}|^{-\gamma_{1}})$ for $z\in V(\theta’)$$(0<\theta’<\theta)$ . So our interest is to find asolutionofthe equation$u(z)\in \mathcal{O}_{temp,c’}(V(\theta^{l}))$$L(z, \partial)u(z)=f(z)\in \mathcal{O}_{temp,c}(U(\theta)))$for some polydisc $V\subset U$ and constants$c’$

and $0<\theta’<\theta$ .

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Let us give conditions $(C_{i})(0\leq i\leq p^{*}-1)$ . For fixed , $0\leq i\leq p^{*}-2$
.
and
for
$i$

$(C_{i})$$j_{\alpha}=0$$\alpha\in\triangle \mathrm{o}(i)$$\chi_{L,i}(0, \xi’)\not\equiv 0$

For $i=p^{*}-1$$(C_{p^{*}-1})$$0$$|\alpha’|\leq l_{p^{*}-1}$.

for$ce\in\{\alpha\in \mathrm{N}^{d+1},\cdot e_{L,\alpha}=e(p^{*}-1)\}$and$\chi_{L,p^{*}-1}(0, \xi’)\not\equiv$

Our main existence theorem ishold for all $0\leq i\leq p^{*}-1$ . Let
Theorem 2.2. Suppose $p^{*}\geq 2$ andand be a constant with $0< \theta’<\min\{\theta, \pi/2\gamma_{1}\}$ . Then
there is a solutionof (Eq) for some polydisc andconstant .
$(C_{i})$$f(z)\in \mathcal{O}_{temp,c}(U(\theta))$$\theta’$$V$$u(z)\in \mathcal{O}_{temp,c’}(V(\theta’))$

$a$

$c’$

We note that the opening angle 0’ of sectorial region is restricted by
We need two theorems in order to show Theorem 2.2. One is$\gamma_{1}$.

Theorem 2.3. Suppose $p^{*}\geq 2$ andsatisfies $(C_{p-1}*)$ . Let $f(z)\in$
and 0’ be a constant with $0< \theta’<\min\{\theta, \pi/2\gamma_{p-1}*\}$ . Then
there is afor some polydisc and a constant such.
that (Rf)(z) $:=(L(z, \partial)v(z)-f(z))\sim \mathrm{O}$ in$L(z, \partial)$$\mathcal{O}_{temp,c}(U(\theta))$$V$$v(z)\in \mathcal{O}_{temp,c},(V(\theta’))$$c’$$A_{\{\gamma_{\mathrm{p}^{*}-1}\}(V(\theta’))}$The other isandTheorem 2.4. Suppose
satisfies
for $i=0,1$ , p’. Then for any $0< \theta’<\min\{\theta, \pi/2\gamma_{1}\}$ there
2and letsatisfying $L(z, \partial)u(z)=f(z)$ inisfor some
polydisc .
$p^{*}\geq 2$$L(z, \partial)$$(C_{i})$$\cdots$$f(z)\in A_{\{\gamma_{p^{*}-1}\}}(U(\theta))$$V(\theta’)$$u(z)\in A_{\{\gamma_{\mathrm{p}^{*}-1}\}}(V(\theta’))$$V$

Theorem 2.4 is given in [8] and [9], where we considered the existence ofsolutions with asymptotic expansion under the condition that $f(z)$ in (Eq)
has an asymptotic expansion. We exclude $p^{*}=1$ in the preceding theorems,
however, we have from results in [4]Theorem 2.5. Suppose $p^{*}=1$ andThen there is a solution
and a constant$(C_{0})$.
holds. Letof (Eq) for some polydisc$f(z)\in \mathcal{O}_{temp,c}(U(\theta))$$V$$u(z)\in \mathcal{O}_{temp,c’}(V(\theta))$$c’$

The operators of Fuchsian type (see [1]) satisfy the conditions of Theorem

2.5.
Example. Let(2.1)$L(z, \partial)=\partial_{1}^{5}+A_{1}(z)\partial_{1}^{3}\partial_{0}+A_{2}(z)\partial_{0}^{2}$,$z=(z_{0}, z_{1})\in \mathbb{C}^{2}$,

117, $B_{i}(0)\neq 0$ for $i=1,2$ . According to thewhere $A_{i}(z)=z_{0}^{j_{i}}B_{i}(z)$ ,values of and , several cases occur. However the conditions in Theorem
2.2 or the conditions in Theorem 2.5 hold for any case. So $L(z, \partial)u(z)=f(z)$has always asolution $u(z)$ with tempered singularities in asectorial region
for $f(z)$ with tempered singularities.
$j_{i}\in \mathrm{N}$$j_{2}$$j_{1}$\S 3

Outline of the proof of Theroem 2.3.

In order to find $v(z)$ in Theorem 2.3 we construct aparametrix of.
The method of construction of the parametrix is amodification of that in [6].
We may assume $e(p^{*}-1)=0$ and be aconstant with $0<\theta_{0}<\pi/2\gamma_{p^{*}-1}$ .
$v(z)=(Gf)(z)$ is constructed of the form$L(z, \partial_{z})$$\theta_{0}$(Gf) (z) $:= \int_{S}G(z, w)f(w)dw$ ,

(3.1)

where

$S$

is achain in$V(\theta_{0})$$w=(w_{0}, w_{1}, \ldots, w_{d})=(w_{0}, w’)$,

. The kernel $G(z, w)$ has the form

.

(3.2)

$G(z, w)= \frac{1}{2\pi i}\int_{\lambda_{0}}^{\infty}z_{0}^{\lambda}w_{0}^{-\lambda-1}K(z, w’, \lambda)d\lambda$

We can find $K(z, w’, \lambda)$ with the following:
1. $K(z, w’, \lambda)$ is holomorphic $\{z_{0};0<|z_{0}|<r_{0}, |\arg z_{0}|<\theta_{0}\}\cross\{(z’, w’);|z_{j}|$$r_{1}<r_{2}<|w_{j}|<r_{3},1\leq j\leq d\}$ and holomorphic in Ain some infinite
region.

2. $K(z, w’, \lambda)$ has an asymptotic expansion$K(z, w’, \lambda)\sim\hat{K}(z, w’, \lambda)=\sum_{n=0}^{\infty}k_{n}(z, w’, \lambda)z_{0}^{n}$

where

3.

$\hat{K}(z, w’, \lambda)$$\hat{K}(z, w’, \lambda)$

is aformal power series of$z_{0}$,.

satisfies formally$L(z, \partial)(z_{0}^{\lambda}\hat{K}(z, w’, \lambda))=\frac{z_{0}^{\lambda}}{(2\pi i)^{d}}\prod_{j=1}^{d}\frac{1}{(w_{j}-z_{j})}$.

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As for $G(z, w)$ we have
(3.3)

$L(z, \partial)G(z, w)=\delta(z, w)+R(z, w)$ ,

where

(3.4)

$\{\begin{array}{l}\delta(z,w)=\frac{\mathrm{l}}{(2\pi i)^{d+1}}(\int_{\lambda_{0}}^{\infty}z_{0}^{\lambda}w_{0}^{-\lambda-1}d\lambda)\prod_{j=1}^{d}\frac{\mathrm{l}}{(w_{j}-z_{j})}|R(z,w)|\leq C\mathrm{e}\mathrm{x}\mathrm{p}(-c|z_{0}|^{-\gamma_{p^{*}-1}})\end{array}$

It follows from (3.3) and (3.4) that (Rf)(z) $=L(z, \partial)v(z)-f(z)$ satisfies theconclusions of Theorem 2.3.

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Hamada, Y., Leray, J. et Wagschal, C., Syst\‘eme d’equation aux deriv\’ees partielles \‘a caracteristic multiples; probleme de Cauchy ramifie;
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domain, Publ. RIMS Kyoto Univ., 36 (2000), 457-482$\overline{\mathrm{O}}$$\overline{\mathrm{O}}$

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66-72.
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