数理解析研究所講究録 1261 巻 2002 年 192-201

192

Fuchsian PDE with applications to normal
forms of resonant vector fields\dag er

中央大学

・

経済学部

吉野 正史 ( Masafimi Yoshino ) \dag er \dag er

Faculty of Economioe, Chuo University, Higashinakano Hachioji,
Tokyo 192-0393, Japan

Motivations
Let$t\in \mathrm{C}$or. We consider Fuchsiandifferential equa, where
us an polynomial of one variable. We call$t\in \mathrm{C}$$\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{i}\cdot \mathrm{a}\mathrm{r}\mathrm{y}$tions
an incidencial polynomial of . We consider the solvability of theequation $Pu=f(t)$ , where $f(t)$ is analytic at the origin $t=0$ .
If the “non-log condition”$P \equiv p(t\frac{d}{ae})$$p(\zeta)$$p(\zeta)$

$P$

(1)

$p(\zeta)\neq 0$

for$\zeta=0,1,2$,$\ldots$

is fufftled $Pu=f$ has an analytic solution. Indeed, the solution isconstructffi by a method of indeterninate coefficients if we expand
in Taylor series.
Now, let us consider the case where a“non-log condition” is not
fulfiled. For the sake of simplicity, we consider under the condition
$u$

(2)

$\exists\zeta_{0}\in \mathrm{Z}_{+}=\{0,1,2, \ldots\}$,$p(\zeta_{0})=0$.

Remark. lf there exists
such that $p(\zeta_{1})=0$ Condition (2)is aspecial case where the difference of characteristic exponents haveintegral difference.
$\zeta_{1}\in \mathrm{Z}$By Robenius, the fundamental solutions contain a functionof the form
Ais acertain constant. It follows that thesolution of $Pu=f(t)$ is singular, or has finite differentiability.
Question
What happens in the case of nonlinear partial differential equations ofFuchs type ?
In order to answer to this question, we first introduce aclass of$\mathrm{t}\mathrm{h}\infty \mathrm{r}\mathrm{e}\mathrm{m}$$t^{\lambda}\log t,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$

$u$

$u$

$\mathrm{s}\mathrm{e}\succ \mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{d}$\dag er

Supported by Grant-in Aid for Scientific Raeearch (No. 11640183), Ministry of
Educations, Scienoe and Culture, Japan and by Chuo University, Tokyo, Japan.
\dag er \dag er
Pax.: $+81-0426- 74-3425$-mad adress:$B$$\mathrm{y}\mathrm{o}\mathrm{e}\mathrm{h}\mathrm{i}\cdot \mathrm{o}\mathrm{m}@\mathrm{t}\mathrm{a}\mathrm{m}\mathrm{a}\mathrm{c}\mathrm{c}.\mathrm{c}\mathrm{h}\mathrm{u}\triangleright \mathrm{u}.\mathrm{a}\mathrm{c}.\mathrm{j}\mathrm{p}$193
Fuchsian partial differential equations which appear from geometi
problems.
We also cite related works by Tahara, Mandai, Yamane and ’)mazawa.

Vector fields with an isolated singular poin
We consider(3)$\mathcal{X}(x)=\sum_{j=1}^{n}a_{j}(x)\frac{\partial}{\partial x_{j}}$where$a_{j}(x)$,$x=(x_{1}, \ldots, x_{n})$,is asmooth function of . We assume
$x$

(4)

$\mathcal{X}(0)=0$,

and the origin $x=0$ is an isolated singular point of .
by acoordinate change
We want to linearize$\mathcal{X}$$\mathcal{X}(x)$$x=y+v(y)$ ,(5)$v=O(|y|^{2})$ .

We write(6)$\mathcal{X}(x)=x\Lambda\frac{\partial}{\partial x}+R(x)\frac{\partial}{\partial x}\equiv X(x)\frac{\partial}{\partial x}$,$\frac{\partial}{\partial x}=(\frac{\partial}{\partial x_{1}’}\cdots,\frac{\partial}{\partial x_{n}})$$X(x)=x\Lambda+R(x)$ ,

(7)

where
(8)

$R(x)=(R_{1}(x), \ldots, R_{n}(x))$

and Ais an
Noting that$n\cross n$,$R(x)=O(|x|^{2})$ ,constant matrix.

$X(x) \frac{\partial}{\partial x}=X(y+v(y))\frac{\partial y}{\partial x}\frac{\partial}{\partial y}=X(y+v(y))(\frac{\partial x}{\partial y})^{-1}\frac{\partial}{\partial y}$

the linearizability condition implies$X(y+v)(1+\partial_{y}v)^{-1}=y\Lambda$.

It follows that(9).

$(y+v)\Lambda+R(y+v)=y\Lambda(1+\partial_{y}v)=y\Lambda+y\Lambda\partial_{y}v$,

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Therefore
$(*)$

$v$

solves theed homology equation$\mathrm{s}(\succ \mathrm{c}\mathrm{a}\mathrm{n}$$\mathcal{L}v\equiv yAdyV-v\Lambda=R(y+v(y))$,$v=(v_{1}, \ldots,v_{n})$.

Therefore we have
Eq. (’) has a solutionchange $x=y+v(y)$ .

$v$

if and only$\dot{\iota}f\mathcal{X}$is linearized by a coordinate

Expression of ahomology equation

We calculate the form of$\mathcal{L}$in case Ais adiagonal matrix. Namely weassume(10)$\mathrm{A}=(\begin{array}{lll}\lambda_{1} 0 \ddots 0 \lambda_{n}\end{array})$

Because we have$y \Lambda\partial_{y}=\sum_{k=1}^{n}\lambda_{k}y_{k^{\frac{\partial}{\partial y_{k}}}}$

we have(11).

$\mathcal{L}v=(\begin{array}{lll}\Sigma\lambda_{k}y_{k}\frac{\partial}{\partial y_{k}}-\lambda_{1} 0 \ddots 0 \Sigma\lambda_{k}y_{k}\frac{\partial}{\partial y_{k}}-\lambda_{n}\end{array})(\begin{array}{l}v_{1}\vdots v_{n}\end{array})$

Remark. The homology equation
is aspecial case of totally characteristic Fuchsian PDE. (cf. Tahara [4]). We also cite Shirai [3].
$(^{*})$

Non-log condition and anon resonantcondition
For simplicity, we consider the above example. The indicial polynomial
is defined by(12)$\sum_{k=1}^{n}\lambda_{k}\zeta_{k}-\lambda_{\mathrm{j}}$,$(j=1, \ldots,n)$.

Non resonant conditionis said to be non-resonant if$\mathcal{L}$(13)$\sum_{k=1}^{n}\lambda_{k}\alpha_{k}-\lambda_{j}\neq 0$

for Vo$\in(\alpha_{1}, \ldots, \alpha_{n})\in \mathrm{Z}_{+}^{n}$,$|\alpha|\geq 2$.

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Non resonant condition impies the existence of aformal
deed, we have

solution. In-.

$\mathcal{L}(\sum_{\alpha}v_{\alpha}y^{\alpha})=\sum_{\alpha}(\sum_{k=1}^{n}\lambda_{k}\alpha_{k}-\Lambda)v_{\alpha}y^{\alpha}$

exist on aset of formal power series if anon-resonant conHence
condition isdition is fulfilled. It should be noted that anon-resonance$\mathcal{L}^{-1}$anon$\log$

condition.

Two theorems concerning the solvability of homology
equationsby
As to the solvability of (’), probably the first result was obtained
Poincare’ in 19th century. He introduced a s0-caUed Poincar\’e condition.
condition
Then the middle of 20th century, Siegel introduced a Siegel
under a Siegel condition.
and he essentially showed the solvability ofsolvabih.ty
On the other hand, in the real domain, Sternberg showed theof (’) in aclass of smooth functions without any diophantine condition.
resonant
He essentially assumed the nonresonance condition. As to thecase, Hartman showed the solvability of (’) in aclass of continuousfunctions. Our result is closely related to Hartman’s theorem.
$(^{*})$

Sternberg’s theorem
Suppose that ahyperbolic condition(14)$Re\lambda_{k}\neq 0$,$k=1$ , $\ldots,n$

is filfilled. Moreover, assume that anon resonant condition is satis
fied. Then, Eq. $(’)$ has asmooth solution.
condiSternberg’s theorem shows the solvability of (’) under non$\log$tion.

Grobman- Hartman’s theorem
If the hyperbolicity condition is satisfied, Eq.(*) has acontinuous solution.
is defined by aweak solution.
Remark. Acontinuous solution ofHartman’s theorem treats the case where anon-log condition is notsatisfied.
$(^{*})$The Object of the Stud196
We consider the case where anon-log condition is not satisfied. The
typical example is avolumn preserving vector field,.
We want to solve (’) in areal domain in aclass of finetely differentiable
functions, which corresponds to Hartman’s theorem in aclass. Thisis closely related to the construction of asingular solution in acomplex
domain.
Remark. Geometrically, the resonance does not vanish under aformalchange of variables. Because the solvabilty of (’) implies that thechange of variables $x=y+v(y)$ linearizes the given vector field, theresonance also vanish under the change of varibles. This implies that$R$ .
does not necessarily have afomal solution.
$\lambda_{1}+\cdots+\lambda_{n}=0$$C^{\ell}$$(^{*})$

Heuristic Statement of Resultstheorem$C^{\ell}$Hartman-

For the sake of simplicity, we$\mathrm{w}\mathrm{i}\mathrm{U}$state the special case of our theorem.

Theorem Assume that$(k =1, \ldots, n)$ are nonzero real number,
hyperbolicity). Then Eq. $(^{*})hae$ asolution for acertain
determined by an indicial polynomials.
$\lambda_{k}$$C^{\ell}$$(a$$\ell\geq 0$

Idea of the Proof.
Why Picard’s iteration does not work ?
Firstly, we note the loss of derivatives of. In fact, even if there, we have aloss of derivatives. In order to see this, let us
existsconsider$\mathcal{L}^{-1}$$\mathcal{L}^{-1}$$(t \frac{d}{dt}-\lambda)u=g(t)$,${\rm Re}\lambda<0$.

The solution is given by.

$u(t)= \int_{0}^{1}\sigma^{-\lambda-1}g(\sigma t)d\sigma$

Clearly, we do not gain the derivatives.
On the other hand, in order to define the right-hand side of$R(y+v)$ one needs derivatives of . Indeed, Sobolev’s embedding$v$$(^{*})$,$\mathrm{t}\mathrm{h}\triangleright$

orem implies:
if $0\leq m<k-n/p<m+1$ and $0\leq\alpha<k$ $-m-n/p<1$ , it folows
that.
$W^{kp},(\Omega)\epsilonarrow C^{m,\alpha}(\circ\Omega$

Here$W^{k\mathrm{p}},(\Omega)$is the space of distributions whose derivatives up to orde

197
is a H\"older space, namely.
is in the Lebesgue space
the set of functions with derivatives up to $m$ has H\"older exponent .
Therefore the iteration scheme $v=\mathcal{L}^{-1}R(y+v)$ does not seem to converge.
In view of this we need to employ aNash- Moser scheme, arapidly
convergent iteration scheme.
$L^{p}(\Omega)$$k$$C^{m,\alpha}(\overline{\Omega})$$\alpha$Rapidly Convergent Iteration Scheme
1. We need asmoothing operator which has not asmoothing effecttransversal to the singular locus of the equation, $yj=0$ , $(j=1, \ldots, n)$ .

2. The crucial step of the Nash-Moser iteration scheme is to solve aat $v=w$ is given bylinearized equation. The linearized equation of$(^{*})$$\mathcal{L}v-\nabla R(y+w)v$.

We note that is singular or does not have regularity. The solvabilityof linear Fuchsian partial differential equations with singular coefficientsseems open.
In order to handle these problems we use aMellin transform, and aNash-Moser iteration scheme of tangential type.
$w$

Statement of the Theorem
Mellin Transform
Let $N\geq 1$ be an integer. Let $f(x)=(f_{1}(x), \ldots, f_{N}(x))$ be an integrable, and let us define aMellin transformfunction on$\hat{f}(\zeta)(\zeta\in\Gamma+i\mathrm{R}^{n})$$\mathrm{R}_{+}^{n}$by$\hat{f}(\zeta)=\int_{\mathrm{R}_{+}^{n}}f(x)x^{\zeta-e}dx$$e=(1, \ldots, 1)$ .

,The inverse Mellin transform is given by$f(x)=M^{-1}( \hat{f})(\zeta)=\frac{1}{(2\pi i)^{n}}\int_{\mathrm{R}^{n}}\hat{f}(\eta+i\xi)x^{-\eta-i\xi}d\xi$where$\eta\in\Gamma$, $x_{j}>0$ , $j=1$ , $\ldots,n$ ,is chosen so that the integral converges.

Definition of afunction space
be an open set. We define, and letLet
set of holomorphic vector-valued functions$\sigma\geq 0$$\Gamma\subset \mathrm{R}^{n}$$v(\zeta)=(v_{1}(\zeta), \ldots, v_{N}(\zeta))$,$H_{\sigma}\equiv H_{\sigma,\Gamma}$$\zeta=\eta+i\xi\in\Gamma+i\mathrm{R}^{n}$as the198
such that$||v||_{\sigma,\Gamma}:= \sup_{\eta\in\Gamma}\int_{\mathrm{R}^{n}}\langle\zeta\rangle^{\sigma}|v(\zeta)|d\xi<\infty$

,

where
$\langle\zeta\rangle=1$

The space$H_{\sigma,\Gamma}$$+ \sum_{j=1}^{n}|\zeta_{\mathrm{j}}|$,$|v( \zeta)|=(\sum_{j=1}^{N}|v_{j}(\zeta)|^{2})^{1/2}$

is aBanach space with the normLetbe the inverse Melin transform ofdefined by$\mathcal{H}_{\sigma,\Gamma}$$H_{\sigma,\Gamma}$$||\cdot||_{\sigma T}$.

. The norm of$||u||_{\mathcal{H}_{\sigma,\Gamma}}\equiv||u||_{\sigma,\Gamma}:=||M(u)||_{H_{\sigma,\Gamma}}$$\mathcal{H}_{\sigma,\Gamma}$is.

We define an incidencial polynomial by$p( \zeta)=-\sum_{\mathrm{j}=1}^{n}\zeta_{j}\lambda_{j}I-\Lambda$,

where I is an identity matrix.
We say thatat the origin if
being identicallyequal to 1in some neighborhood of the origin such that
$R\in \mathcal{H}_{\nu,\Gamma}$$\exists\psi\in C_{0}^{\infty}(\mathrm{R}^{n})$$M(\psi R)\in \mathcal{H}_{\nu,\Gamma}$.

Then we have

Theorem Suppose that there exist $C>0$ and an open bounded set ,$\Gamma$$0\in\Gamma\subset \mathrm{R}^{n}$such that$|p(\eta+i\xi)|>C>0$ ,Let$\sigma\geq 1$$\forall\eta\in\Gamma,\forall\xi\in \mathrm{R}^{n}$.

be an integer. Then there exists $\nu\geq 0$ such that, if$R\in \mathcal{H}_{\nu,\Gamma}$

at the origin,$\Phi\cdot(^{*})$

and$\nabla R_{j}\in \mathcal{H}_{\nu,\Gamma}$

has asolution,$j=1$ ,

$v\in \mathcal{H}_{\sigma,\Gamma’}$$\ldots$,

for every$n$$\Gamma’\subset\subset\Gamma$.

Remark The set determine the vanishing order of. Henceexpresses the smoothness up to the set $y_{j}=0$ $(j=1, \ldots, n)$ , because
we have the interior regularity, $x_{\mathrm{j}}>0j=1$ , , .
In order to construct asolution in some neighborhood of the origin,
we construct solutions in the domain$(j=1, \ldots,n)$ . Then wepatch up these solutions$\Gamma$$v\in \mathcal{H}_{\sigma,\Gamma’}$$\Gamma$$\ldots$$\pm y_{\mathrm{j}}\geq 0$

$n$

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Further extensions
We will briefly mention how the above theorem is extended to moregeneral systems. We consider $N(N\geq 1)$ system of equations for the
unknown vector $v=(v_{1}, \ldots, v_{N})$where $1\leq s\leq m$ are integers and$j=1$ ,,

$p_{j}(\delta)u_{j}+a_{j}(x, \delta^{\alpha}u;|\alpha|\leq s)=0$$\delta_{j}=\partial/\partial xj$$\delta^{\alpha}=\delta_{1}^{\alpha_{1}}\cdots\delta_{n}^{\alpha_{n}}$$\ldots$, $N$ ,,,

is apolynomial of . The nonlinear term $aj(x, z)$ ,and, whereis supposed to be real-valued and smooth inneighborhood of the origin $z=0$ .
Then we have the same assertion as to the above theorem.

$z=(z_{\alpha}^{j})$$\zeta$$p_{j}(\zeta)$$\mathrm{R}^{n}\cross\Omega$$\Omega$

is a

Example. We consider Monge-Amp\‘ere operator$M(u)=u_{xx}u_{yy}-u_{xy}^{2}+kxyu_{xy}+cu$. Here is real constantin some neighborhood of the origin
and is acomplex constant.
Let $u_{0}=x^{2}y^{2}$ and $f_{0}=M(u_{0})$ . We want to solve$k$$(x, y)\in \mathrm{R}^{2}$$c$$M(u_{0}+v)=f_{0}(x, y)+g(x,y)$ ,in$\mathrm{R}^{2}$,

where $g(x,y)$ is agiven function. This equation is related to find asurface with aprescribed Gaussian curvature. The general theory does
not apply this equation because of the degeneracy of .
The incidencial polynomial is given by$u_{0}$$p(\zeta)=-2\zeta_{1}(\zeta_{1}+1)$ $-2\zeta_{2}(\zeta_{2}+1)-(k-8)\zeta_{1}\zeta_{2}-c$.

Our theorem shows that If $4<k<12$ and $c=iK$ , $K\gg 1$ there exists
asolution of the above equation.
$v$

Proof of the Theorem
Definition of asmoothing operator innear the origin $x=0$ . Let $N\geq 1$,, $0\leq\phi$Letbe positive integers. We set
and let$\mathcal{H}_{s,\Gamma}$$\phi$$\in C_{0}^{\infty}(\mathrm{R}^{\mathrm{n}})$$\leq 1$$\phi$$\equiv 1$$\ell\geq 1$$\psi_{N}(\zeta)=\exp(\frac{1}{N^{2\tau}}\sum_{j=1}^{n}\zeta_{j}^{2\tau})$,

200
and define$\chi_{N}^{\ell}=\int_{\mathrm{R}^{*}}.\{\psi_{N}(\zeta)($ $e^{-\sigma\zeta/N}-\mathrm{I}$where$\chi_{N}^{\ell}(\zeta)$$(- \frac{\sigma\zeta}{N})^{\nu}\frac{1}{\nu!})+(1 -\psi_{N}(\zeta))e^{-\sigma\zeta/N}\}dc$

is an odd integer such thatis an entire function of (and real,$2\tau\geq\ell$$\tau$$\overline{\chi_{N}^{\ell}(\zeta)}=\chi_{N}^{\ell}(\overline{\zeta})$

We define asmoothing operator$S_{N}$. We can easily see that.

by

$S_{N}v:=M^{-1}(\chi_{N+1}^{\ell}(\zeta)\hat{v}(\zeta))$,$v\in \mathcal{H}_{s,\Gamma}$

where
denotes the Mellin transform of , andinverse Mellin transform.
$\hat{v}(\zeta)$$v$,$M^{-1}$denotes theProof of the Theorem
Let 1 $<\tau<2$ and $d>1$ be the constants chosen later. Let$(k=0,1,2, \ldots)$ be asmoothing operator defined above with $N+1=$. We define$S_{k}$$\mu_{k}:=d^{r^{k}}$$G(v)=\mathcal{L}v-R(y+v)$ .

Let

be the linearized operator of $G$ at $v=w$ . We define $g_{0}=G(0)$ .

$L_{w}$

Iterative scheme
We construct an approximate sequence$w_{0}=0$,$w_{k\dagger 1}=w_{k}+S_{k}\rho_{k}$Estimates
There exist$\exists\nu$,$\exists\kappa$

and,$L_{w_{k}}\rho_{k}=g_{k}$$\exists c>0$,$\nu>\kappa$$\{w_{k}\}$by, $g_{k}=-G(w_{k})$ , $k=0,1$ , 2,$\ldots$$>1$ such that$||g_{k}||_{0,\Gamma}\leq c\mu_{k}^{-\kappa}d^{\kappa}||g_{0}||_{\nu+1,\Gamma}$.

If we can show this estimate we see that $g_{k}arrow 0$ as $karrow\infty$ andthat
is aCauchy sequence. It follows that $w:= \lim_{k}w_{k}$ satisfies$\{w_{k}\}$$G(w)=0$ .

Step 1A prion estimate of .
There exists $C>0$ independent of$w_{k}$$k$such that, for $j=1$ ,$||w_{j}||_{\ell,\Gamma}\leq Cd^{\kappa}||g_{0}||_{\nu+1,\Gamma}$,

if$\ell<\kappa$$+s$ ,$\ldots$, $k+1$

201$||w_{j}||_{\ell,\Gamma}\leq C\mu_{j-1}^{\ell+1-\kappa-s}d^{\kappa}||g_{0}||_{\nu+1,\Gamma}$

Step 2A priori estimate of
There exists $C>0$ independent of

,

if

$l$

$\geq\kappa+s$.

$g_{k}$$k$such that$||g_{k}||_{\nu,\Gamma}\leq Cd^{\kappa}||g_{0}||_{\nu+1,\Gamma}(1 +\mu_{k}^{(\nu+m+n+2-\kappa-s)/\tau})$.

Using these estimates we can show the desired estimate. The constants
and are determined by the equation.
$\tau$$d$References
1. N. S. Madi and M. Yoshino, Uniqueness and solvability of nonlinear
Fuchsian equations, Bull Sc. math., 114 (1990), 41-60.
2. T. Mandai, The method of Probenius to Fuchsian partial differential
equations, J. Math. Soc. Japan 52 (2000), 645-672.
3. A. Shirai, Convergence of formal solutions of singular first ordernonlinear partial differential equations of totally characte tic type,preprint.
4. H. Tahara, Solvability of nonlinear totally characteristic type partial
differential equations when resonances occur, in preparation.
5. H. Yamane, Nonlinear Singular First Order Partial Differential
Equations Whose Characteristic Exponent Takes a Positive Integral
Value, Publ. RIMS, Kyoto Univ., 33 (1997), 801-811.
6. H. Yamazawa, Singular Solutions of the Briot-Bouquet Type Partial
Differential Equations, preprint.
$\dot{m}$2000 AMS Mathematics Subject Classification: primary 35G20 secondary$37\mathrm{C}10,37\mathrm{J}40$