Characteristic Cauchy problems and solutions of formal power series

Ouchi, Sunao

Annales de l'Institut Fourier, Tome 33 (1983) no. 1, pp. 131-176.

Soit $L(z,{\partial }_z)=({\partial }_{z_0}{)}^k-A(z,{\partial }_z)$ un opérateur linéaire différentiel à coefficients holomorphes, où

$$A(z,{\partial }_z)=\sum _{j=0}^{k-1}{A}_j(z,{\partial }_{z^{\prime }}\left)\right({\partial }_{z_0}{)}^j,\mathrm{ord}.A(z,{\partial }_z)=m>k$$

et

$$z=(z_0,z^{\prime })\in C^{n+1}.$$

On considère le problème de Cauchy aux données holomorphes

$$L(z,{\partial }_z)u\left(z\right)=f\left(z\right),\left({\partial }_{z_0}{)}^{i}u\right(0,z^{\prime })={\widehat{u}}_i(z^{\prime }\left)\right(0\le i\le k-1).$$

On peut facilement obtenir une solution formelle $\widehat{u}\left(z\right)={\sum }_{n=0}^{\infty }{\widehat{u}}_n(z^{\prime }\left)\right(z_0{)}^n/n!$, mais en général elle diverge. On montre sous certaines conditions que pour un secteur arbitraire $S$ d’ouverture moindre qu’une constante déterminée par $L(z,{\partial }_z)$, il y a une fonction $u_S\left(z\right)$ holomorphe sauf sur $\{z_0=0\}$, telle que $L(z,{\partial }_z)u_S\left(z\right)=f\left(z\right)$ et $u_S\left(z\right)\sim \widehat{u}\left(z\right)$ quand $z_0\to 0$ dans $S$.

Let $L(z,{\partial }_z)=({\partial }_{z_0}{)}^k-A(z,{\partial }_z)$ be a linear partial differential operator with holomorphic coefficients, where

$$A(z,{\partial }_z)=\sum _{j=0}^{k-1}{A}_j(z,{\partial }_{z^{\prime }}\left)\right({\partial }_{z_0}{)}^j,\mathrm{ord}.A(z,{\partial }_z)=m>k$$

and

$$z=(z_0,z^{\prime })\in C^{n+1}.$$

We consider Cauchy problem with holomorphic data

$$L(z,{\partial }_z)u\left(z\right)=f\left(z\right),\left({\partial }_{z_0}{)}^{i}u\right(0,z^{\prime })={\widehat{u}}_i(z^{\prime }\left)\right(0\le i\le k-1).$$

We can easily get a formal solution $\widehat{u}\left(z\right)={\sum }_{n=0}^{\infty }{\widehat{u}}_n(z^{\prime }\left)\right(z_0{)}^n/n!$, bu in general it diverges. We show under some conditions that for any sector $S$ with the opening less that a constant determined by $L(z,{\partial }_z)$, there is a function $u_S\left(z\right)$ holomorphic except on $\{z_0=0\}$ such that $L(z,{\partial }_z)u_S\left(z\right)=f\left(z\right)$ and $u_S\left(z\right)\sim \widehat{u}\left(z\right)$ as $z_0\to 0$ in $S$.

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