Characteristic Cauchy problems and solutions of formal power series
Annales de l'Institut Fourier, Volume 33 (1983) no. 1, pp. 131-176.

Let $L\left(z,{\partial }_{z}\right)=\left({\partial }_{{z}_{0}}{\right)}^{k}-A\left(z,{\partial }_{z}\right)$ be a linear partial differential operator with holomorphic coefficients, where

 $A\left(z,{\partial }_{z}\right)=\sum _{j=0}^{k-1}{A}_{j}\left(z,{\partial }_{{z}^{\prime }}\right)\left({\partial }_{{z}_{0}}{\right)}^{j},\phantom{\rule{3.33333pt}{0ex}}\mathrm{ord}.A\left(z,{\partial }_{z}\right)=m>k$

and

 $z=\left({z}_{0},{z}^{\prime }\right)\in {C}^{n+1}.$

We consider Cauchy problem with holomorphic data

 $L\left(z,{\partial }_{z}\right)u\left(z\right)=f\left(z\right),\phantom{\rule{3.33333pt}{0ex}}\left({\partial }_{{z}_{0}}{\right)}^{i}u\left(0,{z}^{\prime }\right)={\stackrel{^}{u}}_{i}\left({z}^{\prime }\right)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\left(0\le i\le k-1\right).$

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

Soit $L\left(z,{\partial }_{z}\right)=\left({\partial }_{{z}_{0}}{\right)}^{k}-A\left(z,{\partial }_{z}\right)$ un opérateur linéaire différentiel à coefficients holomorphes, où

 $A\left(z,{\partial }_{z}\right)=\sum _{j=0}^{k-1}{A}_{j}\left(z,{\partial }_{{z}^{\prime }}\right)\left({\partial }_{{z}_{0}}{\right)}^{j},\phantom{\rule{3.33333pt}{0ex}}\mathrm{ord}.A\left(z,{\partial }_{z}\right)=m>k$

et

 $z=\left({z}_{0},{z}^{\prime }\right)\in {C}^{n+1}.$

On considère le problème de Cauchy aux données holomorphes

 $L\left(z,{\partial }_{z}\right)u\left(z\right)=f\left(z\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left({\partial }_{{z}_{0}}{\right)}^{i}u\left(0,{z}^{\prime }\right)={\stackrel{^}{u}}_{i}\left({z}^{\prime }\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left(0\le i\le k-1\right).$

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

@article{AIF_1983__33_1_131_0,
author = {Ouchi, Sunao},
title = {Characteristic {Cauchy} problems and solutions of formal power series},
journal = {Annales de l'Institut Fourier},
pages = {131--176},
publisher = {Institut Fourier},
volume = {33},
number = {1},
year = {1983},
doi = {10.5802/aif.907},
mrnumber = {85g:35014},
zbl = {0494.35017},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.907/}
}
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Ouchi, Sunao. Characteristic Cauchy problems and solutions of formal power series. Annales de l'Institut Fourier, Volume 33 (1983) no. 1, pp. 131-176. doi : 10.5802/aif.907. http://www.numdam.org/articles/10.5802/aif.907/

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