# ANNALES DE L'INSTITUT FOURIER

Characteristic Cauchy problems and solutions of formal power series
Annales de l'Institut Fourier, Volume 33 (1983) no. 1, p. 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},
publisher = {Imprimerie Louis-Jean},
volume = {33},
number = {1},
year = {1983},
pages = {131-176},
doi = {10.5802/aif.907},
zbl = {0494.35017},
mrnumber = {85g:35014},
language = {en},
url = {aif.centre-mersenne.org/item/AIF_1983__33_1_131_0}
}

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. https://aif.centre-mersenne.org/item/AIF_1983__33_1_131_0/

[1] Y. Hamada, The singularity of solutions of Cauchy problem, Publ. Res. Inst. Math. Sci., 5 (1969), 21-40. | MR 40 #3056 | Zbl 0203.40702

[2] Y. Hamada, Problème analytique de Cauchy à caractéristiques multiples dont les données de Cauchy ont des singularités polaires, C.R.A.S., Paris, Ser. A, 276 (1973), 1681-1684. | MR 48 #4482 | Zbl 0256.35012

[3] Y. Hamada, J. Leray et C. Wagschal, Systèmes d'équations aux dérivées partielles à caractéristiques multiples ; Problème de Cauchy ramifié ; hyperbolicité partielle, J. Math. Pure Appl., 55 (1971), 297-352. | MR 55 #8572 | Zbl 0307.35056

[4] H. Komatsu, Irregularity of characteristic elements and construction of null solutions, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 23 (1976), 297-342. | MR 54 #5596 | Zbl 0443.35009

[5] S. Mizohata, On kowalewskian systems, Russian Math. Survey, 29, vol. 2 (1974), 223-235. | MR 53 #6063 | Zbl 0305.35003

[6] S. Ōuchi, Asymptotic behaviour of singular solutions of linear partial differential equations in the complex domain, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 27 (1980), 1-36. | MR 81m:35019 | Zbl 0441.35020

[7] S. Ōuchi, An integral representation of singular solutions of linear partial differential equations in the complex domain, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 27 (1980), 37-85. | MR 82e:35002 | Zbl 0439.35019

[8] S. Ōuchi, Characteristic Cauchy problems and solutions of formal power series, Proc. Japan Acad., vol. 56 (1980), 372-375. | MR 83g:35024 | Zbl 0467.35001

[9] J. Persson, On the Cauchy problem in Cn with singular data, Mathematiche, 30 (1975), 339-362. | MR 56 #16128 | Zbl 0359.35048

[10] C. Wagschal, Problème de Cauchy analytique à données méromorphes, J. Math. Pure Appl., 51 (1972), 375-397. | MR 50 #784 | Zbl 0242.35016

[11] W. Wasow, Asymptotic expansions for ordinary differential equations, Interscience Publishers, 1965. | MR 34 #3041 | Zbl 0133.35301