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

Soit L(z,∂ z )=(∂ z 0 ) k -A(z,∂ z ) un opérateur linéaire différentiel à coefficients holomorphes, où

A(z,∂z)=∑j=0k-1Aj(z,∂z′)(∂z0)j, ord .A(z,∂z)=m>k

et

z=(z0,z′)∈Cn+1.

On considère le problème de Cauchy aux données holomorphes

L(z,∂z)u(z)=f(z),(∂z0)iu(0,z′)=u^i(z′)(0≤i≤k-1).

On peut facilement obtenir une solution formelle u ^(z)=∑ n=0 ∞ u ^ n (z ′ )(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,∂ z ), il y a une fonction u S (z) holomorphe sauf sur {z 0 =0}, telle que L(z,∂ z )u S (z)=f(z) et u S (z)∼u ^(z) quand z 0 →0 dans S.

Let L(z,∂ z )=(∂ z 0 ) k -A(z,∂ z ) be a linear partial differential operator with holomorphic coefficients, where

A(z,∂z)=∑j=0k-1Aj(z,∂z′)(∂z0)j, ord .A(z,∂z)=m>k

and

z=(z0,z′)∈Cn+1.

We consider Cauchy problem with holomorphic data

L(z,∂z)u(z)=f(z),(∂z0)iu(0,z′)=u^i(z′)(0≤i≤k-1).

We can easily get a formal solution u ^(z)=∑ n=0 ∞ u ^ n (z ′ )(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,∂ z ), there is a function u S (z) holomorphic except on {z 0 =0} such that L(z,∂ z )u S (z)=f(z) and u S (z)∼u ^(z) as z 0 →0 in 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},
     address = {Grenoble},
     volume = {33},
     number = {1},
     year = {1983},
     doi = {10.5802/aif.907},
     zbl = {0494.35017},
     mrnumber = {85g:35014},
     language = {en},
     url = {https://aif.centre-mersenne.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, Tome 33 (1983) no. 1, pp. 131-176. doi : 10.5802/aif.907. https://aif.centre-mersenne.org/articles/10.5802/aif.907/

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