Characteristic Cauchy problems and solutions of formal power series
Annales de l'Institut Fourier, Tome 33 (1983) no. 1, p. 131-176
Soit L(z, z )=( z 0 ) k -A(z, z ) un opérateur linéaire différentiel à coefficients holomorphes, oùA(z,z)=j=0k-1Aj(z,z)(z0)j, ord .A(z,z)=m>ketz=(z0,z)Cn+1.On considère le problème de Cauchy aux données holomorphesL(z,z)u(z)=f(z),(z0)iu(0,z)=u^i(z)(0ik-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, whereA(z,z)=j=0k-1Aj(z,z)(z0)j, ord .A(z,z)=m>kandz=(z0,z)Cn+1.We consider Cauchy problem with holomorphic dataL(z,z)u(z)=f(z),(z0)iu(0,z)=u^i(z)(0ik-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},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {33},
     number = {1},
     year = {1983},
     pages = {131-176},
     doi = {10.5802/aif.907},
     zbl = {0494.35017},
     mrnumber = {85g:35014},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1983__33_1_131_0}
}
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. http://www.numdam.org/item/AIF_1983__33_1_131_0/

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