Characteristic Cauchy problems and solutions of formal power series

Ouchi, Sunao. "Characteristic Cauchy problems and solutions of formal power series." Annales de l'institut Fourier 33.1 (1983): 131-176. <http://eudml.org/doc/74567>.

@article{Ouchi1983,
abstract = {Let $L(z,\partial _z)=(\partial _\{z_0\})^k-A(z,\partial _z)$ be a linear partial differential operator with holomorphic coefficients, where\begin\{\}A(z,\partial \_z)=\sum ^\{k-1\}\_\{j=0\}A\_j(z,\partial \_\{z^\{\prime \}\}) (\partial \_\{z\_0\})^j,~\{\rm ord\}.A(z,\partial \_z)=m&gt;k\end\{\}and\begin\{\}z=(z\_0,z^\{\prime \})\in C^\{n+1\}.\end\{\}We consider Cauchy problem with holomorphic data\begin\{\}L(z,\partial \_z)u(z)=f(z),~(\partial \_\{z\_0\})^iu(0,z^\{\prime \})= \hat\{u\}\_i(z^\{\prime \})~~ (0\le i\le k-1).\end\{\}We can easily get a formal solution $\hat\{u\}(z)=\sum ^\infty _\{n=0\}\hat\{u\}_n(z^\{\prime \})(z_0)^n/n!$, 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 $u_S(z)$ holomorphic except on $\lbrace z_0=0\rbrace $ such that $L(z,\partial _z)u_S(z)=f(z)$ and $u_S(z)\sim \hat\{u\}(z)$ as $z_0\rightarrow 0$ in $S$.},
author = {Ouchi, Sunao},
journal = {Annales de l'institut Fourier},
keywords = {holomorphic coefficients; Cauchy problem; holomorphic data; formal solution},
language = {eng},
number = {1},
pages = {131-176},
publisher = {Association des Annales de l'Institut Fourier},
title = {Characteristic Cauchy problems and solutions of formal power series},
url = {http://eudml.org/doc/74567},
volume = {33},
year = {1983},
}

TY - JOUR
AU - Ouchi, Sunao
TI - Characteristic Cauchy problems and solutions of formal power series
JO - Annales de l'institut Fourier
PY - 1983
PB - Association des Annales de l'Institut Fourier
VL - 33
IS - 1
SP - 131
EP - 176
AB - Let $L(z,\partial _z)=(\partial _{z_0})^k-A(z,\partial _z)$ be a linear partial differential operator with holomorphic coefficients, where\begin{}A(z,\partial _z)=\sum ^{k-1}_{j=0}A_j(z,\partial _{z^{\prime }}) (\partial _{z_0})^j,~{\rm ord}.A(z,\partial _z)=m&gt;k\end{}and\begin{}z=(z_0,z^{\prime })\in C^{n+1}.\end{}We consider Cauchy problem with holomorphic data\begin{}L(z,\partial _z)u(z)=f(z),~(\partial _{z_0})^iu(0,z^{\prime })= \hat{u}_i(z^{\prime })~~ (0\le i\le k-1).\end{}We can easily get a formal solution $\hat{u}(z)=\sum ^\infty _{n=0}\hat{u}_n(z^{\prime })(z_0)^n/n!$, 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 $u_S(z)$ holomorphic except on $\lbrace z_0=0\rbrace $ such that $L(z,\partial _z)u_S(z)=f(z)$ and $u_S(z)\sim \hat{u}(z)$ as $z_0\rightarrow 0$ in $S$.
LA - eng
KW - holomorphic coefficients; Cauchy problem; holomorphic data; formal solution
UR - http://eudml.org/doc/74567
ER -