# Convergence of formal solutions of first order singular partial differential equations of nilpotent type

• Volume: 97, Issue: 1, page 91-99
• ISSN: 0137-6934

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## Abstract

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Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: $Pu\left(x,y,z\right):=\left(y{\partial }_{x}-z{\partial }_{y}\right)u\left(x,y,z\right)=f\left(x,y,z\right){\in }_{x,y,z}$, where $P=y{\partial }_{x}-z{\partial }_{y}{:}_{x,y,z}{\to }_{x,y,z}$. For this equation, our aim is to characterize the solvability on ${}_{x,y,z}$ by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.

## How to cite

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Masatake Miyake, and Akira Shirai. "Convergence of formal solutions of first order singular partial differential equations of nilpotent type." Banach Center Publications 97.1 (2012): 91-99. <http://eudml.org/doc/281602>.

@article{MasatakeMiyake2012,
abstract = {Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: $Pu(x,y,z): = (y∂_x - z∂_y)u(x,y,z) = f(x,y,z) ∈ _\{x,y,z\}$, where $P = y∂_x - z∂_y: _\{x,y,z\} → _\{x,y,z\}$. For this equation, our aim is to characterize the solvability on $_\{x,y,z\}$ by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.},
author = {Masatake Miyake, Akira Shirai},
journal = {Banach Center Publications},
keywords = {nilpotent vector field},
language = {eng},
number = {1},
pages = {91-99},
title = {Convergence of formal solutions of first order singular partial differential equations of nilpotent type},
url = {http://eudml.org/doc/281602},
volume = {97},
year = {2012},
}

TY - JOUR
AU - Masatake Miyake
AU - Akira Shirai
TI - Convergence of formal solutions of first order singular partial differential equations of nilpotent type
JO - Banach Center Publications
PY - 2012
VL - 97
IS - 1
SP - 91
EP - 99
AB - Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: $Pu(x,y,z): = (y∂_x - z∂_y)u(x,y,z) = f(x,y,z) ∈ _{x,y,z}$, where $P = y∂_x - z∂_y: _{x,y,z} → _{x,y,z}$. For this equation, our aim is to characterize the solvability on $_{x,y,z}$ by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.
LA - eng
KW - nilpotent vector field
UR - http://eudml.org/doc/281602
ER -

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