ALGEBRAIC ANALYTICAL STUDY OF SHEAVES AND INFINITE ORDRE DIFFERENTIAL EQUATIONS

滑轮和无限阶微分方程的代数分析研究

• 批准号: 13640154
• 负责人: ISHIMURA Ryuichi
• 金额: $ 2.43万
• 依托单位: CHIBA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640154/
• 关键词: Algebraic analysis; pseudo-differential equations; infinite ordre differential equations; partial differential equations; Microlocal study of sheaves; differential-difference equations; convolution equations; Cauchy problem

The aims of this research were as follows :[1] To generalize the theorem for Cauchy problem of micro-differential equations in the complex domain, obtained by the head investigator, to the sistem of pseudo-differential equations.[2] To characterize the automorphism of the sheaf of holomorphic functions by the infinite ordre differential operators, without assuming the continuity.[3] To generalize the results concerning the existence and the analytic continuation for the single convolution equation in the complex domain to the system.At first, by using the cohomological method, we have defined a natural class of non-local pseudo-differential operators containing any linear differential-difference operators. And furthermore, we gave the composition of two such operators and also the operation to holomorphic functions. We proved the one to one correspondance between the operators and their symbols and finally, defining the characteristic set for the non-local pseudo-differential operator, we proved the invertibility theorem for the non-local pseudo-differential operator.

[1]将研究组长关于复域上微微分方程Cauchy问题的定理推广到伪微分方程系统。[2]用无穷阶微分算子刻划全纯函数层的自同构，而不要求连续性。[3]为了将复域上单卷积方程的存在性和解析延拓的结果推广到系统中，首先利用上同调方法定义了一类包含任意线性微分-差算子的自然非局部伪微分算子。并进一步给出了这两个算子的合成以及对全纯函数的运算。证明了算子与符号之间的一一对应关系，最后定义了非局部伪微分算子的特征集，证明了非局部伪微分算子的可逆性定理.

项目成果

Y.HINO, S.MURAKAMI, T.NAITO, V.M.NGUYEN: "A variation-of-constants formula for abstract functional differential equations in the phase space"Journal of Differential Equations. 179. 336-355 (2002)

Y.HINO、S.MURAKAMI、T.NAITO、V.M.NGUYEN：“相空间中抽象泛函微分方程的常数变分公式”微分方程杂志。

O.LIESS, Y.OKADA, N.TOSE: "Hartogs phenomena for maicrofunctions with holomorphic parameters"Publications RIMS, Kyoto University. 37(2). 221-238 (2001)

O.LIESS、Y.OKADA、N.TOSE：“具有全纯参数的宏函数的 Hartogs 现象”出版物 RIMS，京都大学。

R.ISHIMURA: "Non-local pseudo-differential operators"Journal de Mathematiques pures et appliquees. 81. 1241-1276 (2002)

R.ISHIMURA：“非局部伪微分算子”Journal de Mathematiques pures et appliquees。

HINO Y., MURAKAMI S., NAITO T. and NGUYEN V.M.,: "A variation-of-constants formula for abstract functional differential equations in the phase space"Journal of Differential Equations. 179. 336-355 (2002)

HINO Y.、MURAKAMI S.、NAITO T. 和 NGUYEN V.M.，：“相空间中抽象泛函微分方程的常数变分公式”微分方程杂志。

T.AOKI, T.KAWAI, T.KOIKE, Y.TAKEI: "On the exact WKB analysis of operators admitting infinitely many phases"Advances in Mathematics. (to appear).

T.AOKI、T.KAWAI、T.KOIKE、Y.TAKEI：“关于允许无限多相的算子的精确 WKB 分析”数学进展。