一个奇异偏微分方程的形式解的单项可和性的研究
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摘要
20世纪以来，随着人们对于解析偏微分方程的发散级数解的研究以及对于这些发散级数解的意义的探索，经典的渐近展开及可和性理论得到了很好的发展，现在成为了应用背景更为广泛及研究前景更为广阔的Gevrey渐近展开理论及可和与多重可和理论。
本文第一部分：通过比较系数法验证某一奇异偏微分方程存在唯一形式幂级数解，给出该级数解的具体形式，利用形式Borel变换证明此形式幂级数解关于某单项式是1-Gevrey的。
第二部分：利用泛函分析中的不动点定理以及Gevrey渐近理论及可和理论中的部分结论证明上述形式解关于某单项式是1-可和的。
Since the20th century, with the study on divergent power series solution and the exploringof the meaning of them, the classical asymptotic expansion theory and summability theory havebeen developed rapidly. And now they become Gevrey asymptotic expansion theory andmultisummability theory, which have more extansive used and wider vistas of research.
Part one: We prove the existence and uniqueness of the formal power series solution to asingular partial differential equation by comparing coeffients of an identity, given the specificform of the series solution, we construct holomorphic solution of the problem with the help ofthe formal Borel transform, and prove that the formal power series solution is1-Gevrey.
Part two: We use the fixed point theorem and part of the results in Gevrey asymptotictheory and summability theory to prove that the above formal solution is1-summable in amonomial.
本文第一部分：通过比较系数法验证某一奇异偏微分方程存在唯一形式幂级数解，给出该级数解的具体形式，利用形式Borel变换证明此形式幂级数解关于某单项式是1-Gevrey的。
第二部分：利用泛函分析中的不动点定理以及Gevrey渐近理论及可和理论中的部分结论证明上述形式解关于某单项式是1-可和的。
Since the20th century, with the study on divergent power series solution and the exploringof the meaning of them, the classical asymptotic expansion theory and summability theory havebeen developed rapidly. And now they become Gevrey asymptotic expansion theory andmultisummability theory, which have more extansive used and wider vistas of research.
Part one: We prove the existence and uniqueness of the formal power series solution to asingular partial differential equation by comparing coeffients of an identity, given the specificform of the series solution, we construct holomorphic solution of the problem with the help ofthe formal Borel transform, and prove that the formal power series solution is1-Gevrey.
Part two: We use the fixed point theorem and part of the results in Gevrey asymptotictheory and summability theory to prove that the above formal solution is1-summable in amonomial.
引文
[1]郇中丹，黄海洋，偏微分方程，北京：高等教育出版社，2004.
[2]钟玉泉，复变函数论，北京：高等教育出版社，2004.
[3] W. Walter， An elementary proof of the Cauchy–Kowalevsky theorem， Amer.Math.Monthly，92，1985，115–126.
[4]李家春，周显初，数学物理中的渐近方法，北京：科学出版社，2002.
[5] W. Balser，W. B. Jurkat，and D. A. Lutz，Birkhoff invariants and Stokes multipliers formeromorphic linear differential equations，J. of Math. Analysis and Appl.，71，1979，48–94.
[6] B. L. J. Braaksma，Multisummability and Stokes multipliers of linear meromorphicdifferential equations，J. Differential Equations，92，1991，45–75.
[7] O. Costin，On Borel summation and Stokes phenomena for Rank-1nonlinear systems ofordinary differential equations，Duke Math. J.，93，1998，289–344.
[8] Y. Sibuya，Gevrey Asymptotics and Stokes Multipliers in Differential Equation andComputation， Algebra. M. F. Singer. ed. Academic Press，1991，131–147.
[9] H. L. Turrittin，Stokes multipliers for asymptotic solutions of a certain differential equationtransform, Amer. Math. Soc.，68，1950，304–329.
[10] R. Sch fke，The connection problem for two neighboring regular singular points of generallinear complex ordinary differential equations，SIAM J. Math. Anal.，11，1980，863–875.
[11] R. Sch fke，A connection problem for a regular and an irregular singular point of complexordinary differential equations，SIAM J. Math. Anal.，15，1984，253–271.
[12] J. P. Ramis and Y. Sibuya.，Hukuhara’s domains and fundamental existence anduniqueness theorems for asymptotic solutions of Gevrey type，Asymptot. Anal.，2，1989，39–94.
[13] J. P. Ramis and Y. Sibuya, A new proof of multisummability of formal solutions of nonlinear meromorphic differential equations，Annal. Inst. Fourier Grenoble，44，1994，811–848.
[14] W. Balser, W. B. Jurkat, and D. A. Lutz，A general theory of invariants for meromorphicdifferential equations; Part I, formal invariants，Funkcialaj Ekvacioj，22，1979，197–221.
[15] W. Balser, W. B. Jurkat, and D. A. Lutz，A general theory of invariants for meromorphicdifferential equations; Part II, proper invariants，Funkcialaj Ekvacioj，22，1979，257–283.
[16] B. L. J. Braaksma and A. Schuitman，Some classes of Watson transforms and relatedintegral equations for generalized functions，SIAM J. of Math. Analysis，7，1976，771–798.
[17] W. Balser， Formal Power Series and Linear Systems of Meromorphic OrdinaryDifferential Equations，Springer，New York，2000.
[18] W. Balser, B. L. J. Braaksma, J. P. Ramis, and Y. Sibuya，Multisummability of formalpower series solutions of linear ordinary differential equations，Asymptotic Analysis，5，1991，27–45.
[19] W. Balser， A different characterization of multisummable power series， preprintUniversit t Ulm，1990.
[20] W. Balser，Summation of formal power series through iterated Laplace transform，Universit t Ulm，preliminary version，1990.
[21] H. L. Turrittin，Convergent solutions of ordinary linear homogeneous differential equationsin the neighborhood of an irregular singular point，Acta Math，93，1955，27–66.
[22] M. Canalis-Durand, J. P. Ramis, R. Sch fke, and Y. Sibuya，Gevrey solutions of singularlyperturbed differential and difference equations，tech. rep. IRMA Strasbourg，1999，Accepted by J. reine und angew. Math.
[23] M. Canalis-Durand，J. Mozo-fernández and R. sch fke，Monomial summability and doublysingular differential equations，J. Differential Equations，233，2007，485-511.
[24]张鸿庆，泛函分析，大连：大连理工出版社，2007.
[25]丁同仁，李承治，常微分方程教程，第二版，北京：高等教育出版社，1991.
[26]王高雄，周之铭，常微分方程，2006.
[27]熊金城，点击拓扑讲义，北京：高等教育出版社，2003.
[28]时宝，王兴平等，泛函分析引论及其应用，北京：国防工业出版社，2006.
[29]郑维行，王声望，实变函数与泛函分析概要，北京：高等教育出版社，2006.
[30]吴翊，屈田兴，应用泛函分析，长沙：国防科技大学出版社，2002.
[31]胡冠章，王殿军，应用近世代数，北京：清华大学出版社，2006.
[2]钟玉泉，复变函数论，北京：高等教育出版社，2004.
[3] W. Walter， An elementary proof of the Cauchy–Kowalevsky theorem， Amer.Math.Monthly，92，1985，115–126.
[4]李家春，周显初，数学物理中的渐近方法，北京：科学出版社，2002.
[5] W. Balser，W. B. Jurkat，and D. A. Lutz，Birkhoff invariants and Stokes multipliers formeromorphic linear differential equations，J. of Math. Analysis and Appl.，71，1979，48–94.
[6] B. L. J. Braaksma，Multisummability and Stokes multipliers of linear meromorphicdifferential equations，J. Differential Equations，92，1991，45–75.
[7] O. Costin，On Borel summation and Stokes phenomena for Rank-1nonlinear systems ofordinary differential equations，Duke Math. J.，93，1998，289–344.
[8] Y. Sibuya，Gevrey Asymptotics and Stokes Multipliers in Differential Equation andComputation， Algebra. M. F. Singer. ed. Academic Press，1991，131–147.
[9] H. L. Turrittin，Stokes multipliers for asymptotic solutions of a certain differential equationtransform, Amer. Math. Soc.，68，1950，304–329.
[10] R. Sch fke，The connection problem for two neighboring regular singular points of generallinear complex ordinary differential equations，SIAM J. Math. Anal.，11，1980，863–875.
[11] R. Sch fke，A connection problem for a regular and an irregular singular point of complexordinary differential equations，SIAM J. Math. Anal.，15，1984，253–271.
[12] J. P. Ramis and Y. Sibuya.，Hukuhara’s domains and fundamental existence anduniqueness theorems for asymptotic solutions of Gevrey type，Asymptot. Anal.，2，1989，39–94.
[13] J. P. Ramis and Y. Sibuya, A new proof of multisummability of formal solutions of nonlinear meromorphic differential equations，Annal. Inst. Fourier Grenoble，44，1994，811–848.
[14] W. Balser, W. B. Jurkat, and D. A. Lutz，A general theory of invariants for meromorphicdifferential equations; Part I, formal invariants，Funkcialaj Ekvacioj，22，1979，197–221.
[15] W. Balser, W. B. Jurkat, and D. A. Lutz，A general theory of invariants for meromorphicdifferential equations; Part II, proper invariants，Funkcialaj Ekvacioj，22，1979，257–283.
[16] B. L. J. Braaksma and A. Schuitman，Some classes of Watson transforms and relatedintegral equations for generalized functions，SIAM J. of Math. Analysis，7，1976，771–798.
[17] W. Balser， Formal Power Series and Linear Systems of Meromorphic OrdinaryDifferential Equations，Springer，New York，2000.
[18] W. Balser, B. L. J. Braaksma, J. P. Ramis, and Y. Sibuya，Multisummability of formalpower series solutions of linear ordinary differential equations，Asymptotic Analysis，5，1991，27–45.
[19] W. Balser， A different characterization of multisummable power series， preprintUniversit t Ulm，1990.
[20] W. Balser，Summation of formal power series through iterated Laplace transform，Universit t Ulm，preliminary version，1990.
[21] H. L. Turrittin，Convergent solutions of ordinary linear homogeneous differential equationsin the neighborhood of an irregular singular point，Acta Math，93，1955，27–66.
[22] M. Canalis-Durand, J. P. Ramis, R. Sch fke, and Y. Sibuya，Gevrey solutions of singularlyperturbed differential and difference equations，tech. rep. IRMA Strasbourg，1999，Accepted by J. reine und angew. Math.
[23] M. Canalis-Durand，J. Mozo-fernández and R. sch fke，Monomial summability and doublysingular differential equations，J. Differential Equations，233，2007，485-511.
[24]张鸿庆，泛函分析，大连：大连理工出版社，2007.
[25]丁同仁，李承治，常微分方程教程，第二版，北京：高等教育出版社，1991.
[26]王高雄，周之铭，常微分方程，2006.
[27]熊金城，点击拓扑讲义，北京：高等教育出版社，2003.
[28]时宝，王兴平等，泛函分析引论及其应用，北京：国防工业出版社，2006.
[29]郑维行，王声望，实变函数与泛函分析概要，北京：高等教育出版社，2006.
[30]吴翊，屈田兴，应用泛函分析，长沙：国防科技大学出版社，2002.
[31]胡冠章，王殿军，应用近世代数，北京：清华大学出版社，2006.