Study of singularities in partial differential equations

2001 Fiscal Year Final Research Report Summary

• Project/Area Number: 12640189
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Sophia University
• Principal Investigator: TAHARA Hidetoshi Sophia University, Faculty of Science and Technology, Professor, 理工学部, 教授 (60101028)
• Co-Investigator(Kenkyū-buntansha): YOSHINO Kunio Sophia University, Faculty of Science and Technology, Lecturer, 理工学部, 講師 (60138378) ; UCHIYAMA Koichi Sophia University, Faculty of Science and Technology, Professor, 理工学部, 教授 (20053689) ; OUCHI Sunao Sophia University, Faculty of Science and Technology, Professor, 理工学部, 教授 (00087082) ; YAMANE Hideshi Chiba Institute of Technology, Faculty of Technology, Lecturer, 工学部, 講師 (80286145) ; OKADA Yasunori Chiba University, Faculty of Science, Ass. Professor, 理学部, 助教授 (60224028)
• Project Period (FY): 2000 – 2001
• Keywords: nonlinear PDE / linear PDE / complex domain / singularity / holomorphic solution / Fuchsiar type

Research Abstract

1. Nonlinear totally characteristic type partial differential equations were studied in the complex domain and the following results were obtained :
(1) Unique existence result of the holomorphic solution was proved under Poincare condition and non-resonance condition.
(2) The solvability was proved also in the case where non resonance condition is not satisfied.
2. Solutions with singularities on a hypersurface were studied in linear and nonlinear partial differential equations and the following results were obtained :
(1) Solvability of a particular class of linear partial differential equations Puf was proved in the space of functions with regular singularities.
(2) By using the integral representation of homogeneous solutions, the asymptotic behavior was determined.
(3) Branching of singularities was studied in linear Fuchsian partial differential equations.
(4) Non-existence of a class of singularities was proved in nonlinear Fuchsian partial differential equations.
3. The possibility of the analytic continuation of holomorphic solutions to nonlinear partial differential equations was studied and the non-existence of some class of singularities was proved.
4. A class of p-elliptic differential equations was studied and the structure of regular singularities was determined.

Research Products

• [Publications] H.Chen, H.Tahara: "On the holomorphic solutions of non-linear totally characteristic equations"Math.Nachr.. 219. 89-96 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] H.Tahara: "On the singular solutions of nonlinear singular partial differential equations"J.Math.Soc.Japan. 53. 711-729 (2001)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] H.Chen, Z.Luo, H.Tahara: "Formal solutions of nonlinear first order totally characteristic type PDE with irregular singularities"Ann.Inst.fouvier. (to appear).
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] H.Tahara: "On the singularitie of solutions of nonlinear partial differential equations in the complex domain"Micvolocal Anaysis and Fourier Analysis. (to appear).
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] T.Mandai, H.Tahara: "Structure of solutions to Fuchsian systems of partial differential equations"Nagoya Math.J.. (to appear).
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] J.E.C.Lope, H.Tahara: "On the analytic continuation of solutions to nonlinear partial differential equation"J.Math.Pure et Appl.. (to appear).
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] S.Ouchi: "Growth property and slowly in creasing behaviour of singulan solutions of linear partial differential equations in the complex domain"J.Math.Soc.Japan. 52. 767-792 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] S.Ouchi: "Asymptotic expansion of singular solutions and the characteristic polygon of linear partial differential equations in the complex domain"Publ.RIMS, Kyoto Univ.. 36. 457-482 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] K.Uchiyama: "Graphical illustration of Stokes phenomenon of integrals with saddles"Toward the exact WKB analysis of differential equation, linear and non-linear. 87-95 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] K.Balachardran, K.Uchiyama: "Existence of local solutions of quasi-linear integrodifferetid equation in Banach spaces"Appl.Anal.. 76. 1-8 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] K.Balachandran, K.Uchiyama: "Existence of solutions of nonlinear integrodifferential equations of Sobolev type wit nonlocal condition in Barach spaces"Proc.Indian Acad.Sci.. 110. 225-232 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] O.Liess, Y.Okada, N.Tose: "Hartogo' phenomena for microfurctions with holomorpghic parameters"Publ.RIMS, Kyoto Univ.. 37. 221-238 (2001)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] R.Ishimura, J.Okada, Y.Okada: "Continuation of holomorphic solutions to convolution equations, in complex domains"Annales Polonici Mathematici. 74. 105-115 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] Y.Okada: "On the characteristics for convolution equations in tube domain"J.Math.Soc.Japan. 52. 535-544 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] H.Yamane: "Ramified Cauchy problem for a class if Fuchsian operators with tangent characteristics"J.Math.Pune et Appl.. 79. 271-294 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] H.Yamane: "Global Fuchsian Cauchy problem"J.Math.Sci.Univ.Tokyo. 7. 147-162 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] H.Yamane: "Fourter-Ehrenpreis integral representation of harmonic functions in four variables"(preprint).
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] H.Hirata, Miao Changxing: "Space-time estimates of linear flow and application to some nonlinear integro-differential equations corresponding to fractional order time derluative"Advances in Differential Equations. 7. 217-236 (2002)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] M.Tsuji: "Special reflexive modules and the first Cher class"Leefurenotes in pure and appl.Math.. 214. 549-581 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] M.Tsuji: "Dimca hypersurfaces and Nagata automorphisms"Math.Nachr.. 218. 175-184 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] M.Tsuji: "Deformation of singularities via quivers"Proc.Second ISAAC Congres. 839-848 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] M.Tsuji: "Deformation spaces of quotient surface singularities"Rroc.8th Int.Collog.on Complex Analysis. 238-243 (2000)
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] H.Chen - H.Tahara: "On the holomorphic solutions of non-linear totally characteristic equations"Math. Nachr.. 219. 89-96 (2000)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] H.Tahara: "On the singular solutions of nonlinear singular partial differential equations"J. Math. Soc. Japan. 53. 711-729 (2001)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] H.Chen - Z.Luo - H.Tahara: "Formal solutions of nonlinear first order totally characteristic type PDE with irregular singularities"Ann. Inst. Fourier.. (to appear in).
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] H.Tahara: "On the singularities of solutions of nonlinear partial dif ferential equations in the complex domain, to appear in "Microlocal Analysis and Fourier Analysis""(to the published from World Sci.).
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] T.Mandai - H.Tahara: "Strncture of solutions to Fuchsian systems of partial differential equations"Nagoya. J. Math.. to appear in.
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] J.E.C.Lope - H.Tahara: "On the analytic continuation of solutions to nonlinear partial differential equations"J. Math. Pure et Appl.. to appear in.
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] S.Ouchi: "Growtn property and slowly increasing behavior of sin gular solutions of linear partial differential equations in the complex clomain"J. Math. Soc. Japan. 52-3. 767-792 (2000)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] S.Ouchi: "Asymptotic expansion of singular solutions and the char acteristic polygon of linear partial differential equations in the com plex domain"Publ. RIMS. Kyoto Univ.. 36-4. 457-482 (2000)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] K.Uchiyama: "Graphical illustration of Stokes phenomenon of inte grals with saddles"in "Toward the exact WKB analysis of differen tial equations, linear or non-linear" ed. by C. J. Howls, T. Kawai and Y. Takei, Kyoto U. P., Kyoto. 87-95
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] K.Balachandran - K.Uchiyama: "Existence of local solutions of quasi-linear integroclifferential equations in Banach spaces"Appl. Anal. 76. 1-8 (2000)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] K.Balachandran - K.Uchiyama: "Existence of solutions of nonlinear iritegroclifferential equations of Sobolev type with nonlocal condi tion in Banach spaces"Proc. Indian Acad. Sci. 110. 225-232 (2000)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] O.Liess - Y.Okada - N.Tose: "Hartogs' phenomena for.microfunc tions with holomorphic paramcters"Publ. RIMS. Kyoto Univ. 37(2). 221-238 (2001)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] R.Ishimura - J.Okada - Y.Okada ;: "Continuation of holomorphic solutions to convolution equations in complex domains"Annales Polonici Mathernatici. 74. 105-115 (2000)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] Y.Okada: "On the characteristics for convolution equations in tube domains"J. Math. Soc. Japan. 52(3). 535-544 (2000)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] H.Yamane: "Ramified Cauchy problem for a class of Fuchsian opera tors with tangent characteristics"Journal de Matheinatiques Pures et Appliquees. 79(3). 271-294 (2000)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] H.Yamane: "Global Fuchsian Cauchy problem, Journal of Mathe-matical Sciences"Univ. of Tokyo. 7. 147-162 (2000)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] H.Yamane: "Fourier-Ehrenprels integral representation of harmonicfunctions in four variables"preprint.
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] H.Yamane: "fourier-Ehrenpreis integral representation of harmonic functions"preprint.
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] H. Hirata-Miao Ohangxing ;: "Space-time estimates of linear flow and application to some nonlinear integro-differential equations corre-sponding to fractional order time derivative/Advances in Differen-"Advances in Differen-tial Equations,. 7, No.2. 217-236 (2002)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] M. Tsuji: "Special reflexive modules and the first Chern class, Finite or Infinite Dimensional Complex Analysis"LecluTe Notes in Pure and Applied Mathematics, 214, Marcel Dekker Inc, (New York-Basel). Chapter 57. 549-581 (2000)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] M. Tsuji: "Dimca hypersurfaces and Nagata automorphisms, Math-ernatische Nachrichten"WILEY-VCH Verlag Berlin GmbH. 218. 175-184 (2000)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] M. Tsuji: "Deformation of singularities via quivers"proceedings of the Second ISAAC Congress, Kluwer Academic Publisher (Dortrect-Boston- London). Chapter 95. 839-848 (2000)
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] M. Tsuji: "Deformation spaces of quotient surface singularities"Pro ceedings of the eighth international colloquium on complex analy sis,. 238-243 (2000)
  • Description: 「研究成果報告書概要(欧文)」より