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2002 Fiscal Year Final Research Report Summary

Research on canonical forms to nonlinear elliptic boundary value problems and the global structure of all solutions including singular solutions

Research Project

Project/Area Number 12640225
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionRyukoku University

Principal Investigator

YOTSUTANI Shoji  Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (60128361)

Co-Investigator(Kenkyū-buntansha) MORITA Yoshihisa  Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (10192783)
NINOMIYA Hirokazu  Ryukoku Univ., Faculty of Sci. and Tech., Associate Professor, 理工学部, 助教授 (90251610)
YANAGIDA Eiji  Tohoku Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80174548)
IKEDA Tsutomu  Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (50151296)
OKA Hiroe  Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (20215221)
Project Period (FY) 2000 – 2002
Keywordselliptic equation / radial solution / canonical form / cross-diffusion / reaction diffusion / nonlocal
Research Abstract

First, a head investigator S. Yotsutani have obtained canonical forms and structure theorems for radial solutions to semilinear elliptic problems with Y. Kabeya and E. Yanagida.
Radial solutions of semilinear elliptic problems satisfy some boundary value problems for second order differential equations. It is seen that the boundary value problems can be reduced to a canonical form with the Dirichlet, Neumann or Robin boundary condition after suitable change of variables. We get structure theorems to canonical forms to equations with power nonlinearities and various boundary conditions. By using these theorems, it is possible to study the properties of radial solutions of semilinear elliptic equations in a systematic way, and make clear unknown structure of various equations.
Second, it is possible through the canonical forms to convert results for one problem to that of others, and moreover, to find an original methods to investigate singular solutions. As a concrete example, S. Yotsutani clarified the structure of solutions including singular solutions in a unit ball with H. Myogahara and E. Yanagida.
As related problems, we are investigating a limiting equation to a cross-diffusion equation that appears in mathematical biology. We showed that it has different kinds of singular solutions with Y. Lou and W.-M. Ni. This problem is a nonlocal nonlinear elliptic boundary problem, for which no method was known to solve it. We discovered a new method. There are a lot of interesting problems for which the method is applicable. A problem of the Ossen's spiral flow is one of them.

  • Research Products

    (16 results)

All Other

All Publications (16 results)

  • [Publications] H.Ikeda, K.Kondo, H.Okamoto, S.Yotsutani: "On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows"Communication on Pure and Applied Analysis. (to appear). (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Lou, W.-M.Ni, S.Yotsutani: "On a Limiting System in the Lotka-Volterra Competition With Cross-Diffusion"Discrete and Continuous Dynamical Systems. (to appear). (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Kabeya, E.Yanagida, S.Yotsutani: "Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems"Communication on Pure and Applied Analysis. 1. 85-102 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Myogahara, E.Yanagida, S.Yotsutani: "Structure of positive radial solutions for semilinear Dirichlet problems on a ball"Funkcial.Ekvac.. 45. 1-21 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Kabeya, H.Norishita, H.Ninomiya: "Imperfect bifurcations arising from elliptic boundary value problems. Nonlinear Anal"Nonlinear Anal.Ser.A : Theory Methods. 48. 663-684 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Jinbo, Y.Morita: "Ginzburg-Landau Equation with magnetic effect in a thin domain"Calc.Var.Partial Differential Equations. 15. 325-352 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H. Ikeda, K. Kondo, H. Okamoto and S. Yotsutani: "On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows"Communication on Pure and Applied Analysis. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Lou, W.-M. Ni and S. Yotsutani: "On a Limiting System in the Lotka-Volterra Competition with Cross-Diffusion"Discrete and Continuous Dynamical Systems. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Kabeya, E. Yanagida and S. Yotsutani: "Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems"Communication on Pure and Applied Analysis. 1. 85-102 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H. Myogahara, E. Yanagida and S. Yotsutani: "Structure of positive radial solutions for semilinear Dirichlet problems on a ball"Funkcial. Ekvac.. 45. 1-21 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Kabeya, H. Morishita and H. Ninomiya: "Imperfect bifurcations arising from elliptic boundary value problems"Nonlinear Anal. Ser. A: Theory Methods. 48. 663-684 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Jinbo and Y. Morita: "Ginzburg-Landau Equation with magnetic effect in a thin domain"Calc. Var. Partial Differential Equations. 15. 325-352 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] E. Yanagida and S. Yotsutani: "A unified approach to the structure of radial solutions for semilinear elliptic problems"Japan J. Indust. Appl. Nath.. 18. 503-519 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Kabeya, E. Yanagida and S. Yotsutani: "Global structure of solutions for equations of Brezis-Nirenberg type on the unit ball"Proc. Roy. Soc. Edinburgh Sect. A. 131. 647-665 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H. Morishita, E. Yanagida and S. Yotsutani: "Structural change of solutions for a scalar curvature equation"Differential Integral Equations. 14. 273-288 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Yotsutani: "Canonical form related with radial solutions of semilinear elliptic equations and its applications"Taiwanese J. Math.. 5. 507-517 (2001)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2004-04-14  

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