2007 Fiscal Year Final Research Report Summary
Studies on construction of solutions of nonlinear elliptic equations arising from Gaugetheories and on the asymptotic behavior of their heat flows
| Project/Area Number |
17540186
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Osaka City University |
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Principal Investigator |
TAKAHASHI Futoshi Osaka City University, Graduate School of Science, associate professor (10374901)
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| Co-Investigator(Kenkyū-buntansha) |
SUZUKI Takashi Osaka University, Graduate School of Engineering Science, professor (40114516)
KOZONO Hideo Tohoku University, Graduate School of Science, professor (00195728)
OGAWA Takayoshi Tohoku University, Graduate School of Science, professor (20224107)
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| Project Period (FY) |
2005 – 2007
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| Keywords | Gauge Theory / Nonlinear Elliptic Equations / Critical Nonlinearity |
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| Research Abstract |
Head investigator, F. Takahashi, in cooperation with other investigators, has made studies on the construction, asymptotic behavior, and blow-up analysis of solutions to various nonlinear elliptic equations arising from Gauge theories which were proposed by physicists.
Also we have studied the long time asymptotics of time global solutions, or blow up behavior of time local solutions to heat flows associated with the above nonlinear elliptic equations. Such nonlinear elliptic equations have variational structures, and have strong relations to the critical inequalities such as Sobolev, or Trudinger-Moser inequality on compact manifolds or on bounded domains in Euclidean spaces. They also have quantized blow-up mechanisms and exhibit mass-concentration phenomena commonly.
In the former half of the term of our project, we have established the existence of solutions to some mean filed equations which come from the statistical mechanics of many vortices with a neutral orientation in a perfect fluid. Our study has become a trigger of other studies of the equilibrium mean field equations, and now, this is one of the most active area in the fields.
In the latter half, head investigator has begun to study the blow up analysis and some qualitative properties of blowing-up solutions to nonlinear elliptic equations with the critical Sobolev exponents. These studies lead to the current research project of my own.
In summary, we have clarified the relations between various properties of blowing-up solutions and those of singular limits, pictured typical mass or energy quantization phenomena, and established many analytical tools which will be useful to the future studies of this kind, through our research project.
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