2011 Fiscal Year Final Research Report
New development and its application of Aubry-Mather theory for Hamilton-Jacobi equations
| Project/Area Number |
21540168
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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 |
Basic analysis
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| Research Institution | University of Toyama |
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Principal Investigator |
YASUHIRO Fujita 富山大学, 大学院・理工学研究部(理学), 教授 (10209067)
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| Co-Investigator(Renkei-kenkyūsha) |
HITOSHI Ishii 早稲田大学, 教育総合科学学術院, 教授 (70102887)
KATSUSHI Ohmori 富山大学, 人間発達学部, 教授 (20110231)
KATSUYUKI Ishii 神戸大学, 海事科学研究科, 准教授 (40232227)
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| Project Period (FY) |
2009 – 2011
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| Keywords | 関数方程式 / ハミルトン-ヤコビ方程式 / オーブリー・マザー理論 / 対数型ソボレフの不等式 |
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| Research Abstract |
About the study of this Kakenhi, I have obtained several results related with the Aubry-Mather theory and talked about these results in several conferences and seminars. These results are published in some journals.
The first result is to clarify the relation between the quotient Aubry sets and uniqueness sets for minimization formula for Hamilton-Jacobi equations. The second one is to provide a new proof of classical inequalities by using a comparison theorem for the Aubry set of Hamilton-Jacobi equations. The third one is to derive an optimal logarithmic Sobolev inequality with Lipschitz constant. In the proof of this inequality, an asymptotic solution of the Aubry-Mather theory for a Hamilton-Jacobi equation is used. The fourth one is to investigate a rate of convergence appearing in the asymptotic behavior of a viscosity solution to the Cauchy problem for the Hamilton-Jacobi equation with quadratic gradient term. I showed that the semiconvexity property of this Hamiltonian is an important factor which determines this rate. Here, the Aubry set is closely related with the semiconvexity property of this Hamiltonian.
As a conclusion, I think that I have done a complete job about the study of this Kakenhi by using the Aubry-Mather theory.
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