2003 Fiscal Year Final Research Report Summary
Study on Nonlinear Evolution Equations and Nonlinear Elliptic Equations
| Project/Area Number |
12440051
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Waseda University |
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Principal Investigator |
OTANI Mitsuharu Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (30119656)
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| Co-Investigator(Kenkyū-buntansha) |
ISHII Hitoshi Waseda University, School of Education, Professor, 教育学部, 教授 (70102887)
TANAKA Kazunaga Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (20188288)
YAMADA Yoshio Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (20111825)
SAKAGUCHI Shigeru Ehime University, Faculty of Science, Professor, 理学部, 教授 (50215620)
SUZUKI Takashi Osaka University, Graduate School of Engineering Science, Professor, 大学院・基礎工学研究科, 教授 (40114516)
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| Project Period (FY) |
2000 – 2003
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| Keywords | Nonlinear evolution equation / Nonlinear elliptic equation / Nonlinear PDE / Method of variation / Subdifferential operator |
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| Research Abstract |
(1)"L^∞-energy method" is invented. This assures the high differentiablity of solutions of quasilinear parabolic equations. By this method, the existence of W^<1. ∞>-solutions for a general doubly nonlinear parabolic equations and the open problem : "porous medium equations admit C^∞-solutions?" is solved affirmatively. Recent studies suggest that this gives a quite powerful tool for various problems.
(2)"The theory of nonmonotone perturbations for subdifferentials " is extended to Banach space setting. By this theory, we can treat the existence and regularity of solutions for degenerate parabolic equations in a more natural way than Galerkin' s method and open problems, left unsolved in the usual way, were solved.
(3)A Concentration Compactness (CC) theory with partial symmetry is given. The usual CC theory is known to be useful to analyze the problem with lack of compactness. On the other hand, the high symmetry such as the radial symmetry often recovers the compactness. It is studied how the partial symmetry not enough to recover compactness is reflected to CC theory. By this theory, the existence of nontrivial solutions is proved for some quasilinear elliptic equations in infinite cylindrical domains.
(4)The classical "Principle of Symmetric Criticality (PSC)" by R.Palais assures that under suitable conditions, critical points in the subspace with the symmetry give real critical points in the whole space, but is restricted to the system with variational structures. PSC is extended to a more general theory which covers the elliptic systems without full symmetry or evolution equations including time evolution terms.
(5)A new degree theory is established. It can teat mutivuled operators including subdifferential operators and cover nonlinear PDE with various multivaluedness nature.
(6)The theory of nonmonotone perturbations for subdifferentials is ameliorated to cover the initial-boundary value problems and time periodic problems for magneto-micropolar fluid equations.
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