1998 Fiscal Year Final Research Report Summary
Qualitative studies of solutions to elliptic equations in unbounded domains
| Project/Area Number |
09640192
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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 |
解析学
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| Research Institution | Hiroshima University |
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Principal Investigator |
USAMI Hiroyuki Hiroshima University Faculty of Integrated Arts & Sciences, assistant professor, 総合科学部, 助教授 (90192509)
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| Co-Investigator(Kenkyū-buntansha) |
NAITO Yuki Kobe University Faculty of Engineering assistant professor, 工学部, 助教授 (10231458)
NAKAYAMA Hiromichi Hiroshima University Faculty of Integrated Arts & Sciences, assistant professor, 総合科学部, 助教授 (30227970)
SHIBATA Tetsutaro Hiroshima University Faculty of Integrated Arts & Sciences, assistant protessor, 総合科学部, 助教授 (90216010)
NAITO Manabu Ehime University Faculty of Science, Professor, 理学部, 教授 (00106791)
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| Project Period (FY) |
1997 – 1998
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| Keywords | elliptic equation / degenerate Laplacian / Liouville-type theorem / Sturm-Liouville problem / oscillation / eigenvalue problem / symmetry of solutions / elliptic system |
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| Research Abstract |
(1) Oscillation criteria for elliptic equations : Effective oscillation criteria are established for second-order quasilinear elliptic equations whose leading terms are degenerate Laplacians. Our method is based on comparison principles and asymptotic theory for quasilinear ordinary differential equations. For one-dimensional case, useful information about numbers of zeros of solutions is obtained via the generalized Prufer transformation.
(2) Liouville-type theorems and nonexistence of positive solutions of BVPs : Lioville-type theorems are established for quasilinear elliptic equations whose leading terms are degenerate Laplacians and (generalized) mean curvature operators. Our results can be regarded as a natural extension of classical Liouville theorem.
(3) Symmetry of positive solutions for elliptic problems : We prove by means of the moving plane method or moving sphere method that positive solutions of elliptic equations of certain types are radially symmetric. Useful information about self-similar solutions of parabolic problems can be derived from our results.
(4) Two-parameter eigenvalue problems : Two-parameter nonlinear Sturm-Liouville problems are considered. The existence of the variational eigenvalues is established. Asymptotic formulas of eigenvalues and eigenfunctions are obtained.
(5) Asymptotic theory for solutions of elliptic systems : Semilinear elliptic systems are considered. We establish nonexistence criteria of positive solutions, Liouville-type theorems, and oscillation criteria.)
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