2004 Fiscal Year Final Research Report Summary
On the structure of singular solutions for degenerate elliptic equations
| Project/Area Number |
14540151
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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 | IBARAKI UNIVERSITY |
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Principal Investigator |
HORIUCHI Toshio Ibaraki Univ., Fac.Of Science, Professor, 理学部, 教授 (80157057)
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| Co-Investigator(Kenkyū-buntansha) |
ONISHI Kazuei Ibaraki Univ., Fac.Of Science, Professor, 理学部, 教授 (20078554)
SHIMOMURA Katunori Ibaraki Univ., Fac.Of Science, Associate Professor, 理学部, 助教授 (00201559)
ANDO Hiroshi Ibaraki Univ., Fac.Of Science, Lecturer, 理学部, 講師 (60292471)
NAKAI Eiichi Osaka Educational Univ., Fac.Of Education, Associate Professor, 教育学部, 助教授 (60259900)
SATO Tokushi Tohoku Univ., Graduate School, Institute of Science, Assistant, 大学院・理学研究科, 助手 (00261545)
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| Project Period (FY) |
2002 – 2004
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| Keywords | degenerate elliptic equation / singular solution / P-harmonic / nonlinear elliptic equation / minimal solution / Hardy inequality / Rellich inequality / missing terms |
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| Research Abstract |
1.On the structue of singular solutions for degenerate ellptic equations :
(1)We studied p-harmomic equation with strong positive nonlinear terms in the right-hand side systematically, and established the unique existence results of minimal solutions. Moreover we studied very well linearized operators at the minimal solutions. Proving fundamental properties of the linearized operators, we made clear the coercivity and positivity of the operator and applied them to construct the theory of Bifurcation.
(2)We studied various type of generalized Hardy-Sobolev-Rellich inequalities, and improved them by finding out sharp missing terms. We applied them to the study of Blow-up solutions.
2.On the regularity of solutions for genuinely degenerated elliptic equations :
The existence of bounded solutions for degenerate elliptic equations were studied. To study further regularities, multiplicative Sobolev inequalities with weghts were established.
3.On the variational problems with critical nonlinear terms and singular solutions :
Using the results in the above, singular variational problems were investigated.
4.The potential theory for degenerate elliptic operators :
Multi-parabolic operator was studied and generalized mean-value property was established.
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