On the regularity of solutions for degenerate elliptic equations
Grant-in-Aid for Scientific Research (C)
|Allocation Type||Single-year Grants |
|Research Institution||IBARAKI UNIVERSITY |
HORIUCHI Toshio IBARAKI Univ.Fuc.of Science Prof., 理学部, 教授 (80157057)
SHIMOMURA Katsunori IBARAKI Univ.Fuc.of Science lectuel, 理学部, 講師 (00201559)
OSHIMA Hideaki IBARAKI Univ.Fuc.of Science Prof., 理学部, 教授 (70047372)
TAKANO Katsuo IBARAKI Univ.Fuc.of Science Prof., 理学部, 教授 (30007827)
ONOSE Hiroshi IBARAKI Univ.Fuc.of Science Prof., 理学部, 教授 (80007559)
ONISHI Kuzuei IBARAKI Univ.Fuc.of Science Prof., 理学部, 教授 (20078554)
荷見 守助 茨城大学, 理学部, 教授 (60007549)
田村 英男 茨城大学, 理学部, 教授 (30022734)
|Project Period (FY)
1997 – 1998
Completed (Fiscal Year 1998)
|Budget Amount *help
¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 1998: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1997: ¥1,400,000 (Direct Cost: ¥1,400,000)
|Keywords||degenerate elliptic equation / regularity / removable singularity / Sobolev inequality / ポテンシャル論 / ソボレフのうめ込み定理 / キャパシティ / 非線形楕円型方程式 / 解の有界性 / 特異点の除去可能性 / 解の数値解析 / うめ込み定理|
1. On the Removable singularities of solutions for degenerate ellptic equations :
(1) When the principal part of the operator is linear, we have succeeded in establishing a sufficient condition in order to remove singularities of solutions of degenerate elliptic equations with absorption terms. Moreover, if the set on which the operator may be degenerate is smooth enough, then this condition is also necessary.
(2) We have extended our results on the linear operators to the case when the principal part is nonlinear. In order to overcome the difficulties caused from nonlinearlity, we used the conjugate function (Legendre transform) of the absorption term and modified capacities.
2. On the quasilinear elliptic equations with critical nonlinear terms :
(1) Quasilinear elliptic equations with critical nonlinear terms were successfully investigated, and as an application, the best constants for various types of weighted Sobolev inequalities with weights were determined. We also showed that the best constants and the existence of extremal functions essentially depend on the degeneracy of the operators in a very subtle way.
(2) The existence of bounded solutions for degenerate elliptic equations were studied. To study further regularities, multiplicative Sobolev inequalities with weights were established.
Report (3 results)
Research Products (27 results)