On the regularity of solutions for degenerate elliptic equations

简并椭圆方程解的正则性

• 批准号: 09640153
• 负责人: HORIUCHI Toshio
• 金额: $ 1.54万
• 依托单位: IBARAKI UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640153/
• 关键词: 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.

1.关于退化椭圆型方程解的可去性：(1)当算子的主体为线性时，我们成功地建立了一个充分条件来消除带吸收项的退化椭圆型方程解的可去性。此外，如果算子退化的集合足够光滑，那么这一条件也是必要的。(2)我们把我们关于线性算子的结果推广到了主部是非线性的情况。为了克服非线性带来的困难，我们使用了吸收项和修正电容的共轭函数(勒让德变换)。关于具临界非线性项的拟线性椭圆型方程：(1)成功地研究了具临界非线性项的拟线性椭圆型方程，并作为应用，确定了各类带权的加权Sobolev不等式的最佳常数。我们还证明了最佳常数和极值函数的存在性本质上依赖于算子的退化程度。(2)研究了退化椭圆型方程有界解的存在性。为了进一步研究正则性，建立了带权的乘性Sobolev不等式。

项目成果

K.Shirota, G.Nakamura and K.Onishi: "Numerical impedance tomography" Theoretical and Applied Mechanics. 46. 317-330 (1997)

K.Shirota、G.Nakamura 和 K.Onishi：“数值阻抗断层扫描”理论与应用力学。

Horiuchi, Toshio: "Removable singularities of solutions to degenerate semilinear elliptic equations and theirapplications. (Japanese)" Potential theory and related fields (Japanese) (Kyoto, 1997) .R.I.M.S. No.1016. 8-21 (1997)

Horiuchi, Toshio：“退化半线性椭圆方程解的可去除奇点及其应用。（日语）”势理论和相关领域（日语）（京都，1997 年）.R.I.M.S.

堀内 利郎: "Notes on removable singularities for a certain class of semilinear degenerate elliptic equations" Math.J.Ibaraki Univ. 29. 31-39 (1997)

Toshiro Horiuchi：“关于某类半线性简并椭圆方程的可去除奇点的注释” Math.J.Ibaraki Univ. 29. 31-39 (1997)

堀内利郎: "Best constant in weighted sobolev irejuality with weights boing powers of distance from the on gin" Journal on Inequaty and Applications. 1. 275-292 (1997)

Toshiro Horiuchi：“加权 sobolev 不规则性中的最佳常数，其重量与杜松子酒的距离”，《不平等与应用杂志》，1. 275-292 (1997)。

堀内利郎: "Existence of bounded solutions for semilimear degenerate elliptic equations with absorption term" Proc.Japan Acad.Ser.A Math.Sci.74・3. 43-45 (1998)

Toshiro Horiuchi：“带有吸收项的半线性简并椭圆方程的有界解的存在性”Proc.Japan Acad.Ser.A Math.Sci.74・3（1998）。