The research on degenerate elliptic partial differential equations by the method of real analysis

简并椭圆偏微分方程的实分析方法研究

• 批准号: 11640150
• 负责人: HORIUCHI Toshio
• 金额: $ 2.24万
• 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: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640150/
• 关键词: degeneration alliptic; Singularity; Sobolev inequality; Hardy inequality; P-harmonic; potential theory; non-linear eliptic; ハーディー不等式; p調和方程式; 非線形楕円型; 除去可能な特異点; ソボレフの不等式; ハーディーの不等式; 非線型楕円型方程式; 特異点の除去可能性; うめ込み定理; 解の正則性

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 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 weights were established.3. On the quasilinear elliptic equations with critical nonlinear terms: 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.4. Missing terms in the generalized Hardy inequalities were investigated in connection with 3.

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)。关于真退化椭圆方程解的正则性：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。关于具有临界非线性项的拟线性椭圆型方程：成功地研究了具有临界非线性项的拟线性椭圆型方程，并作为应用，确定了各种带权的加权Soblev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本质上依赖于算子的简并性。在文献3的基础上，研究了广义Hardy不等式中的缺失项。

项目成果

堀内 利郎: "Some remarks on Kato is inequality"Journal of inequalities and applications. 6. 29-36 (2000)

Toshiro Horiuchi：“关于加藤的一些评论是不平等”不平等与应用杂志 6. 29-36 (2000)。

Hoshiro, Toshihiko.: "Mourre's method and smoothing properties of dispersive equations."Comm. Math. Phys.. 202, no. 2. 255-265 (1999)

Hoshiro, Toshihiko.：“莫雷方法和色散方程的平滑特性。”Comm。

堀内利郎: "Removable singularities for semilinear degenerate elliptic equations and its applications"J.Math.of Kyoto Univi.. 40・1. 13-35 (2000)

堀内敏郎：“半线性简并椭圆方程的可移奇点及其应用”京都大学数学杂志 40・1（2000）

堀内 利郎: "Some remarks on Kato's inequality"J.of Inequalities and applications. (未定). (2000)

Toshiro Horiuchi：“关于加藤不等式的一些评论”J.of 不等式和应用（待定）。

Hoshiro, Toshihiko.: "On the estimates for Helmholtz operator."Tsukuba J. Math.. 23, no. 1. 131-149 (1999)

Hoshiro, Toshihiko.：“关于亥姆霍兹算子的估计。”Tsukuba J. Math.. 23，no。