Studies on nonlinear partial differential equations by potential analysis

利用势分析研究非线性偏微分方程

• 批准号: 19740062
• 负责人: HIRATA Kentaro
• 金额: $ 2.47万
• 依托单位: Akita University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Young Scientists (B)
• 财政年份: 2007
• 资助国家: 日本
• 起止时间: 2007 至 2009
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-19740062/
• 关键词: 実解析; 関数方程式; 優調和関数; 境界挙動; 半線形楕円型方程式; Helmholtz方程式; 調和測度; John領域; Green関数; p-Laplace方程式; 一様領域; 準一様領域; Martin核; 境界増大度; 非線形楕円型方程式

In a bounded smooth domain, we established a suitable boundary growth estimate for positive superharmonic functions satisfying a nonlinear inequality. Moreover, we proved the existence of positive solutions of semilinear elliptic equations that are comparable to the Poisson kernel. Also, we gave a sufficient condition for Dirichlet boundary data to guarantee the existence of positive solutions of singular semilinear elliptic equations. In regard to potential theory, we studied a doubling property of harmonic measure, an estimate for the product of the Green function and the Martin kernel, and the boundary behavior of Martin kernels.

在有界光滑区域中，我们建立了满足一个非线性不等式的正超调和函数的一个合适的边界增长估计。此外，我们还证明了与泊松核相当的半线性椭圆型方程正解的存在性。给出了奇异半线性椭圆型方程Dirichlet边值问题正解存在的一个充分条件。在位势理论方面，我们研究了调和测度的倍增性质，格林函数与Martin核的乘积的估计，以及Martin核的边界性质。

项目成果

Boundary behavior of solutions of the Helmholtz equation

亥姆霍兹方程解的边界行为

• 发表时间: 2009
• 期刊: Canadian Mathematical Bulletin Vol.52
• 作者: Witold Karwowski;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;安田公美;安田公美;安田公美;Kumi Yasuda;安田公美;安田公美;Kumi Yasuda;安田公美;Kumi Yasuda;K. Hirata
• 通讯作者: K. Hirata

Boundary behaviour of quotients of Martin kernels

Martin 核商的边界行为

• 发表时间: 2007
• 期刊: Proceedings of the Edinburgh Mathematical Society 50
• 作者: Kentaro;Hirata
• 通讯作者: Hirata

The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations

超调和函数的边界增长与非线性椭圆方程的正解

• 发表时间: 2008
• 期刊: Mathematische Annalen Vol.340
• 作者: H. Aikawa;K. Hirata;K. Hirata;K. Hirata
• 通讯作者: K. Hirata

Boundary behavior of quotients of Martin kernels

Martin 核商的边界行为

• 发表时间: 2007
• 期刊: Proceedings of the Edinburgh Mathematical Society Vol.50
• 作者: Kentaro;Hirata;K. Hirata;K. Hirata;K. Hirata
• 通讯作者: K. Hirata

Limits at infinity of superharmonic functions and solutions of semilinear elliptic equations of Matukuma type

超调和函数的无穷大极限及Matukuma型半线性椭圆方程的解

• 发表时间: 2009
• 期刊: Potential Analysis Vol.30
• 作者: Witold Karwowski;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;Kumi Yasuda;安田公美;安田公美;安田公美;Kumi Yasuda;安田公美;安田公美;Kumi Yasuda;安田公美;Kumi Yasuda;K. Hirata;K. Hirata;K. Hirata
• 通讯作者: K. Hirata