On the linearization of quasilinear degenerate elliptic equations and the structure of singular solutions

拟线性简并椭圆方程的线性化及奇异解的结构

• 批准号: 17540146
• 负责人: HORIUCHI Toshio
• 金额: $ 2.37万
• 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: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540146/
• 关键词: degenerate elliptic equation; singular solution; P-harmonic; weighted Sobolev ineauality; linearization; Hardy ineouality; Rellich inequality; missing terms; 最良定数; 変分問題; ハーディの不等式; 非線形楕円型方程式; P-調和作用素; ハーディーの不等式; 最小解

1. On the structure of singular solutions for degenerate elliptic equations:(1) We studied various types of Hardy-Sobolev-Rellich inequalities, and improved them by finding out sharp missing terms. We applied them to the study of Blow-up solutions.(2) We studied the weighted Hardy-Sobolev inequalities. For the Caffarelli-Kohn-Nirenberg type weight functions, the inequalities have been improved by adding sharp remainder terms. In particular for the critical weight case and the supercritical case new CKN-inequalities were obtained.(3) We studied CKN-inequalities and extended them to the critical and supercritical case. This gives not only new imbedding theorems but also new framework for the PDE problems. We also established Existence and Nonexistence results on the extremals, asymptotic behaviors of the best constant and some qualitative properties of extremals. We applied our inequalities to study the boundary value problems having singular potential with the best constant as its coefficient in nontrivial functional framework.2. On the linearization of quasi-linear elliptic PDE.P-harmonic equation with strong positive nonlinear terms in the right-hand side has been studied systematically and established the unique existence results of minimal solutions. Moreover we studied very well linearized operators at the minimal solutions to made clear the coercivity and positivity of the operator and to construct the theory of Bifurcation.

1.关于退化椭圆型方程奇异解的结构：（1）研究了各种类型的Hardy-Sobolev-Rellich不等式，并通过寻找尖锐的缺失项改进了它们。我们将它们应用于Blow-up解的研究。(2)研究了加权Hardy-Sobolev不等式.对于Caffarelli-Kohn-Nirenberg型权函数，通过添加尖锐的余项改进了不等式。特别是对于临界权情形和超临界情形，得到了新的CKN-不等式。(3)研究了CKN-不等式，并将其推广到临界和超临界情形。这不仅给出了新的嵌入定理，而且为偏微分方程问题提供了新的框架。建立了极值的存在性和不存在性、最佳常数的渐近性态以及极值的一些定性性质。在非平凡泛函框架下，我们应用这些不等式研究了具有奇异位势且系数为最佳常数的边值问题.关于拟线性椭圆型偏微分方程的线性化问题，系统地研究了右端具有强正非线性项的P-调和方程，建立了极小解的唯一存在性结果。此外，我们还研究了在极小解处线性化得很好的算子，以明确算子的正性和非负性，从而建立分支理论。

项目成果

Kato's inequalities for degenerate quasilinear elliptic operators

简并拟线性椭圆算子的加藤不等式

• 发表时间: 2008
• 期刊: Kyungpook Mathematical Journal 48
• 作者: Kevin Burrage;Yoshio Komori(発表者);堀内利郎
• 通讯作者: 堀内利郎

重み付きハーディー・ソボレフの不等式の精密化について

关于加权Hardy-Sobolev不等式的改进

• 发表时间: 2007
• 作者: Nishio;Masaharu;安藤 広
• 通讯作者: 安藤 広

On the weighted Hardy-Sobolev inequalities

关于加权 Hardy-Sobolev 不等式

• 发表时间: 2007
• 作者: Nishio;Masaharu;安藤 広;H. Ando
• 通讯作者: H. Ando

Construction of an atomic decomposition for functions with compact support

构造具有紧凑支持的函数的原子分解

• 发表时间: 2006
• 期刊: J. of Math. Analysis and Applications Vol.313
• 作者: Norio Kikuchi;Eiichi Nakai;Naohito Tomita;Kozo Yabuta and Tsuyoshi Yoneda;中井 英一;Eiichi Nakai
• 通讯作者: Eiichi Nakai

茨城大学研究者情報総覧

茨城大学研究员信息一览