Various functional inequalities and their applications to the variational problems

各种函数不等式及其在变分问题中的应用

• 批准号: 24840024
• 负责人: WADADE Hidemitsu
• 金额: $ 1.91万
• 依托单位: Gifu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Research Activity Start-up
• 财政年份: 2012
• 资助国家: 日本
• 起止时间: 2012-08-31 至 2014-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-24840024/
• 关键词: 関数不等式; 楕円型偏微分方程式; 変分問題

Our main purpose of the research lies in studying Sobolev type inequalities and the corresponding variational problems. Especially, we concern the critical embeddings which appear in Trudinger-Moser type inequalities, Gagliardo-Nirenberg type interpolation inequalities and so on. In a period of this fund, we investigated the existence of maximizer associated with the Trudinger-Moser type inequalities of the scaling invariant form obtained by Adachi-Tanaka in 1999, and actually proved the existence of a maximizer. On the other hand, it was known that the similar type Trudinger-Moser inequalities on the whole space which do not satisfy the scaling invariance. Ishiwata (2010) proved that the existence of maximizers for this inequality heavily depends on the dimension. Based on this fact, we considered the variational structure and clarified the relation between the scaling invariance and the existence of a maximizer.

我们研究的主要目的是研究Sobolev型不等式及其相应的变分问题。特别地，我们讨论了出现在Trudinger-Moser型不等式、Gagliardo-Nirenberg型插值不等式等中的临界嵌入。在该基金的一段时间内，我们研究了Adachi-Tanaka在1999年得到的具有尺度不变形式的Trudinger-Moser型不等式的极大化子的存在性，并实际证明了极大化子的存在性。另一方面，已知整个空间上的类似类型的Trudinger-Moser不等式不满足尺度不变性。Ishiwata(2010)证明了这个不等式的极大值的存在性在很大程度上依赖于维度。基于这一事实，我们考虑了变分结构，并阐明了标度不变性与极大值子的存在之间的关系。

项目成果

On a singular gap between Sobolev-Besov-Lorentz spaces in the limiting case

极限情况下 Sobolev-Besov-Lorentz 空间之间的奇异间隙

• 发表时间: 2012
• 作者: Ichikawa H;K Kawai;S Yamamoto;and M Kameyama;Hidemitsu Wadade
• 通讯作者: Hidemitsu Wadade

Optimal embeddings on critical S obolev-Lorentz spaces into general ized Morrey spaces

临界 Sobolev-Lorentz 空间到广义 Morrey 空间的最优嵌入

• 发表时间: 2012
• 期刊: Advances in Mathematical Sciences and Applic ations
• 作者: Tomoyuki Suzuki;and Hidemitsu Wadade
• 通讯作者: and Hidemitsu Wadade

MINIMIZING PROBLEMS FOR THE HARDY-SOBOLEV TYPE INEQUALITY WITH THE SINGULARITY ON THE BOUNDARY

• DOI: 10.2748/tmj/1332767341
• 发表时间: 2012-03
• 期刊: Tohoku Mathematical Journal
• 影响因子: 0.5
• 作者: Changshou Lin;H. Wadade

Logarithmic Hard y inequalities on Sobolev-Lorentz -Zygmund spaces in the limiting case

极限情况下 Sobolev-Lorentz-Zygmund 空间上的对数 Hard y 不等式

• 发表时间: 2013
• 作者: Kawai K;and T Tsuchiya;渡辺優;S. Ma;和田出秀光
• 通讯作者: 和田出秀光

Logarithmic Hardy inequalities on Sobolev-Lorentz-Zygmund spaces in the limiting case

极限情况下 Sobolev-Lorentz-Zygmund 空间上的对数 Hardy 不等式

• 发表时间: 2013
• 作者: Yohei Fujishima;and Kazuhiro Ishige;Shouhei Ma;渡辺優;Hidemitsu Wadade
• 通讯作者: Hidemitsu Wadade