Nonlinear harmonic analysis and partial differential equations of Lane-Emden and Riccati type

非线性调和分析以及 Lane-Emden 和 Riccati 型偏微分方程

• 批准号: 0901083
• 负责人: Phuc Nguyen
• 金额: $ 11万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901083&HistoricalAwards=false
• 关键词: Nonlinear; harmonic; analysis; partial; differential

Abstract (Nguyen, 0901083)This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project will investigate several problems stemming from a class of quasi-linear and fully nonlinear elliptic equations of Lane-Emden and Riccati type with singular coefficients and measure data, and with nonlinear source terms that involve both solutions and their derivatives. Various methods from nonlinear potential theory, harmonic analysis, and the theory of partial differential equations are proposed to obtain sharp a priori estimates for solutions and their derivatives, to derive capacitary inequalities, and to establish the boundedness of the related nonlinear singular operators. As a consequence, complete characterizations of solvability will be found in different explicit terms including geometric (capacitary) or potential theoretic terms. Moreover, removable singularities of solutions and a more general class of operators and nonlinearities related to subelliptic geometry and nonstandard Sobolev spaces will also be considered.The proposed research is closely related to central problems in nonlinear partial differential equations that require hard analysis and major tools from different branches of mathematics, such as harmonic analysis, nonlinear potential theory, control theory, and differential geometry. Many of the ideas to be developed in this project are new, and they promise to bring new results to bear on applications to diverse problems in both pure and applied mathematics. These applications involve the development of models to describe various phenomena that are studied in physics and engineering, including heat transfer, mass transfer, and fluid flow. The proposed research is tightly integrated with the teaching of graduate students and the interaction of the principal investigator with a network of researchers worldwide, which promotes the exchange of results and ideas between diverse scientific communities.

摘要（阮，0901083）该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助。本计画将研究一类具奇异系数与量测资料之Lane-Emden与Riccati型拟线性与全非线性椭圆型方程，其非线性源项包含解与其导数。从非线性势理论，调和分析，偏微分方程理论的各种方法被提出来获得尖锐的先验估计的解决方案和他们的衍生物，推导出容量不等式，并建立相关的非线性奇异算子的有界性。因此，完整的可解性特征将在不同的明确条款，包括几何（电容）或潜在的理论条款。 此外，解的可移除奇异性和更一般的一类算子以及与次椭圆几何和非标准Sobolev空间相关的非线性也将被考虑。拟议的研究与非线性偏微分方程中的中心问题密切相关，这些问题需要来自不同数学分支的硬分析和主要工具，如调和分析，非线性势理论，控制理论和微分几何。在这个项目中开发的许多想法都是新的，他们承诺带来新的结果，以承担在纯数学和应用数学的各种问题的应用。这些应用涉及模型的开发，以描述物理和工程中研究的各种现象，包括传热，传质和流体流动。拟议的研究与研究生的教学以及首席研究员与全球研究人员网络的互动紧密结合，这促进了不同科学界之间的成果和想法交流。