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

摘要（Nguyen, 0901083）该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本项目将研究一类具有奇异系数和测量数据的Lane-Emden和Riccati型拟线性和完全非线性椭圆方程的几个问题，这些方程具有非线性源项，涉及解及其导数。从非线性势理论、调和分析和偏微分方程理论中，提出了各种方法来获得解及其导数的先验估计，推导电容不等式，以及建立相关非线性奇异算子的有界性。因此，在不同的显式术语中，包括几何（电容）或势理论术语，将发现可解性的完整表征。此外，解的可移动奇点和更一般的一类算子以及与亚椭圆几何和非标准Sobolev空间有关的非线性也将被考虑。所提出的研究与非线性偏微分方程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，如谐波分析，非线性势理论，控制理论和微分几何。在这个项目中发展的许多想法都是新的，它们有望在纯数学和应用数学的各种问题上带来新的结果。这些应用涉及模型的发展，以描述在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，这促进了不同科学团体之间的结果和思想交流。

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.