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

摘要(阮氏，0901083)该奖项是根据《2009年美国复苏和再投资法案》(公法第111-5条)资助的。这个项目将研究一类拟线性和完全非线性的Lane-Emden和Riccati型椭圆型方程的几个问题，该方程具有奇异系数和测量数据，并且具有涉及解及其导数的非线性源项。从非线性位势理论、调和分析和偏微分方程组理论出发，提出了各种方法来获得解及其导数的精确先验估计，导出容量不等式，以及建立相关的非线性奇异算子的有界性。因此，可以用不同的显式项，包括几何(容量)项或势理论项，找到可解性的完整特征。此外，还将考虑解的可去奇性以及与次椭圆几何和非标准Sobolev空间相关的更一般的一类算子和非线性。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困难分析和主要工具，如调和分析、非线性位势理论、控制理论和微分几何。在这个项目中要开发的许多想法都是新的，它们承诺将带来新的结果，使其适用于纯数学和应用数学中的各种问题。这些应用包括开发模型来描述在物理和工程中研究的各种现象，包括热传递、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世界各地的研究人员网络的互动紧密结合在一起，这促进了不同科学界之间的结果和思想的交流。

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.