Nonlinear Equations, Weighted Norm Inequalities, and Best Constants in Harmonic Analysis

摘要：本文研究了一类非线性偏微分和积分方程及其相关的加权范数不等式。一个系统的使用将作出的功能空间内在地与所涉及的问题，并矢线性和非线性模型，非线性潜在的理论，拟度量空间的齐次和非齐次型（没有加倍财产），以及其他手段的现代分析。主要目标是完全表征可解性问题（即，找到必要和匹配的充分条件），并得到了在低阶项和数据上具有非常一般系数的非线性椭圆和抛物偏微分方程解的精确估计，以及具有一般核的非线性积分算子解的精确估计，同时也将考虑具有一般势（可能是复值分布）的Schrodinger型算子的线性问题.对所涉及的常数进行良好的控制是本研究的重要组成部分。特别是，将特别注意出现在调和分析和算子理论的相关问题中的最佳常数不等式，所提出的研究中考虑的许多问题是由数学物理，控制理论和随机过程的研究所激发的。其结果是，尖锐的不等式和可解性的标准将被发现的方程和运营商，描述重要的现象与线性和非线性源出现在量子物理，流体流动，传热，电磁问题。

ABSTRACT:The proposed research is concerned with the investigation of certain classes of nonlinear partial differential and integral equations, andrelated weighted norm inequalities. A systematic use will be made of function spaces intrinsically connected with the problems involved, dyadic linear and nonlinear models, nonlinear potential theory, quasimetrics on spaces of homogeneous and nonhomogeneous type (without the doubling property), and other means of modern analysis. The main goal is to characterize completely the solvability problem (i.e., find necessary and matching sufficient conditions), and obtain sharp estimates for solutions of nonlinear elliptic and parabolicPDEs with very general coefficients at the lower order terms and data, as well as for nonlinear integral operators with general kernels.Linear problems for operators of Schrodinger type with general potentials (possibly complex valued distributions) will be considered as well. A good control of the constants involved is an essential part of this study. In particular, a special attention will be paid to best constant inequalities which appear in related problems of harmonic analysis and operator theory.Many questions considered in the proposed research are motivated bystudies in mathematical physics, control theory, and stochastic processes. As a result, sharp inequalities and criteria of solvability will be found for equations and operators which describe important phenomena with linear and nonlinear sources appearing in quantum physics, fluid flow, heat transfer, and electromagnetism problems.