Partial Differential Equations and Real Harmonic Analysis

9706497 Gutierrez 本项目的主要内容是研究具有非光滑系数的非散度型退化椭圆方程和抛物方程解的性质。这样的系数可能为零，也可能为无穷大，或者两者兼而有之。 一个主要的目标是研究的有效性的Harnack原则的非负解和估计的二阶导数的解决方案。虽然可测系数的散度型方程的理论是近年来的研究热点，但非散度型方程的理论却发展得很少。非发散方程在概率论、控制论和金融学中是很自然的。 在这个方向上已知的结果涉及光滑系数或严格椭圆的情况下，但这些情况下开发的方法不适用于提出的问题。 在某些情况下，研究线性方程所需的几何是由蒙赫-安培方程给出的，这是一个完全非线性的偏微分方程。 这种方法需要研究的形状和不变性性质的水平集的解决方案的蒙赫-安培方程，进一步的工作将继续沿着这些路线。 将用于解决这组问题的基础和方法是通过Calderon-Zygmund分解的最大值原理，局部化和非线性变体。 这个数学研究是在偏微分方程，线性和非线性领域。 这些方程是将数学应用于物理世界的主要经典工具。 该项目与欧几里德空间中的调和分析有着密切的联系，这是一个在二十世纪后半叶蓬勃发展的主题，已成为提供有关偏微分方程解的定性和定量信息的不可或缺的工具。 该项目中考虑的一些方程用于大气和海洋流动的模型以及描述气体在多孔介质中的扩散。

9706497 Gutierrez The problems in this project concentrate on the study of the behavior of solutions of degenerate elliptic and parabolic equations in non-divergence form with non-smooth coefficients. Such coefficients may either vanish, be infinite, or both. A primary goal is to study the validity of the Harnack principle for nonnegative solutions and estimates of the second derivatives of solutions. Though the theory for divergence form equations with measurable coefficients has been the focus of research during recent years, the corresponding theory for non-divergence form equations is much less developed. Non-divergence equations are natural in Probability, Control Theory, and Finance. The results known in this direction concern either smooth coefficients or the strictly elliptic case, but the methods developed for those cases do not apply to the problems proposed. In some cases the geometry needed to study the linear equation is given by the Monge-Ampere equation, a fully nonlinear partial differential equation. This approach requires the study of the shape and invariance properties of the level sets of solutions to the Monge-Ampere equation, and further work will continue along these lines. The basis and methodology that will be used to solve this set of problems are via the maximum principle, localization, and nonlinear variants of the Calderon-Zygmund decomposition. This mathematical research is in the field of partial differential equations, linear and nonlinear. These equations are the principal classical tool of the applications of mathematics to the physical world. The project has a strong connection with Harmonic Analysis in Euclidean space, a subject that has flourished during the second half of the twentieth century and that has become an indispensable tool to provide qualitative and quantitative information about the solutions of partial differential equations. Some equations considered in the project are used in models of atmospheric a nd oceanic flows and in the description of the diffusion of a gas in a porous medium.