Partial Differential Equations and Real Harmonic Analysis

9706497古铁雷斯，这个项目中的问题集中在研究具有非光滑系数的非散度形式的退化椭圆和抛物型方程的解的行为。这样的系数可以为零，也可以为无穷大，或者两者兼而有之。主要目的是研究非负解的Harnack原理的有效性以及解的二阶导数的估计。虽然具有可测系数的散度型方程的理论是近年来研究的热点，但相应的非散度型方程的理论发展较少。无散度方程在概率论、控制论和金融学中是很自然的。在这个方向上已知的结果要么涉及光滑系数，要么涉及严格椭圆情况，但为这些情况开发的方法不适用于所提出的问题。在某些情况下，研究线性方程所需的几何形状由Monge-Ampere方程给出，这是一个完全非线性的偏微分方程式。这种方法需要研究Monge-Ampere方程解的水平集的形状和不变性，并将沿着这些方向继续进行进一步的工作。用于解决这一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。这项数学研究是在偏微分方程领域，线性和非线性的。这些方程是将数学应用于物理世界的主要经典工具。该项目与欧几里得空间的调和分析有很强的联系，这是一个在二十世纪后半叶蓬勃发展的学科，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。项目中考虑的一些方程被用在大气和海洋流动的模型中，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.