Mathematical Sciences: Fourier Analysis and Partial Differential Equations

Fefferman 9311692这个项目将集中在调和分析和偏微分方程式方面正在进行的几个研究主题。特别是，将继续推广乘积空间上的奇异积分理论。这里的目的是分析与欧几里得空间的各种伸缩群交换的算子的性质。到目前为止，完整的理论只适用于经典膨胀和乘积膨胀。该理论的演变与卡尔德龙-齐格蒙德理论有很大不同。研究非光滑系数椭圆型方程问题的工作也将继续进行。特别地，我们将讨论具有Lp系数的Dirichlet问题的可解性。这项研究的一个重要目标是确定一个人可以在多大程度上扰动方程的系数，并且仍然保证解仍然保持在相同的勒贝格空间中。本着同样的精神，我们将研究系数只被假定为可测的非散度形式的椭圆算子。所使用的方法是用具有光滑系数的方程的解来逼近解，并确定该过程是否导致唯一的极限，而不考虑原始算子的平滑。*偏微分算子是研究一般形式的偏微分方程解的基础。通过深入抽象的方法分析这些运算符，可以得到有关相关方程类的令人惊讶的完整信息。目前的研究集中在光滑性假设仅限于最基本的算子上。通过这种方式，结果反映了微分方程式试图表示的更真实的物理世界的图景。

Fefferman 9311692 This project will focus on several ongoing research themes in harmonic analysis and partial differential equations. In particular work will continue on the extension of the theory of singular integrals on product spaces. Here the objective is to analyze the properties of operators which commute with various groups of dilations of Euclidean space. So far, a complete theory is available only for the classical dilations and product dilations. The theory evolving is quite different from the Calderon-Zygmund theory. Work will also continue on problems from elliptic equations with non-smooth coefficients. In particular, the solvability of the Dirichlet problem with Lp coefficients will be treated. An important goal in this research is to determine the extent that one can perturb coefficients of the equation and still guarantee that the solutions still remain in the same Lebesgue space. In the same spirit, elliptic operators in nondivergence form with coefficients which are only assumed to be measurable will be studied. The approach used will be that of approximating solutions by solutions to equations with smooth coefficients and determine whether or not the process leads to unique limits regardless of the smoothing of the original operator. *** Partial differential operators form a basis for studying the generic forms of partial differential equations. The analysis of these operators through deep abstract methods leads to surprisingly complete information about related classes of equations. Present research concentrates on operators in which smoothness assumptions are limited to the bare essentials. In this way, results reflect a more realistic picture of the physical world which the differential equations seek to represent.