Mathematical Sciences: Partial Differential Equations and Harmonic Analysis

这个项目代表了数学研究的四个领域。总的来说，这项工作反映了调和分析方法在偏微分方程组和算子理论问题中的应用。一个主题涉及现在被称为Fatou定理的一类非线性椭圆型方程，包括重要的p-拉普拉斯方程。对于非负解，目标是确定解具有非相切边值的程度。一个主要目标是获得关于解的基本估计，这些解取决于边界的形状，但不取决于方程中系数的光滑性。本着同样的精神，还将研究定义在非光滑区域上的二阶椭圆组和高阶椭圆型算子的存在性问题。这里，通常指的是出现在物理世界中，但没有非常平滑的边界的域。例如，允许使用角和边。只是在过去的十年里，在理解一个人可以在多大程度上建立一个良好的存在和规律性理论方面，才取得了任何重大进展。在前文中，所获得的主要结果使用了经典的双层势论和现代算符理论相结合的方法。为上述目的开发的运营者的工作将继续进行。这里的重点将是分析运营商的频谱，并从中读取关于运营商的信息。众所周知，经典算符的谱是由实数组成的，要研究的猜想是它位于一个长度为原点一半的对称区间内。最后，我们将研究二阶非散度型椭圆算子的Dirichlet问题。虽然自上世纪50年代初的S以来，关于这类算子的理论已经有了很大的发展，但人们一直假设算子的系数是光滑的。这项工作打破了传统，试图定义当算子仅由有界系数组成时，人们所说的合理解是什么意思。尽管人们必须期待具有不连续性和分布导数的解决方案，但这项工作的应用潜力非常高。

Four areas of mathematical research are represented in this project. In general terms, the work reflects the application of methods of harmonic analysis to problems in the theory of partial differential equations and operators. One theme concerns what are now called Fatou theorems for classes of nonlinear elliptic equations, including the important p-Laplacian. The object is to determine, for nonnegative solutions, the extent to which the solution has nontangential boundary values. A primary goal is to obtain basic estimates on solutions which depend on the shape of the boundary but do not depend on the smoothness of the coefficients in the equation. In the same spirit, work will also be done investigating existence questions for second order elliptic systems and higher order elliptic operators defined in nonsmooth domains. By this, one usually means domains which occur in the physical world, but which do not have very smooth boundaries. Corners and edges are allowed for example. It is only in the last decade that any major progress has been made toward understanding the extent to which one may establish a good existence and regularity theory. In the preceding, the major results obtained used techniques of classical double layer potential theory mixed with modern operator theory. Work will continue on the operators developed for the above purposes. The focus here will be to analyze the spectrum of the operators and read from it information about the operators. The spectrum is known to consist of real numbers for classical operators and the conjecture to be studied is that it lies in a symmetric interval of length one-half about the origin. Finally, work will be done on the Dirichlet problem for second order nondivergence form elliptic operators. Although a highly developed theory for such operators has existed since the early 1950's, there has always been a standing assumption that the coefficients of the operator be smooth. This work breaks with that tradition in seeking to define what one would mean by a reasonable solution when the operator is formed from merely bounded coefficients. Even though one must expect solutions with discontinuities and distributional derivatives, the potential for application of this work is very high.