Mathematical Sciences: Partial Differential Equations and Harmonic Analysis

数学研究的四个领域在这本书中有代表性。 项目 总的来说，这项工作反映了 调和分析方法在偏微分方程理论中的应用 微分方程和算子。 一个主题涉及到现在所谓的法图定理， 类的非线性椭圆方程，包括重要的 p-Laplacian 目的是确定，对于非负的 解决方案，解决方案具有非切向的程度 边界值。 一个主要目标是获得基本估计数， 取决于边界形状但不 取决于方程中系数的平滑度。 本着同样的精神，也将进行调查工作， 二阶及更高阶椭圆型方程组解的存在性问题 定义在非光滑域上的阶椭圆算子 这样， 一个通常是指发生在物理世界中的域， 它们的边界并不平滑。 角和边是 例如允许。 只是在过去的十年里， 在理解这一问题的程度方面取得了重大进展， 可以建立一个好的存在性和规律性理论。 在前面的文章中，我们使用了一些技术， 经典双层势理论与现代 算子理论 将继续开发操作程序， 用于上述目的。 这里的重点将是分析 频谱的运营商，并从中读取有关的信息 运营商 已知频谱由真实的数组成， 经典算子和猜想要研究的是，它 位于一个关于原点的长度为二分之一的对称区间内。 最后，工作将完成的狄利克雷问题， 二阶非发散型椭圆算子 虽然 这种算子的高度发达的理论自1970年以来就已经存在。 早在20世纪50年代，人们就一直假设， 算子的系数是平滑的。 这项工作打破了 在试图定义一个人的意思时， 当运营商仅由 有界系数 即使人们必须期待解决方案， 不连续性和分布导数， 这项工作的应用非常高。

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.