Mathematical Sciences: "Partial Differential Equations"

该奖项支持三名研究人员在各种 问题的各个方面结合了几个复杂的变量， 偏微分方程一个主要的重点将是 二阶正则性的继续研究 微分算子，复导数d-杆， 一个函数是全纯的， 限于流形或域边界的关联算子 称为d-bar-B运算符。 规律性的研究 次椭圆不等式 规律性分析， 通过这些不平等现象， 在分析的意义上，以及无限的， 可微性与规律性有关，也可能是 研究它的最强动机是如何 平滑投影后保留的函数边界值 全纯函数空间的一个例子。 一个具体 投影的表达式是通过对 伯格曼核。 这个项目的第二个目的是了解 Bergman核在各种不同的 域，尤其是有限类型的域。三分之一，更多一些 几何工作线涉及层次的正则性理论 表面，晶体生长问题和退化方程。 微分方程解的水平面，特别是 那些分析热的粘性解的水平集的 方程将被研究。 最基本的问题之一 ”这是一个由德Giorgi，寻求。 表面的规则性作为参数的函数， 微分方程 偏微分方程是 物理科学中的数学建模。 的现象 包括连续变化，如运动、材料 和能量都服从某些普遍规律， 可以用相互作用和关系来表达 偏导数之间的关系 数学的关键作用不是 来陈述这些关系，而是为了提取定性的 和量化的意义。

This award supports three investigators working on various aspects of problems combining several complex variables and partial differential equations. One primary focus will be continuing research on the regularity of two first order differential operators, the complex derivative d-bar which measures the degree to which a function is holomorphic and the associated operator restricted to manifolds or domain boundaries called the d-bar-b operator. The study of regularity is governed by sub-elliptic inequalities. Regularity is analyzed both locally and globally through these inequalities, and will be considered both in the analytic sense as well as that of infinite differentiability. Related to regularity, and possibly one of the strongest motivations for studying it, is the question of how smooth the boundary values of a function remain after projection onto the space of holomorphic functions on a domain. A concrete expression for the projection is given by integration against the Bergman kernel. A second thrust of this project is to understand the boundary behavior of the Bergman kernel on various kinds of domains, especially those of finite type. A third, somewhat more geometric line of work concerns the regularity theory of level surfaces, problems of crystal growth and degenerate equations. Level surfaces of solutions of differential equations, expecially those which analyze the level sets of viscosity solutions of heat equations will be studied. Among the most fundamental questions to be considered is one due to De Giorgi, which seeks the regularity of the surfaces as a function of a parameter in the differential equation. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.