Mathematical Sciences: "Partial Differential Equations"

该奖项支持三位研究人员在多个复杂变量和偏微分方程组合问题的各个方面进行研究。一个主要的焦点将是继续研究两个一阶微分算子的正则性，即测量函数全纯程度的复导数d-bar和限制于流形或域边界的相关算子d-bar-b算子。正则性的研究是由次椭圆不等式支配的。通过这些不等式分析了局部和全局的正则性，并将在解析意义和无穷可微性意义上加以考虑。与正则性相关的，可能是研究正则性的最强烈动机之一，是一个函数的边界值在投影到一个域上的全纯函数空间后保持多光滑的问题。投影的具体表达式由对伯格曼核的积分给出。这个项目的第二个重点是理解Bergman核在各种域上的边界行为，特别是那些有限类型的域。第三，更几何的工作线涉及水平面的正则性理论，晶体生长和退化方程的问题。本文将研究微分方程解的水平面，特别是热方程黏性解的水平集。要考虑的最基本的问题之一是由De Giorgi引起的问题，它寻求曲面作为微分方程中参数的函数的规律性。偏微分方程是物理科学中数学建模的基础。涉及连续变化的现象，例如在运动、材料和能量中看到的现象，都遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。数学的关键作用不是说明关系，而是从中提取定性和定量的意义。

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.