Mathematical Sciences: Partial Differential Equations and Harmonic Analysis

该奖项将支持数学分析的三个领域的研究。这项工作的主题是围绕定义在具有非光滑边界的区域上的偏微分方程组和与此类方程相关的反问题。第一个领域涉及在非光滑区域上逼近经典层势的逆的计算算法。人们感兴趣的是确定用于逼近由层势产生的边界积分算子的逆的数学证明的数值格式。这些势与二阶偏微分方程组的边值问题有关。第二个研究方向是非光滑系数的非线性椭圆型方程和线性抛物型方程的Fatou理论和位势理论。其目标是准确地确定在区域内部定义的解在沿非相切方向逼近边界时如何变化。最终的目标是建立非光滑区域上函数类的精确有界性估计，其形式为涉及二阶或更高阶椭圆算子的乘法范数不等式，在向量=值函数的情况下涉及Stokes算子的函数。在曾经被认为遥不可及的问题上取得稳步进展是可能的。除了智力上的挑战，这些问题还有一个实际的方面--物理世界中大多数可以用微分方程建模的现象都带有粗糙的边界条件。这种类型的研究只验证通过直觉或通过计算获得的信息。

Research in three areas of mathematical analysis will be supported by this award. The theme of the work revolves around partial differential equations defined on domains with nonsmooth boundaries and inverse problems related to such equations. The first areas concerns computational algorithms for approximating the inverse of classical layer potentials on nonsmooth domains. One is interested in determining mathematically justified numerical schemes for approximating inverses of boundary integral operators arising from layer potentials. These potentials are associated with boundary value problems for second order partial differential equations and systems. A second line of study concerns Fatou theory and potential theory for nonlinear elliptic equations and linear parabolic equations with nonsmooth coefficients. The goal is to determine exactly how solutions, defined on the interior of a domain, change as the boundary is approached in a nontangential direction. The final goal is to establish sharp boundedness estimates for classes of functions on nonsmooth domains in the form of multiplicative norm inequalities involving a second or higher order elliptic operator on the functions of involving the Stokes operator in the case of a vector=valued function.The analysis of partial differential equations on nonsmooth domains is an subject of intense recent activity in mathematical research. It has been possible to make steady headway on problems once thought to be out of reach. Aside from their intellectual challenge, these problems have a practical side -most phenomena in the physical world which can be modeled by differential equations are burdened with rough boundary conditions. This type of research validates information only obtained intuitively or through computation.