Mathematical Sciences: Partial Differential Equations and Harmonic Analysis

在数学分析的三个领域的研究将是 支持这个奖项。 作品的主题围绕 非光滑区域上的偏微分方程 边界和与这些方程相关的逆问题。 的 第一个领域涉及用于近似 非光滑区域上经典层势的逆。 一个人感兴趣的是确定数学上合理的 边界积分近似逆的数值格式 由层势产生的算子。 这些潜力是 二阶边值问题 偏微分方程与系统 的第二线 研究了非线性的Fatou理论和位势理论 椭圆方程和线性抛物方程 非光滑系数 我们的目标是确定 定义在域内部的解，随着 在非切向方向上接近边界。 最终 目标是建立类的尖锐有界性估计 非光滑域上乘性范数形式的函数 涉及二阶或高阶椭圆算子的不等式 关于涉及Stokes算子的函数， 向量=值函数。偏微分的分析 非光滑区域上的方程是一个最近很热门的课题 数学研究活动。 已经有可能使 在曾经被认为遥不可及的问题上取得稳步进展。一边 从他们的智力挑战，这些问题有一个 物理世界中最实际的现象， 由微分方程建模， 边界条件 这种类型的研究验证信息 只能通过直觉或计算获得。

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.