Mathematical Sciences: Partial Differential Equations in Nonsmooth Domains

这项拨款所支持的数学工作是在偏微分方程组领域。更具体地说，将研究椭圆型和抛物型系统及高阶方程的非光滑边界域的边值问题。此外，还将开展均质化工作。我们感兴趣的一个特殊问题涉及到勒贝格空间中具有边界数据的弹性静力学椭圆系统和抛物系统。在四维或更大的维度中，边界数据的幂范数范围将产生解是未知的。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。

The mathematical work supported by this grant lies in the area of systems of partial differential equations. More specifically, boundary value problems in domains with nonsmooth boundaries for elliptic and parabolic systems and higher order equations will be investigated. In addition work on homogenization will be carried out. One of the particular problems of interest involves elliptic systems of elastostatics and parabolic systems with boundary data in the Lebesgue spaces. It is not known, in dimension four and greater, what range of power norms of boundary data will yield solutions. 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.