Mathematical Sciences: Partial Differential Equations Under Minimal Smoothness Conditions

该奖项支持偏微分方程式领域的数学研究。这项工作涉及具有相对粗糙边界的区域中的边值问题，也就是允许导数不连续的边界，例如具有角点的边界。椭圆型方程和抛物型方程都将被讨论。尽管在过去的几年里，这一领域已经做了大量的工作，但对混合边值问题的关注却很少。特别要注意正则性问题：如果已知边界分量是某一阶的可积，那么人们想要确定梯度的最佳幂积分。第二项工作集中在椭圆算子的谱性质上。这里的问题是在由一个空腔和一个由细管连接的无限外部区域组成的区域中定位光谱共振。这类研究的重点是量化算符可以在多大程度上由各个部分上的单独椭圆算子建模。最后，将继续研究涉及反边值问题的问题。在更实际的情况下，人们想知道物体表面的电荷等量在多大程度上唯一地决定了物体表面的电流。这项工作的目标是研究潜在的微分算子可能具有非光滑系数的情况。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。

This award supports mathematical research in the area of partial differential equations. The work concerns boundary value problems in domains with relatively coarse boundaries, that is boundaries on which discontinuities in the derivative are allowed such as those with corners. Both elliptic and parabolic equations will be treated. Although considerable work has been done in the area in the past few years, little attention has been given to mixed boundary value problems. In particular attention will be given to the question of regularity: if the boundary components are known to be integrable of a certain order then one would like to establish the best power integral of the gradient. A second line of work focuses on spectral properties of elliptic operators. Here the problem is that of locating spectral resonances in domains which consists of a cavity joined to an unbounded exterior domain by a thin tube. The point of this type of investigation is to quantify the extent to which the operator may be modelled by separate elliptic operators on the individual parts. Finally, work will continue on questions involving inverse boundary value problems. In its more practical setting, one wants to know the extent to which quantities such as charge on the surface of a body uniquely determines current across the surface. The goal of this work is to look at cases where the underlying differential operator may have nonsmooth coefficients. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.