Mathematical Sciences: Singular Integrals and Parabolic Partial Differential Equations

9400782霍夫曼该奖项支持对奇异积分理论和偏微分方程问题的数学研究，这些问题与非光滑区域上抛物型方程的边值问题有关，其边界允许随时间变化。这些问题部分是由物理考虑引起的，例如材料随时间的膨胀和收缩或以其他方式改变形状，部分是由数学考虑引起的。人们最终想要处理的时变域在自然意义上是Lipschitz域的抛物线类比，它在近年来的椭圆理论中扮演着重要的角色。该项目的主要目标是通过层势解决这方面的初始边值问题。在可能使用或研究的工具中，有Rellich恒等式、摄动技术、热量测量估计、多线性和非线性奇异积分的分析以及与粗糙奇异积分理论相关的思想圈。偏微分方程构成了对物理世界进行数学建模的基础。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。在研究这些方程的现代方法中，有奇异积分的应用以及相关的强大的调和分析技术。***

9400782 Hofmann This award supports mathematical research on problems in singular integral theory and partial differential equations which arise in connection with boundary value problems for parabolic equations on non-smooth domains, whose boundaries are allowed to vary with time. The problems are motivated in part by physical considerations, such as the expansion and contraction or otherwise changing of shape of materials over time, and in part by mathematical considerations. The time-varying domains which one would like to ultimately treat are in a natural sense theparabolic analogues of the Lipschitz domains which have played a prominent role in the elliptic theory in recent years. The main goal of the project is to solve initial boundary values problems in this context by means of layer potentials. Among the tools which are likely to be employed or studied are the Rellich identities, perturbation techniques, caloric measure estimates, analysis of multilinear and nonlinear singular integrals and the circle of ideas connected with theory of rough singular integrals. Partial differential equations form a basis for mathematicalmodeling 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. Among the modern approaches to the study of these equations is the application of singular integrals and the associated powerful techniques of harmonic analysis. ***