Mathematical Sciences: Singular Integrals and Parabolic Partial Differential Equations

数学科学：奇异积分和抛物型偏微分方程

• 批准号: 9400782
• 负责人: Steven Hofmann
• 金额: $ 5万
• 依托单位: Wright State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9400782&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Singular; Integrals; Parabolic

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. ***

小行星9400782 该奖项支持奇异积分理论和偏微分方程中的问题的数学研究，这些问题与非光滑域上抛物方程的边界值问题有关，其边界允许随时间变化。 这些问题的动机部分是物理上的考虑，例如材料的膨胀和收缩或形状随时间的变化，部分是数学上的考虑。 时变域，其中一个人想最终治疗的是在自然意义上的抛物类似物的Lipschitz域已发挥了突出的作用，在椭圆理论在最近几年. 该项目的主要目标是解决在这种情况下，通过层潜力的初边值问题。 其中的工具可能被采用或研究的Rellich身份，扰动技术，热量措施估计，分析多线性和非线性奇异积分和圆的想法与理论的粗糙奇异积分。 偏微分方程是物理世界数学模型的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。 在现代的方法来研究这些方程是应用奇异积分和相关的强大技术的调和分析。 ***