Mathematical Sciences: Maximum Principles and Dilation Invariant Estimates for Sobolev and Dirichlet Problems

数学科学：索博列夫和狄利克雷问题的极大原理和膨胀不变估计

• 批准号: 9401354
• 负责人: Gregory Verchota
• 金额: $ 6万
• 依托单位: Syracuse 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=9401354&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Maximum; Principles; Dilation

9401354 Verchota This project in mathematical analysis focuses on the application of methods of harmonic analysis to the study of partial differential equations. Work will be done on the formation of estimates of solutions to boundary value problems of elliptic operators with constant coefficients. The tools employed include singular integrals and Sobolev and Hardy spaces on lower dimensional Lipschitz surfaces. The lower dimensional theory, which is just developing, seeks to yield new maximum principles for higher order equations and systems. One of the underlying goals of the research is that of understanding which properties of dilation invariant solutions might be considered operator independent. Classical theory fails to distinguish those properties of solutions that are operator dependent from those that are domain dependent. Boundary value problems such as the Dirichlet problem are to be generalized in an algebraic way to higher order operators and systems that may not conform to intuition on how values at the boundary should control values inside a domain. 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. ***

小行星9401354 这个数学分析项目的重点是调和分析方法在偏微分方程研究中的应用。 工作将完成对形成的估计的解决方案的椭圆算子的常系数边值问题。 所采用的工具包括奇异积分和Sobolev和哈代空间的低维Lipschitz曲面。 低维理论是一个正在发展的理论，它试图为高阶方程和系统提供新的最大值原理。 研究的一个基本目标是了解哪些性质的膨胀不变的解决方案可能被认为是运营商无关的。 经典理论不能区分那些依赖于算子的解的性质和那些依赖于域的解的性质。 边界值问题，如狄利克雷问题，是广义的代数方式高阶运营商和系统，可能不符合直觉的价值观，在边界应如何控制值内的一个领域。 偏微分方程是物理世界数学建模的基础。 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。 ***