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

[401354 . Verchota]这个数学分析项目的重点是将调和分析方法应用于偏微分方程的研究。本文将讨论常系数椭圆算子边值问题解的估计的形成。所使用的工具包括奇异积分以及低维Lipschitz曲面上的Sobolev和Hardy空间。正处于发展阶段的低维理论试图为高阶方程和系统提供新的最大值原理。研究的潜在目标之一是理解膨胀不变解的哪些性质可以被认为是算子独立的。经典理论无法区分算子相关解和域相关解的性质。边值问题，如狄利克雷问题，将以代数的方式推广到高阶算子和系统，这些算子和系统可能不符合直觉，即边界上的值应该如何控制域内的值。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创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

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. ***