Problems in Analysis Related to Symmetrization

与对称化相关的分析问题

• 批准号: 9801282
• 负责人: Albert Baernstein
• 金额: $ 8.24万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801282&HistoricalAwards=false
• 关键词: Problems; Analysis; Related; Symmetrization

Proposal: DMS-9801282 Principal Investigator: Albert Baernstein II Abstract: Professor Baernstein is writing a book on the theory and practice of symmetrization, in which particular symmetrization tools such as inequalities for Dirichlet integrals and convolutions will be obtained in a unified way from certain "main inequalities." The tools will then be applied to prove a number of results in real and complex analysis and in partial differential equations. Baernstein is also investigating particular extremal problems in which symmetries of various kinds play a role. Examples include finding the exact values of the Bloch and Landau constants in complex function theory, determining L-p norms of k-plane transforms, and investigating a conjectural integral inequality for gradients of vector fields in the plane whose truth would tell us the L-p norm of a singular integral operator related to quasiconformal mappings, and whose falsity would confirm a conjecture of C.B. Morrey in the calculus of variations which asserts that rank-one convex functions need not be quasisconvex. The prototypical result on which Baernstein's project is modelled is the isoperimetric inequality. In simplest form, this inequality asserts that if a plot of land is to be enclosed by a fence of specified length, then to make the area inside the fence as large as possible, one should lay out the fence in a circle. Each of the unsolved problems mentioned in the first paragraph can be interpreted as an "extremal problem" of the same flavor as the isoperimetric inequality, in that the expected extremals are the competitors which appear to have the most symmetry. The challenge to the researcher is to devise mathematical tools which enable one to prove in a rigorous fashion that the expected extremals really are extremal, or to discover "counterexamples" which show that the established guess for the extremals was wrong. The problems mentioned above come from mathematical fields with connections to various domains of application . Phenomena encountered in the search for exact Bloch and Landau constants are related to crystallography, k-plane transforms occur in tomography, and quasiconformal mappings are related to elasticity theory and to problems about optimal mixtures of materials.

建议：DMS-9801282首席研究员：阿尔伯特·巴恩斯坦二世摘要：巴恩斯坦教授正在写一本关于对称化理论和实践的书，在书中，特定的对称化工具，如狄利克莱积分和卷积的不等式，将以统一的方式从某些“主要不等式”获得。然后，这些工具将被用来在真实和复杂的分析以及偏微分方程式中证明一些结果。巴恩斯坦还在研究特殊的极端问题，在这些问题中，各种对称性都起着作用。例如，求复变函数论中Bloch常数和Landau常数的精确值，确定k平面变换的L-p范数，以及研究平面上向量场梯度的一个猜想积分不等式，它的真假将告诉我们与拟共形映射有关的奇异积分算子的L-p范数，它的假将证实C.B.Morrey在变分中的一个猜想，该猜想断言一阶凸函数不必是拟凸的。巴恩斯坦的项目所依据的典型结果是等周不等。在最简单的形式中，这种不平等断言，如果一块土地要被指定长度的栅栏围住，那么为了使栅栏内的面积尽可能大，人们应该将栅栏布置成一个圆圈。第一段中提到的每一个未解决的问题都可以被解释为一个与等周不等式感相同的“极值问题”，因为预期的极值是似乎具有最大对称性的竞争者。研究人员面临的挑战是设计数学工具，使人们能够以严格的方式证明预期的极值确实是极端的，或者发现表明对极值的既定猜测是错误的“反例”。上述问题来自与各种应用领域相关的数学领域。在寻找精确的Bloch和Landau常数时遇到的现象与结晶学有关，在层析成像中发生k平面变换，而准共形映射与弹性理论和最佳材料混合物的问题有关。