Problems in Analysis Related to Symmetrization

提案：DMS-9801282主要研究者：Albert Baernstein II 摘要：Baernstein教授正在写一本关于对称化的理论和实践的书，其中特定的对称化工具，如Dirichlet积分和卷积的不等式，将以统一的方式从某些“主要不等式”中获得。“这些工具将被应用于证明真实的和复杂的分析和偏微分方程中的一些结果。贝恩斯坦还调查特别极端的问题，其中各种对称性发挥作用。例子包括在复变函数论中找到Bloch和朗道常数的精确值，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代数积分不等式，其真值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，而其假值将证实C. B. Morrey在变分法中断言秩一凸函数不一定是拟凸的。 Baernstein的项目所依据的原型结果是等周不等式。在最简单的形式中，这个不等式断言，如果一块土地被指定长度的围栏包围，那么为了使围栏内的面积尽可能大，应该将围栏布置成圆形。第一段中提到的每个未解决的问题都可以被解释为与等周不等式具有相同风味的“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值，或者发现“反例”，表明对极值的既定猜测是错误的。上面提到的问题来自数学领域，与各种应用领域有联系。在寻找精确的布洛赫常数和朗道常数时遇到的现象与晶体学有关，k平面变换发生在层析成像中，准共形映射与弹性理论和材料的最佳混合物问题有关。

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.