Mathematical Sciences: Analysis Related to Symmetrization

在这个项目中，将继续开展旨在发展全面和统一的对称化理论的数学研究。将对称性应用于数学分析中的问题，代表了在非常一般的情况下解决极值问题的最强大的范例之一。在对称性和极大正则性的条件下，基本不等式往往会达到极值。等周不等式或许就是这种现象的最好例子。这项工作试图利用最近发现的一个重要的不等式；一个包含分析中使用的一些最基本关系的不等式。这些是与梯度积分、三重卷积和拉普拉斯算子有关的不等式，在微分方程组理论、位势理论和解析函数理论中发挥重要作用的三个积分不等式--仅举几例。具体应用将涉及到函数论中的布洛赫常数和朗道常数问题。这些是(归一化)解析函数范围内最大圆盘的度量。另一个可能用对称化方法解决的问题是固支板的最低本征值问题。振动膜的类似问题早在多年前就已为人所知。在前一种情况下(双调和方程支配)，有一个微妙的问题，涉及到最低本征值的符号变化，必须首先解决。本课题还将讨论复多项式的极值问题、等角映射的导数的幂的可积性以及拟共形映射的面积扭曲等问题。

A continuation of mathematical research aimed at the development of a comprehensive and unified theory of symmetrization will be carried out in this project. The application of symmetry to problems in mathematical analysis represents one of the most powerful paradigms for the resolution of extremal problems in very general contexts. It is often the case that fundamental inequalities reach their extreme values under conditions of symmetry and great regularity. The isoperimetric inequality is perhaps the best example of such an event. This work seeks to exploit a recently discovered master inequality; an inequality containing some of the most basic relationships used in analysis. These are inequalities relating integrals of gradients, triple convolutions and the Laplace operator, three integral inequalities which play essential roles in the theory of differential equations, potential theory and the theory of analytic functions - to name a few. Specific applications will touch on the problem of the Bloch and Landau constants from function theory. These are measures of the largest disc in the range of (normalized) analytic functions. Another question which may yet be resolved by symmetrization is that of the lowest eigenvalue of a clamped plate. The analogous question for the vibrating membrane has been known for years. In the former case (the biharmonic equation governs), there is a delicate matter involving the change of sign in the lowest eigenvalue which must first be resolved. Extremal problems for complex polynomials, integrability of powers of the derivatives of confomal maps and area distortion of quasiconformal maps will also be treated in this project.