Mathematical Sciences: Analysis Related to Symmetrization

数学研究的继续，旨在 发展一个全面和统一的理论， 在这个项目中将进行对称化。 的 对称性在数学分析中的应用 代表了一个最有力的范例， 在一般情况下的极值问题。 往往是 基本不平等达到极端值的情况 在对称性和高度规则性的条件下。 的 等周不等式也许是最好的例子， 活动 这项工作旨在利用最近发现的大师 不平等;包含一些最基本的不平等 分析中使用的关系。 这些不平等关系到 梯度积分、三重卷积和拉普拉斯 算子，三个积分不等式，发挥重要作用 在微分方程理论、势能理论和 解析函数理论-仅举几例。 具体的应用将触及布洛赫问题 和朗道常数。 这些是衡量 在（标准化）解析函数范围内的最大圆盘。 另一个可以通过对称化解决的问题是 固支板的最低特征值的最小特征值。 的类似 振动膜的问题已经知道很多年了。 在 前一种情况（双调和方程支配），存在一个 微妙的问题，涉及最低的符号变化， 必须首先解决的特征值。 极值问题 复多项式，导数幂的可积性，可积性 共形映射的面积畸变和拟共形映射的面积畸变 也在这个项目中得到了处理。

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.