Extremal Problems in Complex Analysis and Potential Theory

DMS 0412908 Alexander Solynin复数分析和位势理论中的极端问题提出者将继续他在极端问题上的工作。这项工作遵循几个途径来解决最近的一些具有挑战性的问题以及一些旧的未解决的问题。通过应用J.Jenkins的极值划分理论，我们将确定所有具有规定的边界组合且带有比例调和测度的单连通区域系统。我们还计划使用二次微分来解决确定理想导电流体的液滴形状的问题，这些液滴通过与表面张力平衡的静电力和压力力保持平衡。我们计划研究有关调和测度的几个问题。例如，我们想要研究当原始域因插入给定集合中的障碍而受损时，寻找拉普拉斯方程解的最小“损伤”的问题。我们还计划使用二次微分技术继续与R.Barnard一起研究R.Robinson 1949年关于单叶函数经典类S中Robinson算子的单叶半径的猜想。1990年，我们通过构造有界域D到其对称域D*的第一个连续对称化，验证了G.Polya和G.Szego在1951年提出的猜想。我们计划将我们早先关于对称化的一些结果推广到关于任意维超平面的Steiner对称化的情形。我们还计划用我们的极化变换来解决“逮捕一个布朗囚犯需要多少个布朗警察？”的鞅问题。我们还建议研究Poincare度量、调和度量和凝聚器容量的自由边界问题，以及保角映射中的最小面积和最小周长问题以及对称化中的其他问题。

DMS 0412908Alexander SolyninExtremal Problems in Complex Analysis and Potential TheoryABSTRACTThe proposer will continue his work on extremal problems. This work follows several avenues to attack a number of recent challenging problems as well as some old unsolved ones. By applying J. Jenkins' theory of extremal partitions, we will determine all systems of simply connected domains, which have a prescribed combinatorics of boundaries and carry proportional harmonic measures. We also plan to use quadratic differentials to solve the problem of determining the shape of droplets of perfectly conducting fluid that are held in equilibrium by electrostatic and pressure forces balanced against surface tension. We plan to study several problems concerning harmonic measures. For example, we want to study the problem of finding the minimal ``damage'' to solutionsof the Laplace equation, when the original domain is damaged by inserting an obstacle from a given set. We also plan to use quadratic differential techniques to continue work with R. Barnard to study R. Robinson's 1949 conjecture concerning the radius of univalence of the Robinson operator in the classical class S of univalent functions. In 1990 we verified a conjecture of G. Polya and G. Szego made in 1951 by constructing the first continuous symmetrization transforming a bounded domain D into its symmetrized domain D*. We plan to work on extending some of our earlier results on symmetrization to the case of Steiner symmetrization with respect to hyperplanes of any dimension k. We also plan to use our polarization transformation to resolve the Martingale problem "How many Brownian policemen does it take to arrest a Brownian prisoner? We also propose to investigate free boundary problems for the Poincare metric, harmonic measure, and capacity of a condenser, as well as the minimal area and minimal perimeter problems in conformal mapping and other problems in symmetrization.