Extremal Problems in Complex Analysis and Potential Theory

DMS 0412908亚历山大Solynin极值问题在复杂的分析和潜在的理论摘要提议者将继续他的工作在极值问题。 这项工作遵循几个途径来攻击一些最近的挑战性问题，以及一些老的未解决的。 通过应用J.Jenkins的极值分割理论，我们将确定所有单连通域的系统，这些系统具有预定的边界组合并带有比例调和测度。我们还计划使用二次微分来解决确定完美导电流体的液滴形状的问题，这些流体通过静电和压力与表面张力平衡而保持平衡。 我们打算研究几个有关调和测度的问题。例如，我们想研究的问题，找到最小的“损害”的解决方案的拉普拉斯方程，当原来的域被损坏插入障碍从一个给定的集合。我们还计划使用二次微分技术来继续研究R。巴纳德研究R。罗宾逊1949年关于罗宾逊算子在经典单叶函数类S中的单叶半径的猜想。1990年我们验证了G. Polya和G. Szego在1951年通过构造第一个连续对称化，将有界域D变换为它的对称化域D*。我们计划将我们以前关于对称化的一些结果推广到关于任意维数k的超平面的Steiner对称化的情形。我们还计划使用我们的极化变换来解决鞅问题“需要多少布朗警察才能逮捕一个布朗囚犯？ 我们还提出了研究Poincare度量、调和测度和电容器的自由边界问题，以及保角映射中的最小面积和最小周长问题和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.