Extremal Problems in Complex Analysis and Potential Theory

亚历山大·索利尼复分析与势理论中的极值问题[摘要]申请者将继续他在极值问题上的研究。这项工作遵循几个途径来解决一些最近具有挑战性的问题以及一些旧的未解决的问题。通过应用詹金斯的极值划分理论，我们将确定所有单连通域系统，这些系统具有规定的边界组合并携带比例调和测度。我们还计划使用二次微分来解决确定完美导电流体液滴形状的问题，这些液滴由静电和压力力与表面张力平衡保持平衡。我们计划研究几个有关谐波措施的问题。例如，我们想研究的问题是，当从给定集合中插入障碍物破坏原始域时，如何找到对拉普拉斯方程解的最小“损害”。我们还计划利用二次微分技术继续与R. Barnard合作，研究R. Robinson在1949年关于经典单价函数S类中Robinson算子的一价半径的猜想。1990年，我们通过构造第一个连续对称，验证了G. Polya和G. Szego在1951年提出的一个猜想，该连续对称将有界域D转化为它的对称域D*。我们计划将我们早期关于对称的一些结果扩展到关于任何维度k的超平面的斯坦纳对称的情况下。我们还计划使用我们的极化变换来解决鞅问题“逮捕一个布朗囚犯需要多少布朗警察？”我们还研究了庞加莱度规、调和测度和电容的自由边界问题，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.