The 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.
