Abstract

Since its invention by Jacob Steiner in 1838, symmetrization method had found numerous applications in mathematics and its applications. As well known, Steiner introduced his symmetrization when searching for a geometric tool to solve the classical isoperimetric problem on the maximal area for planar regions with fixed perimeter. After Steiner, many outstanding mathematicians, including George Po´lya, Gabor Szego¨, James Jenkins, Igor Mityuk, Al Baernstein II, Vladimir Dubinin, introduced new symmetrizing procedures, each time targeting some unsolved extremal problem.

In the first part of this talk, I will introduce some of these transformations and discuss their applications to extremal problems in Complex Analysis and Potential Theory, which motivated the invention of these transformations. Then I will discuss my recent results on capacities and distribution of heat in systems of n balls, on the problem of distribution of heat on long pipes heated along some surface areas, and on the problem on the “squeezing function” that has application to spaces of analytic functions of one and several complex variables. My solution of each of these problems uses certain geometric symmetrization-type transformation. A few remaining open and challenging extremal problems, which may require new symmetrization-type transformations will be also discussed.