Harmonic measure and its application to steady state head distribution

In this dissertation we study harmonic measure of boundary sets of planar domains and its applications to the distribution of heat. In the first chapter, we introduce main definitions and present basic properties of harmonic measure. In particular, we explain how harmonic measure of a set E on the boundary of a domain D describes steady-state distribution of heat on D when E is kept at temperature 1 while the remaining part of the boundary of D is kept at temperature 0. In the second chapter, we study several problems on the heat distribution when a heater of given length is sliding along the boundary of a given domain. In the third chapter, we discuss few problems how harmonic measure, or, equivalently, steady-state distribution of heat, changes when a flexible or rigid heater of fixed length is installed inside a given domain in such a way that it touches its boundary.