The relation between hele-shaw flow and complex monge-ampère equation

In this thesis, the relation between the Hele-Shaw flow which is a real two dimensional flow of an incompressible fluid, also known as two dimensional Laplacian growth, and complex Monge-Ampère equation is examined by following the path of the works of Julius Ross and David Witt Nyström. Using these connections a proof for existence and uniquness for the Hele-Shaw flow with varying permeability when starting from a single point (injection point) are presented. Additionally, it is shown that the Hele-Shaw flow with varying permeability is an inverse potential problem, inverse logarithmic potential problem, when starting from a smooth Jordan domain. Thus the existence and uniqueness for a such Hele-Shaw flow are observed.