The Hele-Shaw problem with surface tension in a half-plane: A model problem

Consider the Hele-Shaw problem with surface tension in the half-plane {y1>0} when at time t=0 the domain Ω(t) lies partly on the line y1=0, and partly in {y1>0}. In order to establish existence of a solution to this free boundary problem we need to study the (linear) model problem when the Ω(t) is a fixed angular domain. In this paper we consider this model problem and establish existence of a solution satisfying sharp weighted Hölder estimates. These estimates will be used in subsequent work to solve the full Hele-Shaw problem.