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

In this paper, we consider the Hele-Shaw problem in a 2-dimensional fluid domain Ωt which is constrained to a half-plane. The boundary of Ωt consist of two components: Γ0t which lies on the boundary of the half-plane, and Γt which lies inside the half-plane. On Γt we impose the classical boundary conditions with surface tension, and on Γ0t we prescribe the normal derivative of the fluid pressure. At the point where Γ0t and Γt meet, there is an abrupt change in the boundary condition giving rise to a singularity in the fluid pressure. We prove that the problem has a unique solution with smooth free boundary Γt for some small time interval.