THE HELE-SHAW PROBLEM AND AREA-PRESERVING CURVE-SHORTENING MOTIONS

We prove existence, locally in time, of a solution of the following Hele-Shaw problem: Given a simply connected curve contained in a smooth bounded domain OMEGA, find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in OMEGA and whose boundary value on the curve equals its curvature. We show that this motion is a curve-shortening motion which does not change the area of the region enclosed by the curve. In case OMEGA is the whole plane R2, we also show that if the initial curve is close to an equilibrium curve, i.e., to a circle, then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity.