On geometrical properties of free boundaries in the Hele–Shaw flows moving boundary problem

In the article we discuss the geometrical properties of the moving boundary for two basic cases in the plain problem of the Hele–Shaw flows: for the inner problem for the flows in a bounded simply connected domain; and for the exterior problem for dynamics of an aerofoil connected with the flows in the exterior part of a bounded simply connected domain. We prove the invariance of the properties of starlikeness in case of the inner problem of pumping; of convexity in case of the exterior problem of tightening of an aerofoil. We also adduce some examples for the problem of tightening where the corresponding properties of starlikeness, convexity and close-to-convexity are not inherited by the moving boundary.