A Proof of the Convexity of the Free Boundary for Porous Flow Through a Rectangular Dam Using the Maximum Principle

We consider the steady two-dimensional flow under gravity ofwater from one reservoir (on the left) to a lower reservoir(on the right) through a porous rectangular isotropic homogeneousdam with impervious bottom. Because of gravity the water doesnot flow through the entire dam and the dam is dry near itsupper right corner. The interface separating the dry and wetregions of the dam is a free boundary. Recently, Friedman &Jensen (1977) have proved that the free boundary is convex.We give a different proof which uses only the maximum principleand its generalizations.