A Proof of the Convexity of the Free Boundary for Porous Flow Through a Rectangular Dam Using the Maximum Principle
C. W. CRYER · IMA Journal of Applied Mathematics · 1980
We consider the steady two-dimensional flow under gravity of water from one reservoir (on the left) to a lower reservoir (on the right) through a porous rectangular isotropic homogeneous dam with impervious bottom. Because of gravity the water does not flow through the entire dam and the dam is dry near its upper right corner. The interface separating the dry and wet regions 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 principle and its generalizations.