Equilibrium of membranes elastically supported at the edges

The problem considered is that of finding the statical deflection of a stretched membrane, subjected to a uniform pressure on one side and elastically supported at the edges. The deflection of the membrane is supposed small and the problem reduces to solving Laplace’s equation with mixed boundary conditions. Solutions are given for the cases where the bounding curve of the membrane is (i) an equilateral triangle, and (ii) a rectangle.