Equilibrium States of Inextensible Membranes under Gravity

We discuss the possible shapes and stress distributions of an inextensible membrane acted on by a gravitational force. It is shown that there are smooth shapes which cannot be an equilibrium solution under gravity for any stress distribution. In addition there are infinitely many shapes which are equilibrium solutions under gravity and for which the stress distribution is unique. Finally, there are infinitely many solution surfaces – each consisting of straight line generators parallel to the direction of gravity. These solutions are possible under infinitely many different stress distributions. In several cases we will solve the equations explicitly to find actual shapes and stress distributions. Mathematics Subject Classifications (2000) 74.