Existence theorems for thin inflated wrinkled membranes subjected to a   hydrostatic pressure
  In this paper, we establish rigorous existence theorems for a mathematical model of a thin inflated wrinkled membrane that is subjected to a shape dependent hydrostatic pressure load. We are motivated by the problem of determining the equilibrium shape of a strained high altitude large scientific balloon. This problem has a number of unique features. The balloon is very thin (30 micron), especially when compared with its diameter (over 100 meters). Unlike a standard membrane, the balloon is unable to support compressive stresses and will wrinkle or form folds of excess material. Our approach can be adapted to a wide variety of inflatable membranes, but we will focus on two types of high altitude balloons, a zero-pressure natural shape balloon and a super-pressure pumpkin shaped balloon. We outline the shape finding process for these two classes of balloon designs, formulate the problem of a strained balloon in an appropriate Sobolev space setting, establish rigorous existence theorems using direct methods in the calculus of variations, and present numerical studies to complement our theoretical results. 
Frank Baginski, Michael Barg, William Collier