  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.
