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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 (20-30 µm), 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.

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