On inflating closed mylar shells

We discuss the inflating of a closed thin shell made of inextensible flexible material like mylar. The problem is to determine the extremal form of the shell, when it is inflated to the maximal possible volume. We introduce a variational problem which describes the inflating of rotationally symmetric shells. The main result presents a criteria for a rotationally symmetric shell to admit volume increasing deformations without surface stretching. Moreover explicit solutions are found for cylindrical and biconical shells.