  We formulate and study two mathematical models of a thermoforming process
involving a membrane and a mould as implicit obstacle problems. In particular,
the membrane-mould coupling is determined by the thermal displacement of the
mould that depends in turn on the membrane through the contact region. The two
models considered are a stationary (or elliptic) model and an evolutionary (or
quasistatic) one. For the first model, we prove the existence of weak solutions
by solving an elliptic quasi-variational inequality coupled to elliptic
equations. By exploring the fine properties of the variation of the contact set
under non-degenerate data, we give sufficient conditions for the existence of
regular solutions, and under certain contraction conditions, also a uniqueness
result. We apply these results to a series of semi-discretised problems that
arise as approximations of regular solutions for the evolutionary or
quasistatic problem. Here, under certain conditions, we are able to prove
existence for the evolutionary problem and for a special case, also the
uniqueness of time-dependent solutions.
