  The goal of this work is to analyze a model for the rate-independent
evolution of sets with finite perimeter. The evolution of the admissible sets
is driven by that of a given time-dependent set, which has to include the
admissible sets and hence is to be understood as an external loading. The
process is driven by the competition between perimeter minimization and
minimization of volume changes. \par In the mathematical modeling of this
process, we distinguish the adhesive case, in which the constraint that the
(complement of) the `external load' contains the evolving sets is penalized by
a term contributing to the driving energy functional, from the brittle case,
enforcing this constraint. The existence of Energetic solutions for the
adhesive system is proved by passing to the limit in the associated
time-incremental minimization scheme. In the brittle case, this
time-discretization procedure gives rise to evolving sets satisfying the
stability condition, but it remains an open problem to additionally deduce
energy-dissipation balance in the time-continuous limit. This can be obtained
under some suitable quantification of data.
  The properties of the brittle evolution law are illustrated by numerical
examples in two space dimensions.
