  We introduce and analyze a lower envelope method (LEM) for the tracking of
interfaces motion in multiphase problems. The main idea of the method is to
define the phases as the regions where the lower envelope of a set of functions
coincides with exactly one of the functions. We show that a variety of complex
lower-dimensional interfaces naturally appear in the process. The phases
evolution is then achieved by solving a set of transport equations. In the
first part of the paper, we show several theoretical properties, give
conditions to obtain a well-posed behaviour, and show that the level set method
is a particular case of the LEM. In the second part, we propose a LEM-based
numerical algorithm for multiphase shape optimization problems. We apply this
algorithm to an inverse conductivity problem with three phases and present
several numerical results.
