  We investigate the existence of minimizers of variational models with
Eulerian-Lagrangian formulations. We consider energy functionals depending on
the deformation of a body, defined on its reference configuration, and an
Eulerian map defined on the unknown deformed configuration in the actual space.
Our existence theory moves beyond the purely elastic setting and accounts for
material failure by addressing free-discontinuity problems where both
deformations and Eulerian fields are allowed to jump. To do so, we build upon
the work of Henao and Mora-Corral regarding the variational modeling of
cavitation and fracture in nonlinear elasticity. Two main settings are
considered by modeling deformations as Sobolev and SBV-maps, respectively. The
regularity of Eulerian maps is specified in each of these two settings
according to the geometric and topological properties of the deformed
configuration. We present some applications to specific models of liquid
crystals, phase transitions, and ferromagnetic elastomers. Effectiveness and
limitations of the theory are illustrated by means of explicit examples.
