  The behavior of a surface energy $\mathcal F(E,u)$, where $E$ is a set of
finite perimeter and $u\in L^1(\partial^* E, \mathbb R_+)$ is studied. These
energies have been recently considered in the context of materials science to
derive a new model in crystal growth that takes into account the effect of
atoms freely diffusing on the surface (called adatoms), which are responsible
for morphological evolution through an attachment and detachment process.
Regular critical points, existence and uniqueness of minimizers are discussed
and the relaxation of $\mathcal F$ in a general setting under the $L^1$
convergence of sets and the vague convergence of measures is characterized.
This is part of an ongoing project aimed at an analytical study of diffuse
interface approximations of the associated evolution equations.
