  We introduce a weak formulation of the non-parametric prescribed mean
curvature equation with measure data and show the existence and several
properties of $BV$ solutions under natural assumptions on the prescribed
measure. Our approach does not rely on approximate or viscosity-type solutions.
It requires combining various ingredients, including Anzellotti's pairing
theory for divergence-measure fields and its recent developments, a refinement
of Anzellotti-Giaquinta approximation, and convex duality theory. We also prove
a Gamma-convergence result valid for suitable smooth approximations of the
prescribed measure, and a maximum principle for continuous weak solutions. We
finally construct some examples of non-uniqueness, showing at the same time the
need for the continuity assumption in the maximum principle and an unexpected
feature of weak solutions.
