  We consider, in a first instance, the class of boundaries of sets with
locally finite perimeter whose (weakly defined) mean curvature is $g \nu$, for
a given continuous positive ambient function $g$, and where $\nu$ denotes the
inner normal. It is well-known that taking limits in the sense of varifolds
within this class is not possible in general, due to the appearence of "hidden
boundaries", that is, portions (of positive measure with even multiplicity) on
which the (weakly defined) mean curvature vanishes, so that $g$ does not
prescribe the mean curvature in the limit. As a special instance of a more
general result, we prove that locally uniform $L^q$-bounds on the (weakly
defined) second fundamental form, for $q>1$, in addition to the customary
locally uniform bounds on the perimeters, lead to a compact class of boundaries
with mean curvature prescribed by $g$.
  The proof relies on treating the boundaries as 'oriented integral varifolds',
in order to exploit their orientability feature (that is lost when treating
them as (unoriented) varifolds). Specifically, it relies on the formulation and
analysis of a weak notion of curvature coefficients for oriented integral
varifolds (inspired by Hutchinson's work).
  This framework gives (with no additional effort) a compactness result for
oriented integral varifolds with curvature locally bounded in $L^q$ with $q>1$
and with mean curvature prescribed by any $g\in C^0$ (in fact, the function can
vary with the varifold for which it prescribes the mean curvature, as long as
there is locally uniform convergence of the prescribing functions). Our notions
and statements are given in a Riemannian manifold, with the oriented varifolds
that need not arise as boundaries (for instance, they could come from two-sided
immersions).
