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We consider, in a first instance, the class of boundaries of sets withlocally finite perimeter whose (weakly defined) mean curvature is $g \nu$, fora given continuous positive ambient function $g$, and where $\nu$ denotes theinner normal. It is well-known that taking limits in the sense of varifoldswithin this class is not possible in general, due to the appearence of "hiddenboundaries", that is, portions (of positive measure with even multiplicity) onwhich the (weakly defined) mean curvature vanishes, so that $g$ does notprescribe the mean curvature in the limit. As a special instance of a moregeneral result, we prove that locally uniform $L^q$-bounds on the (weaklydefined) second fundamental form, for $q>1$, in addition to the customarylocally uniform bounds on the perimeters, lead to a compact class of boundarieswith 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 treatingthem as (unoriented) varifolds). Specifically, it relies on the formulation andanalysis of a weak notion of curvature coefficients for oriented integralvarifolds (inspired by Hutchinson's work).  This framework gives (with no additional effort) a compactness result fororiented 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 canvary with the varifold for which it prescribes the mean curvature, as long asthere is locally uniform convergence of the prescribing functions). Our notionsand statements are given in a Riemannian manifold, with the oriented varifoldsthat need not arise as boundaries (for instance, they could come from two-sidedimmersions).

Hypersurfaces with mean curvature prescribed by an ambient function: compactness results