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We establish lower semicontinuity results for perimeter functionals withmeasure data on $\mathbb{R}^n$ and deduce the existence of minimizers to thesefunctionals with Dirichlet boundary conditions, obstacles, orvolume-constraints. In other words, we lay foundations of a perimeter-basedvariational approach to mean curvature measures on $\mathbb{R}^n$ capable ofproving existence in various prescribed-mean-curvature problems with measuredata. As crucial and essentially optimal assumption on the measure data weidentify a new condition, called small-volume isoperimetric condition, whichsharply captures cancellation effects and comes with surprisingly manyproperties and reformulations in itself. In particular, we show that thesmall-volume isoperimetric condition is satisfied for a wide class of$(n{-}1)$-dimensional measures, which are thus admissible in our theory. Ouranalysis includes infinite measures and semicontinuity results on very generaldomains.

Isoperimetric conditions, lower semicontinuity, and existence results for perimeter functionals with measure data