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

Abstract

We establish lower semicontinuity results for perimeter functionals with measure data on ${\mathbb{R}}^n$ and deduce the existence of minimizers to these functionals with Dirichlet boundary conditions, obstacles, or volume-constraints. In other words, we lay foundations of a perimeter-based variational approach to mean curvature measures on ${\mathbb{R}}^n$ capable of proving existence in various prescribed-mean-curvature problems with measure data. As crucial and essentially optimal assumption on the measure data we identify a new condition, called small-volume isoperimetric condition, which sharply captures cancellation effects and comes with surprisingly many properties and reformulations in itself. In particular, we show that the small-volume isoperimetric condition is satisfied for a wide class of $(n-1)$-dimensional measures, which are thus admissible in our theory. Our analysis includes infinite measures and semicontinuity results on very general domains.

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