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 $${\mathbb {R}}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> 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 $${\mathbb {R}}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> 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 $$(n{-}1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional measures, which are thus admissible in our theory. Our analysis includes infinite measures and semicontinuity results on very general domains.