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

created by schmidt on 17 Feb 2023
modified on 15 Mar 2023

[BibTeX]

Submitted Paper

Inserted: 17 feb 2023
Last Updated: 15 mar 2023

Year: 2023

ArXiv: 2302.13396 PDF

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.