Calculus of Variations and Geometric Measure Theory

T. Schmidt

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


Submitted Paper

Inserted: 17 feb 2023
Last Updated: 15 mar 2023

Year: 2023

ArXiv: 2302.13396 PDF


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.