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

## TL;DR

This paper develops a variational framework for perimeter functionals with measure data, establishing lower semicontinuity and existence of minimizers under new isoperimetric conditions, applicable to a broad class of measures.

## Contribution

It introduces the small-volume isoperimetric condition, a novel criterion ensuring lower semicontinuity and existence results for perimeter functionals with measure data.

## Key findings

- The small-volume isoperimetric condition is satisfied by many measures, including infinite measures.
- The framework applies to general domains and includes semicontinuity results.
- Existence of minimizers is proved for problems with boundary conditions, obstacles, or volume constraints.

## 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.