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

TL;DR

The paper develops a BV-based variational framework for perimeter functionals with measure data, introducing a generalized functional ${\mathscr P}_{\mu_+,\mu_-}[A;\Omega]$ and a sharp small-volume isoperimetric condition to ensure lower semicontinuity and the existence of minimizers. By establishing semicontinuity on ${\mathbb R}^n$ and extending to general domains and Dirichlet problems, obstacles, and volume constraints, it provides a robust foundation for a perimeter-driven approach to prescribed-mean-curvature problems with measure data. The work identifies wide classes of admissible measures, including many singular and rectifiable $(n-1)$-dimensional measures, for which the theory applies and even proves optimal density bounds. Through a comprehensive treatment of ICs, capacity arguments, and Gauss–Green-type trace constructions, the results offer new semicontinuity by cancellation and extend known nonparametric results to the parametric perimeter setting. The findings have potential implications for existence results in mean curvature measures and for variational models in geometric analysis with measure-data right-hand sides.

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.

Figures (5)

• Figure 1: An illustration of the decisive cancellation effect in ${\mathds{R}}^2$: A sequence $(A_k)_{k\in{\mathds{N}}}$ forms thinner and thinner tentacles around a $1$d portion of $\mathop{\mathrm{spt}}\nolimits\mu$, but in the limit $A_\infty^+$ does not cover this portion anymore.
• Figure 2: The sets $A_k$, which converge locally in measure on ${\mathds{R}}^2$ to $\emptyset$, in case $n=2$, $B={[{-}1,0]}$, $v=1$.
• Figure 3: A set $S$ which cuts off the tentacle of Figure \ref{['fig:tentacle']} in the sense of Lemma \ref{['lem:good-ext-approx']} (for mildly small $\varepsilon$).
• Figure 4: A minimizer $A$ in the obstacle problem \ref{['eq:obst-min']} for $n=2$, some smooth $I\Subset{\mathds{R}}^2$, $O={\mathds{R}}^2$, $\mu_+\equiv0$, and $\mu_-=\sqrt2{\mathcal{H}}^1\mathop{}\nolimits({\mathds{R}}{\times}\{0\})$.
• Figure 5: An illustration of $\sigma_E$ and $\sigma$, which differ by reversing the arrows inside $E$.

Theorems & Definitions (111)

• Definition 1.1: small-volume isoperimetric condition
• Theorem 1.2: lower semicontinuity on full space; prototypical case
• Theorem 1.3: existence in obstacle and prescribed-volume problems
• Theorem 1.4: divergence criterion for the small-volume IC
• Theorem 1.5: small-volume IC for rectifiable ${\mathcal{H}}^{n-1}$-measures
• Proposition 1.6: small-volume IC for the sum of singular measures
• Theorem 2.1: coarea formula for Lipschitz functions
• Lemma 2.2: compactness from perimeter bounds
• Lemma 2.3: lower semicontinuity of the perimeter
• Theorem 2.4: De Giorgi's structure theorem; partial statement