Regularity for Minimizers of a Planar Partitioning Problem with Cusps

TL;DR

This paper proves regularity properties of minimizers in a planar partitioning problem with cusps, showing that boundaries are smooth curves with finitely many constant curvature segments, and characterizes minimizers under certain conditions.

Contribution

It establishes regularity results for minimizers of a soap bubble-like partitioning problem with cusps, including boundary smoothness and junction behavior, and characterizes minimizers for small perturbations.

Findings

Boundaries are $C^{1,1}$ with finitely many constant curvature curves.

Cusps occur only at junctions involving the set $G$.

Minimizers for small $ ext{delta}$ are perturbations of the zero-$ ext{delta}$ case with wetted triple junctions.

Abstract

We study the regularity of minimizers for a variant of the soap bubble cluster problem: \begin{align*} \min \sum_{\ell=0}^N c_{\ell} P( S_\ell)\,, \end{align*} where $c_\ell>0$, among partitions $\{S_0,\dots,S_N,G\}$ of $\mathbb{R}^2$ satisfying $|G|\leq \delta$ and an area constraint on each $S_\ell$ for $1\leq \ell \leq N$. If $\delta>0$, we prove that for any minimizer, each $\partial S_{\ell}$ is $C^{1,1}$ and consists of finitely many curves of constant curvature. Any such curve contained in $\partial S_{\ell} \cap \partial S_{m}$ or $\partial S_\ell \cap \partial G$ can only terminate at a point in $\partial G \cap \partial S_\ell \cap \partial S_{m}$ at which $G$ has a cusp. We also analyze a similar problem on the unit ball $B$ with a trace constraint instead of an area constraint and obtain analogous regularity up to $\partial B$. Finally, in the case of equal coefficients $c_\ell$, we completely characterize minimizers on the ball for small $\delta$: they are perturbations of minimizers for $\delta=0$ in which the triple junction singularities, including those possibly on $\partial B$, are ``wetted" by $G$.