Regularity for Minimizers of a Planar Partitioning Problem with Cusps

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 $|G|\le \delta$ and an area constraint on each $S_{\ell }$ for $1\le \ell \le N$. 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 $G$.

References in corpus (3)

• Collapsing and the convex hull property in a soap film capillarity model
• Smoothness of collapsed regions in a capillarity model for soap films
• Plateau's problem as a singular limit of capillarity problems