Regularity for Minimizers of a Planar Partitioning Problem with Cusps

TL;DR

The article establishes sharp regularity results for planar multi-chamber minimizers with a small wet region $G$ of area $|G|\le δ$, proving that chamber boundaries are $C^{1,1}$ and that interfaces are finite unions of constant-curvature arcs meeting cusp points on $\partial G$; it extends these results to a ball with a trace constraint and shows, in the equal-weight case, that for small $δ$ the minimizers are perturbations of the dry problem with triple junction singularities wetted by $G$. Central to the analysis is a blow-up/monotonicity framework that classifies tangent cones at interfacial points and yields local graphical/curvature structure, which in turn implies global regularity and convexity of chambers. For δ=0 on the ball, the paper provides a complete blow-up classification with explicit angle relations among participating chambers, and for small δ with equal coefficients a detailed resolution describes how cusp singularities are wetted by three-circle-arc configurations. The results bridge classical planar soap-film regularity with thickness/wetting effects and yield a precise, quantifiable description of how a small wet region stabilizes otherwise singular 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 $|G|\leq δ$ and an area constraint on each $S_\ell$ for $1\leq \ell \leq N$. If $δ>0$, 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 $δ$: they are perturbations of minimizers for $δ=0$ in which the triple junction singularities, including those possibly on $\partial B$, are ``wetted" by $G$.

Figures (4)

• Figure 1.1: On the left is a minimizing cluster $\mathcal{S}^0$ for the $\delta=0$ problem on the ball with chambers $S_\ell^0$. On the right is a minimizer $\mathcal{S}^\delta$ for small $\delta$, with $|G^\delta|=\delta$. Near the triple junctions of $\mathcal{S}^0$, $\partial G^\delta$ consists of three circular arcs meeting in cusps; see Theorem \ref{['resolution for small delta corollary']}.
• Figure 1.2: On the left is a double bubble-type configuration $\mathcal{S}^\delta$ with singularities wetted by $G^\delta$. $\mathcal{S}^\delta$ can be approximated by $\tilde{\mathcal{S}}^\delta$, where $\tilde{G}^\delta$ wets the entire interface.
• Figure 2.1: Both the sets $E$ and $E^h$ have the same trace on $\partial Q'$, and $P(E^h;\mathrm{int}\,Q')<P(E;\mathrm{int}\,Q')$ because $E$ has vertical slices which are not intervals.
• Figure 3.1: Here $h$ jumps at $x_i$, $1\leq i\leq 5$, and $\{h=1\}$ on the arcs $a_1^1$ and $a_2^1$. Equation \ref{['circular segment containment']} states that for any minimizer $\mathcal{S}$ with this boundary data, $S_1^{(1)}$ must contain the regions bounded by $a_j^1$ and the chords $c_j^1$ for $j=1,2$.

Theorems & Definitions (58)

• Theorem 1.1: Regularity on $\mathbb{R}^2$ for $\delta>0$
• Remark 1.2: Interpretation of $G^\delta$
• Remark 1.3: Constraint on $G^\delta$
• Theorem 1.4: Regularity on the Ball for $\delta>0$
• Remark 1.5
• Theorem 1.6: Regularity on the Ball for $\delta=0$
• Remark 1.7
• Theorem 1.8: Resolution for Small $\delta$ on the Ball
• Remark 1.9: Wetting of Singularities
• Remark 1.10: Triple Junctions for Vector Allen-Cahn