Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.11865 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929686908502016 |
|---|---|
| author | Novack, Michael |
| author_facet | Novack, Michael |
| contents | 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 $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_11865 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Regularity for Minimizers of a Planar Partitioning Problem with Cusps Novack, Michael Analysis of PDEs 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 $G$. |
| title | Regularity for Minimizers of a Planar Partitioning Problem with Cusps |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2305.11865 |