An Infinite Double Bubble Theorem
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913866978426880 |
|---|---|
| author | Bronsard, Lia Novack, Michael |
| author_facet | Bronsard, Lia Novack, Michael |
| contents | The classical double bubble theorem characterizes the minimizing partitions of $\mathbb{R}^n$ into three chambers, two of which have prescribed finite volume. In this paper we prove a variant of the double bubble theorem in which two of the chambers have infinite volume. Such a configuration is an example of a (1,2)-cluster, or a partition of $\mathbb{R}^n$ into three chambers, two of which have infinite volume and only one of which has finite volume. A $(1,2)$-cluster is locally minimizing with respect to a family of weights $\{c_{jk}\}$ if for any $B_r(0)$, it minimizes the interfacial energy $\sum_{j<k} c_{jk} \mathscr{H}^n(\partial \mathscr{X}(j) \cap \partial\mathscr{X}(k) \cap B_r(0))$ among all variations with compact support in $B_r(0)$ which preserve the volume of $\mathscr{X}(1)$. For $(1,2)$ clusters, the analogue of the weighted double bubble is the weighted lens cluster, and we show that it is locally minimizing. Furthermore, under a symmetry assumption on $\{c_{jk}\}$ that includes the case of equal weights, the weighted lens cluster is the unique local minimizer in $\mathbb{R}^n$ for $n\leq 7$, with the same uniqueness holding in $\mathbb{R}^n$ for $n\geq 8$ under a natural growth assumption. We also obtain a closure theorem for locally minimizing $(N,2)$-clusters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08063 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Infinite Double Bubble Theorem Bronsard, Lia Novack, Michael Analysis of PDEs Differential Geometry Metric Geometry The classical double bubble theorem characterizes the minimizing partitions of $\mathbb{R}^n$ into three chambers, two of which have prescribed finite volume. In this paper we prove a variant of the double bubble theorem in which two of the chambers have infinite volume. Such a configuration is an example of a (1,2)-cluster, or a partition of $\mathbb{R}^n$ into three chambers, two of which have infinite volume and only one of which has finite volume. A $(1,2)$-cluster is locally minimizing with respect to a family of weights $\{c_{jk}\}$ if for any $B_r(0)$, it minimizes the interfacial energy $\sum_{j<k} c_{jk} \mathscr{H}^n(\partial \mathscr{X}(j) \cap \partial\mathscr{X}(k) \cap B_r(0))$ among all variations with compact support in $B_r(0)$ which preserve the volume of $\mathscr{X}(1)$. For $(1,2)$ clusters, the analogue of the weighted double bubble is the weighted lens cluster, and we show that it is locally minimizing. Furthermore, under a symmetry assumption on $\{c_{jk}\}$ that includes the case of equal weights, the weighted lens cluster is the unique local minimizer in $\mathbb{R}^n$ for $n\leq 7$, with the same uniqueness holding in $\mathbb{R}^n$ for $n\geq 8$ under a natural growth assumption. We also obtain a closure theorem for locally minimizing $(N,2)$-clusters. |
| title | An Infinite Double Bubble Theorem |
| topic | Analysis of PDEs Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2401.08063 |