تقرير
On a Free-Endpoint Isoperimetric Problem in $\mathbb{R}^2$
| العنوان: | On a Free-Endpoint Isoperimetric Problem in $\mathbb{R}^2$ |
|---|---|
| المؤلفون: | Alama, Stanley, Bronsard, Lia, Vriend, Silas |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics Mathematical Physics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, Mathematical Physics, Mathematics - Classical Analysis and ODEs, Mathematics - Optimization and Control, 49Q10 (Primary), 49J05 (Secondary) |
| الوصف: | Inspired by a planar partitioning problem involving multiple improper chambers, this article investigates using classical techniques what can be said of the existence, uniqueness, and regularity of minimizers in a certain free-endpoint isoperimetric problem. By restricting to curves which are expressible as graphs of functions, a full existence-uniqueness-regularity result is proved using a convexity technique inspired by work of Talenti. The problem studied here can be interpreted physically as the identification of the equilibrium shape of a sessile liquid drop in half-space (in the absence of gravity). This is a well-studied variational problem whose full resolution requires the use of geometric measure theory, in particular the theory of sets of finite perimeter, but here we present a more direct, classical geometrical approach. Conjectures on improper planar partitioning problems are presented throughout. Comment: 17 pages, 4 figures. Accepted with minor revisions to the proceedings of Anisotropic Isoperimetric Problems & Related Topics (AIPRT) 2022 |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2304.10531 |
| رقم الأكسشن: | edsarx.2304.10531 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |