Inspired by a planar partitioning problem involving multiple unbounded chambers, this thesis investigates using classical techniques what can be said of the existence, uniqueness, and regularity of minimizers in a certain free-endpoint isoperimetric problem. In two cases, 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. A crash course on the theory required for the modern statement of the equilibrium shape theorem is presented in an appendix. / Thesis / Master of Science (MSc)
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/28471 |
| Date | January 2023 |
| Creators | Vriend, Silas |
| Contributors | Alama, Stanley, Bronsard, Lia, Mathematics and Statistics |
| Source Sets | McMaster University |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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