The isoperimetric inequality on Euclidean space and on surfaces
This master’s thesis focuses on the analysis of the isoperimetric problem, which involves finding shapes that enclose the maximum volume with a given boundary area. We examine this problem both in Euclidean space and on hypersurfaces. In the Euclidean setting, we review several distinct proofs using...
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| Tipo de recurso: | tesis de maestría |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/416461 |
| Acceso en línea: | https://hdl.handle.net/2117/416461 |
| Access Level: | acceso abierto |
| Palabra clave: | Curves Graph theory Isoperimetric problem Isoperimetric inequality Corbes Grafs, Teoria de Classificació AMS::05 Combinatorics::05C Graph theory Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | This master’s thesis focuses on the analysis of the isoperimetric problem, which involves finding shapes that enclose the maximum volume with a given boundary area. We examine this problem both in Euclidean space and on hypersurfaces. In the Euclidean setting, we review several distinct proofs using different approaches, including symmetrization techniques, geometric measure theory, and partial differential equations. We also discuss the consequences and applications of these results, particularly the Faber-Krahn inequality, which states that among all domains with a fixed volume, the ball minimizes the first Dirichlet eigenvalue of the Laplacian. For the isoperimetric problem on hypersurfaces, we introduce the Michael-Simon and Allard inequality, which is a modified version of the isoperimetric inequality that account for surface curvature. |
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