Periodic stationary sets for the fractional perimeter with a general kernel
The aim of this master's thesis is to construct some periodic sets in the plane with constant nonlocal (or fractional) mean curvature induced by general anisotropic kernels. Inspired by the article \cite{bib1} by Cabré, Moustapha, Solà-Morales and Weth, we employ a Lyapunov-Schmidt reduction te...
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| Tipo de recurso: | tesis de maestría |
| Fecha de publicación: | 2023 |
| 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:2117/392899 |
| Acceso en línea: | https://hdl.handle.net/2117/392899 |
| Access Level: | acceso abierto |
| Palabra clave: | Differential equations, Partial Kernel functions Nonlocal mean curvature fractional perimeter nonlocal Delaunay cylinders fractional Laplacian minimal surfaces Equacions en derivades parcials Kernel, Funcions de Classificació AMS::35 Partial differential equations Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | The aim of this master's thesis is to construct some periodic sets in the plane with constant nonlocal (or fractional) mean curvature induced by general anisotropic kernels. Inspired by the article \cite{bib1} by Cabré, Moustapha, Solà-Morales and Weth, we employ a Lyapunov-Schmidt reduction technique. This approach yields a family of periodic perturbations of an infinite band. To understand the framework and the tools used in the procedure, we introduce fractional Sobolev spaces and the fractional Laplacian. Furthermore, after reviewing the classical theory of minimal surfaces, we extrapolate these concepts into a nonlocal context, obtaining the fractional perimeter and the nonlocal mean curvature. Additionally, we prove the existence of solutions to the Dirichlet problem for the fractional Laplacian. |
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