The Dirichlet problem for the fractional Laplacian
The main object under study in this thesis is the fractional Laplacian. In this work, we review the proof of existence and uniqueness of weak solutions to the Dirichlet problem for this operator. We also prove that such weak solutions are in fact smooth, and show some qualitative properties related...
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| Tipo de recurso: | tesis de maestría |
| Fecha de publicación: | 2025 |
| 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:2117/423445 |
| Acceso en línea: | https://hdl.handle.net/2117/423445 |
| Access Level: | acceso abierto |
| Palabra clave: | Functional analysis Differential equations, Partial Partial Differential Equations Nonlocal operators Fractional Laplacian Anàlisi funcional Equacions diferencials parcials Classificació AMS::35 Partial differential equations::35R Miscellaneous topics involving partial differential equations Classificació AMS::35 Partial differential equations::35A General theory Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | The main object under study in this thesis is the fractional Laplacian. In this work, we review the proof of existence and uniqueness of weak solutions to the Dirichlet problem for this operator. We also prove that such weak solutions are in fact smooth, and show some qualitative properties related to the fractional Laplacian: maximum and comparison principles, and Harnack's inequality. To achieve this, we introduce key ideas like nonlocality and the necessary tools from functional analysis and the calculus of variations. |
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