Some constructions for the fractional Laplacian on noncompact manifolds
We give a definition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by Caffarelli-Silvestre. While this definition in the compact case is straightforward, in the noncompact setting one needs to have a precise control of the behavior of the metric at...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2015 |
| 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:2117/85051 |
| Acceso en línea: | https://hdl.handle.net/2117/85051 https://dx.doi.org/10.4171/RMI/850 |
| Access Level: | acceso abierto |
| Palabra clave: | Fractional differential equations Fractional Laplacian non-compact manifolds hyperbolic space extension problem Fourier symbol singular kernel heat kernel sobolev inequalities riemannian-manifolds conformal laplacians schrodinger-equation h-n regularity operators Equacions diferencials el·líptiques Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | We give a definition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by Caffarelli-Silvestre. While this definition in the compact case is straightforward, in the noncompact setting one needs to have a precise control of the behavior of the metric at infinity and geometry plays a crucial role. First we give explicit calculations in the hyperbolic space, including a formula for the kernel and a trace Sobolev inequality. Then we consider more general noncompact manifolds, where the problem reduces to obtain suitable upper bounds for the heat kernel. |
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