Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space
In this paper, we study Hardy–Sobolev inequalities on hypersurfaces of Rn+1, all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael–Simon and Allard, in our codimension one framework....
| Autores: | , |
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| Formato: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Recursos: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/388185 |
| Acesso em linha: | https://hdl.handle.net/2117/388185 https://dx.doi.org/10.1142/S0219199721500632 |
| Access Level: | acceso abierto |
| Palavra-chave: | Inequalities on hypersurfaces Michael–Simon Sobolev inequality Carron Hardy inequality Mean curvature Classificació AMS::26 Real functions::26D Inequalities Classificació AMS::46 Associative rings and algebras::46E Linear function spaces and their duals Classificació AMS::53 Differential geometry::53A Classical differential geometry Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Resumo: | In this paper, we study Hardy–Sobolev inequalities on hypersurfaces of Rn+1, all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael–Simon and Allard, in our codimension one framework. Using their ideas, but simplifying their presentations, we give a quick and easy-to-read proof of the inequality. Next, we establish two new Hardy inequalities on hypersurfaces. One of them originates from an application to the regularity theory of stable solutions to semilinear elliptic equations. The other one, which we prove by exploiting a “ground state” substitution, improves the Hardy inequality of Carron. With this same method, we also obtain an improved Hardy or Hardy–Poincaré inequality. |
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