Symmetry properties for the extremals of the Sobolev trace embedding
In this article we study symmetry properties of the extremals for the Sobolev trace embedding H1(B(0, µ)) ,→ Lq(∂B(0, µ)) with 1 ≤ q ≤2(N − 1)/(N − 2) for different values of µ. These extremals u are solutions of the problem {∆u = u in B(0, µ), ∂u_∂η = λ|u|q−2u on ∂B(0, µ). We find that, for 1 ≤ q &...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2004 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/110302 |
| Acceso en línea: | http://hdl.handle.net/11336/110302 |
| Access Level: | acceso abierto |
| Palabra clave: | NONLINEAR BOUNDARY CONDITIONS SOBOLEV TRACE EMBEDDING https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | In this article we study symmetry properties of the extremals for the Sobolev trace embedding H1(B(0, µ)) ,→ Lq(∂B(0, µ)) with 1 ≤ q ≤2(N − 1)/(N − 2) for different values of µ. These extremals u are solutions of the problem {∆u = u in B(0, µ), ∂u_∂η = λ|u|q−2u on ∂B(0, µ). We find that, for 1 ≤ q < 2(N − 1)/(N − 2), there exists a unique normalized extremal u, which is positive and has to be radial, for µ small enough. For the critical case, q = 2(N−1)/(N−2), as a consequence of the symmetry properties for small balls, we conclude the existence of radial extremals. Finally, for 1 < q ≤ 2, we show that a radial extremal exists for every ball. |
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