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 Δu=uinB(0,μ),∂u∂η= λ|u|q-2uon∂B(0,μ). We find that, for 1≤q<2(N-1)/(N-2), there...
| Autores: | , , |
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| Tipo de documento: | artigo |
| Estado: | Versão publicada |
| Data de publicação: | 2004 |
| País: | Argentina |
| Recursos: | Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales |
| Repositório: | Biblioteca Digital (UBA-FCEN) |
| Idioma: | inglês |
| OAI Identifier: | paperaa:paper_02941449_v21_n6_p795_Bonder |
| Acesso em linha: | http://hdl.handle.net/20.500.12110/paper_02941449_v21_n6_p795_Bonder |
| Access Level: | Acceso aberto |
| Palavra-chave: | Nonlinear boundary conditions Sobolev trace embedding Bessel functions Boundary value problems Eigenvalues and eigenfunctions Mathematical models Problem solving Theorem proving Boundary conditions |
| Resumo: | 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 Δu=uinB(0,μ),∂u∂η= λ|u|q-2uon∂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. © 2004 Elsevier SAS. All rights reserved. |
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