Symmetry properties for the extremals of the Sobolev trace embedding

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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...

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Detalles Bibliográficos
Autores principales: Bonder, J.F., Dozo, E.L., Rossi, J.D.
Formato: Artículo publishedVersion
Lenguaje:Inglés
Publicado: 2004
Materias:
Nonlinear boundary conditions
Sobolev trace embedding
Bessel functions
Boundary value problems
Eigenvalues and eigenfunctions
Mathematical models
Problem solving
Theorem proving
Boundary conditions
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_02941449_v21_n6_p795_Bonder
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
Descripción
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 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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