Symmetry and symmetry breaking for minimizers in the trace inequality
We consider symmetry properties of minimizers in the variational characterization of the best constant in the trace inequality C∥u∥^p_L^q(∂B_ρ) ≤∥u∥^p_W1,p(Bρ) in the ball Bρ of radius ρ. When p is fixed minimizers in this problem can be radial or nonradial depending on the parameters q and ρ. We pr...
| Autores: | , |
|---|---|
| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2011 |
| País: | Argentina |
| Recursos: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/110168 |
| Acesso em linha: | http://hdl.handle.net/11336/110168 |
| Access Level: | acceso abierto |
| Palavra-chave: | SYMMETRY BREAKING SOBOLEV TRACE INEQUALITY NONLINEAR BOUNDARY CONDITION https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Resumo: | We consider symmetry properties of minimizers in the variational characterization of the best constant in the trace inequality C∥u∥^p_L^q(∂B_ρ) ≤∥u∥^p_W1,p(Bρ) in the ball Bρ of radius ρ. When p is fixed minimizers in this problem can be radial or nonradial depending on the parameters q and ρ. We prove that there is a global radial function u0 > 0, with u0 independent of q, such that any radial minimizer is a multiple of the restriction of u0 to Bρ. Next we prove that if either q or ρ is sufficiently large then the minimizers are nonradial. In the case when p = 2 we consider a generalization of the minimization problem and improve some of the above symmetry results. We also present some numerical results describing the exact values of q and ρ for which radial symmetry breaking occurs. |
|---|