Symmetry and symmetry breaking for minimizers in the trace inequality

Abstract

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.