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 vertical bar vertical bar u vertical bar vertical bar(p)(Lq(partial derivative BNp)) <= vertical bar vertical bar u vertical bar vertical bar(p)(W1,p (B rho)) in the ball B-rho of radius rho. When p is fixed, minimizers in this problem can be radial or non-radial depending on the parameters q and p. We prove that there is a global radial function u(0) > 0, with u(0) independent of q, such that any radial minimizer is a multiple of the restriction of uo to B-rho. Next, we prove that if either q or rho is sufficiently large, then the minimizers are non-radial. 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 rho for which radial symmetry breaking occurs.