A one dimensional problem related to the symmetry of minimisers for the Sobolev trace constant in a ball

Abstract

The symmetry of minimisers for the best constant in the trace inequality in a ball, $S_q(\rho)=\inf_{u\in W^{1,p}(B_\rho)} \|u\|^p_{W^{1,p}(B_\rho)}/ \|u\|^{p}_{L^q(\partial B(\rho))}$ has been studied by various authors. Partial results are known which imply radial symmetry of minimisers, or lack thereof, depending on the values of trace exponent $q$ and the radius of the ball $\rho$. In this work we consider a one dimensional analogue of the trace inequality and the corresponding minimisation problem for the best constant. We describe the exact values of $q$ and $\rho$ for which minimisers are symmetric. We also consider the behaviour of minimisers as the symmetry breaking threshold for $q$ and $\rho$ is breached, and show a case in which both symmetric and nonsymmetric minimisers coexist.