### 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.

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.

#### Keywords

Trace inequality; symmetry; symmetry breaking

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