Optimality and stability of the radial shapes for the Sobolev trace constant

In this work we establish the optimality and the stability of the ball for the Sobolev trace operator $W^{1,p}(\mathrm{\Omega })\hookrightarrow L^q(\mathrm{\partial }\mathrm{\Omega })$ among convex sets of prescribed perimeter for any $1<p<+\mathrm{\infty }$ and $1\le q\le p$. More precisely, we prove that the trace constant ${\sigma }_{p,q}$ is maximal for the ball and the deficit is estimated from below by the Hausdorff asymmetry. With similar arguments, we prove the optimality and the stability of the spherical shell for the Sobolev exterior trace operator $W^{1,p}({\mathrm{\Omega }}_0\setminus \bar{\mathrm{\Theta }})\hookrightarrow L^q(\mathrm{\partial }{\mathrm{\Omega }}_0)$ among open sets obtained removing from a convex set ${\mathrm{\Omega }}_0$ a suitably smooth open hole $\mathrm{\Theta }\subset \subset {\mathrm{\Omega }}_0$, with ${\mathrm{\Omega }}_0\setminus \bar{\mathrm{\Theta }}$ satisfying a volume and an outer perimeter constraint.