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 $W^{1,p}(Ω)\hookrightarrow L^q(\partialΩ)$ among convex sets of prescribed perimeter for any $1< p <+\infty$ and $1\le q\le p$. More precisely, we prove that the trace constant $σ_{p,q}$ 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 $W^{1,p}(Ω_0\setminus\overlineΘ)\hookrightarrow L^q(\partialΩ_0)$ among open sets obtained removing from a convex set $Ω_0$ a suitably smooth open hole $Θ\subset\subsetΩ_0$, with $Ω_0\setminus\overlineΘ$ satisfying a volume and an outer perimeter constraint.

💡 Research Summary

The paper investigates two isoperimetric‑type optimization problems for Sobolev trace embeddings and provides quantitative stability results.

1. Optimality of the ball for the Sobolev trace constant
   For a bounded Lipschitz domain (\Omega\subset\mathbb R^{d}) and parameters (1<p<\infty), (1\le q\le p), the Sobolev trace constant (\sigma_{p,q}(\Omega)) is defined as the best constant in the inequality
   \