Published Paper
Inserted: 30 apr 2025
Last Updated: 4 mar 2026
Journal: Nonlinear Analysis
Year: 2026
Doi: https://doi.org/10.1016/j.na.2026.114086
Abstract:
In this work we establish the optimality and the stability of the ball for the Sobolev trace operator $W^{1,p}(\Omega)\hookrightarrow L^q(\partial\Omega)$ 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 $\sigma_{p,q}$ 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 $W^{1,p}(\Omega_0\setminus\overline{\Theta})\hookrightarrow L^q(\partial\Omega_0)$ among open sets obtained removing from a convex set $\Omega_0$ a suitably smooth open hole $\Theta\subset\subset\Omega_0$, with $\Omega_0\setminus\overline{\Theta}$ satisfying a volume and an outer perimeter constraint.
Keywords: p-Laplace operator, Sobolev trace inequality, Quantitative spectral inequality
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