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

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GSTW05 - Emerging Horizons in Geometric Spectral Theory: an ECRs workshop

In this work we establish the optimality and the stability of the ball for the Sobolev trace operator $W(\Omega)\hookrightarrow Lq(\partial\Omega)$ 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 $W(\Omega_0\setminus\overline{\Theta})\hookrightarrow Lq(\partial\Omega_0)$ 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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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