Estimates of the best Sobolev constant of the embedding of $BV(\Omega )$ into $L^1(\partial \Omega )$ and related shape optimization problems

Abstract

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality ${\lambda }_1(\Omega )\Vert u{\Vert }_{L^1(\partial \Omega )}\le \Vert u{\Vert }_{W^{1,1}(\Omega )}$ that are independent of $\Omega$. This estimates generalize those of \cite{BS} concerning the $p$-Laplacian to the case $p=1$. We apply our results to prove existence of an extremal for this embedding. We then study an optimal design problem related to ${\lambda }_1$, and eventually compute the shape derivative of the functional $\Omega \to {\lambda }_1(\Omega )$. As a consequence, we obtain that a ball of ${\mathbb{R}}^n$ of radius $n$ is critical for volume-preserving deformations.