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

Abstract

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality $λ_1(Ω)\|u\|_{L^1(\partialΩ)} \le \|u\|_{W^{1,1}(Ω)}$ that are independent of $Ω$. 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 $λ_1$, and eventually compute the shape derivative of the functional $Ω\toλ_1(Ω)$. As a consequence, we obtain that a ball of $\R^n$ of radius $n$ is critical for volume-preserving deformations.