Estimates of the best Sobolev constant of the embedding of BV(Ω) into L1(∂Ω)L1(∂Ω) and related shape optimization problems

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality λ1(Ω)‖u‖L1(∂Ω)≤‖u‖W1,1(Ω)λ1(Ω)‖u‖L1(∂Ω)≤‖u‖W1,1(Ω) that are independent of ΩΩ. These estimates generalize those of [J. Fernandez Bonder, N. Saintier, Estimates for the Sobolev trace constant with critical exponent and applications, Ann. Mat. Pura. Aplicata (in press)] concerning the pp-Laplacian to the case p=1p=1.We apply our results to prove the existence of an extremal for this embedding. We then study an optimal design problem related to λ1λ1, and eventually compute the shape derivative of the functional Ω→λ1(Ω)Ω→λ1(Ω).