Around the best constants for the Sobolev trace map from W 1,1 (Ω) into L 1 (∂Ω)

In this article, we prove that the best constant for the Sobolev trace map from W^1,1( $\mathbb{R}$ ^N−1× $\mathbb{R}$ ⁺) into L¹( $\mathbb{R} $ ^N−1×{0}) is 1. We deduce from it that when Ω is a bounded open set whose boundary is piecewise C¹, the best first constant for the Sobolev trace map γ from W^1,1(Ω) into L¹(∂Ω) is also 1, in the sense that: for any ε>0, there exists B_ε>0 such that for any u in W^1,1(Ω), ∫_∂Ω|u|≤(1+ε)∫_Ω|∇u|+B_ε∫_Ω|u|.

Independently, we prove that the best second constant for Sobolev trace map from W^1,1(Ω) into L¹(∂Ω) is |∂Ω|/|{Ω}| (where |∂Ω| denotes the (N−1)‐dimensional measure of ∂Ω and |{Ω}| the N‐dimensional measure of Ω) when Ω is a connected open bounded set; in other words, there exists A>0 such that for any u in W^1,1(Ω): ∫_∂Ω|u|≤A∫_Ω|∇u|+ $\dfrac{|\partial{\Omega}|}{|{\Omega}|}$ ∫_Ω|u|.