The best constant for the Sobolev trace embedding from image into image Original Research Article

In this paper we study the best constant, λ1(Ω)λ1(Ω) for the trace map from W1,1(Ω)W1,1(Ω) into L1(∂Ω)L1(∂Ω). We show that this constant is attained in BV(Ω)BV(Ω) when λ1(Ω)<1λ1(Ω)<1. Moreover, we prove that this constant can be obtained as limit when p↘1p↘1 of the best constant of W1,p(Ω)↪Lp(∂Ω)W1,p(Ω)↪Lp(∂Ω). To perform the proofs we will look at Neumann problems involving the 1-Laplacian, Δ1(u)=div(Du/|Du|)Δ1(u)=div(Du/|Du|).