Around the best constants for the Sobolev trace map from W^1,1(Ω) into L¹(∂Ω)

Article type: Research Article

Authors: Motron, Mélissa

Affiliations: Université de Cergy‐Pontoise, Département de Mathématiques, site de Saint‐Martin, 2, avenue Adolphe Chauvin, 95302 Cergy‐Pontoise cedex, France

Abstract: In this article, we prove that the best constant for the Sobolev trace map from W1,1($\mathbb{R}$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 W1,1(Ω) into L1(∂Ω) is also 1, in the sense that: for any ε>0, there exists Bε>0 such that for any u in W1,1(Ω), ∫∂Ω|u|≤(1+ε)∫Ω|∇u|+Bε∫Ω|u|. Independently, we prove that the best second constant for Sobolev trace map from W1,1(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 W1,1(Ω): ∫∂Ω|u|≤A∫Ω|∇u|+ $\dfrac{|\partial{\Omega}|}{|{\Omega}|}$∫Ω|u|.

Journal: Asymptotic Analysis, vol. 29, no. 1, pp. 69-90, 2002