A characterization of differentiability for the best trace Sobolev constant function (Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations)

Abstract

Let 1 < p < N and let Ω be a smooth bounded domain in RN. In this paper we show some regularity results for the best constant Sq of the trace Sobolev embedding W1, p(Ω) ↪ Lq(∂Ω), considering that Sq is a function of q. We prove that Sq is absolutely continuous, thus S'q = d/dq Sq exists a.e. q ∈ [1, p*], p*= p(N-1)/N-p. We give a characterization on a set where S'q exists. These are natural extensions of the recent work by Ercole for the best constant of the Sobolev embedding W1, p 0(Ω) ↪ Lq(Ω) for q∈ [1, p*], p* = Np/N-p.