Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries

In this paper, we establish some sharp Sobolev trace inequalities on n-dimensional, compact Riemannian manifolds with smooth boundaries. More specifically, let q = 2(n - 1)/(n - 2), 1/S = inf {∫ |∇u|² : ∇u ∈ L²(R₊ⁿ), ∫ |u|^q = 1}. We establish for any Riemannian manifold with a smooth boundary, denoted as (M, g), that there exists some constant A = A(M, g) > 0, (∫_dM|u|^q ds_g)^2/q < or = to S ∫_M |∇_gu|² dv_g + A ∫_dMu² ds_g, for all u ∈ H¹ (M). The inequality is sharp in the sense that the inequality is false when S is replaced by any smaller number. © 1997 John Wiley & Sons, Inc.