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. Mor e specifically, let q=2(n-1)/(n-2), 1/S=inf{integral(Rn+)\del u\(2) : del u is an element of L-2(R-+(n)), integral(partial derivative Rn+) \ u\(q)=1}. We establish for any Riemannian manifold with a smooth bound ary, denoted as (M, g), that there exists some constant A = A(M,g) > 0 , (integral(partial derivative M) \u\(q) ds(g))(2/q) less than or equa l to S integral(M) \del(g)u\(2) dv(g) + A integral(partial derivative M) u(2) ds(g), for all u is an element of H-1(M). The inequality is sh arp in the sense that the inequality is false when S is replaced by an y smaller number. (C) 1997 John Wiley & Sons, Inc.