On the best Sobolev inequalities on Riemannian manifolds with boundaries

Let $ (M,g) $ (M,g) be a smooth n-dimensional compact Riemannian manifold with totally geodesic boundary $ \partial M $ partial derivative M, $ n\ge 2 $ n >= 2. We prove a sharp Sobolev inequality of $ W<^>{1,p}(M) $ W1,p(M) for all 1<p<n. When p = 2 and $ n\ge 3 $ n >= 3, this inequality was proved by Li-Zhu in [Sharp Sobolev inequalities involving boundary terms. Geom Funct Anal. 1998;8:59-87].