In this paper, we examine the boundary $L^2$ term of the sharp Sobolev trace inequality $\|u\|_{ L^{ q}(\pa M)}^2\leq S \|\nabla_g u\|_{ L^2(M)}^2 +A(M,g)\|u\|^2_{ L^2(\pa M)}$ on Riemannian manifolds $(M,g)$ with boundaries $\pa M$, where $q=\frac{ 2(n-1)}{ n-2}$, $S$ is the best constant and $A(M,g)$ is some positive constant depending only on $M$ and $g$. We obtain a sharp trace inequality involving the mean curvature in a remainder term, which would fail in general once the mean curvature is replaced by any smaller function.
Categories: math.AP