Let (M,∂M,g) be a compact Riemannian manifold with boundary. As a generalization of the Yamabe invariant with boundary Y(M,∂M,g) , we define the kth Yamabe invariant with boundary Yk(M,∂M,g). We prove some of its properties and study when it can be attained by the generalized metric. We also prove a version of conformal Schwarz lemma on (M,∂M,g)by using the Yamabe flow with boundary.
