On the Yamabe invariant of certain compact manifolds with boundary

We generalize Kobayashi’s connected-sum inequality to the $\lambda$-Yamabe invariants. As an application, we calculate the $\lambda$-Yamabe invariants of $\mathrm{\#}{m}_1{\mathbb{R}\mathbb{P}}^n\mathrm{\#}{m}_2({\mathbb{R}\mathbb{P}}^{n-1}\times S^1)\mathrm{\#}l{H}^n\mathrm{\#}k{S}_{+}^n$, for any $\lambda \in [0,1]$, $n\ge 3$, provided $k+l\ge 1$. As a corollary, we prove that ${\mathbb{R}\mathbb{P}}^n$ minus finitely many disjoint $n$-balls have the same $\lambda$-Yamabe invariants as the hemi-sphere, which forms an interesting contrast with the famous Bray-Neves results on the Yamabe invariants of ${\mathbb{R}\mathbb{P}}^3$.