Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds

Given a domain $\mathrm{\Omega }$ of ${\mathbb{R}}^{m+1}$ and a $k$-dimensional non-degenerate minimal submanifold $K$ of $\text{pa}\mathrm{\Omega }$ with $1\le k\le m-1$, we prove the existence of a family of embedded constant mean curvature hypersurfaces which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\mathrm{\partial }\mathrm{\Omega }$ perpendicularly.