Let $M$ be a smooth manifold whose boundary is the disjoint union of two compact manifolds, $\partial M=M_0\cup M_1$. I wish to find a Morse function $f:M\to [-1,1]$ such that $f^{-1}(-1)=M_0, f^{-1}(1)=M_1$ and such that there's an open neighbourhood of $\partial M$ containing no critical points.
My first idea is to "mix" two Morse functions: one away from the boundary and one near the boundary. Find a "tubular" neighbourhood $\partial M\subset N \cong \partial M\times [0,1)$, which we can decompose into $M_0\times [-1,0)$ and $M_1\times (0,1]$. On this $N$ we define $f$ as the "vertical" coordinate. Then, we can use any Morse function on $M-N$ and "smooth" it out to $0$ using a bump function near its boundary.
Unfortunately I can't quite find a rigorous argument to make this work. Would anyone have any ideas?