## diffeomorphism on manifolds with boundary

Hi,

I'm confused with some differential properties on manifold with boundary,

here is my problem :

Let $K$ be a compact subset of $\mathbb{R}^{n}$ (with non empty interior, if $n=2$ $K$ is "flat").

I suppose $K$ to be a manifold with boundary.

Let $D$ be a manifold with boundary in $\mathbb{R}^{m}$, $m\geq n$.

Consider a $\mathcal{C}^{1}$ function $f:K\to D$ such that $f$ is one to one and $J_{f}(x)>0$ on $K$.

I would like to show a global inversion theorem to say that $f$ is a $\mathcal{C}^{1}$ - diffeomorphism beetween the two manifolds with boundary, however i didn't succeed building a diffeomorphic extension $\overline{f}$ of $f$ on a an open neighbourhood $V$ of $K$.

Do you know if the global inversion theorem holds in the manifolds with boundary case (and where i could find it)?

If yes, do we also have such result when the compacts $K$ and $D$ are manifolds with corners?