**Setting**


Suppose I have two bounded open domains $\Omega' \subset \Omega \subset \mathbb{R}^n$ (I'm particularly interested in case n = 2 or n = 3). We assume that all boundaries of domains are $C^\infty$-smooth and that inner domain is lying properly (with it closure) inside the outer: $\bar{\Omega'} \subset \Omega$.


Suppose I'm given a smooth function $f \in C^\infty\left(\overline{\Omega \setminus \Omega'}; \mathbb{R}\right)$. We could assume that $L^\infty$ norm of all derivatives of $f$ is bounded.


**Question**


Is it possible to continue it smoothly inside $\Omega'$? What are the constructive ways to do it (or may be just with finite smoothness)? 


**Example**


If we seek for just a continuous continuation of $f$, then we could put a "rubber film" over inner domain, or, formally, continue $f$ with the solution of the following [Poisson equation](http://en.wikipedia.org/wiki/Poisson%2527s_equation):
$\Delta u = 0$ in $\Omega'$ with Dirichlet boundary conditions: $u\_{\partial \Omega'} = f\_{\partial \Omega'}$.


**Update** By constructive I mean "numerically friendly", i.e. easy to code.


References and comments are appretiated! 



Continuation of a smooth function
This answer may not be practically useful to you, but I think it's nice from a conceptual point of view. The extension could be done in two steps. I'm presuming that you are defining a smooth function on the closed set $K=\overline{\Omega\setminus\Omega'}$ as one that is smooth on a small neighborhood $K'$ (which I'll also take as closed) $K$.


First, you could extend $f$ continuously from the closed set $K'$ to all of $\Omega$. Second, you could smooth the extension, preserving it on the interior of $K$. The continuous extension is a direct application of the [Tietze extension theorem](http://en.wikipedia.org/wiki/Tietze_extension_theorem), while the smoothing is an application of the [Steenrod approximation theorem](http://ncatlab.org/nlab/show/Steenrod-Wockel+approximation+theorem).


