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Suppose that $M$ is a manifold with boundary $\partial M$ and that there is a continuous map $f: M \rightarrow N$ (where $N$ is a smooth manifold). Whitney's approximation theorem states that $f$ can be $C^0$-approximated by a smooth map $g$, which can be made to agree with $f$ on $\partial M$ if $f$ is already smooth on $\partial M$. I am interested in dropping the condition that $f$ is smooth on $\partial M$.

Question: Can $f$ be approximated by map $g: M \rightarrow N$ which is smooth in $M\backslash \partial M$ and equal to $f$ on $\partial M$?

So in this case, $g$ can be quite bad on $\partial M$ but smooth in $M\backslash \partial M$. A sample case is $M = D^n, N = D^m$.

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    $\begingroup$ Yes, you can, and you can use Whitney's original theorem to prove it. Hint: Use an exhaustion of the interior of $M$ by smooth properly embedded codimension zero submanifolds with boundary. $\endgroup$ Commented 2 days ago
  • $\begingroup$ @MoisheKohan Thanks! Do you mind making this an official answer, adding some more details if possible? $\endgroup$ Commented 2 days ago

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Here is a sketch in the general case, you should be able to fill in the details. Let $M$ be a smooth manifold. Consider an exhaustion $M_k$ of $int(M)$ by an increasing sequence of smooth compact codimension 0 submanifolds with boundary. Define $A_k:= M_k\setminus int(M_{k-1})$.

Given your map $f$, you construct (using Whitney's approximation) a smooth $\epsilon$-approximation inductively. Start with $M_1$ and find a smooth map $f_1: M\to N$ such that $d(f, f_1)<\epsilon/2$. (Here and below, $d$ comes from a fixed metric on $N$.) Then consider the restriction of $f_1$ to $\partial M_1\subset A_1$ and $f|\partial M_2$. Using Whitney again, you find a smooth map $f_2: M_2\to N$ which agrees with $f_1$ on $M_1$ and $d(f_2|A_2, f|A_2)< \epsilon/2 + \epsilon/4$, while $d(f_2|\partial M_2, f|\partial M_2)<\epsilon/4$. Then repeat for $M_3$ so that $$ d(f_3, f)< \epsilon/2 + \epsilon/4 + \epsilon/8, $$ $$ d(f_3|A_3, f|A_3)<\epsilon/4 +\epsilon/8, d(f_3|\partial M_3, f|\partial M_3)<\epsilon/8 $$ Continue inductively. In the end, you obtain the required approximation.

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I think I can say something for the case your $M$ is a surface with boundary and $N$ open in the real line.

I don't know if you have ever heard of the Dirichlet Boundary problem. It says the following:

Consider $D$ an open and connected set on the plane with smooth boundary. Assume we have a function $u$ continuous on $\partial{D}$. Then, there exists a function $f$ defined on $D \cup \partial{D}$, s.t.: $f=u$ on $\partial{D}$, $f$ is continuous on $D \cup \partial{D}$ and smooth inside $D$.

As locally, any manifold with boundary, near a boundary point looks like $ \mathbb{H}= \{(x,y) : y \geq 0 \}$, locally for $f$ restricted on a neighborhoud like that, can be extended to an open set around the boundary.

Then, I would say you can "glue" these functions defined on a local coordinate of every point just using partitions of unity. So you have extended $f$ to some smooth function defined on an open set around $\partial{D}$.

Again using partitions of unity, you may extend this new function to the whole interior of $M$ and we are done in the case of $2-$dimensional manifolds with boundary.

For bigger dimensions or different $N$'s it probably is also true.

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