Any homeomorphism of a compact surface can be approximated by diffeomorphisms. Is there a parametrized version of this result, where the parameter space is an $n$-disk?
In other words, if $S$ is a compact surface, can every continuous map $D^n\to\mathrm{Homeo}(S, \mathrm{rel}\, \partial S)$ be approximated by a continuous map $D^n\to\mathrm{Diff}(S, \mathrm{rel}\, \partial S)$? Here the diffeomorphism group has $C^0$ topology, so that it is a subspace of the homeomorphism group.
I am mainly interested in $S=D^2$.
The proof in the unparametrized version (written by Munkres his 1960 Annals paper) goes by approximating the homeomorphism by a PL homeomorphism and smoothing the corners. This seems problematic when $n$ is large.
EDIT: Let me add references implementing the first step in Misha's comments, and also discuss what happens in higher dimensions. As Misha says, one can answer the question as follows.
Approximate a family of homeomorphisms of the surface by a family of PL homeomorphisms.
Note that PL homeomorphisms are quasiconformal ($1$-skeleton is removable for quasiconformal maps).
Smooth the dilatations of the PL homeomorphisms by convolution (or other method).
Solve the Beltrami equation for these dilatations (here one may have to normalize the homeomorphisms to fix 3 points to get a unique solution for each dilatation).
All of this should be done relatively i.e. assume that the approximation is given on a subcomplex of $D^n$, and try to extend it to the whole $D^n$.
Let me discuss step 1. Hamstrom's paper linked in comments does not contain the relative version, even though she surely knew it as is clear from proofs). A nice relative version can be found in this paper by Yagasaki, see Proposition 3.1, where he deals with homeomorphisms that fix a compact (possiby empty) subpolyhedron $X$ of the surface, and where he find a self-homotopy of $\mathrm{Homeo}(S; \mathrm{rel}\, X)$ that starts at the identity and instantly moves the space into the subspace of PL homeomorphisms.
Is there a similar approximation result in higher dimensions? It would probably have to be PDE-free.
It turns out there is a general isotopy approximation theorem in Kirby-Siebenmann's book (p.78). Their result approximates topological isotopies $D^n\to \mathrm{Homeo}(M)$ by CAT ($=$ PL or Diff) isotopies. Here $M$ is a compact manifold, possibly with boundary, such that $\dim(M)\neq 4\neq\dim(\partial M)$. Then an approximation always exists, and it can be done relatively as follows. Suppose $C$ is a closed subset of $M$ and $\Lambda$ is a retract of $D^n$. Any topological isotopy that is CAT near $D^n\times C\cup\Lambda\times M$ can be approximated by a CAT isotopy rel $D^n\times C\cup \Lambda\times M$.
The retriction that $\Lambda$ is a retract is annoying, e.g. given a sphere in $\mathrm{Diff}(M)$ that is null-homotopic in $\mathrm{Homeo}(M)$, it is natural to ask if we can one move the null-homotopy into $\mathrm{Diff}(M)$. To do so one can try to proceed as follows. Assume CAT=Diff. Decompose $\partial D^n$ into the union of two disks, one of which will be our set $\Lambda$ and the other $\Lambda^\prime$ is very small (in the sense to be determined). Suppose we are given a topological isotopy that is smooth on $\partial D^n\times M$. Now smoothly approximate it relative to $\Lambda\times M$. The $\Lambda^\prime$-portion will move but if $\mathrm{Diff}(M)$ is locally contractible as a subspace of $\mathrm{Homeo}(M)$, then using smallness of $\Lambda^\prime$ one can arrange the approximation to extend the one on $\Lambda\cup\Lambda^\prime$.
The question remains whether $\mathrm{Diff}(M)$ is locally contractible in $C^0$ topology. This may depend on the dimension of $M$.
