This question is triggered by a talk by Pierre Bousquet, who considered related questions (but not quite what I ask below).

Take a classical algebraic topological result, like the inexistence of retraction map $f:D^2\to \partial D^2$. Can we lower the regularity hypothesis (i.e., replace continuity with something weaker, or at least something not implying continuity) and still get a result?

Let me be more precise:

For which values of $p$ Does it exist a map $f:D^2\to \partial D^2$ in $W^{1,p}$ such that the trace of $f$ on the boundary is the identity?

In the same spirit:

For which values of $s,p$ must each map $f:D^2 \to D^2$ in $W^{s,p}$ have an almost fixed point in some sense (e.g. a sequence $x_n\to x$ such that $f(x_n)\to x$).