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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 $f(x_n)\to x$).

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It is difficult to define the values on a set of measure zero of a "function" in an arbitrary Sobolev space. For example, the boundary of the disc, or countable sets (e.g. the values of a sequence) have Lebesgue measure zero in $\mathbb{R}^n$ for $n>1$. This happens because elements of Sobolev spaces are really equivalence classes of functions identified if they differ on a set of measure zero. As such, one has to be very careful to describe what one means by their values on a set of measure zero. – Ricardo Andrade May 9 '13 at 9:28
@Ricardo Andrade: you are of course right, but one can take the trace - I edited the question accordingly. – Benoît Kloeckner May 9 '13 at 11:31
@Benoît: Indeed. Somehow, I forgot about the trace... Thank you very much for the clarification. – Ricardo Andrade May 9 '13 at 15:25

This does not answer your specific questions, but degree theory has been extended to certain classes of non-continuous functions (including some Sobolev spaces) by Brezis and Nirenberg. See

Brezis, H., & Nirenberg, L. (1995). Degree theory and BMO; part I: Compact manifolds without boundaries. Selecta Mathematica, New Series, 1(2), 197-263.

Brezis, H., & Nirenberg, L. (1996). Degree theory and BMO; part II: Compact manifolds with boundaries. Selecta Mathematica, New Series, 2(3), 309-368.

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Here is a more recent survey by P. Mironescu of these topics… – Liviu Nicolaescu May 9 '13 at 19:40

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