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Let $\Omega$ be a bounded, open, simply connected subset of $\mathbb R^n$ with Lipschitz boundary.

Question: Does every function in the Sobolev space $W^{1,1} (\Omega)$ admit a representative whose graph in $\Omega \times \mathbb R$ has a path connected component whose projection to $\Omega$ has full measure in $\Omega$?

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    $\begingroup$ I would check Evans and Gariepy. The answer by Hajlasz here mentions a theorem in "Approach 1" which comes very close to what you want. $\endgroup$
    – Willie Wong
    Commented Apr 19, 2021 at 14:25
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    $\begingroup$ ACL ("absolutely continuous on lines") characterisation of weakly differentiable functions should do the job, should it not? $\endgroup$
    – Mateusz Kwaśnicki
    Commented Apr 19, 2021 at 14:26
  • $\begingroup$ Oh, I suppose so - with the newest stipulation that there needs to only be a full measure path connected component, then ACL should work. $\endgroup$
    – Nate River
    Commented Apr 19, 2021 at 14:27

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As pointed out by Mateusz Kwaśnicki, the answer is yes.

Applying the ACL characterisation of Sobolev functions (see, for example page 36 here: https://math.aalto.fi/~jkkinnun/files/sobolev_spaces.pdf) to each of the coordinates successively, we obtain a full measure subset $S$ of $\Omega$ whereby any two points in the graph of $f$ over $S$ are joined by the graphs of at most $n$ absolutely continuous functions (that are parallel to the coordinate axes). So in particular the graph over $S$ is path connected, as required.

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