Ntalampekos Questions 1.7 and 1.8 on monotone Sobolev functions

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let $1<p<\infty$ , and let $u\in C(\Omega)\cap W^{1,p}(\Omega)$ be continuous and Lebesgue monotone. Determine whether $u$ can be approximated uniformly and strongly in $W^{1,p}(\Omega)$ by smooth monotone functions, with the prescribed Sobolev boundary values and without increasing the $p$ -Dirichlet energy. Determine also whether the corresponding level-set assertions hold: for almost every $t$ , is $\{u=t\}$ locally a finite- $\mathcal{H}^{n-1}$ -measure embedded $(n-1)$ -dimensional topological submanifold, and do the associated stronger topological and exceptional-set conclusions hold? In the planar case, determine whether the approximation can additionally be achieved by local smoothing near isolated critical points of $p$ -harmonic functions.

References

Primary source

arXiv

Additional references

Monotone Sobolev functions: approximation, critical points, and level sets — arXiv — Deguang Zhong

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A new paper gives several positive and negative answers, but does not completely settle all parts of the two questions.

The questions concern approximation of continuous monotone Sobolev functions by smooth monotone functions, together with related level-set and topological assertions. Ntalampekos’s 2019 work already established substantial planar approximation and level-set results.

Known results

In the plane, continuous monotone $W_{\mathrm{loc}}^{1,p}$ functions admit uniform and Sobolev-norm approximation by smooth monotone functions (Ntalampekos, 2019).
For almost every level $t$ , the level set is locally of finite Hausdorff $1$ -measure and is an embedded $1$ -dimensional topological submanifold (Ntalampekos, 2019).

September 2026 mixed result

Deguang Zhong’s preprint reports planar local smoothing and several higher-dimensional approximation and level-set assertions, while disproving stronger topological and exceptional-set conclusions. The result is substantive progress, but the abstract describes mixed answers rather than a complete resolution; the claims are unverified here.

Current status (as of September 2026): planar results are established and Zhong reports additional affirmative and negative results, but the full scope of Questions $1.7$ and $1.8$ is not completely settled.

Sources

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