Guerra–Tione Question 5.5 on gradient mappings

Guerra--Tione Question 5.5 asks whether, for $u\in W^{2,n}_{\mathrm{loc}}(\Omega)$ with $\det D^2u\ge\delta>0$ almost everywhere, the gradient map $Du$ must be open and discrete. The question is linked to a conjecture of Šverák from 1992.

Known results

• Šverák, 1992: the corresponding index-constancy conjecture was proved in dimensions $n\le3$.
• Constancy of the index for gradient mappings, 2025: proves almost-everywhere constancy of $\operatorname{ind}(D^2u)$ for $u\in W^{2,\infty}_{\mathrm{loc}}$ under the determinant lower bound, in every dimension.
• In dimension $2$, an explicit $W^{2,\infty}$ example shows that replacing $\det D^2u\ge\delta$ by $\det D^2u>0$ can fail.

August 2026 counterexample

Zhong and Deguang claim an explicit $u\in W^{2,n}_{\mathrm{loc}}(\Omega)$ for every $n\ge4$ with $\det D^2u\ge\delta>0$ and $D^2u>0$ almost everywhere, while $Du$ collapses a line segment; hence $Du$ is neither open nor discrete. The preprint says its construction does not extend to $n=3$, where a logarithmic divergence appears. The claim has not been independently verified in the retrieved sources.

Current status (as of August 2026): A preprint claims the question is false for every $n\ge4$, but that counterexample remains unverified, and the $n=3$ case is open.