Rivière’s regularity question for critical n-Laplace systems with antisymmetric potentials

Rivière asked whether every weak solution of the critical $n$-Laplace system with antisymmetric potential $\Omega\in L^n$ must be continuous. The question was explicitly left unanswered in earlier work.

Known results

• Martino and Schikorra (2023): continuity under stronger Lorentz assumptions, including $\Omega\in L^{(n,2)}$ plus a curl condition in $L^{(n,1)}$.
• Martino and Schikorra (2023): continuity for $n$-harmonic maps when $\nabla u\in L^{(n,2)}$.

2026 claimed counterexample

A preprint claims that for every $n\ge 3$ there are discontinuous $u\in W^{1,n}\cap L^\infty$ and antisymmetric $\Omega\in L^n$ solving the system distributionally, with endpoint behavior $\nabla u,\Omega\in L^{(n,q)}$ exactly when $q>2$. For $n\ge 4$, it additionally claims $\Omega=\nabla\Xi$ with $\Xi\in W^{1,n}$. The article says ChatGPT contributed ideas and a first draft; human validation and final decisions were retained.

Current status (as of August 2026): Rivière’s unrestricted $L^n$ continuity assertion has a claimed explicit counterexample, but its validity remains unverified; the stronger-assumption results are settled, while related weakly $n$-harmonic-map and higher-dimensional $H$-system questions remain open.