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

Let $n>2$. We construct a map $U\in W^{1,n}(B^n,\mathbb{R}^{n+2})$ that is discontinuous at the origin and smooth on the punctured ball $B^n \setminus \{0\}$, together with an antisymmetric potential $\Omega\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n)$ such that $-\mathrm{Div}(|\nabla U|^{n-2}\nabla U)=\Omega\cdot |\nabla U|^{n-2}\nabla U$ in $D'(B^n)$. This gives a negative answer to a regularity question posed by Rivière. Our potential admits the Lorentz-space regularity $\Omega \in \bigcap_{q>2}L^{(n,q)} \setminus L^{(n,2)}$. In addition for given $1<p<\infty$ we can enforce $\nabla U \in L^{(n,p)}$ but $\nabla U \notin L^{(n,1)}$. The construction does not give a counterexample to regularity for weakly $n$-harmonic maps or for higher-dimensional $H$-systems. The example was generated by ChatGPT 5.6 Sol on August 5, 2026. The work itself was written by the author and thoroughly reviewed to ensure its correctness.

For every $n>2$, the paper constructs a bounded map $U\in W^{1,n}(B^n,\mathbb{R}^{n+2})$, smooth on $B^n\setminus\{0\}$ but discontinuous at the origin, together with an antisymmetric potential $$ \Omega\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n) $$ such that $$ -\mathrm{Div}\bigl(|\nabla U|^{n-2}\nabla U\bigr) = \Omega\cdot|\nabla U|^{n-2}\nabla U \qquad\text{in }D'(B^n). $$ Moreover, the potential satisfies the sharper Lorentz-space regularity $$ \Omega\in\bigcap_{q>2}L^{(n,q)}\setminus L^{(n,2)}, $$ and, for any prescribed $1<p<\infty$, the construction can be arranged so that $$ \nabla U\in L^{(n,p)} \qquad\text{but}\qquad \nabla U\notin L^{(n,1)}. $$ This gives a negative answer to Rivière’s general regularity question for critical $n$-Laplace systems with antisymmetric $L^n$ potentials: antisymmetry and critical $L^n$ control alone do not imply continuity.