A Discontinuous Solution of the Critical n-Laplace System with Antisymmetric Potential

Abstract

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 $D^{\prime }(B^n)$. 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.