General regularity of weakly n-harmonic maps in the critical Sobolev class

Determine the general regularity of weakly n-harmonic maps belonging to the critical Sobolev space W^{1,n}, without assuming additional Lorentz-space integrability or structural hypotheses.

Background

The paper discusses higher-dimensional analogues of the two-dimensional regularity theory for harmonic maps. Under additional assumptions, such as L^{(n,2)}-integrability of the antisymmetric potential together with a transformed-potential condition, continuity can be obtained. For weakly n-harmonic maps, related arguments yield continuity when the gradient additionally satisfies abla u\in L^{(n,2)}.

The authors explicitly distinguish this unresolved problem from the antisymmetric-potential system addressed in the paper. Their discontinuous example answers the broader Rivire-type system negatively, but it is not a counterexample to the regularity of weakly n-harmonic maps, so the general critical-space regularity problem remains unresolved.