Regularizing properties of n-Laplace systems with antisymmetric potentials in Lorentz spaces

We show continuity of solutions $u\in W^{1,n}(B^n,{\mathbb{R}}^N)$ to the system $$-\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{n-2}\mathrm{\nabla }u)=\mathrm{\Omega }\cdot \mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{n-2}\mathrm{\nabla }u$$ when $\mathrm{\Omega }$ is an $L^n$-antisymmetric potential – and additionally satisfies a Lorentz-space assumption. To obtain our result we study a rotated n-Laplace system $$-\mathrm{d}\mathrm{i}\mathrm{v}(Q\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{n-2}\mathrm{\nabla }u)=\tilde{\mathrm{\Omega }}\cdot \mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{n-2}\mathrm{\nabla }u,$$ where $Q\in W^{1,n}(B^n,SO(N))$ is the Coulomb gauge which ensures improved Lorentz-space integrability of $\tilde{\mathrm{\Omega }}$. Because of the matrix-term $Q$, this system does not fall directly into Kuusi-Mingione’s vectorial potential theory. However, we adapt ideas of their theory together with Iwaniec’ stability result to obtain $L^{(n,\mathrm{\infty })}$-estimates of the gradient of a solution which, by an iteration argument leads to the regularity of solutions. As a corollary of our argument we see that $n$-harmonic maps into manifolds are continuous if their gradient belongs to the Lorentz-space $L^{(n,2)}$ – which is a trivial and optimal assumption if $n=2$, and the weakest assumption to date for the regularity of critical $n$-harmonic maps, without any added differentiability assumption. We also discuss an application to H systems.