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

Abstract

We show continuity of solutions $u \in W^{1,n}(B^n,\mathbb {R}^N)$ to the system

$$\begin{aligned} -\textrm{div} (|\nabla u|^{n-2} \nabla u) = \Omega \cdot |\nabla u|^{n-2} \nabla u \end{aligned}$$

when $\Omega$ is an $L^n$-antisymmetric potential – and additionally satisfies a Lorentz-space assumption. To obtain our result we study a rotated n-Laplace system

$$\begin{aligned} -\textrm{div} (Q|\nabla u|^{n-2} \nabla u) = {\tilde{\Omega }} \cdot |\nabla u|^{n-2} \nabla u, \end{aligned}$$

where $Q \in W^{1,n}(B^n,SO(N))$ is the Coulomb gauge which ensures improved Lorentz-space integrability of ${\tilde{\Omega }}$. 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 $L^{(n,\infty )}$-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 prove a corresponding result for n-Laplace H-systems.