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 \[ -{\rm div} (|\nabla u|^{n-2} \nabla u) = Ω\cdot |\nabla u|^{n-2} \nabla u \] 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 \[ -{\rm div} (Q|\nabla u|^{n-2} \nabla u) = \tildeΩ \cdot |\nabla u|^{n-2} \nabla u, \] 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 discuss an application to H systems.

Added application to H-systems

Cited by in corpus (1)

• Optimal quantitative stability of the Möbius group of the sphere in all dimensions