{
  "id": "2305.05961",
  "version": "v1",
  "published": "2023-05-10T08:11:08.000Z",
  "updated": "2023-05-10T08:11:08.000Z",
  "title": "Regularizing properties of n-Laplace systems with antisymmetric potentials in Lorentz spaces",
  "authors": [
    "Dorian Martino",
    "Armin Schikorra"
  ],
  "categories": [
    "math.AP"
  ],
  "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) = \\Omega \\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) = \\tilde{\\Omega} \\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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-10T08:11:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "antisymmetric potential",
      "lorentz spaces",
      "regularizing properties",
      "harmonic maps",
      "kuusi-mingiones vectorial potential theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}