{
  "id": "1604.05461",
  "version": "v1",
  "published": "2016-04-19T07:55:47.000Z",
  "updated": "2016-04-19T07:55:47.000Z",
  "title": "Horizontal $α$-Harmonic Maps",
  "authors": [
    "Francesca Da Lio",
    "Tristan Rivière"
  ],
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "Given a $C^1$ planes distribution $P_T$ on all ${\\mathbb R}^m$ we consider {\\em horizontal $\\alpha$-harmonic maps}, $\\alpha\\ge 1/2$, with respect to such a distribution. These are maps $u\\in H^{\\alpha}({{\\mathbb R}}^k,{{\\mathbb R}}^m)$ satisfying $P_T\\nabla u=\\nabla u$ and $P_T(u)(-\\Delta)^{\\alpha}u=0$ in ${\\mathcal D}'({{\\mathbb R}}^k).$ If the distribution of planes is integrable then we recover the classical case of $\\alpha$-harmonic maps with values into a manifold. In this paper we shall focus our attention to the case $\\alpha=1/2$ in dimension $1$ and $\\alpha=2$ in dimension $2$ and we investigate the regularity of the {\\em horizontal $\\alpha$-harmonic maps}. In both cases we show that such maps satisfy a Schr\\\"odinger type system with an antisymmetric potential, that permits us to apply the previous results obtained by the authors. Finally we study the regularity of {\\em variational $\\alpha$-harmonic} maps which are critical points of $\\|(-\\Delta)^{\\alpha/2} u\\|^2_{L^2}$ under the constraint to be tangent (horizontal) to a given planes distribution. We produce a convexification of this variational problem which permits to write it's Euler Lagrange equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-19T07:55:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58E20",
      "35R11",
      "53C17",
      "35B65",
      "35S99",
      "49Q05"
    ],
    "keywords": [
      "harmonic maps",
      "horizontal",
      "planes distribution",
      "euler lagrange equations",
      "maps satisfy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}