Horizontal $\alpha$-Harmonic Maps

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 ${\mathcal{D}}^{\prime }({\mathbb{R}}^k).$ 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 Schrödinger 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 $\Vert (-\Delta {)}^{\alpha /2}u{\Vert }_{L^2}^2$ 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.

Cited by in corpus (1)

• On energy-critical half-wave maps into ${\mathbb{S}}^2$