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 ${\mathcal D}'({{\mathbb R}}^k).$ 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.