The fractional Hopf differential and a weak formulation of stationarity for the half Dirichlet energy

Abstract We obtain a weak formulation of the stationarity condition for the half Dirichlet energy, which can be expressed in terms of a fractional quantity, related to the trace of the Hopf differential of the harmonic extension of the original map. As an application we show that conformal harmonic maps from the disc are precisely the harmonic extensions of stationary points of the half Dirichlet energy on the circle. We also derive a Noether theorem and a Pohozaev identity for stationary points of the half Dirichlet energy.