Regularity for a class of non-linear elliptic systems

where ~ is a smooth positive function satisfying the ellipticity condition o(Q) + 2~'(Q)q > 0, V denotes the gradient, and Vs[2 = ~ = 1 Vskl 2. This type of system arises as the EulerLagrange equations for the stat ionary points of an energy integral which has an intrinsic definition on maps between two Riemannian manifolds; the equations are therefore of geometric interest. However, the method of proof also applies to the equations of non-linear Hodge theory, which have been studied by L. M. and R. B. Sibner 19]. These are systems of equations for a closed p-form so, deo = 0 and