  We prove the existence of minimisers for a family of models related to the
single-slip-to-single-plane relaxation of single-crystal, strain-gradient
elastoplasticity with $L^p$-hardening penalty. In these relaxed models, where
only one slip-plane normal can be activated at each material point, the main
challenge is to show that the energy of geometrically necessary dislocations is
lower-semicontinuous along bounded-energy sequences which satisfy the
single-plane condition, meaning precisely that this side condition should be
preserved in the weak $L^p$-limit. This is done with the aid of an 'exclusion'
lemma of Conti \& Ortiz, which essentially allows one to put a lower bound on
the dislocation energy at interfaces of (single-plane) slip patches, thus
precluding fine phase-mixing in the limit. Furthermore, using div-curl
techniques in the spirit of Mielke \& M\"uller, we are able to show that the
usual multiplicative decomposition of the deformation gradient into plastic and
elastic parts interacts with weak convergence and the single-plane constraint
in such a way as to guarantee lower-semicontinuity of the (polyconvex) elastic
energy, and hence the total elasto-plastic energy, given sufficient ($p>2$)
hardening, thus delivering the desired result.
