  In the paper we introduce a boundary value problem for a G_{2} structure on a
7-manifold with boundary, with prescribed 3-form on the boundary. We make some
general observations about this problem and then study in more detail
reductions to 4 and 3 dimensions by imposing symmetry. The 3-dimensional
reduction admits a dual variational formulation in terms of solutions of the
real Monge-Ampere equation
