  We show that the $G_{2}$ holonomy equation on a manifold with boundary, with
prescribed 3-form on the boundary, is elliptic. The main point is to set up a
suitable linear elliptic boundary value problem. This result leads to a
deformation theory. In particular we establish the existence of certain $G_{2}$
cobordisms between two small deformations of a Calabi-Yau 3-fold.
