  The large-N limit of the two-dimensional U$(N)$ Yang-Mills theory on an
arbitrary orientable compact surface with boundaries is studied. It is shown
that if the holonomies of the gauge field on boundaries are near the identity,
then the critical behavior of the system is the same as that of an orientable
surface without boundaries with the same genus but with a modified area. The
diffenece between this effective area and the real area of the surface is
obtained and shown to be a function of the boundary conditions (holonomies)
only. A similar result is shown to hold for the group SU$(N)$ and other simple
groups.
