  In this work we study solutions of the prescribed mean curvature equation
over a general domain that do not necessarily attain the given boundary data.
To such a solution, we can naturally associate a current with support in the
closed cylinder above the domain and with boundary given by the prescribed
boundary data and which inherits a natural minimizing property. Our main result
is that its support is a $C^{1,\alpha}$ manifold-with-boundary, with boundary
equal to the prescribed boundary data, provided that both the initial domain
and the prescribed boundary data are of class $C^{1,\alpha}$.
