  We show the existence of Yang--Mills--Higgs (YMH) fields over a Riemann
surface with boundary where a free boundary condition is imposed on the section
and a Neumann boundary condition on the connection. In technical terms, we
study the convergence and blow-up behavior of a sequence of Sacks-Uhlenbeck
type $\alpha$-YMH fields as $\alpha\to 1$. For $\alpha>1$, each $\alpha$-YMH
field is shown to be smooth up to the boundary under some gauge transformation.
This is achieved by showing a regularity theorem for more general coupled
systems, which extends the classical results of Ladyzhenskaya-Ural'ceva and
Morrey.
