  We study min-max theory for area functional among hypersurfaces constrained
in a smooth manifold with boundary. A Schoen-Simon-type regularity result is
proved for integral varifolds which satisfy a variational inequality and
restrict to a stable minimal hypersurface in the interior. Based on this, we
show that for any admissible family of sweepouts $\Pi$ in a compact manifold
with boundary, there always exists a closed $C^{1,1}$ hypersurface with
codimension$\geq 7$ singular set in the interior and having mean curvature
pointing outward along boundary realizing the width $\textbf{L}(\Pi)$.
