Min-max minimal hypersurfaces with obstacle

We study min-max theory for the 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 restricts 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,C-1 hypersurface with codimension >= 7 singular set in the interior and having mean curvature pointing outward along boundary realizing the width L(Pi).