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 integral varifolds which satisfy variational inequality and restricts to stable minimal hypersurface interior. Based on this, we show that any admissible family of sweepouts $$\Pi $$ compact boundary, there always exists closed $$C^{1,1}$$ codimension $$\ge 7$$ singular set interior having mean curvature pointing outward along boundary realizing width $$\mathbf {L}(\Pi )$$ .