  We prove a local splitting theorem for three-manifolds with mean convex
boundary and scalar curvature bounded from below that contain certain locally
area-minimizing free boundary surfaces. Our methods are based on those of
Micallef and Moraru. We use this local result to establish a global rigidity
theorem for area-minimizing free boundary disks. In the negative scalar
curvature case, this global result implies a rigidity theorem for solutions of
the Plateau problem with length-minimizing boundary.
