  We investigate the regularity of the free boundaries in the 3 elastic
membranes problem.
  We show that the two free boundaries corresponding to the coincidence regions
between consecutive membranes are $C^{1,\log}$-hypersurfaces near a regular
intersection point. We also study two types of singular intersections. The
first type of singular points are locally covered by a
$C^{1,\alpha}$-hypersurface. The second type of singular points stratify and
each stratum is locally covered by a $C^1$-manifold.
