Free boundary regularity in the triple membrane problem

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. In the physical dimension $d=2$, these fully resolve the free boundary regularity in this problem.