Optimal boundary regularity at triple junctions

Establish whether, for Brakke flows near a static multiplicity-one triple junction C and represented as normal graphs f_i over the half-planes of C with common boundary graph ξ along the spine, the sheets f_i are at least C^2 up to the boundary \overline{\Omega_i} ∩ graph ξ (and, more generally, whether optimal regularity up to the singular set holds), at least when the forcing field u is smooth or vanishes.

Background

The main theorem yields C^{1,α} regularity for the common boundary graph ξ and C^{1,α} regularity up to the boundary for the sheets f_i, with higher interior regularity depending on the smoothness of the forcing term u. In stationary (elliptic) problems, analogous free-boundary configurations are known to be real-analytic up to the boundary, but the divergence-form structure used there is absent for mean curvature flow.

The authors highlight that bridging the gap from the established C^{1,α} boundary regularity to C^{2,α} (and beyond) for the moving free boundary and the sheets is currently not known, even when u is smooth or zero.