  We demonstrate that Radon measures which arise as the limit of the
Modica-Mortola measures associated to phase-fields with uniformly bounded
diffuse area and Willmore energy may be singular at the boundary of a domain
and discuss implications for practical applications. We furthermore give
partial regularity results for the phase-fields $u_\epsilon$ at the boundary in
terms of boundary conditions and counterexamples without boundary conditions.

Analysis of PDEs