On the Boundary Regularity of Phase-Fields for Willmore's Energy

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.