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 with 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 _ε at the boundary in terms of boundary conditions and counterexamples without boundary conditions. [ABSTRACT FROM AUTHOR]