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Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)

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On the boundary regularity of phase-fields for Willmore's energy
Part of
Patrick W. Dondl, Stephan Wojtowytsch
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 149 / Issue 4 / August 2019

Published online by Cambridge University Press:

27 December 2018, pp. 1017-1035

Print publication:

August 2019
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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 uε at the boundary in terms of boundary conditions and counterexamples without boundary conditions.