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On the boundary regularity of phase-fields for Willmore's energy

Part of: Miscellaneous topics in calculus of variations and optimal control Manifolds Elliptic equations and systems

Published online by Cambridge University Press:  27 December 2018

Patrick W. Dondl  and
Stephan Wojtowytsch
Patrick W. Dondl
Affiliation:
Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität, Freiburg, Hermann-Herder-Str. 10 79104 Freiburg i. Br., Germany (patrick.dondl@mathematik.uni-freiburg.de)
Stephan Wojtowytsch
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA (swojtowy@andrew.cmu.edu)
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Abstract

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.

Keywords

Willmore energyphase-fieldboundary regularity

MSC classification

Primary: 35J67: Boundary values of solutions to elliptic equations
Secondary: 49Q20: Variational problems in a geometric measure-theoretic setting 49Q10: Optimization of shapes other than minimal surfaces 49N60: Regularity of solutions 35J15: Second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 4 , August 2019 , pp. 1017 - 1035
Copyright
Copyright © Royal Society of Edinburgh 2018 

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