Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T19:49:25.290Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Manifolds Miscellaneous topics in calculus of variations and optimal control 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)
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alessandroni, R. and Kuwert, E. Local solutions to a free boundary problem for the Willmore functional. Calc. Var. PDE 55 (2016), Article:24.Google Scholar
2Bellettini, G. and Paolini, M. Approssimazione variazionale di funzionali con curvatura. Seminario di Analisi Matematica, Dipartimento di Matematica dell'Università di Bologna. (1993).Google Scholar
3Bretin, E., Masnou, S. and Oudet, E. Phase-field approximations of the Willmore functional and flow. Numerische Mathematik 131 (2015), 115–171.Google Scholar
4Brezis, H. Functional analysis, Sobolev spaces and partial differential equations (New York: Universitext. Springer, 2011).Google Scholar
5De Giorgi, E. Some remarks on Γ-convergence and least squares method. In Composite media and homogenization theory (Trieste, 1990),pp. 135–142 (Boston, MA, Boston, MA: Birkhäuser Boston, 1991).Google Scholar
6Dondl, P. W. and Wojtowytsch, S. Uniform convergence of phase-fields for Willmore's energy. Calc. Var. PDE 56 (2017).Google Scholar
7Dondl, P. W., Lemenant, A. and Wojtowytsch, S. Phase field models for thin elastic structures with topological constraint. Arch. Ration. Mech. Anal. 223 (2017), 693–736.Google Scholar
8Gilbarg, D. and Trudinger, N. S. Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn (Berlin: Springer-Verlag, 1983).Google Scholar
9Modica, L. The gradient theory of phase transitions and the minimal interface criterion. Arch Ration Mech Anal 98 (1987), 123–142.Google Scholar
10Modica, L. and Mortola, S. Un esempio di Γ-convergenza. Boll. Un. Mat. Ital. B (5) 14 (1977), 285–299.Google Scholar
11Nezza, E. D., Palatucci, G. and Valdinoci, E. Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521–573, 04.Google Scholar
12Röger, M. and Schätzle, R. On a modified conjecture of De Giorgi. Math. Z. 254 (2006), 675–714.Google Scholar
13Savin, O. and Valdinoci, E.. Γ-convergence for nonlocal phase transitions. Ann. Inst. H. Poincaré. Anal. Non Linéaire, 29 (2012), 479–500.Google Scholar