Uniform regularity and convergence of phase-fields for Willmore’s energy

Dondl, Patrick W.; Wojtowytsch, Stephan

Abstract

We investigate the convergence of phase fields for the Willmore problem away from the support of a limiting measure μμ . For this purpose, we introduce a suitable notion of essentially uniform convergence. This mode of convergence is a natural generalisation of uniform convergence that precisely describes the convergence of phase fields in three dimensions. More in detail, we show that, in three space dimensions, points close to which the phase fields stay bounded away from a pure phase lie either in the support of the limiting mass measure μμ or contribute a positive amount to the limiting Willmore energy. Thus there can only be finitely many such points. As an application, we investigate the Hausdorff limit of level sets of sequences of phase fields with bounded energy. We also obtain results on boundedness and LpLp -convergence of phase fields and convergence from outside the interval between the wells of a double-well potential. For minimisers of suitable energy functionals, we deduce uniform convergence of the phase fields from essentially uniform convergence.

Citation

Dondl, P. W., & Wojtowytsch, S. (2017). Uniform regularity and convergence of phase-fields for Willmore’s energy. Calculus of Variations and Partial Differential Equations, 56(4), Article 90. https://doi.org/10.1007/s00526-017-1178-9

• Journal Article Type: Article
• Acceptance Date: May 4, 2017
• Online Publication Date: Jun 5, 2017
• Publication Date: Jun 5, 2017
• Deposit Date: Jul 18, 2017
• Publicly Available Date: Jun 5, 2018
• Journal: Calculus of Variations and Partial Differential Equations
• Print ISSN: 0944-2669
• Electronic ISSN: 1432-0835
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 56
• Issue: 4
• Article Number: 90
• DOI: https://doi.org/10.1007/s00526-017-1178-9