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

Uniform regularity and convergence of phase-fields for Willmore’s energy We investigate the convergence of phase fields for the Willmore problem away from the support of a limiting measure $$\mu$$ μ . 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 $$\mu$$ μ 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 $$L^p$$ L p -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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

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

, Volume 56 (4) – Jun 5, 2017
22 pages

/lp/springer_journal/uniform-regularity-and-convergence-of-phase-fields-for-willmore-s-d3oB0gwn0D
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1178-9
Publisher site
See Article on Publisher Site

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