Uniform Regularity and Convergence of PhaseFields for Willmore's Energy
Abstract
We investigate the convergence of phase fields for the Willmore problem away from the support of a limiting measure $\mu$. 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 $\mu$ 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 $L^p$convergence of phase fields and convergence from outside the interval between the wells of a doublewell potential. For minimisers of suitable energy functionals, we deduce uniform convergence of the phase fields from essentially uniform convergence.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1512.08641
 Bibcode:
 2015arXiv151208641D
 Keywords:

 Mathematics  Analysis of PDEs;
 49Q20;
 49Q10;
 49N60;
 35J15;
 35J35;
 74G65