Uniform Regularity and Convergence of Phase-Fields for Willmore's Energy

TL;DR

This paper studies the convergence behavior of phase fields related to Willmore's energy in three dimensions, introducing a new notion of uniform convergence to better understand their limiting properties.

Contribution

It introduces a natural generalization of uniform convergence for phase fields in three dimensions and analyzes their convergence away from the support of the limiting measure.

Findings

Points near which phase fields stay bounded away from a pure phase are either in the support of the limiting measure or contribute to the energy.

Finitely many such points exist in three dimensions.

Results on Hausdorff limits, boundedness, and $L^p$-convergence of phase fields.

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 double-well potential. For minimisers of suitable energy functionals, we deduce uniform convergence of the phase fields from essentially uniform convergence.