On a conjecture of De Giorgi about the phase-field approximation of the Willmore functional

TL;DR

This paper proves that De Giorgi's conjecture on the phase-field approximation of the Willmore functional holds with a zero coefficient, extending previous partial results and providing new insights into the limit measures under energy control.

Contribution

The paper confirms De Giorgi's original conjecture with k=0, showing the phase-field approximation converges as predicted without the need for the modified term.

Findings

De Giorgi's conjecture holds with k=0 in the phase-field approximation.

The limit measures exhibit specific properties under energy bounds.

The result extends previous partial proofs in 2 and 3 dimensions.

Abstract

In 1991 De Giorgi conjectured that, given $\lambda >0$, if $\mu_\varepsilon$ stands for the density of the Allen-Cahn energy and $v_\varepsilon$ represents its first variation, then $\int [v_\varepsilon^2 + \lambda] d\mu_\varepsilon$ should $\Gamma$-converge to $c\lambda \mathrm{Per}(E) + k \mathcal{W}(\Sigma)$ for some real constant $k$, where $\mathrm{Per}(E)$ is the perimeter of the set $E$, $\Sigma=\partial E$, $\mathcal{W}(\Sigma)$ is the Willmore functional, and $c$ is an explicit positive constant. A modified version of this conjecture was proved in space dimensions $2$ and $3$ by R\"oger and Sch\"atzle, when the term $\int v_\varepsilon^2 \, d\mu_\varepsilon$ is replaced by $ \int v_\varepsilon^2 {\varepsilon}^{-1} dx$, with a suitable $k>0$. In the present paper we show that, surprisingly, the original De Giorgi conjecture holds with $k=0$. Further properties on the limit measures obtained under a uniform control of the approximating energies are also provided.