De Giorgi's phase-field approximation conjecture for the Willmore functional

Let $n\geq 2$ be an integer, let $E\subset \mathbb{R}^n$ have boundary $\Sigma:=\partial E$ of class $C^2$, and let $\Omega\subset \mathbb{R}^n$ be open. For $\lambda>0$, define

$$\mathcal{DG}_\eps(u,\Omega):=\int_{\Omega} \left[\left(2\eps\Delta u-\frac{\sin u}{\eps} \right)^2+\lambda\right] \left[\eps|\nabla u|^2 + \frac{1-\cos u}{\eps} \right]\\,dx$$

for $u\in W^{2,1}(\Omega)$ and set $\mathcal{DG}_\eps(u):=+\infty$ otherwise. Here $\chi_E$ is the characteristic function of $E$, $H$ is the mean curvature of $\Sigma$, and $c=8\sqrt{2}$. De Giorgi's conjecture. There exists $k\in\mathbb{R}$ such that

$$\Gamma(L^1(\Omega))-\lim_{\eps\to0^+}\mathcal{DG}_\eps(2\pi\chi_E,\Omega)=c\lambda\mathcal{H}^{n-1}(\Sigma\cap\Omega)+k\int_{\Sigma\cap\Omega}H^2\,d\mathcal{H}^{n-1}.$$

The conjecture proposed a phase-field approximation of a multiple of the Willmore functional, with the perimeter term arising from the Allen–Cahn energy. The paper's abstract states that the original conjecture in fact holds with $k=0$, so the conjecture is solved.