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

In 1991 De Giorgi conjectured that, given $\lambda >0$, if ${\mu }_{\epsilon }$ stands for the density of the Allen-Cahn energy and $v_{\epsilon }$ represents its first variation, then $\int [v_{\epsilon }^2+\lambda ]d{\mu }_{\epsilon }$ should $\mathrm{\Gamma }$-converge to $c\lambda \mathrm{P}\mathrm{e}\mathrm{r}(E)+k\mathcal{W}(\mathrm{\Sigma })$ for some real constant $k$, where $\mathrm{P}\mathrm{e}\mathrm{r}(E)$ is the perimeter of the set $E$, $\mathrm{\Sigma }=\mathrm{\partial }E$, $\mathcal{W}(\mathrm{\Sigma })$ 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 Röger and Schätzle, when the term $\int v_{\epsilon }^{2}d{\mu }_{\epsilon }$ is replaced by $\int v_{\epsilon }^2{\epsilon }^{-1}dx$, 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.