Published Paper

Inserted: 10 jun 2022
Last Updated: 14 apr 2023

Journal: Arch. Rational Mech. Anal.
Volume: 247
Year: 2023
Doi: 10.1007/s00205-023-01870-z

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 of the limit measures obtained under a uniform control of the approximating energies are also provided.