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

Abstract

In 1991 De Giorgi conjectured that, given $λ>0$, if $μ_\varepsilon$ stands for the density of the Allen-Cahn energy and $v_\varepsilon$ represents its first variation, then $\int [v_\varepsilon^2 + λ] dμ_\varepsilon$ should $Γ$-converge to $cλ\mathrm{Per}(E) + k \mathcal{W}(Σ)$ for some real constant $k$, where $\mathrm{Per}(E)$ is the perimeter of the set $E$, $Σ=\partial E$, $\mathcal{W}(Σ)$ 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 $\int v_\varepsilon^2 \, dμ_\varepsilon$ 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.