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

In 1991 De Giorgi conjectured that, given λ>0, if με stands for the density of the Allen-Cahn energy and vε represents its first variation, then ∫[v2ε+λ]dμε should Γ-converge to cλPer(E)+kW(Σ) for some real constant k, where Per(E) is the perimeter of the set E, Σ=∂E, 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 ∫v2εdμε is replaced by ∫v2εε−1dx, 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.