On a conjecture of 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 ∫ [v2 ε + λ]dμε should Γ-converge to cλ Per(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 ∫ v2 ε dμε is replaced by∫ v2 ε ε−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 of the limit measures obtained under a uniform control of the approximating energies are also provided.