On a Conjecture of De Giorgi About the Phase-Field Approximation of the Willmore Functional

Abstract

AbstractIn 1991 De Giorgi conjectured that, given $$\lambda >0$$ λ > 0 , if $$\mu _\varepsilon $$ μ ε stands for the density of the Allen-Cahn energy and $$v_\varepsilon $$ v ε represents its first variation, then $$\int [v_\varepsilon ^2 + \lambda ] d\mu _\varepsilon $$ ∫ [ v ε 2 + λ ] d μ ε should $$\Gamma $$ Γ -converge to $$c\lambda {\text {Per}}(E) + k \mathcal {W}(\Sigma )$$ c λ Per ( E ) + k W ( Σ ) for some real constant k, where $${\text {Per}}(E)$$ Per ( E ) is the perimeter of the set E, $$\Sigma =\partial E$$ Σ = ∂ E , $$\mathcal {W}(\Sigma )$$ 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 $$\int v_\varepsilon ^2 \, d\mu _\varepsilon $$ ∫ v ε 2 d μ ε is replaced by $$ \int v_\varepsilon ^2 {\varepsilon }^{-1} dx$$ ∫ v ε 2 ε - 1 d x , with a suitable $$k>0$$ k > 0 . In the present paper we show that, surprisingly, the original De Giorgi conjecture holds with $$k=0$$ k = 0 . Further properties of the limit measures obtained under a uniform control of the approximating energies are also provided.