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

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 ] dmu _varepsilon should Gamma -converge to clambda {text {Per}}(E) + k mathcal {W}(Sigma ) for some real constant k, where {text {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 Röger and Schätzle, when the term int v_varepsilon ^2 , dmu _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.