In 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.