| Contents |
| We prove the one-dimensional character of minimizers of the relaxation ${\mathcal J}_\omega$ of the functional
$$
\int_{{\mathbb R} \times \omega} (\sqrt{1+|\nabla v|^2} -1)\, dx
+ \int_{{\mathbb R} \times \omega} F(v) \, dx
$$
on the set ${\mathcal E}_\omega$ of all functions $v : {\mathbb R} \times \omega\to {\mathbb R}$ having bounded variation in any cylinder $ {]-T,T[} \times\omega $ and
satisfying
$$
\lim_{x_1\to \pm\infty} v(x_1, x_2, \dots, x_N) = \pm 1
\quad \hbox{uniformly a.e. with respect to $(x_2, \dots, x_{N} ) \in \omega$.}
$$
Here $\omega\subset {\mathbb R}^{N-1} $ is an open bounded set and
$F : {\mathbb R} \to [0,+\infty[$ is a double-well potential,
a typical example being $F(s) = \frac{1}{4}(1-s^2)^2$.
The result gives a partial positive answers to Gibbons' conjecture with the Laplace operator replaced by the curvature operator (Gibbon's conjecture is a variant of De Giorgi's conjecture on the rigidity of the solutions of the stationary Allen-Cahn equation $\Delta v = F'(v)$ in ${\mathbb R}^N$).
Minimizers of ${\mathcal J}_\omega$ represent the transitions between phase states of a substance within the van der Waals-Cahn-Hilliard gradient theory of phase transitions, when the model is characterized for small gradients by linear gradient-flux relations and the increase of gradients is expected to slow down and ultimately to approach saturation at large gradients. |
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