Uniqueness of the minimizer for a random non-local functional with double-well potential in $d\le 2$

Dirr, Nicolas; Orlandi, Enza

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 3, pp. 593-622.

We consider a small random perturbation of the energy functional

$${\left[u\right]}_{H^s(\Lambda ,{ℝ}^d)}^2+\underset{\Lambda }{\int }W\left(u\left(x\right)\right)\mathrm{d}x$$

for $s\in (0,1)$, where the non-local part ${\left[u\right]}_{H^s(\Lambda ,{ℝ}^d)}^2$ denotes the total contribution from $\Lambda \subset {ℝ}^d$ in the $H^s\left({ℝ}^d\right)$ Gagliardo semi-norm of u and W is a double well potential. We show that there exists, as Λ invades ${ℝ}^d$, for almost all realizations of the random term a minimizer under compact perturbations, which is unique when $d=2$, $s\in (\frac{1}{2},1)$ and when $d=1$, $s\in [\frac{1}{4},1)$. This uniqueness is a consequence of the randomness. When the random term is absent, there are two minimizers which are invariant under translations in space, $u=\pm 1$.

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