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

Dirr, Nicolas; Orlandi, Enza

Uniqueness of the minimizer for a random non-local functional with double-well potential in

d \leq 2

Dirr, Nicolas ; Orlandi, Enza

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 593-622 / Harvested from Numdam

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Résumé

We consider a small random perturbation of the energy functional ${[u]}_{H^{s} (Λ, ℝ^{d})}^{2} + \int_{Λ} W (u (x)) d x$ for $s \in (0, 1)$ , where the non-local part ${[u]}_{H^{s} (Λ, ℝ^{d})}^{2}$ denotes the total contribution from $Λ \subset ℝ^{d}$ in the $H^{s} (ℝ^{d})$ 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, $u = \pm 1$ .

Publié le : 2015-01-01

MR 3353702 | Zbl 1320.35355

DOI : https://doi.org/10.1016/j.anihpc.2014.02.002
Classification: 35R60, 80M35, 82D30, 74Q05

@article{AIHPC_2015__32_3_593_0,
     author = {Dirr, Nicolas and Orlandi, Enza},
     title = {Uniqueness of the minimizer for a random non-local functional with double-well potential in $ d\leq 2$
      },
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {32},
     year = {2015},
     pages = {593-622},
     doi = {10.1016/j.anihpc.2014.02.002},
     zbl = {1320.35355},
     mrnumber = {3353702},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2015__32_3_593_0}
}

Dirr, Nicolas; Orlandi, Enza. Uniqueness of the minimizer for a random non-local functional with double-well potential in $ d\leq 2$
      . Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) pp. 593-622. doi : 10.1016/j.anihpc.2014.02.002. http://gdmltest.u-ga.fr/item/AIHPC_2015__32_3_593_0/

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