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

 ${\left[u\right]}_{{H}^{s}\left(\Lambda ,{ℝ}^{d}\right)}^{2}+\underset{\Lambda }{\int }W\left(u\left(x\right)\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$
for $s\in \left(0,1\right)$, where the non-local part ${\left[u\right]}_{{H}^{s}\left(\Lambda ,{ℝ}^{d}\right)}^{2}$ 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 $d=2$, $s\in \left(\frac{1}{2},1\right)$ and when $d=1$, $s\in \left[\frac{1}{4},1\right)$. 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, $u=±1$.

DOI : https://doi.org/10.1016/j.anihpc.2014.02.002
Classification : 35R60,  80M35,  82D30,  74Q05
Mots clés : Random functionals, Phase segregation in disordered materials, Fractional Laplacian
@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},
pages = {593--622},
publisher = {Elsevier},
volume = {32},
number = {3},
year = {2015},
doi = {10.1016/j.anihpc.2014.02.002},
mrnumber = {3353702},
zbl = {1320.35355},
language = {en},
url = {www.numdam.org/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) no. 3, pp. 593-622. doi : 10.1016/j.anihpc.2014.02.002. http://www.numdam.org/item/AIHPC_2015__32_3_593_0/

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