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

Dirr, Nicolas; Orlandi, Enza

doi:10.1016/j.anihpc.2014.02.002

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) no. 3, pp. 593-622.

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

MR Zbl

DOI : 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},
     zbl = {1320.35355},
     mrnumber = {3353702},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.02.002/}
}

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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/articles/10.1016/j.anihpc.2014.02.002/

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