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

[u] H s (Λ, d ) 2 + ΛW(u(x))dx
for s(0,1), where the non-local part [u] H s (Λ, d ) 2 denotes the total contribution from Λ d in the H s ( d ) Gagliardo semi-norm of u and W is a double well potential. We show that there exists, as Λ invades d , for almost all realizations of the random term a minimizer under compact perturbations, which is unique when d=2, s(1 2,1) and when d=1, s[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=±1.

DOI : 10.1016/j.anihpc.2014.02.002
Classification : 35R60, 80M35, 82D30, 74Q05
Mots clés : Random functionals, Phase segregation in disordered materials, Fractional Laplacian
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     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},
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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://archive.numdam.org/articles/10.1016/j.anihpc.2014.02.002/

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