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, Volume 32 (2015) no. 3, p. 593-622
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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 : https://doi.org/10.1016/j.anihpc.2014.02.002
Classification:  35R60,  80M35,  82D30,  74Q05
Keywords: 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},
     publisher = {Elsevier},
     volume = {32},
     number = {3},
     year = {2015},
     pages = {593-622},
     doi = {10.1016/j.anihpc.2014.02.002},
     mrnumber = {3353702},
     zbl = {1320.35355},
     language = {en},
     url = {http://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, Volume 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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