Abstract
We consider a small random perturbation of the energy functional [u]Hs(Λ,Rd)2+∫ΛW(u(x))dx for s∈(0,1), where the non-local part [u]Hs(Λ,Rd)2 denotes the total contribution from Λ⊂Rd in the Hs(Rd) Gagliardo semi-norm of u and W is a double well potential. We show that there exists, as Λ invades Rd, for almost all realizations of the random term a minimizer under compact perturbations, which is unique when d=2, s∈(12,1) and when d=1, s∈[14,1). 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.
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