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

Abstract

We consider a small random perturbation of the energy functional [u]_{H^{s}(\Lambda ,\mathbb{R}^{d})}^{2} + \int \limits_{\Lambda }W\left(u(x)\right)\:\mathrm{d}x for s \in (0,1) , where the non-local part [u]_{H^{s}(\Lambda ,\mathbb{R}^{d})}^{2} denotes the total contribution from \Lambda \subset \mathbb{R}^{d} in the H^{s}(\mathbb{R}^{d}) Gagliardo semi-norm of u and W is a double well potential. We show that there exists, as Λ invades \mathbb{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 .

Locations

• Annales de l Institut Henri Poincaré C Analyse Non Linéaire - View - PDF
• ORCA Online Research @Cardiff (Cardiff University) - View - PDF