Communications in Mathematical Sciences

Volume 9 (2011)

Number 2

Unique minimizer for a random functional with double-well potential in dimension 1 and 2

Pages: 331 – 351

DOI: https://dx.doi.org/10.4310/CMS.2011.v9.n2.a1

Authors

Nicolas Dirr (Department of Mathematical Sciences, University of Bath, United Kingdom)

Enza Orlandi (Dipartimento di Matematica, Università di Roma Tre, Roma, Italy)

Abstract

We add a random bulk term, modelling the interaction with the impurities of the medium, to a standard functional in the gradient theory of phase transitions consisting of a gradient term with a double well potential. We show that in d≤2 there exists, for almost all the realizations of the random bulk term, a unique random macroscopic minimizer. This result is in sharp contrast to the case when the random bulk term is absent. In the latter case there are two minimizers which are (in law) invariant under translations in space.

Keywords

random functionals, phase segregation in disordered materials

2010 Mathematics Subject Classification

35R60, 74Q05, 80M35, 82D30

Published 21 December 2010