Universal properties of stable states of a free energy model with small parameters

We consider the problem of minimizing functionals of the form , for small , subject to the constraint . (Here is typically a double well potential.) We study structural properties of minimizers that are shared by all minimizers of the problem, for sufficiently small values of . It is shown that the average ‘mass’ and ‘energy’ of minimizers, over intervals of length , is almost uniformly distributed (depending on ), for all sufficiently small . Here is a constant depending on the degree of ‘uniformity’, but independent of . Received May 5, 1996 / Accepted October 28, 1996