*Accepted Paper*

**Inserted:** 5 jan 2002

**Last Updated:** 9 dec 2002

**Journal:** Interfaces and free boundaries

**Volume:** 5

**Pages:** 1-20

**Year:** 2003

**Abstract:**

The paper deals with the asymptotic behaviour (as $\epsilon\to0$) of a family of integral functionals in the framework of phase separation. In order to obtain a selection criterion for the minima of the usual double-well, non-convex free energy involving the phase variable $u$, we add a gradient term in a new variable $v$ which is related to $u$ through the $L^2$-distance between $u$ and $v$, weighted by a coefficient $\alpha$. We prove that the limit as $\epsilon\to0$ is a minimal area model with a surface tension of nonlocal form. The well-known Modica-Mortola constant can be recovered in this setting as a limit case when $\alpha\to+\infty$.

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