*Accepted Paper*

**Inserted:** 19 sep 2007

**Journal:** Ann. Inst. H. Poincaré Anal. Non Linéaire

**Year:** 2007

**Abstract:**

We study a family of singular perturbation problems of the kind
$$ \inf \left\{\frac{1}{\varepsilon}\int_{\Omega} f(u , \varepsilon \nabla u, \varepsilon \rho) \, dx \ :\ \int_{\Omega} u = m_{0} \;,\ \int_{\Omega} \rho = m_{1\right\}\;,$$
}
where $u$ represents a fluid density and the nonnegative energy density $f$ vanishes only for $u=\alpha$ or $u=\beta$.
The novelty of the model is the additional variable $\rho\ge 0$ which is also unknown and interplays with the gradient of $u$ in the formation of interfaces. Under mild assumptions on $f$, we characterize the limit energy as $\varepsilon\to 0$ and find for each $f$ a transition energy (well defined when $u\in BV(\Omega;\{\alpha,\beta\})$ and $\rho$ is a measure) which depends on the $n-1$ dimensional density of the measure $\rho$ on the jump set of $u$. An explicit formula is also given.

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