*Published Paper*

**Inserted:** 4 nov 2009

**Last Updated:** 25 mar 2011

**Journal:** Calc. Var. Partial Differential Equations

**Volume:** 41

**Number:** 1-2

**Pages:** 127-150

**Year:** 2011

**Abstract:**

We consider the problem of the asymptotic description, as $\varepsilon$ tends to zero, of the functionals $F^k_\varepsilon$ introduced by Coleman and Mizel in the theory of nonlinear second-order materials\ie
\begin{equation**}
F ^{k}_{\varepsilon}(u):=\int_{I} \left(\frac{W(u)}{\varepsilon}-k\,\varepsilon\,(u')^{2+\varepsilon}^{3}(u'')^{2\right)\,} dx,\quad u\in W^{{2,2}}(I),
\end{equation**}
where $k>0$ and $W\colon\mathbb{R}\to[0,+\infty)$ is a double-well potential with two potential wells of level zero at $a,b\in\mathbb{R}$.
By proving a new nonlinear interpolation inequality,
we show that there exists a positive constant $k_0$ such that, for $k<k_0$ and for a class of potentials $W$,
$F^k_\varepsilon$ $\Gamma (L^1)$-converges to
\begin{equation

**Keywords:**
Second-order singular perturbation, phase transitions, nonlinear interpolation, $\Gamma$-convergence

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