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+\varepsilon3}(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} F^k(u):=m_k \, \#(S(u)),\quad u \in BV(I;\{a,b\}), \end{equation} where $*m*_k$ is a constant depending on $W$ and $k$. Moreover, in the special case of the classical potential $W(s)=\frac{(s^2-1)^2}{4}$, we provide an upper bound on the values of $k$ such that the minimizers of $F_\varepsilon^k$ cannot develop oscillations on some fine scale, thus improving previous estimates by Mizel, Peletier and Troy.

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

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