Asymptotic analysis of a second-order singular perturbation model for phase transitions

Cicalese, Marco; Spadaro, Emanuele; Zeppieri, Caterina
May 2011
Calculus of Variations & Partial Differential Equations;May2011, Vol. 41 Issue 1/2, p127
Academic Journal
We study the asymptotic behavior, as $${\varepsilon}$$ tends to zero, of the functionals $${F^k_\varepsilon}$$ introduced by Coleman and Mizel in the theory of nonlinear second-order materials; i.e.,where k > 0 and $${W:\mathbb{R}\to[0,+\infty)}$$ 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 such that, for k < k, and for a class of potentials W, $${(F^k_\varepsilon)}$$ Γ( L)-converges towhere m is a constant depending on W and k. Moreover, in the special case of the classical potential $${W(s)=\frac{(s^2-1)^2}{2}}$$, 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 and a lower bound on the values for which oscillations occur, the latter improving a previous estimate by Mizel, Peletier and Troy.


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