A Note on Calculation of Asymptotic Energy for a Functional of Ginzburg-Landau Type with Externally Imposed Lower-Order Oscillatory Term in One Dimension

Raguž, Andrija

Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 1125-1142

Résumé

In this note we consider the Ginzburg-Landau functional \begin{equation*}I^\epsilon_a(v) = \int_0^1(\epsilon^2 v''^2(s) + W(v'(s)) + a(\epsilon^{-\beta}s(v^2(s)) \, ds\end{equation*} where $\beta > 0$ and a is 1-periodic. We determine how (rescaled) minimal asymptotic energy associated to $I^\epsilon_a$ depends on parameter $\beta > 0$ as $\epsilon \o 0$. In particular, our analysis shows that minimizers of $I_{a}^{\epsilon}$ are nearly $\epsilon^{1/3}$-periodic.