Geodesic
A Note on Calculation of Asymptotic Energy for a Functional of Ginzburg-Landau Type with Externally Imposed Lower-Order Oscillatory Term in One Dimension
Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 1125-1142.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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.
In questa nota consideriamo il funzionale di Ginzburg-Landau \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*} ove $\beta > 0$ e $a$ è 1-periodica. Mostreremo come la minima energia asintotica (ridimensionata) associata a $I^\epsilon_a$ dipenda dal parametro $\beta > 0$ per $\epsilon \to 0$. In particolare, la nostra analisi mostra che i minimizzatori di $I^\epsilon_a$ sono quasi $\epsilon^{1/3}$-periodici. 
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Raguž, Andrija. A Note on Calculation of Asymptotic Energy for a Functional of Ginzburg-Landau Type with Externally Imposed Lower-Order Oscillatory Term in One Dimension. Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 1125-1142. http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_3_a44/

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