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

Andrija Raguž

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 3, page 1125-1142
  • ISSN: 0392-4033

Abstract

top
In this note we consider the Ginzburg-Landau functional I a ϵ ( v ) = 0 1 ( ϵ 2 v ′′ 2 ( s ) + W ( v ( s ) ) + a ( ϵ - β s ( v 2 ( s ) ) d s where β > 0 and a is 1-periodic. We determine how (rescaled) minimal asymptotic energy associated to I a ϵ depends on parameter β > 0 as ϵ ø 0 . In particular, our analysis shows that minimizers of I a ϵ are nearly ϵ 1 / 3 -periodic.

How to cite

top

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 dell'Unione Matematica Italiana 10-B.3 (2007): 1125-1142. <http://eudml.org/doc/290378>.

@article{Raguž2007,
abstract = {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.},
author = {Raguž, Andrija},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {1125-1142},
publisher = {Unione Matematica Italiana},
title = {A Note on Calculation of Asymptotic Energy for a Functional of Ginzburg-Landau Type with Externally Imposed Lower-Order Oscillatory Term in One Dimension},
url = {http://eudml.org/doc/290378},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Raguž, Andrija
TI - A Note on Calculation of Asymptotic Energy for a Functional of Ginzburg-Landau Type with Externally Imposed Lower-Order Oscillatory Term in One Dimension
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/10//
PB - Unione Matematica Italiana
VL - 10-B
IS - 3
SP - 1125
EP - 1142
AB - 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.
LA - eng
UR - http://eudml.org/doc/290378
ER -

References

top
  1. ALBERTI, G. - MÜLLER, S., A new approach to variational problems with multiple scales, Comm. Pure Appl. Math., 54 (2001), 761-825. Zbl1021.49012MR1823420DOI10.1002/cpa.1013
  2. BALL, J. M., A version of the fundamental theorem for Young measures, in PDE's and Continuum Models of Phase Transitions (M. Rascle and al., eds.), Lecture Notes in Physics, 344, Springer, Berlin1989. MR1036070DOI10.1007/BFb0024945
  3. CHOKSI, R., Scaling laws in microphase separation of diblock copolymers, J. Nonlinear Sci., 11 (2001), 223-236. Zbl1023.82015MR1852942DOI10.1007/s00332-001-0456-y
  4. DALMASO, G., An Introduction to Γ -convergence, Progress in Nonlinear Differential Equations, Birkhauser, Boston1993. MR1201152DOI10.1007/978-1-4612-0327-8
  5. KOHN, R. V. - MÜLLER, S., Branching of twins near an austensite-twinned-martensite interface, Philosophical Magazine A, 66 (1992), 697-715. 
  6. MODICA, L. - MORTOLA, S., Un esempio di Γ -convergenca, Boll. Un. Mat. Ital. (5), 14-B (1977), 285-299. MR445362
  7. MÜLLER, S., Singular perturbations as a selection criterion for minimizing sequences, Calc. Var., 1 (1993), 169-204. MR1261722DOI10.1007/BF01191616
  8. OHTA, T. - KAWASAKI, K., Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632. 
  9. RAGUZ, A., Relaxation of Ginzburg-Landau functional with 1-Lipschitz penalizing term in one dimension by Young measures on micropatterns, Asymptotic Anal., 41 (3,4) (2005), 331-361. Zbl1095.49013MR2128001
  10. YOUNG, L. C., Lectures on the calculus of variations and optimal control theory, Chelsea, 1980. MR259704

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.