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On stability of asymptotic energy for a functional of Ginzburg‐Landau type with ε‐dependent weakly‐star convergent 1‐Lipschitz penalizing term
Author(s) -
Raguž Andrija
Publication year - 2009
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200910240
Subject(s) - lipschitz continuity , term (time) , mathematics , energy functional , relaxation (psychology) , dimension (graph theory) , exponential stability , function (biology) , star (game theory) , type (biology) , energy (signal processing) , stability (learning theory) , pure mathematics , mathematical analysis , physics , quantum mechanics , computer science , nonlinear system , psychology , social psychology , ecology , statistics , evolutionary biology , biology , machine learning
We apply the approach developed in the paper G. Alberti, S. Müller: A new approach to variational problems with multiple scales, Comm. Pure Appl. Math. 54 , 761‐825 (2001) to calculate rescaled asymptotic energy associated to certain Ginzburg‐Landau functional in one dimension. We generalize results from the paper A. Raguž: Relaxation of Ginzburg‐Landau functional with 1‐Lipschitz penalizing term in one dimension by Young measures on micropatterns, Asymptotic Anal. 41(3,4) , 331‐361 (2005), where original functional was penalized by 1‐Lipschitz function g. In this note we consider the case when such penalizing functions depend on small parameter ε as their derivatives oscillate with period equal to ε to the power of γ for some γ > 0. We show that there are three distinctive cases of γ which lead to different asymptotic energy. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)