On stability of asymptotic energy for a functional of Ginzburg‐Landau type with ε‐dependent weakly‐star convergent 1‐Lipschitz penalizing term

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)