Γ‐convergence for one‐dimensional Ginzburg‐Landau functional with generalized Lipschitz penalizing term

Abstract

We use the approach developed in the paper G. Alberti, S. Muller: A new approach to variational problems with multiple scales, Comm. Pure Appl. Math. 54, 761-825 (2001) to obtain Γ-convergence for a class of Ginzburg-Landau functionals I ε (v), where v = v(s) is appropriate Sobolev function. 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 = g(s). In this note we prove Γ-convergence when g = g(s, v(s), v (s)) under suitable growth conditions imposed on g.