Relaxation of Ginzburg–Landau functional with 1‐Lipschitz penalizing term in one dimension by Young measures on micropatterns

Article type: Research Article

Authors: Raguž, Andrija

Affiliations: Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia E‐mail: [email protected]

Abstract: In this paper we study asymptotic behavior as ε→0 of Ginzburg–Landau functional Iε(v):=∫Ω(ε2v″2(s)+W(v′(s))+a(s)(v(s)+g(s))2) ds for v∈Hper2(Ω), where Ω⊆R is a bounded open interval, W is a non‐negative continuous function vanishing at ±1, a∈L1(Ω), and g is 1‐Lipschitz. Our consideration follows the approach introduced in the original paper by G. Alberti and S. Müller (Comm. Pure Appl. Math. 54 (2001), 761–825), where the case g=0 was studied. We show that their program can be modified in the case of functional Iε: we define suitable relaxation of Iε and prove a Γ‐convergence result in the topology of the so‐called Young measures on micropatterns. Moreover, we identify a unique minimizing measure for the functional in the limit, which is the unique translation‐invariant measure supported on the orbit of a particular periodic sawtooth function having minimal period and slope dependent on a derivative of g.

Keywords: Young measures, relaxation, Ginzburg–Landau functional, Gamma convergence

Journal: Asymptotic Analysis, vol. 41, no. 3-4, pp. 331-361, 2005