Relaxation of Ginzburg–Landau functional with 1‐Lipschitz penalizing term in one dimension by Young measures on micropatterns
Andrija Raguž · Asymptotic Analysis · 2005
In this paper we study asymptotic behavior as ε→0 of Ginzburg–Landau functional I ε (v):=∫ Ω (ε 2 v″ 2 (s)+W(v′(s))+a(s)(v(s)+g(s)) 2 ) ds for v∈H per 2 (Ω), where Ω⊆R is a bounded open interval, W is a non‐negative continuous function vanishing at ±1, a∈L 1 (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 I ε : we define suitable relaxation of I ε 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.