engleski

Relaxation of Ginzburg-Landau functional with 1-Lipschitz penalizing term in one dimension by Young measures on micro-patterns

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.

Asymptotic analysis ; Young measures ; Ginzburg–Landau functional ; gamma convergence

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