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

Abstract

In this paper we study asymptotic behavior as ε→0 of Ginzburg–Landau functional

I_ε(v):=∫_Ω(ε²v″²(s)+W(v′(s))+a(s)(v(s)+g(s))²) ds

for v∈H_per²(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 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.