Remarks on Γ-Convergence of Penalized Functionals of Ginzburg-Landau Type in One Dimension

Abstract

In this note we study the Ginzburg-Landau functional

$$ I^\varepsilon (v): = \int_\Omega {(\varepsilon ^2 v''(s)^2 + W(v'(s)) + a(s)(v(s) - g(s))^2 )ds} $$

for υ ∈ H ²_per (Ω). Ω (−R is a bounded open set, a ∈ L^∞(Ω), a ≥ α > 0 and

$$ g \in C^1 (\bar \Omega ),|g'| < 1 $$

. W is non-negative continuous function such that W(ξ) = 0 iff ξ ∈ {−1, 1}.

In view of the approach of Alberti and Müller in [1], we formulate the relaxation and minimization problem related to the functional I ^ε and we discuss the choice of relaxation and blowup procedure adjusted to capture two characteristic small scales associated to the minimizing sequences.

Also, we prove Γ-convergence result for the integrands, and we highlight the idea of proof for Γ-convergence for integral functionals induced by the chosen relaxation.