A result in asymptotic analysis for the functional of Ginzburg-Landau type with externally imposed multiple small scales in one dimension

Abstract

Abstract. In this paper we present technical improvement of results in [19]. We study asymptotic behavior of the functionalas ε −→ 0, where a is 1 × 1-periodic. We determine (rescaled) minimal asymptotic energy associated to J ε a,β,γ as ε −→ 0 where β, γ ≥ 0, β+γ > 0.

Γ‐convergence for one‐dimensional Ginzburg‐Landau functional with generalized Lipschitz penalizing term

“…While Γ-convergence result is not available in the case p = 1, we are able to compute asymptotic energies: Theorem 2.2 If p = 1, then E α,g,per = E α∞,g0,per , where α ∞ (s) = 1 2 α(1 + g 2 (s)g 3 (−1)) 2 + 1 2 α(1 + g 2 (s)g 3 (1)) 2 . P r o o f. To begin with, we use suitable approximation of functions g 0 and g 2 by piecewise affine functions (g n 0 ) and piecewise constant functions (g n 2 ) as in [2] and [4]. First we consider the case when functions g n 0 are affine and g n 2 are constant.…”

“…Once we have recovered asymptotic energy for every M > 0 (which follows by Theorem 3.3 in [3] and Theorem 4.17 in [2]), we can pass to the limit as M −→ +∞, getting the asymptotic energy for I ε α,gn . Then, quite in the same way as in [4], we use integration by parts to extended our results to the case of piecewise affine functions g n 0 and piecewise constant functions g n 2 . Finally, we can complete the proof as n −→ +∞ by the dominated convergence theorem.…”

Some properties of asymptotic energy of a class of functionals of Ginzburg‐Landau type endowed with generalized lower‐order oscillatory term in one dimension

“…Similar types of perturbations of the functional of Alberti and Müller were considered in previous papers by the author (cf. [3] and [4]).…”

Relaxation of Ginzburg–Landau functional perturbed by continuous nonlinear lower-order term in one dimension

“…s ∈ (0, 1), the minimizers v ε of (1.2) for sufficiently small ε resemble a particular sawtooth function, and satisfy I ε a0 (v ε ) ≈ E 0 W (ζ)dζ. For results involving different types of lower-order terms see [15][16][17].…”