2021
DOI: 10.1007/s00205-021-01731-7
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Topological Singularities in Periodic Media: Ginzburg–Landau and Core-Radius Approaches

Abstract: We describe the emergence of topological singularities in periodic media within the Ginzburg–Landau model and the core-radius approach. The energy functionals of both models are denoted by $$E_{\varepsilon ,\delta }$$ E ε , δ , where $$\varepsilon $$ ε represent the coherence length (in the Ginzburg–Landau model) or th… Show more

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Cited by 8 publications
(12 citation statements)
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“…We claim that µ = 0 . Indeed, let us denote by v ε the piecewise affine interpolation of v ε and let η : [0, 3] → R be the piecewise affine function such that η = 1 on [0, 1] , η = 0 on [2,3] and η is affine on [1,2] . Then define the Lipschitz function ϕ ∈ C 0,1 c (B 3 ) via ϕ(x) = η(|x|) .…”
Section: Proofs In the Regime θ ε εmentioning
confidence: 99%
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“…We claim that µ = 0 . Indeed, let us denote by v ε the piecewise affine interpolation of v ε and let η : [0, 3] → R be the piecewise affine function such that η = 1 on [0, 1] , η = 0 on [2,3] and η is affine on [1,2] . Then define the Lipschitz function ϕ ∈ C 0,1 c (B 3 ) via ϕ(x) = η(|x|) .…”
Section: Proofs In the Regime θ ε εmentioning
confidence: 99%
“…To this end, given θ ε → 0 , we find the relevant scaling κ ε and we study the Γ -limit of 1 κε E ε . The limit strongly depends on the rate of convergence θ ε → 0 and can be characterized by interfacial-type singularities [23,2,21,27,8,17,20,24,12] (see also [18,5]) or vortex-like singularities [38,4,6,7,19,14,3], possibly coexisting. In this paper we are interested in the following regimes: ε| log ε| θ ε , θ ε ∼ ε| log ε|, and θ ε ε .…”
mentioning
confidence: 99%
“…We shall prove that φ and φ coincide (up to translating φ of an integer multiple of 2π), so that a unique lifting is defined in the annulus A r,R . 3 The proof of this fact is standard: one can consider for every ρ ∈ (0, 1) the scaled triangle T ρ := ρ(T −b(T ))+b(T ) and define the "conical" approximation v…”
Section: Topological Singularitiesmentioning
confidence: 99%
“…, L and ε small enough. To reduce to the setting in Lemma 6.6 we follow an argument introduced, e.g., in [27] or [3]. It is aimed at separating the scales of the radii of the balls charged by µ vε .…”
Section: Ball Constructionmentioning
confidence: 99%
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