Topological Singularities in Periodic Media: Ginzburg–Landau and Core-Radius Approaches
Roberto Alicandro, Andrea Braides, Marco Cicalese, Lucia De Luca, Andrey Lvovich Piatnitski · Archive for Rational Mechanics and Analysis · 2021
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 $$E_{\varepsilon ,\delta }$$ Eε,δ , where $$\varepsilon $$ ε represent the coherence length (in the Ginzburg–Landau model) or the core-radius size (in the core-radius approach) and $$\delta $$ δ denotes the periodicity scale. We carry out the $$\Gamma $$ Γ -convergence analysis of $$E_{\varepsilon ,\delta }$$ Eε,δ as $$\varepsilon \rightarrow 0$$ ε→0 and $$\delta =\delta _\varepsilon \rightarrow 0$$ δ=δε→0 in the $$|\log \varepsilon |$$ |logε| scaling regime, showing that the $$\Gamma $$ Γ -limit consists in the energy cost of finitely many vortex-like point singularities of integer degree. After introducing the scale parameter $$\begin{aligned} \lambda =\min \Bigl \{1,\lim _{\varepsilon \rightarrow 0} {|\log \delta _\varepsilon |\over |\log \varepsilon |}\Bigr \} \end{aligned}$$ λ=min{1,limε→0|logδε||logε|} (upon extraction of subsequences), we show that in a sense we always have a separation-of-scale effect: at scales smaller than $$\varepsilon ^\lambda $$ ελ we first have a concentration process around some vortices whose location is subsequently optimized, while for scales larger than $$\varepsilon ^\lambda $$ ελ the concentration process takes place “after” homogenization.