Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional

Mugnai, Luca

ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 740-753.

We establish some new results about the Γ-limit, with respect to the L¹-topology, of two different (but related) phase-field approximations ${\left\{{ℰ}_{ϵ}\right\}}_{ϵ},{\left\{{\widetilde{ℰ}}_{ϵ}\right\}}_{ϵ}$ of the so-called Euler's Elastica Bending Energy for curves in the plane. In particular we characterize the Γ-limit as ε → 0 of ℰ_ε, and show that in general the Γ-limits of ℰ_ε and ${\widetilde{ℰ}}_{ϵ}$ do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the Γ-limit of ${\widetilde{ℰ}}_{ϵ}$ strictly contains the domain of the Γ-limit of ℰ_ε.

Classification : 49J45, 34K26, 49Q15, 49Q20
Mots clés : Γ-convergence, relaxation, singular perturbation, geometric measure theory

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