Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional
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 {ℰϵ}ϵ,{ℰ˜ϵ}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 ℰε, and show that in general the Γ-limits of ℰε and 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 ℰ˜ϵ strictly contains the domain of the Γ-limit of ℰε.

DOI : 10.1051/cocv/2012031
Classification : 49J45, 34K26, 49Q15, 49Q20
Mots-clés : Γ-convergence, relaxation, singular perturbation, geometric measure theory 
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     author = {Mugnai, Luca},
     title = {Gamma-convergence results for phase-field approximations of the {2D-Euler} {Elastica} {Functional}},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {740--753},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {3},
     year = {2013},
     doi = {10.1051/cocv/2012031},
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     zbl = {1270.49012},
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     url = {https://www.numdam.org/articles/10.1051/cocv/2012031/}
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Mugnai, Luca. Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 740-753. doi : 10.1051/cocv/2012031. https://www.numdam.org/articles/10.1051/cocv/2012031/

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