We establish some new results about the Γ-limit, with respect to the L1-topology, of two different (but related) phase-field approximations 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 do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the Γ-limit of strictly contains the domain of the Γ-limit of ℰε.
Keywords: Γ-convergence, relaxation, singular perturbation, geometric measure theory
@article{COCV_2013__19_3_740_0, 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}, mrnumber = {3092360}, zbl = {1270.49012}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2012031/} }
TY - JOUR AU - Mugnai, Luca TI - Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 740 EP - 753 VL - 19 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012031/ DO - 10.1051/cocv/2012031 LA - en ID - COCV_2013__19_3_740_0 ER -
%0 Journal Article %A Mugnai, Luca %T Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 740-753 %V 19 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012031/ %R 10.1051/cocv/2012031 %G en %F COCV_2013__19_3_740_0
Mugnai, Luca. Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 3, pp. 740-753. doi : 10.1051/cocv/2012031. http://www.numdam.org/articles/10.1051/cocv/2012031/
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