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

ESAIM: Control, Optimisation and Calculus of Variations (2013)

- Volume: 19, Issue: 3, page 740-753
- ISSN: 1292-8119

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topMugnai, Luca. "Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional." ESAIM: Control, Optimisation and Calculus of Variations 19.3 (2013): 740-753. <http://eudml.org/doc/272962>.

@article{Mugnai2013,

abstract = {We establish some new results about the Γ-limit, with respect to the L1-topology, of two different (but related) phase-field approximations $\lbrace \mathcal \{E\}_\rbrace _,\,\lbrace \widetilde\{\mathcal \{E\}\}_\rbrace _$ ℰ ε ε , x10ff65; ℰ ε ε of the so-called Euler’s Elastica Bending Energy for curves in the plane. In particular we characterize theΓ-limit as ε → 0 of ℰε, and show that in general the Γ-limits of ℰεand $\widetilde\{\mathcal \{E\}\}_$ x10ff65; ℰ ε do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of theΓ-limit of $\widetilde\{\mathcal \{E\}\}_$ x10ff65; ℰ ε strictly contains the domain of theΓ-limit of ℰε.},

author = {Mugnai, Luca},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Γ-convergence; relaxation; singular perturbation; geometric measure theory; -convergence},

language = {eng},

number = {3},

pages = {740-753},

publisher = {EDP-Sciences},

title = {Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional},

url = {http://eudml.org/doc/272962},

volume = {19},

year = {2013},

}

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

PB - EDP-Sciences

VL - 19

IS - 3

SP - 740

EP - 753

AB - We establish some new results about the Γ-limit, with respect to the L1-topology, of two different (but related) phase-field approximations $\lbrace \mathcal {E}_\rbrace _,\,\lbrace \widetilde{\mathcal {E}}_\rbrace _$ ℰ ε ε , x10ff65; ℰ ε ε of the so-called Euler’s Elastica Bending Energy for curves in the plane. In particular we characterize theΓ-limit as ε → 0 of ℰε, and show that in general the Γ-limits of ℰεand $\widetilde{\mathcal {E}}_$ x10ff65; ℰ ε do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of theΓ-limit of $\widetilde{\mathcal {E}}_$ x10ff65; ℰ ε strictly contains the domain of theΓ-limit of ℰε.

LA - eng

KW - Γ-convergence; relaxation; singular perturbation; geometric measure theory; -convergence

UR - http://eudml.org/doc/272962

ER -

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