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

Luca Mugnai

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

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

Abstract

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We establish some new results about the Γ-limit, with respect to the L1-topology, of two different (but related) phase-field approximations { ℰ } , { ℰ ˜ } 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 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 ℰ ˜ x10ff65; ℰ ε strictly contains the domain of theΓ-limit of ℰε.

How to cite

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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 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 -

References

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