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

Published online by Cambridge University Press:  03 June 2013

Luca Mugnai
Luca Mugnai*
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany. [email protected]
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Abstract

We establish some new results about the Γ-limit, with respect to the L1-topology, of two different (but related) phase-field approximations \hbox{$\{\mathcal E_\eps\}_\eps,\,\{\widetilde{\mathcalE}_\eps\}_\eps$}{ℰε}ε, {􏽥ℰε}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 \hbox{$\widetilde{\mathcal E}_\eps$}􏽥ℰε do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the Γ-limit of \hbox{$\widetilde{\mathcal E}_\eps$}􏽥ℰε strictly contains the domain of the Γ-limit of ℰε.

Keywords

Γ-convergencerelaxationsingular perturbationgeometric measure theory
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 740 - 753
Copyright
© EDP Sciences, SMAI, 2013

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