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ESAIM: Control, Optimisation and Calculus of Variations (1)

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Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional
Luca Mugnai
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 19 / Issue 3 / July 2013

Published online by Cambridge University Press:

03 June 2013, pp. 740-753

Print publication:

July 2013
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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 \hbox{$\{\mathcal E_\eps\}_\eps,\,\{\widetilde{\mathcalE}_\eps\}_\eps$} ${ℰ_{ε}}_{ε}, {\begin{matrix} _{􏽥} \\ ℰ_{ε} \end{matrix}}_{ε}$ 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{$\widetilde{\mathcal E}_\eps$} $\begin{matrix} 􏽥 \\ ℰ_{ε} \end{matrix}$ 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{$\widetilde{\mathcal E}_\eps$} $\begin{matrix} 􏽥 \\ ℰ_{ε} \end{matrix}$ strictly contains the domain of the Γ-limit of ℰε.