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

Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany
mugnai@mis.mpg.de

Received: 25 November 2010
Revised: 26 January 2012

Abstract

We establish some new results about the Γ-limit, with respect to the L¹-topology, of two different (but related) phase-field approximations $\mathrm{\{}{\mathrm{ℰ}}_{\mathit{\epsilon }}{\mathrm{\}}}_{\mathit{\epsilon }}\mathit{,}\hspace{0.17em}\mathrm{\{}\begin{array}{}{}_{\mathrm{􏽥}}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}{\mathrm{\}}}_{\mathit{\epsilon }}$ 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 ℰ_ε, and show that in general the Γ-limits of ℰ_ε and $\begin{array}{}{\mathrm{􏽥}}_{}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}$ do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the Γ-limit of $\begin{array}{}{\mathrm{􏽥}}_{}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}$ strictly contains the domain of the Γ-limit of ℰ_ε.