GAMMA-CONVERGENCE RESULTS FOR PHASE-FIELD APPROXIMATIONS OF THE 2D-EULER ELASTICA FUNCTIONAL

摘要

We establish some new results about the Γ-limit, with respect to the L~1-topology, of two different (but related) phase-field approximations {ε_ε}_ε, {ε_ε}_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 ε_ε, and show that in general the Γ-limit of ε_ε and ε_ε do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the Γ-limit of ε_ε strictly contains the domain of the Γ-limit of ε_ε.