Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional
ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 3, pp. 740-753.

We establish some new results about the Γ-limit, with respect to the L1-topology, of two different (but related) phase-field approximations { 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 ℰε, and show that in general the Γ-limits of ℰε and ˜ 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 strictly contains the domain of the Γ-limit of ℰε.

DOI: 10.1051/cocv/2012031
Classification: 49J45, 34K26, 49Q15, 49Q20
Keywords: Γ-convergence, relaxation, singular perturbation, geometric measure theory
@article{COCV_2013__19_3_740_0,
     author = {Mugnai, Luca},
     title = {Gamma-convergence results for phase-field approximations of the {2D-Euler} {Elastica} {Functional}},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {740--753},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {3},
     year = {2013},
     doi = {10.1051/cocv/2012031},
     mrnumber = {3092360},
     zbl = {1270.49012},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2012031/}
}
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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, Volume 19 (2013) no. 3, pp. 740-753. doi : 10.1051/cocv/2012031. http://www.numdam.org/articles/10.1051/cocv/2012031/

[1] G. Bellettini, G. Dal Maso and M. Paolini, Semicontinuity and relaxation properties of a curvature depending functional in 2d. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 20 (1993) 247-297. | Numdam | MR | Zbl

[2] G. Bellettini and L. Mugnai, A varifolds representation of the relaxed elastica functional. J. Convex Anal. 14 (2007) 543-564. | MR | Zbl

[3] G. Bellettini and L. Mugnai, Approximation of Helfrich's functional via diffuse interfaces. SIAM J. Math. Anal. 42 (2010) 2402-2433. | MR | Zbl

[4] G. Bellettini and M. Paolini, Approssimazione variazionale di funzionali con curvatura. Seminario Analisi Matematica Univ. Bologna (1993).

[5] A. Braides and R. March, Approximation by Γ-convergence of a curvature-depending functional in visual reconstruction. Commun. Pure Appl. Math. 59 (2006) 71-121. | MR | Zbl

[6] X. Cabré and J. Terra, Saddle-shaped solutions of bistable diffusion equations in all of R2m. J. Eur. Math. Soc. 43 (2009) 819-943. | MR | Zbl

[7] G. Dal Maso, An introduction to Γ-convergence, vol. 8, Progress in Nonlinear Differential Equations and their Applications. Birkhäuser, Boston, MA (1993). | MR | Zbl

[8] H. Dang, P. Fife and L. Peletier, Saddle solutions of the bistable diffusion equation. Z. Angew. Math. Phys. 43 (1992) 984-998. | MR | Zbl

[9] E. De Giorgi, Some remarks on Γ-convergence and least squares method, in Composite media and homogenization theory (Trieste, 1990), MA. Progr. Nonlinear Differ. Eq. Appl. 5 (1991) 135-142. | Zbl

[10] P. Dondl, L. Mugnai and M. Röger, Confined elastic curves. SIAM J. Appl. Math. 71 (2011) 2205-2226. | MR | Zbl

[11] Q. Du, C. Liu, R. Ryham and X. Wang, A phase field formulation of the Willmore problem. Nonlinearity 18 (2005) 1249-1267. | MR | Zbl

[12] Q. Du, C. Liu and X. Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198 (2004) 450-468. | MR | Zbl

[13] J. Hutchinson, C1, α-multiple function regularity and tangent cone behavior for varifolds with second fundamental form in Lp, in Geometric measure theory and the calculus of variations (Arcata, Calif., 1984). Proc. Sympos. Pure Math. Amer. Math. Soc. 44 (1984) 281-306. | MR | Zbl

[14] J. Hutchinson, Second fundamental form for varifolds and the existence of surfaces minimising curvature. Indiana Univ. Math. J. 35 (1986) 281-306. | MR | Zbl

[15] J.S. Lowengrub, A. Rätz and A. Voigt, Phase-field modeling of the dynamics of multicomponent vesicles: spinodal decomposition, coarsening, budding, and fission. Phys. Rev. E 79 (2009) 82C99-92C10. | MR

[16] L. Modica and S. Mortola, Un esempio di Γ − -convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285-299. | MR | Zbl

[17] Y. Nagase and Y. Tonegawa, A singular perturbation problem with integral curvature bound. Hiroshima Math. Journal 37 (2007) 455-489. | MR | Zbl

[18] M. Röger and R. Schätzle. On a modified conjecture of De Giorgi. Math. Z. 254 (2006) 675-714. | Zbl

[19] L. Simon, Proceedings of the Centre for Mathematical Analysis, Australian National University. Centre for Math. Anal., Lectures on Geometric Measure Theory, vol. 3. Australian National Univ., Canberra (1984). | MR | Zbl

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