We establish some new results about the Γ-limit, with respect to the L1-topology, of two different (but related) phase-field approximations 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 do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the Γ-limit of strictly contains the domain of the Γ-limit of ℰε.
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://archive.numdam.org/articles/10.1051/cocv/2012031/} }
TY - JOUR AU - Mugnai, Luca TI - Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 740 EP - 753 VL - 19 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2012031/ DO - 10.1051/cocv/2012031 LA - en ID - COCV_2013__19_3_740_0 ER -
%0 Journal Article %A Mugnai, Luca %T Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 740-753 %V 19 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2012031/ %R 10.1051/cocv/2012031 %G en %F COCV_2013__19_3_740_0
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://archive.numdam.org/articles/10.1051/cocv/2012031/
[1] 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
, and ,[2] A varifolds representation of the relaxed elastica functional. J. Convex Anal. 14 (2007) 543-564. | MR | Zbl
and ,[3] Approximation of Helfrich's functional via diffuse interfaces. SIAM J. Math. Anal. 42 (2010) 2402-2433. | MR | Zbl
and ,[4] Approssimazione variazionale di funzionali con curvatura. Seminario Analisi Matematica Univ. Bologna (1993).
and ,[5] Approximation by Γ-convergence of a curvature-depending functional in visual reconstruction. Commun. Pure Appl. Math. 59 (2006) 71-121. | MR | Zbl
and ,[6] Saddle-shaped solutions of bistable diffusion equations in all of R2m. J. Eur. Math. Soc. 43 (2009) 819-943. | MR | Zbl
and ,[7] An introduction to Γ-convergence, vol. 8, Progress in Nonlinear Differential Equations and their Applications. Birkhäuser, Boston, MA (1993). | MR | Zbl
,[8] Saddle solutions of the bistable diffusion equation. Z. Angew. Math. Phys. 43 (1992) 984-998. | MR | Zbl
, and ,[9] 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] Confined elastic curves. SIAM J. Appl. Math. 71 (2011) 2205-2226. | MR | Zbl
, and ,[11] A phase field formulation of the Willmore problem. Nonlinearity 18 (2005) 1249-1267. | MR | Zbl
, , and ,[12] 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
, and ,[13] 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] Second fundamental form for varifolds and the existence of surfaces minimising curvature. Indiana Univ. Math. J. 35 (1986) 281-306. | MR | Zbl
,[15] Phase-field modeling of the dynamics of multicomponent vesicles: spinodal decomposition, coarsening, budding, and fission. Phys. Rev. E 79 (2009) 82C99-92C10. | MR
, and ,[16] Un esempio di Γ − -convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285-299. | MR | Zbl
and ,[17] A singular perturbation problem with integral curvature bound. Hiroshima Math. Journal 37 (2007) 455-489. | MR | Zbl
and ,[18] On a modified conjecture of De Giorgi. Math. Z. 254 (2006) 675-714. | Zbl
and .[19] 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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