Free Access
Issue
ESAIM: COCV
Volume 27, 2021
Article Number 14
Number of page(s) 28
DOI https://doi.org/10.1051/cocv/2021013
Published online 22 March 2021
  1. F. Alessio, A. Calamai and P. Montecchiari, Saddle-type solutions for a class of semilinear elliptic equations. Adv. Differ. Equ. 12 (2007) 361–380. [Google Scholar]
  2. J.W. Barrett, H. Garcke and R. Nürnberg, A parametric finite element method for fourth order geometric evolution equations. J. Comput. Phys. 222 (2007) 441–462. [Google Scholar]
  3. J.W. Barrett, H. Garcke and R. Nürnberg, Parametric approximation of Willmore flow and related geometric evolution equations. SIAM J. Sci. Comput. 31 (2008) 225–253. [Google Scholar]
  4. J.W. Barrett, H. Garcke and R. Nürnberg, Numerical approximation of gradient flows for closed curves in ℝd. IMA J. Numer. Anal. 30 (2010) 4–60. [Google Scholar]
  5. J.W. Barrett, R. Nürnberg and V. Styles, Finite element approximation of a phase field model for void electromigration. SIAM J. Numer. Anal. 42 (2004) 738–772. [Google Scholar]
  6. G. Bellettini, Variational approximation of functionals with curvatures and related properties. J. Convex Anal. 4 (1997) 91–108. [Google Scholar]
  7. 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. [Google Scholar]
  8. G. Bellettini and L. Mugnai, Characterization and representation of the lower semicontinuous envelope of the elastica functional. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) 839–880. [CrossRef] [MathSciNet] [Google Scholar]
  9. G. Bellettini and L. Mugnai, On the approximation of the elastica functional in radial symmetry. Calc. Var. Partial Differ. Equ. 24 (2005) 1–20. [Google Scholar]
  10. G. Bellettini and L. Mugnai, A varifolds representation of the relaxed elastica functional. J. Convex Anal. 14 (2007) 543–564. [Google Scholar]
  11. G. Bellettini and M. Paolini, Some results on minimal barriers in the sense of De Giorgi applied to driven motion by mean curvature. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 43–67. [Google Scholar]
  12. M. Beneš, K. Mikula, T. Oberhuber and D. Ševčovič, Comparison study for level set and direct Lagrangian methods for computing Willmore flow of closed planar curves. Comput. Vis. Sci. 12 (2009) 307–317. [Google Scholar]
  13. T. Biben, K. Kassner and C. Misbah, Phase-field approach to three-dimensional vesicle dynamics. Phys. Rev. E 72 (2005). [Google Scholar]
  14. A. Bonito, R.H. Nochetto and M.S. Pauletti, Parametric FEM for geometric biomembranes. J. Comput. Phys. 229 (2010) 3171–3188. [Google Scholar]
  15. E. Bretin, F. Dayrens and S. Masnou, Volume reconstruction from slices. SIAM J. Imag. Sci. 10 (2017) 2326–2358. [Google Scholar]
  16. E. Bretin, S. Masnou and É. Oudet, Phase-field approximations of the Willmore functional and flow. Numer. Math. 131 (2015) 115–171. [Google Scholar]
  17. X. Cabré and J. Terra, Saddle-shaped solutions of bistable diffusion equations in all of \2m. J. Eur. Math. Soc. (JEMS) 11 (2009) 819–943. [Google Scholar]
  18. L. Caffarelli, N. Garofalo and F. Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences. Comm. Pure Appl. Math. 47 (1994) 1457–1473. [Google Scholar]
  19. F. Campelo and A. Hernandez-Machado, Dynamic model and stationary shapes of fluid vesicles. Eur. Phys. J. E 20 (2006) 37–45. [CrossRef] [EDP Sciences] [Google Scholar]
  20. A. Dall’Acqua, C.-C. Lin and P. Pozzi, A gradient flow for open elastic curves with fixed length and clamped ends. Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (2017) 1031–1066. [Google Scholar]
  21. A. Dall’Acqua and P. Pozzi, A Willmore-Helfrich L2-flow of curves with natural boundary conditions. Commun. Anal. Geom. 22 (2014) 617–669. [Google Scholar]
  22. H. Dang, P.C. Fife and L.A. Peletier, Saddle solutions of the bistable diffusion equation. Z. Angew. Math. Phys. 43 (1992) 984–998. [Google Scholar]
  23. S. Dasgupta, T. Auth and G. Gompper, Shape and orientation matter for the cellular uptake of nonspherical particles. Nano Lett. 14 (2014) 687–693. [PubMed] [Google Scholar]
  24. E. De Giorgi Some remarks on Γ-convergence and least squares method. In Composite media and homogenization theory (Trieste, 1990). Vol. 5 of Progr. Nonlinear Differential Equations Appl. Birkhäuser Boston, Boston, MA (1991) 135–142. [Google Scholar]
  25. K. Deckelnick, G. Dziuk and C.M. Elliott, Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14 (2005) 139–232. [Google Scholar]
  26. P.W. Dondl, A. Lemenant and S. Wojtowytsch, Phase field models for thin elastic structures with topological constraint. Arch. Ration. Mech. Anal. 223 (2017) 693–736. [Google Scholar]
  27. P.W. Dondl, L. Mugnai and M. Röger, Confined elastic curves. SIAM J. Appl. Math. 71 (2011) 2205–2226. [Google Scholar]
  28. M. Droske and M. Rumpf, A level set formulation for Willmore flow. Interfaces Free Bound. 6 (2004) 361–378. [Google Scholar]
  29. 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. [Google Scholar]
  30. Q. Du, C. Liu and X. Wang, Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comput. Phys. 212 (2006) 757–777. [Google Scholar]
  31. G. Dziuk, E. Kuwert and R. Schätzle, Evolution of elastic curves in ℝn: existence and computation. SIAM J. Math. Anal. 33 (2002) 1228–1245. [Google Scholar]
  32. C.M. Elliott and B. Stinner, Modeling and computation of two phase geometric biomembranes using surface finite elements. J. Comput. Phys. 229 (2010) 6585–6612. [Google Scholar]
  33. S. Esedoḡlu, A. Rätz and M. Röger, Colliding interfaces in old and new diffuse-interface approximations of Willmore-flow. Commun. Math. Sci. 12 (2014) 125–147. [Google Scholar]
  34. M. Fei and Y. Liu, Phase-field approximation of the willmoreflow (2020). [Google Scholar]
  35. M. Franken, M. Rumpf and B. Wirth, A phase field based PDE constrained optimization approach to time discrete Willmore flow. Int. J. Numer. Anal. Model. 10 (2013) 116–138. [Google Scholar]
  36. J.D. Lawrence, A catalog of special plane curves. Dover books on advanced mathematics. Dover Publications (1972). [Google Scholar]
  37. A. Linnér and J.W. Jerome, A unique graph of minimal elastic energy. Trans. Amer. Math. Soc. 359 (2007) 2021–2041. [Google Scholar]
  38. P. Loretiand R. March, Propagation of fronts in a nonlinear fourth order equation. Eur. J. Appl. Math. 11 (2000) 203–213. [Google Scholar]
  39. J. 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) 0311926. [Google Scholar]
  40. L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 357–383. [Google Scholar]
  41. L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations. Commun. Pure Appl. Math. 38 (1985) 679–684. [Google Scholar]
  42. L. Modica and S. Mortola, Un esempio di Γ-convergenza. Boll. Un.Mat. Ital. B 14 (1977) 285–299. [Google Scholar]
  43. L. Modica and S. Mortola, Some entire solutions in the plane of nonlinear Poisson equations. Boll. Un. Mat. Ital. B 17 (1980) 614–622. [Google Scholar]
  44. R. Moser, A higher order asymptotic problem related to phase transitions. SIAM J. Math. Anal. 37 (2005) 712–736. [Google Scholar]
  45. L. Mugnai, Gamma-convergence results for phase-field approximations of the 2D-Euler elastica functional. ESAIM: COCV 19 (2013) 740–753. [EDP Sciences] [Google Scholar]
  46. A. Rätz and M. Röger, A diffuse-interface model accounting for elastic membrane energies with particle–membrane interaction. Inpreparation (2020). [Google Scholar]
  47. M. Röger and R. Schätzle, On a modified conjecture of De Giorgi. Math. Z. 254 (2006) 675–714. [Google Scholar]
  48. R.E. Rusu, An algorithm for the elastic flow of surfaces. Interfaces Free Bound. 7 (2005) 229–239. [Google Scholar]
  49. S. Vey and A. Voigt, AMDiS: adaptive multidimensional simulations. Comput. Visual. Sci. 10 (2007) 57–67. [Google Scholar]
  50. X. Wang, Asymptotic analysis of phase field formulations of bending elasticity models. SIAM J. Math. Anal. 39 (2008) 1367–1401. [Google Scholar]
  51. X. Wang and Q. Du, Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches. J. Math. Biol. 56 (2008) 347–371. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  52. X. Wang, L. Ju and Q. Du, Efficient and stable exponential time differencing Runge-Kutta methods for phase field elastic bending energy models. J. Comput. Phys. 316 (2016) 21–38. [Google Scholar]
  53. T.J. Willmore, Riemannian geometry. Oxford Science Publications, The Clarendon Press Oxford University Press, New York (1993). [Google Scholar]
  54. C. Zwilling, The diffuse interface approximation of the willmore functional in configurations with interacting phaseboundaries (2018). [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.