Free access
Volume 46, Number 1, January-February 2012
Page(s) 59 - 79
Published online 22 July 2011
  1. F. Almgren and J.E. Taylor, Optimal geometry in equilibrium and growth. Fractals 3 (1995) 713–723. Symposium in Honor of B. Mandelbrot. [CrossRef] [MathSciNet]
  2. F. Almgren, J.E. Taylor and L. Wang, Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31 (1993) 387–438. [CrossRef] [MathSciNet]
  3. L. Ambrosio, Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191–246. [MathSciNet]
  4. L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (2005).
  5. 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. [CrossRef] [MathSciNet]
  6. M. Bauer and E. Kuwert, Existence of minimizing Willmore surfaces of prescribed genus. Int. Math. Res. Not. 10 (2003) 553–576. [CrossRef]
  7. T. Baumgart, S.T. Hess and W.W. Webb, Image coexisting fluid domains in biomembrane models coupling curvature and line tension. Nature 425 (2003) 821–824. [CrossRef] [PubMed]
  8. G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J. 25 (1996) 537–566. [MathSciNet]
  9. A. Bonito, R.H. Nochetto and M.S. Pauletti, Parametric FEM for geometric biomembranes. J. Comput. Phys. 229 (2010) 3171–3188. [CrossRef] [MathSciNet]
  10. J.W. Cahn and D.W. Hoffman, A vector thermodynamics for anisotropic surfaces. II. Curved and facetted surfaces. Acta Metall. 22 (1974) 1205–1214. [CrossRef]
  11. T. Chan and L. Vese, A level set algorithm for minimizing the Mumford-Shah functional in image processing, in Proceedings of the 1st IEEE Workshop on Variational and Level Set Methods in Computer Vision (2001) 161–168.
  12. K. Chen, C. Jayaprakash, R. Pandit and W. Wenzel, Microemulsions: A Landau-Ginzburg theory. Phys. Rev. Lett. 65 (1990) 2736–2739. [CrossRef] [PubMed]
  13. P. Cicuta, S.L. Keller and S.L. Veatch, Diffusion of liquid domains in lipid bilayer membranes. J. Phys. Chem. B 111 (2007) 3328–3331. [CrossRef] [PubMed]
  14. U. Clarenz, U. Diewald, G. Dziuk, M. Rumpf and R. Rusu, A finite element method for surface restoration with smooth boundary conditions. Comput. Aided Geom. Des. 21 (2004) 427–445.
  15. M.C. Delfour and J.-P. Zolésio, Shapes and Geometries, Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2001).
  16. H.G. Döbereiner, O. Selchow and R. Lipowsky, Spontaneous curvature of fluid vesicles induced by trans-bilayer sugar asymmetry. Eur. Biophys. J. 28 (1999) 174–178. [CrossRef]
  17. G. Doğan, P. Morin and R.H. Nochetto, A variational shape optimization approach for image segmentation with a Mumford-Shah functional. SIAM J. Sci. Comput. 30 (2008) 3028–3049. [CrossRef] [MathSciNet]
  18. G. Doğan, P. Morin, R.H. Nochetto and M. Verani, Discrete gradient flows for shape optimization and applications. Comput. Meth. Appl. Mech. Eng. 196 (2007) 3898–3914. [CrossRef] [MathSciNet]
  19. M. Droske and M. Bertozzi, Higher-order feature-preserving geometric regularization. SIAM J. Imaging Sci. 3 (2010) 21–51. [CrossRef] [MathSciNet]
  20. G. Dziuk, Computational parametric Willmore flow. Numer. Math. 111 (2008) 55–80. [CrossRef] [MathSciNet]
  21. G. Dziuk, E. Kuwert and R. Schätzle, Evolution of elastic curves in Formula : existence and computation. SIAM J. Math. Anal. 33 (electronic) (2002) 1228–1245.
  22. 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. [CrossRef] [MathSciNet]
  23. W. Helfrich, Elastic properties of lipid bilayers – theory and possible experiments. Zeitschrift Fur Naturforschung C-A J. Biosc. 28 (1973) 693.
  24. M. Hintermüller and W. Ring, A second order shape optimization approach for image segmentation. SIAM J. Appl. Math. 64 (2003/04) 442–467.
  25. M. Hintermüller and W. Ring, An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional. J. Math. Imaging and Vision 20 (2004) 19–42. Special issue on mathematics and image analysis. [CrossRef] [MathSciNet]
  26. J.T. Jenkins, The equations of mechanical equilibrium of a model membrane. SIAM J. Appl. Math. 32 (1977) 755–764. [CrossRef] [MathSciNet]
  27. R. Keriven and O. Faugeras, Variational principles, surface evolution, PDEs, level set methods and the stereo problem. Technical Report 3021, INRIA (1996).
  28. R. Keriven and O. Faugeras, Variational principles, surface evolution, PDEs, level set methods and the stereo problem. IEEE Trans. Image Process. 7 (1998) 336–344. [CrossRef] [MathSciNet] [PubMed]
  29. R. Kimmel and A.M. Bruckstein, Regularized Laplacian zero crossings as optimal edge integrators. IJCV 53 (2003) 225–243. [CrossRef]
  30. E. Kuwert and R. Schätzle, The Willmore flow with small initial energy. J. Differential Geom. 57 (2001) 409–441. [MathSciNet]
  31. E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional. Comm. Anal. Geom. 10 (2002) 307–339. [MathSciNet]
  32. E. Kuwert and R. Schätzle, Removability of point singularities of Willmore surfaces. Ann. Math. 160 (2004) 315–357. [CrossRef]
  33. M. Laradji and O.G. Mouritsen, Elastic properties of surfactant monolayers at liquid-liquid interfaces: A molecular dynamics study. J. Chem. Phys. 112 (2000) 8621–8630. [CrossRef]
  34. M. Leventon, O. Faugeraus and W. Grimson, Level set based segmentation with intensity and curvature priors, in Proceedings of Workshop on Mathematical Methods in Biomedical Image Analysis Proceedings (2000) 4–11.
  35. G.B. McFadden, A.A. Wheeler, R.J. Braun, S.R. Coriell and R.F. Sekerka, Phase-field models for anisotropic interfaces. Phys. Rev. E 48 (1993) 2016–2024. [CrossRef] [MathSciNet]
  36. J. Melenkevitz and S.H. Javadpour, Phase separation dynamics in mixtures containing surfactants. J. Chem. Phys. 107 (1997) 623–629. [CrossRef]
  37. R. Rusu, An algorithm for the elastic flow of surfaces. Interfaces and Free Boundaries 7 (2005) 229–239. [CrossRef] [MathSciNet]
  38. U. Seifert, Configurations of fluid membranes and vesicles. Adv. Phys. 46 (1997) 13–137. [CrossRef]
  39. L. Simon, Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1 (1993) 281–326. [MathSciNet]
  40. G. Simonett, The Willmore flow near spheres. Differential Integral Equations 14 (2001) 1005–1014. [MathSciNet]
  41. J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, Springer Series in Computational Mathematics 16. Springer-Verlag, Berlin (1992).
  42. G. Sundaramoorthi, A. Yezzi, A. Mennucci and G. Sapiro, New possibilities with Sobolev active contours, in Proceedings of the 1st International Conference on Scale Space Methods and Variational Methods in Computer Vision (2007).
  43. J.E. Taylor, Crystalline variational problems. Bull. Amer. Math. Soc. 84 (1978) 568–588. [CrossRef] [MathSciNet]
  44. J.E. Taylor, Mean curvature and weighted mean curvature. Acta Metall. Mater. 40 (1992) 1475–1485. [CrossRef]
  45. J.E. Taylor and J.W. Cahn, Linking anisotropic sharp and diffuse surface motion laws via gradient flows. J. Stat. Phys. 77 (1994) 183–197. [CrossRef]
  46. J.E. Taylor and J.W. Cahn, Diffuse interfaces with sharp corners and facets: Phase field modeling of strongly anisotropic surfaces. Physica D 112 (1998) 381–411. [CrossRef] [MathSciNet]
  47. S.L. Veatch and S.L. Keller, Separation of liquid phases in giant vesicles of ternary mixtures of phospholipids and cholesterol. Biophys. J. 85 (2003) 3074–3083. [CrossRef] [PubMed]
  48. A.A. Wheeler and G.B. McFadden, A ξ-vector formulation of anisotropic phase-field models: 3D asymptotics. Eur. J. Appl. Math. 7 (1996) 367–381.
  49. T.J. Willmore, Total curvature in Riemannian geometry. Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester (1982).

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