PHASE-FIELD MODELS FOR THE APPROXIMATION OF THE WILLMORE FUNCTIONAL AND FLOW ;

This paper is a short account on phase-field approximations of the Willmore functional and the associated L 2 -flows.

[1]  Robert L. Pego,et al.  Front migration in the nonlinear Cahn-Hilliard equation , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[2]  D. A. Singer,et al.  Curve straightening and a minimax argument for closed elastic curves , 1985 .

[3]  Uwe F. Mayer,et al.  A numerical scheme for axisymmetric solutions of curvature-driven free boundary problems, with applications to the Willmore flow , 2002 .

[4]  Luca Mugnai.  GAMMA-CONVERGENCE RESULTS FOR PHASE-FIELD APPROXIMATIONS OF THE 2D-EULER ELASTICA FUNCTIONAL , 2010, 1009.5826.

[5]  Roger Moser,et al.  A Higher Order Asymptotic Problem Related to Phase Transitions , 2005, SIAM J. Math. Anal..

[6]  Yoshihiro Tonegawa,et al.  A singular perturbation problem with integral curvature bound , 2007 .

[7]  Sylvia Serfaty,et al.  Gamma-convergence of gradient flows on Hilbert and metric spaces and applications , 2011 .

[8]  P. Loreti,et al.  Propagation of fronts in a nonlinear fourth order equation , 2000, European Journal of Applied Mathematics.

[9]  Yoshihiro Tonegawa,et al.  Phase field model with a variable chemical potential , 2000, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[10]  G. Bellettini.  Variational approximation of functionals with curvatures and related properties , 1997 .

[11]  Pierluigi Colli,et al.  A Phase-Field Approximation of the Willmore Flow with Volume and Area Constraints , 2012, SIAM J. Math. Anal..

[12]  P. Colli,et al.  A phase-field approximation of the Willmore flow with volume constraint , 2010, 1004.0341.

[13]  Paul C. Fife,et al.  Dynamics of Layered Interfaces Arising from Phase Boundaries , 1988 .

[14]  Gerhard Dziuk,et al.  Evolution of Elastic Curves in Rn: Existence and Computation , 2002, SIAM J. Math. Anal..

[15]  S. Esedoglu,et al.  Colliding interfaces in old and new diffuse-interface approximations of Willmore-flow , 2012, 1209.6531.

[16]  Matthias Röger,et al.  On a Modified Conjecture of De Giorgi , 2006 .

[17]  Reiner Schätzle,et al.  Removability of point singularities of Willmore surfaces , 2004 .

[18]  E. Kuwert,et al.  The Willmore Flow with Small Initial Energy , 2001 .

[19]  Maurizio Paolini,et al.  Semicontinuity and relaxation properties of a curvature depending functional in 2D , 1993 .

[20]  E. D. Giorgi,et al.  Some remarks on Γ-convergence and least squares method , 1991 .

[21]  G. Bellettini,et al.  On the approximation of the elastica functional in radial symmetry , 2005 .

[22]  Giovanni Bellettini,et al.  Approximation of Helfrich's Functional via Diffuse Interfaces , 2009, SIAM J. Math. Anal..

[23]  Maurizio Paolini,et al.  Quasi-optimal error estimates for the mean curvature flow with a forcing term , 1995, Differential and Integral Equations.

[24]  Elie Bretin,et al.  Phase-field approximations of the Willmore functional and flow , 2013, Numerische Mathematik.

[25]  L. Peletier,et al.  Saddle solutions of the bistable diffusion equation , 1992 .