Linearized Stability Analysis of Stationary Solutions for Surface Diffusion with Boundary Conditions

The linearizedstability of stationary solutions to the surface diffusion flow with angle conditions and no-flux conditions as boundary conditions is studied. We perform a linearized stability analysis in which the H-1 -gradient flow structure plays a key role. As a byproduct our analysis also gives a criterion for the stability of critical points of the length functional of curves which come into contact with the outer boundary. Finally, we study the linearized stability of several examples.

[1]  John W. Cahn,et al.  Linking anisotropic sharp and diffuse surface motion laws via gradient flows , 1994 .

[2]  Charles M. Elliott,et al.  The Cahn–Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature , 1996, European Journal of Applied Mathematics.

[3]  Eberhard Zeidler,et al.  Applied Functional Analysis: Applications to Mathematical Physics , 1995 .

[4]  Harald Garcke,et al.  A singular limit for a system of degenerate Cahn-Hilliard equations , 2000, Advances in Differential Equations.

[5]  Paul C. Fife,et al.  Models for phase separation and their mathematics. , 2000 .

[6]  A. Balakrishnan.  Applied Functional Analysis , 1976 .

[7]  J. Keller,et al.  Fast reaction, slow diffusion, and curve shortening , 1989 .

[8]  Moto-Hiko Sato,et al.  Stability of stationary interfaces with contact angle in a generalized mean curvature flow , 1996 .

[9]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[10]  Yoshihito Kohsaka,et al.  Three-phase boundary motion by surface diffusion: stability of a mirror symmetric stationary solution , 2001 .

[11]  Harald Garcke,et al.  On Fully Practical Finite Element Approximations of Degenerate Cahn-Hilliard Systems , 2001 .

[12]  J. Taylor,et al.  Overview no. 113 surface motion by surface diffusion , 1994 .

[13]  Morton E. Gurtin,et al.  On the motion of a phase interface by surface diffusion , 1990 .

[14]  J. Escher,et al.  The surface diffusion flow for immersed hypersurfaces , 1998 .

[15]  Eiji Yanagida,et al.  A stability criterion for stationary curves to the curvature-driven motion with a triple junction , 2003 .

[16]  Peter W. Bates,et al.  Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening , 1990 .

[17]  R. Courant,et al.  Methods of Mathematical Physics, Vol. I , 1954 .

[18]  Harald Garcke,et al.  Exponential stability for a mirror–symmetric three phase boundary motion by surface diffusion , 2003 .

[19]  Eiji Yanagida,et al.  Stability of stationary interfaces in a generalized mean curvature flow , 1993 .

[20]  Peter Sternberg,et al.  Local minimisers of a three-phase partition problem with triple junctions , 1994, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[21]  Mullins-Sekerka motion of small droplets on a fixed boundary , 2000 .

[22]  W. Mullins.  Theory of Thermal Grooving , 1957 .

[23]  Dr. M. G. Worster.  Methods of Mathematical Physics , 1947, Nature.