On the flow of non-axisymmetric perturbations of cylinders via surface diffusion

Abstract

We study the surface diffusion flow acting on a class of general (non--axisymmetric) perturbations of cylinders ${\mathcal{C}}_r$ in ${\mathrm{I}\mathrm{R}}^3$. Using tools from parabolic theory on uniformly regular manifolds, and maximal regularity, we establish existence and uniqueness of solutions to surface diffusion flow starting from (spatially--unbounded) surfaces defined over ${\mathcal{C}}_r$ via scalar height functions which are uniformly bounded away from the central cylindrical axis. Additionally, we show that ${\mathcal{C}}_r$ is normally stable with respect to $2\pi$--axially--periodic perturbations if the radius $r>1$,and unstable if $0<r<1$. Stability is also shown to hold in settings with axial Neumann boundary conditions.

20 pages, 1 figure, submitted for publication

References in corpus (3)

• Surface diffusion flow near spheres
• Anisotropic Function Spaces on Singular Manifolds
• Lifespan theorem for constrained surface diffusion flows

Cited by in corpus (2)

• Function Spaces on Uniformly Regular and Singular Riemannian Manifolds
• Coupled Surface Diffusion and Mean Curvature Motion: Axisymmetric Steady States with Two Grains and a Hole