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 Cr in IR 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 Cr via scalar height functions which are uniformly bounded away from the central cylindrical axis. Additionally, we show that Cr is normally stable with respect to 2π-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.

On quasilinear parabolic equations and continuous maximal regularity

“…Applying our general results to initial data h 0 ∈ bc 1+α (C r ) we extend wellposedness from [12,Proposition 3.2] to surfaces with only one Hölder continuous derivative. Further, we extend [11, Proposition 2.2, 2.3] by restricting to functions h 0 ∈ bc 1+α symm (C r ) exhibiting azimuthal symmetry around the cylinder C r .…”

“…(b) First note that restricting the domains of (A, F 1 , F 2 ) to periodic little-Hölder spaces will maintain the conditions (H1)-(H2) and (4.5), all confirmed in Section 4.1. From the proof of [12,Theorem 4.3] we know that h * is normally stable when r > 1. The conclusion thus follows from Theorem 3.2.…”

“…Considering the setting of Σ = C r ⊂ R 3 , an infinite cylinder of radius r > 0, we work with height functions h(t) : C r → (−r, ∞) which produce so-called axiallydefinable surfaces Γ(h), as in [12]. We show that the resulting surface diffusion flow can be cast as a quasilinear parabolic evolution equation in the form (1.1) with E µ := bc 1+α (Σ) and E β := bc 3+α (Σ), from whence we have µ = 1/4 and β = 3/4 in this setting.…”

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

“…it is geodesically complete, of positive injectivity radius and all covariant derivatives of the curvature tensor are bounded. In particular, every compact manifold without boundary is uniformly regular and the manifolds considered in [20,21] are all uniformly regular.…”

“…(T3) Σ has a tubular neighborhood. (b) All of the manifolds considered in [20,21] are (URT)-hypersurfaces. In particular, the infinite cylinder with radius r > 0,…”