Nonlinear stability of stationary solutions for curvature flow with triple junction

In this paper we analyze the motion of a network of three planar curves with a speed proportional to the curvature of the arcs, having perpendicular intersections with the outer boundary and a common intersection at a triple junction. As a main result we show that a linear stability criterion due to Ikota and Yanagida is also sufficient for nonlinear stability. We also prove local and global existence of classical smooth solutions as well as various energy estimates. Finally, we prove exponential stabilization of an evolving network starting from the vicinity of a linearly stable stationary network.Comment: submitte

On the existence for the Helfrich flow and its center manifold near spheres

The Helfrich variational problem is the minimizing problem of the bending energy among the closed surface with the prescribed area and enclosed volume. This is one of models for shape transformation theory of human red blood cell. Here the associated gradient flow, called the {\em Helfrich flow}, is studied. The existence of this geometric flow is proved locally for arbitrary initial data, and globally near spheres. Furthermore its center manifold near spheres is investigated

Mean curvature flow with triple junctions in higher space dimensions

We consider mean curvature flow of n-dimensional surface clusters. At (n-1)-dimensional triple junctions an angle condition is required which in the symmetric case reduces to the well-known 120 degree angle condition. Using a novel parametrization of evolving surface clusters and a new existence and regularity approach for parabolic equations on surface clusters we show local well-posedness by a contraction argument in parabolic Hoelder spaces.Comment: 31 pages, 2 figure

On the criteria for the stability of unduloids (Regularity and Singularity for Partial Differential Equations with Conservation Laws)

"Regularity and Singularity for Partial Differential Equations with Conservation Laws". June 3～5, 2015. edited by Keiichi Kato, Mishio Kawashita, Masashi Misawa and Takayoshi Ogawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.The stability of steady states for the surface diffusion equation will be studied. In the axisymmetric setting, steady states of surface diffusion equation are the Delaunay surfaces, which are the axisymmetric constant mean curvature surfaces. Unduloid is one of the Delaunay surfaces. In this paper, We consider a linearized stability of unduloids and describe the criteria for the stability of them

Linearized stability analysis of stationary solutions for surface diffusion with boundary conditions

The linearized stability of stationary solutions to the surface diffusion flow with angle and no-flux boundary conditions is studied. In this linearized stability analysis the $H^{-1}$ gradient flow structure plays a key role. As a byproduct the 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. Some examples of linearized stability are studied as well

Surface diffusion with triple junctions: A stability criterion for stationary solutions

We study a fourth order geometric evolution problem on a network of curves in a bounded domain . The flow decreases a weighted total length of the curves and preserves the enclosed volumes. Stationary solutions of the flow are critical points of a partition problem in . In this paper we study the linearized stability of stationary solutions using the H−1-gradient flow structure of the problem. Important issues are the development of an appropriate PDE formulation of the geometric problem and Poincar´e type estimate on a network of curves

Stability analysis of phase boundary motion by surface diffusion with triple junction

The linearized stability of stationary solutions for the surface diffusion flow with a triple junction is studied. We derive the second variation of the energy functional under the constraint that the enclosed areas are preserved and show a linearized stability criterion with the help of the H−1-gradient flow structure of the evolution problem and the analysis of eigenvalues of a corresponding differential operator