Nasrin Arab | Searcharxiv

arXiv2015

Standard Planar Double Bubbles are Stable under Surface Diffusion Flow

Although standard planar double bubbles are stable in the sense that the second variation of the perimeter functional is non-negative for all area-preserving perturbations the question arises whether they are dynamically stable. By presenting connections between these two concepts of stability for double bubbles, we prove that standard planar double bubbles are stable under the surface diffusion flow via the generalized principle of linearized stability in parabolic Hölder spaces.

math.AP↗

arXiv2014

On Convergence of Solutions to Equilibria for Fully Nonlinear Parabolic Systems with Nonlinear Boundary Conditions

Convergence to stationary solutions in fully nonlinear parabolic systems with general nonlinear boundary conditions is shown in situations where the set of stationary solutions creates a $C^2$-manifold of finite dimension which is normally stable. We apply the parabolic Hölder setting which allows to deal with nonlocal terms including highest order point evaluation. In this direction some theorems concerning the linearized systems is also extended. As an application of our main result we prove that the lens-shaped networks generated by circular arcs are stable under the surface diffusion flow.

math.AP↗