Dynamical Stability in Relation to Variational Stability: Double Bubbles in R²

This thesis is devoted to the investigation of the dynamical stability of standard planar double bubbles. By presenting connections between the dynamical stability and variational stability, we prove that standard planar double bubbles are dynamically stable under the surface diffusion flow. This investigation leads us to extend a practical tool the so-called generalized principle of linearized stability (GPLS) to a more general setting. More precisely, convergence to stationary solutions in fully nonlinear parabolic systems with general nonlinear boundary conditions is proved in situations where the set of stationary solutions creates a C2-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 a couple of other useful results on linear parabolic systems are also extended. In addition, as an application of our extended version of GPLS, we prove also that the lens-shaped networks generated by circular arcs are dynamically stable under the surface diffusion flow