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

Zusammenfassung (Englisch)

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 ...

Zusammenfassung (Englisch)

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.

Übersetzung der Zusammenfassung (Deutsch)

In dieser Dissertation wird die dynamische Stabilität von Standard-planaren Doppelblasen untersucht.
Durch Darstellung der Verbindung zwischen der dynamischen Stabilität und der variationellen Stabilität, beweisen wir, dass die Standard-planaren Doppelblasen unter Oberflächen-Diffusion dynamisch stabil sind.

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Zusammenfassung (Englisch)

Zusammenfassung (Englisch)

Übersetzung der Zusammenfassung (Deutsch)

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