Non-linear Stability of Double Bubbles under Surface Diffusion

TL;DR

This paper studies the stability of double bubble configurations under surface diffusion flow, proving their energy minimizers are stable using a Lojasiewicz-Simon inequality despite complex boundary conditions.

Contribution

It establishes the nonlinear stability of double bubbles under surface diffusion with novel boundary conditions derived from a Cahn-Hilliard limit.

Findings

Energy minimizers are stable under surface diffusion.

The stability proof uses a Lojasiewicz-Simon gradient inequality.

Handles complex boundary conditions with non-local tangential parts.

Abstract

We consider the evolution of triple junction clusters driven by the surface diffusion flow. On the triple line we use the boundary conditions derived by Garcke and Novick-Cohen as the singular limit of a Cahn-Hilliard equation with degenerated mobility. These conditions are the concurrency of the triple junction, angle conditions between the hypersurfaces, continuity of the chemical potentials and a flux-balance. For this system we show stability of its energy minimizers, i.e., standard double bubbles.The main argument relies on a Lojasiewicz-Simon gradient inequality. The proof of it differs from others works due to the fully non-linear boundary conditions and problems with the (non-local) tangential part.