2019/10/02 by Harald Garcke, Garcke, Harald, Michael Gößwein +1 · 1 citation
Mathematics · #35K55 #35K93 #35R35 #53C44 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35K55 #msc:35K93 #msc:35R35 #msc:53C44
paper · pdf · doi:10.48550/arxiv.1910.01041
arxiv created 2019/10/07 · arxiv updated 2019/10/08
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.