# On the Surface Diffusion Flow with Triple Junctions in Higher Space Dimensions

## TL;DR

This paper proves short-term existence for the evolution of triple junction clusters driven by surface diffusion flow in higher dimensions, using boundary conditions from a Cahn-Hilliard model and a parabolic H"older approach.

## Contribution

It establishes the short-time existence of surface diffusion flow with triple junctions in higher dimensions using a novel boundary condition framework.

## Key findings

- Short time existence of solutions is proven.
- Boundary conditions are derived as a singular limit of a Cahn-Hilliard model.
- Analytic approach employs a parabolic H"older setting.

## Abstract

We show short time existence for 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 concurrency of the triple junction, angle conditions between the hypersurfaces, continuity of the chemical potentials and a flux-balance. For the existence analysis we first write the geometric problem over a fixed reference surface and then use for the resulting analytic problem an approach in a parabolic H\"older setting.