Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities

Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global $\frac{1}{n+1}$-H\"older continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the H\"older exponent to $\frac12$ in the case of partitions with the same anisotropy and the same mobility and provide a weak comparision result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.