Axially symmetric volume constrained anisotropic mean curvature flow

We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume, we show that the flow exists for all time. Numerical simulations of the curvature flow are presented.