  In this paper, we consider the capillarity-driven evolution of a solid
toroidal island on a flat rigid substrate, where mass transport is controlled
by surface diffusion. This problem is representative of the geometrical
complexity associated with the solid-state dewetting of thin films on
substrates. We apply Onsager's variational principle to develop a general
approach for describing surface diffusion-controlled problems. Based on this
approach, we derive a simple, reduced-order model and obtain an analytical
expression for the rate of island shrinking and validate this prediction by
numerical simulations based on a full, sharp-interface model. We find that the
rate of island shrinking is proportional to the material constants $B$ and the
surface energy density $\gamma_0$, and is inversely proportional to the island
volume $V_0$. This approach represents a general tool for modeling interface
diffusion-controlled morphology evolution.
