The left image is a Boundary Sandpile with initial distribution concentrated at points (34, 0), (-34,0), (0,34), (0,-34) on Z^2, all having mass 40 000, and the boundary capacity is set to 400. The clusters barely survive intersection by one lattice site. On the right, is a Boundary Sandpile having capacity 400, where mass 40 000 is concentrated on each of the sites (33, 0), (-33,0), (0,33), (0,-33).

The pictures indicate instability/jump of Boundary Sandpile process, with regards to the initial mass. Similar situations happen for Bernoulli type free boundary problems.

Paper available at: Discrete_Balayage_and_Boundary_Sandpile