Henrik Shahgholian

Henrik Shahgholian

The first 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. The second image 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

The distribution of accumulated boundary sandpile.

It represents a (discrete) flow moving by the Laplacian of the Greens function on the boundary.

Blue sites have (comparatively) small mass, dark blues have the least amount of mass. Red carries largest amount (comparatively). So, the cooler the color, the smaller is the amount of mass (the range goes from 0=white---dark blue - light blue.... red)

Mini-conference in PDEs

December 11-13, KTH, Stockholm