I am a professor in pure mathematics, with interest in applied problems. I work in the field of partial differential equations and free boundary problems. 


Address: Department of Mathematics, KTH,

Lindstedtsvägen 25,  100 44 Stockholm


Phone: 46-8-7906754
Fax: 46-8-723 17 88
E-mail:
henriksh@kth.se                                       

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)