  We study the most common image and informal description of the optimal
transport problem for quadratic cost, also known as the second boundary value
problem for the Monge--Amp\`{e}re equation -- What is the most efficient way to
fill a hole with a given pile of sand? -- by proving regularity results for
optimal transports between degenerate densities. In particular, our work
contains an analysis of the setting in which holes and sandpiles are
represented by absolutely continuous measures concentrated on bounded convex
domains whose densities behave like nonnegative powers of the distance
functions to the boundaries of these domains.
