Representation of equilibrium solutions to the table problem of growing sandpiles

Abstract

In the dynamical theory of granular matter the so-called table problem consists instudying the evolution of a heap of matter poured continuously onto a bounded domain $\Omega \subset {\mathbb{R}}^2$. The mathematical description of the table problem, at an equilibrium configuration, can be reduced to a boundary value problem for a system of partial differential equations. The analysis of such a system, also connected with other mathematical models such as the Monge-Kantorovich problem, is the object of this paper. Our main result is an integral representation formula for the solution, in terms of the boundary curvature and of the normal distance to the cut locus of $\Omega$.