Granular matter, being soft matter (sand, foam, sludge etc.) as opposed to the classical states of solid, fluid, and gas, poses a variety of unconvential mathematical problems.

We consider slow deposition of granular matter under the influence of gravity, in the absence of wind forces, neglecting inertia. Following the approach of de Gennes and the BCRE model and guided by experiment, we design a model in the form of two hyperbolic partial differential equations for the bulk of material at rest, respecting the angle of repose, and a moving surface layer.

We then consider the problem of appropriate boundary conditions depending on the geometry, e.g., deposition of granular matter in silos, on flat surfaces, on flat surfaces with elevated boundaries, on arbitrary obstacles.

The correponding stationary problems lead to a variety of boundary values problems and obstacle problems for the eikonal equation. Physically meaningful solutions can be characterized in various ways, e.g., by viscosity approaches. Here variational principles are invoked, as well as Perron approaches and finally and most effectively methods based on transport paths.

We underline the striking correspondence between the Dirichlet problems for the Laplace/Poisson equation and the eikonal equation.

An outlook can be given to other time-dependent processes such as superposition of sand heaps and sand ripple formation.