Interaction between degenerate diffusion and shape of domain

We consider the flow of a gas into a bounded tank with smooth boundary; initially, the tank is empty and, at all times, the gas concentration is kept constant on the boundary. The situation is modeled as an initial-boundary value problem for a nonlinear diffusion equation having the property of finite speed of propagation of disturbances from rest (e.g. the porous medium equation). Suppose that, at each fixed time, any ball with fixed radius, contained in the tank and touching its boundary at only one point contains the same amount of gas (thus, that amount only depends on the radius and time fixed). Then the tank must be a ball. The result is based on a short-time asymptotic formula for the gas content of one of such balls: by using the presence of a boundary-layer for short times, it is proved that the first term of such an asymptotic formula depends on the principal curvatures of the tank's boundary at the point touched by the ball. The symmetry result is then a consequence of a version of A.D. Aleksandrov's soap bubble theorem for Weingarten surfaces. The symmetry result is then extended to the case of the evolution p-Laplace equation with p>2.