Interaction between degenerate diffusion and shape of domain

We consider the flow of a gas into a bounded tank Ω with smooth boundary∂ Ω. Initially Ω is empty, and at all times the density of the gas is kept constant on ∂ Ω. Choose a number R > 0 sufficiently small that, for any point x in Ω having distance R from ∂ Ω, the closed ball B with radius R centred at x intersects ∂ Ω at only one point. We show that if the gas content of such balls B is constant at each given time, then the tank Ω must be a ball. In order to prove this, we derive an asymptotic estimate for gas content for short times. Similar estimates are also derived in the case of the evolution p-Laplace equation for p ≥2.