Interaction between degenerate diffusion and shape of domain

Abstract

We consider the flow of a gas into a bounded tank Ω with smooth boundary ∂Q. 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 Q 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.