Nonlinear diffusion with a bounded stationary level surface

TL;DR

This paper proves that in nonlinear diffusion processes with certain boundary conditions, the shape of the container must be spherical, extending classical symmetry results to various geometries and initial conditions.

Contribution

It establishes new symmetry results for nonlinear diffusion with bounded stationary level surfaces, including extensions to spheres and hyperbolic spaces.

Findings

Container boundary must be spherical under specified conditions

Results extend to heat flow in spheres and hyperbolic spaces

Symmetry results hold for various geometries and initial conditions

Abstract

We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class C^2. Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a proper C^2-subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the whole R^N of some substance whose density is initially a characteristic function of the complement of a domain with bounded C^2 boundary, and obtain similar results. These results are also extended to the heat flow in the sphere S^N and the hyperbolic space H^N.