Solutions of elliptic equations with a level surface parallel to the boundary: Stability of the radial configuration

A positive solution of a homogeneous Dirichlet boundary value problem or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of its level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. We show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls \({B_{{r_e}}}\) and \({B_{{r_i}}}\), with the difference r_e-r_i (linearly) controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and elaborates arguments developed by Aftalion, Busca, and Reichel.