On the stability of the Serrin problem

Abstract

We investigate stability issues concerning the radial symmetry of solutions to Serrin's overdetermined problems. In particular, we show that, if u is a solution to Δ u = n in a smooth domain Ω ⊂ Rn, u = 0 on ∂Ω and | D u | is "close" to 1 on ∂Ω, then Ω is "close" to the union of a certain number of disjoint unitary balls.