On Serrin’s overdetermined problem and a conjecture of Berestycki, Caffarelli and Nirenberg
Kelei Wang, Juncheng Wei · Communications in Partial Differential Equations · 2019
This article concerns rigidity results to Serrin’s overdetermined problem in an epigraph{Δu+f(u)=0, in Ω={(x′,xn):xn>φ(x′)},u>0, in Ω,u=0, on ∂Ω,|∇u|=const., on ∂Ω.We prove that up to isometry the epigraph must be a half space and that the solution u must be one-dimensional, provided that one of the following assumptions are satisfied: either n = 2; or φ is globally Lipschitz, or n≤8 and ∂u∂xn>0 in Ω. In view of the counterexample constructed in del Pino, Pacard, Wei in dimensions n≥9 this result is optimal. This partially answers a conjecture of Berestycki, Caffarelli and Nirenberg.