On Serrin's overdetermined problem and a conjecture of Berestycki, Caffarelli and Nirenberg

الأصل · EN

This paper concerns rigidity results to Serrin's overdetermined problem in an epigraph {aligned &Δu+ f(u)=0,inΩ={(x′,xₙ): xₙ>φ(x′)}, &u>0,inΩ, &u=0,on∂Ω, &|∇ u|=const. on ∂Ω. aligned. We prove that up to isometry the epigraph must be an 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 DPW in dimensions n≥ 9 this result is optimal. This partially answers a conjecture of Berestycki, Caffarelli and Nirenberg BCN.