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

This paper concerns rigidity results to Serrin's overdetermined problem in an epigraph $$\{\begin{aligned} &\Delta u+ f(u)=0,\ \ \ {in}\ \Omega=\{(x^\prime,x_n): x_n>\varphi (x^\prime)\},\\ &u>0,\ \ \ {in}\ \Omega,\\ &u=0,\ \ \ {on}\ \partial\Omega,\\ &|\nabla u|=const. {on} \partial\Omega. \end{aligned}.$$ 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 $\varphi$ is globally Lipschitz, or $n \leq 8$ and $\frac{\partial u}{\partial x_n} >0$ in $\Omega$. In view of the counterexample constructed in \cite{DPW} in dimensions $n\geq 9$ this result is optimal. This partially answers a conjecture of Berestycki, Caffarelli and Nirenberg \cite{BCN}.