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

Abstract

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

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