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

TL;DR

This paper proves that under certain conditions, solutions to Serrin's overdetermined problem in an epigraph are one-dimensional and the domain is a half-space, partially confirming a conjecture by Berestycki, Caffarelli, and Nirenberg.

Contribution

The paper establishes rigidity results for Serrin's overdetermined problem in epigraphs, showing the domain must be a half-space and solutions are one-dimensional under specific assumptions.

Findings

Epigraphs must be half-spaces if conditions are met.

Solutions are one-dimensional in the specified settings.

Results are optimal given known counterexamples in higher dimensions.

Abstract

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}.