Serrin's overdetermined problems on epigraphs
Nicolas Beuvin, Alberto Farina
TL;DR
The paper addresses rigidity of Serrin's overdetermined problem for $-\Delta u = f(u)$ in epigraphs $\Omega = \{ x_N > g(x')\}$, with $u>0$, $u=0$ on $\partial\Omega$, and $\partial u/\partial \eta = c$. It develops a unified geometric/monotonicity framework to prove that, under broad conditions (including Allen-Cahn-type nonlinearities and locally Lipschitz $f$ with various sign assumptions at $0$), the only possible domains are affine half-spaces and the solutions are one-dimensional, i.e., $u(x) = u_0(x_N)$. A key contribution is a new monotonicity result valid in any dimension for $f(0) < 0$, from which BCNe-type rigidity is derived for $N \le 3$; this combines moving-planes arguments on epigraphs with blow-up/compactness analyses and energy estimates. The results extend and complement prior work (BCNe, FVarma, Wang-Wei) by handling unbounded solutions and broader epigraph classes, thereby advancing understanding of rigidity phenomena in nonlinear Poisson equations on unbounded domains.
Abstract
In this work we establish some rigidity results for Serrin's overdetermined problem \begin{equation*} \left\{ \begin{array}{cll} - Δu=f(u) & \text{in}& Ω,\newline u > 0& \text{in} & Ω,\newline u=0 & \text{on} & \partial Ω,\newline \dfrac{\partial u}{\partial η} = \mathfrak{c} = const. & \text{on} & \partial Ω, \end{array} \right. \end{equation*} when $Ω\subset \mathbb{R}^N$ is an epigraph (not necessarily globally Lipschitz-continuous) and $u$ is a classical solution, possibly unbounded. In broad terms, our main results prove that $Ω$ must be an affine half-space and $u$ must be one-dimensional, provided the epigraph is bounded from below. These results hold when $f$ is of Allen-Cahn type and $ N \geq 2$ or, alternatively, when $f$ is locally Lipschitz-continuous (with no restriction on the sign of $f(0)$) and $ N \leq 3$. These results partially answer a question raised by Berestycki, Caffarelli and Nirenberg in [1]. Finally, when $f(0) <0$, we also prove a new monotonicity result, valid in any dimension $ N \geq 2$.