Serrin's overdetermined problems on epigraphs

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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is an epigraph (not necessarily globally Lipschitz-continuous) and $u$ is a classical solution, possibly unbounded. In broad terms, our main results prove that $\mathrm{\Omega }$ 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 $N\ge 2$ or, alternatively, when $f$ is locally Lipschitz-continuous (with no restriction on the sign of $f(0)$) and $N\le 3$. 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 $N\ge 2$.