Serrin's overdetermined problems on epigraphs

TL;DR

This paper proves that solutions to Serrin's overdetermined problem on epigraphs are necessarily affine half-spaces with one-dimensional solutions under certain conditions, extending classical symmetry results to broader settings.

Contribution

It establishes new rigidity results for overdetermined problems on epigraphs, including cases with unbounded solutions and less regular domains, partially answering a longstanding open question.

Findings

Epigraphs bounded from below must be affine half-spaces for solutions to exist.

Solutions are necessarily one-dimensional under specified conditions.

New monotonicity results are proved for cases where f(0) < 0.

Abstract

In this work we establish some rigidity results for Serrin's overdetermined problem \begin{equation*} \left\{ \begin{array}{cll} - \Delta u=f(u) & \text{in}& \Omega,\newline u > 0& \text{in} & \Omega,\newline u=0 & \text{on} & \partial \Omega,\newline \dfrac{\partial u}{\partial \eta} = \mathfrak{c} = const. & \text{on} & \partial \Omega, \end{array} \right. \end{equation*} when $\Omega \subset \mathbb{R}^N$ is an epigraph (not necessarily globally Lipschitz-continuous) and $u$ is a classical solution, possibly unbounded. In broad terms, our main results prove that $\Omega$ 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 \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$.