Serrin's overdetermined problems on epigraphs
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910818585542656 |
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| author | Beuvin, Nicolas Farina, Alberto |
| author_facet | Beuvin, Nicolas Farina, Alberto |
| contents | 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$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_04812 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Serrin's overdetermined problems on epigraphs Beuvin, Nicolas Farina, Alberto Analysis of PDEs 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$. |
| title | Serrin's overdetermined problems on epigraphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.04812 |