Overdetermined problems for the rotationally invariant Poisson equation in model manifolds
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arXiv
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| Formato: | Preprint |
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2026
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| author | Greco, Antonio Lucia, Marcello Sicbaldi, Pieralberto |
| author_facet | Greco, Antonio Lucia, Marcello Sicbaldi, Pieralberto |
| contents | We present rigidity results for overdetermined problems associated to the rotationally invariant Poisson equation $-Δ_{g_\mathcal{M}} u = f(r)$ in a model manifold $\mathcal{M} = [0,S) \times_h \mathbb S^{N-1}$ with warping function $h$. The variable $r$ ranges in the interval $[0,S)$, whose endpoint $S$ is positive and possibly infinite. The first part of the paper deals with the problem \[
\begin{array}{ll}
-Δ_{g_\mathcal{M}} {u}=f(r) &\mbox{in $Ω$},
u=φ(r) &\mbox{on $\partial Ω$},
\frac{\partial u}{\partial ν} = κ(r) &\mbox{on $\partial Ω$},
\end{array} \] where $Ω\subset \mathcal{M}$ is a bounded domain containing the point $O \in \mathcal{M}$ corresponding to $r = 0$, $ν$ is the exterior unit normal vector on $\partial Ω$, and $f$, $φ$, $κ$ are three prescribed functions.
In the second part of the paper, we consider a similar overdetermined problem for the exterior Bernoulli problem in a domain $Ω\setminus \overline B_{R_0}(O)$, where $B_{R_0}(O)$ denotes the geodesic ball centered at $O$ with radius $R_0$, within the class of functions that vanish on $\partial B_{R_0}(O)$. In both cases, we give conditions on $f$, $φ$ and $κ$ implying that the solution $u$ is radial and $Ω$ is a geodesic ball centered at $O$. Our results apply in particular to the three space forms $\mathbb{R}^N$, $\mathbb{H}^N$ and $\mathbb{S}^N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18289 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Overdetermined problems for the rotationally invariant Poisson equation in model manifolds Greco, Antonio Lucia, Marcello Sicbaldi, Pieralberto Analysis of PDEs Differential Geometry We present rigidity results for overdetermined problems associated to the rotationally invariant Poisson equation $-Δ_{g_\mathcal{M}} u = f(r)$ in a model manifold $\mathcal{M} = [0,S) \times_h \mathbb S^{N-1}$ with warping function $h$. The variable $r$ ranges in the interval $[0,S)$, whose endpoint $S$ is positive and possibly infinite. The first part of the paper deals with the problem \[ \begin{array}{ll} -Δ_{g_\mathcal{M}} {u}=f(r) &\mbox{in $Ω$}, u=φ(r) &\mbox{on $\partial Ω$}, \frac{\partial u}{\partial ν} = κ(r) &\mbox{on $\partial Ω$}, \end{array} \] where $Ω\subset \mathcal{M}$ is a bounded domain containing the point $O \in \mathcal{M}$ corresponding to $r = 0$, $ν$ is the exterior unit normal vector on $\partial Ω$, and $f$, $φ$, $κ$ are three prescribed functions. In the second part of the paper, we consider a similar overdetermined problem for the exterior Bernoulli problem in a domain $Ω\setminus \overline B_{R_0}(O)$, where $B_{R_0}(O)$ denotes the geodesic ball centered at $O$ with radius $R_0$, within the class of functions that vanish on $\partial B_{R_0}(O)$. In both cases, we give conditions on $f$, $φ$ and $κ$ implying that the solution $u$ is radial and $Ω$ is a geodesic ball centered at $O$. Our results apply in particular to the three space forms $\mathbb{R}^N$, $\mathbb{H}^N$ and $\mathbb{S}^N$. |
| title | Overdetermined problems for the rotationally invariant Poisson equation in model manifolds |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2602.18289 |