Nontrivial domains for the Bernoulli's overdetermined problem with small constant vorticity
Year: 2026
Type: preprint
Abstract: In this paper, we first apply the implicit function theorem to find nontrivial solutions to the following Bernoulli's overdetermined problem \begin{equation} -Δu=γ\,\, \text{in}\,\,Ω\setminus\overline{D}, \,\, u=1,\,\,\partial_\mathbf{n} u=Q\,\,\text{on}\,\,\partial D, \,\,u=0\,\,\text{on}\,\,\partialΩ\nonumber \end{equation} for a fixed boundary $\partialΩ$ and a free boundary $\partial D$ in $\mathbb{R}^{N}$ with $N\geq2$, where $γ$ is a sufficiently small constant, the Bernoulli value $Q$ is determined by the construction, and $Ω\setminus\overline{D}$ is a bounded ring-shaped domain. In dimension $N=n+1$ with $n\geq2$, we further study the existence of a nontrivial unbounded domain $Ω\setminus\overline{D}$ in $\mathbb{R}^N$ such that the Bernoulli's overdetermined problem admits a solution. This unbounded domain we construct is periodic in one variable and radial in the other variables by bifurcation analysis.
Source: arXiv (Cornell University)
Authors: Zhang, Yong, Zhu, Jialong
Institution: Jiangsu University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G6844602229
OpenAlex ID: W7221765247