PreprintCC BY 4.0arXiv
Nontrivial domains for the Bernoulli's overdetermined problem with small constant vorticity
Abstract
In this paper, we first apply the implicit function theorem to find nontrivial solutions to the following Bernoulli's overdetermined problem −Δu=γ in Ω∖D̅, u=1, ∂n u=Q on ∂ D, u=0 on ∂Ωnonumber for a fixed boundary ∂Ω and a free boundary ∂ D in ℝN with N≥2, where γ is a sufficiently small constant, the Bernoulli value Q is determined by the construction, and Ω∖D̅ is a bounded ring-shaped domain. In dimension N=n+1 with n≥2, we further study the existence of a nontrivial unbounded domain Ω∖D̅ in ℝN 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.
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BibTeX
@misc{zhang2026nontrivial,
title = {{Nontrivial domains for the Bernoulli's overdetermined problem with small constant vorticity}},
author = {Yong Zhang and Jialong Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14742},
eprint = {2610.14742},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14742},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nontrivial domains for the Bernoulli's overdetermined problem with small constant vorticity
AU - Yong Zhang
AU - Jialong Zhu
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14742
UR - https://arxiv.org/abs/2610.14742
AB - In this paper, we first apply the implicit function theorem to find nontrivial solutions to the following Bernoulli's overdetermined problem −Δu=γ in Ω∖D̅, u=1, ∂_n u=Q on ∂ D, u=0 on ∂Ωnonumber for a fixed boundary ∂Ω and a free boundary ∂ D in ℝ^N with N≥2, where γ is a sufficiently small constant, the Bernoulli value Q is determined by the construction, and Ω∖D̅ is a bounded ring-shaped domain. In dimension N=n+1 with n≥2, we further study the existence of a nontrivial unbounded domain Ω∖D̅ in 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.
ER -