On the annular overdetermined problems with linear and power nonlinearities in ℝ2
Abstract
In this paper, we study two classes of partially overdetermined elliptic problems on planar annular domains in ℝ2, with a fixed inner circle and a free outer boundary. The Dirichlet values are prescribed to be $1$ and $0$ on the inner and outer boundaries, respectively, while the squared normal derivative is constant on the outer boundary. For the linear law Δu=−μu, we establish local bifurcation under a Dirichlet nonresonance condition. For the power law Δu=u−up with p>1, we take the mean outer radius as the bifurcation parameter and construct large positive radial solutions on thin annuli. The limiting profile solves −V''=Vp on an interval, and the principal mixed Dirichlet--Neumann eigenvalue of its linearization determines a constant κp>0. For every fixed inner radius λ>0 and every sufficiently large integer K, we obtain a critical radius RK=λ+λκp/K+o(K−1) and a smooth local curve of positive solutions with noncircular, even, 2π/K-periodic outer boundaries. The proof combines shape derivatives, spectral estimates, and the Crandall--Rabinowitz bifurcation theorem.
Related
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BibTeX
@misc{wang2026annular,
title = {{On the annular overdetermined problems with linear and power nonlinearities in $\mathbbR^2$}},
author = {Jun Wang and Fei Xu and Yong Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24397},
eprint = {2610.24397},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24397},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the annular overdetermined problems with linear and power nonlinearities in ℝ^2
AU - Jun Wang
AU - Fei Xu
AU - Yong Zhang
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24397
UR - https://arxiv.org/abs/2610.24397
AB - In this paper, we study two classes of partially overdetermined elliptic problems on planar annular domains in ℝ^2, with a fixed inner circle and a free outer boundary. The Dirichlet values are prescribed to be $1$ and $0$ on the inner and outer boundaries, respectively, while the squared normal derivative is constant on the outer boundary. For the linear law Δu=−μu, we establish local bifurcation under a Dirichlet nonresonance condition. For the power law Δu=u−u^p with p>1, we take the mean outer radius as the bifurcation parameter and construct large positive radial solutions on thin annuli. The limiting profile solves −V''=V^p on an interval, and the principal mixed Dirichlet--Neumann eigenvalue of its linearization determines a constant κ_p>0. For every fixed inner radius λ>0 and every sufficiently large integer K, we obtain a critical radius R_K=λ+λκ_p/K+o(K^−1) and a smooth local curve of positive solutions with noncircular, even, 2π/K-periodic outer boundaries. The proof combines shape derivatives, spectral estimates, and the Crandall--Rabinowitz bifurcation theorem.
ER -