On the annular overdetermined problems with linear and power nonlinearities in $\mathbb{R}^2$

In this paper, we study two classes of partially overdetermined elliptic problems on planar annular domains in $\mathbb{R}^2$, 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.