计算机辅助证明平面Berenstein猜想的反例
A computer-assisted counterexample to the planar Berenstein conjecture
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中文总结 AI 辅助
本文借助计算机辅助，将共形固定圆盘等方法适配到Dirichlet端点，构造出26阶二面体对称的非圆盘区域，给出平面Berenstein猜想的反例，否定了无附加限制的该猜想。
中文摘要 AI 辅助
Colbrook与Stepaniants的最新研究给出了平面Pompeiu猜想和Schiffer猜想的首批反例，并引入了共形固定圆盘、圆盘多项式及验证尾项方法。本文将该框架适配到互补Dirichlet端点，从而否定了无附加限制的平面Berenstein猜想。具体而言，我们构造了一个有界单连通区域Ω，其边界为实解析Jordan曲线，且非圆盘；存在k∈(27.4381178838,27.4381198839)及非零实值函数u∈C^ω(Ω的闭包)，满足Ω内的(Δ+k²)u=0，且在∂Ω上u=0、∂_ν u=常数≠0。因此，无附加u的符号假设时，超定Dirichlet-Neumann数据无法唯一刻画圆盘。该区域具有26阶二面体对称性，但既非圆盘也非中心对称，对应的本征函数会变号；等价地，其边界弧长测度满足对所有ω∈S¹，σ_{∂Ω}的傅里叶变换在kω处为0。经共形映射到单位圆盘后，精确支撑恒等式与定量圆盘多项式估计对无穷维尾项实现了严格控制。随后用Newton-Kantorovich论证将存在性问题简化为有限个显式不等式，再用区间算术验证。从Pompeiu-Schiffer问题的扩展并非形式化的：早期构造将两个边界条件纳入单个逆拉普拉斯方程；而本文考虑的Dirichlet端点处，非零Neumann数据要求保留调和源模式，形成耦合的内部-边界系统，涉及完整的零Dirichlet逆及其Neumann迹，还有单独的符号恢复问题。
英文摘要
Recent work of Colbrook and Stepaniants produced the first counterexamples to the planar Pompeiu and Schiffer conjectures and introduced the conformal fixed-disc, disk-polynomial, and validated-tail machinery used here. By adapting this framework to the complementary Dirichlet endpoint, we disprove the unrestricted planar Berenstein conjecture. Specifically, we construct a bounded simply connected domain $Ω$ with real-analytic Jordan boundary, which is not a disc and for which there exist $k\in(27.4381178838,27.4381198839)$ and a nonzero real-valued function $u\in C^ω(\overlineΩ)$ satisfying $(Δ+k^2)u=0$ in $Ω$, with $u=0, \partial_νu=\text{constant}\ne0$ on $\partialΩ$. Thus the overdetermined Dirichlet--Neumann data do not characterize the disc without an additional sign assumption on $u$. The domain has dihedral symmetry of order $26$, but is neither a disc nor centrally symmetric, and the corresponding eigenfunction changes sign. Equivalently, its boundary arclength measure satisfies $\widehat{σ_{\partialΩ}}(kω)=0$ for $ω\in\mathbb S^1$. After conformally transferring to the unit disc, exact support identities and quantitative disk-polynomial estimates yield rigorous control of the infinite-dimensional tail. A Newton--Kantorovich argument then reduces existence to finitely many explicit inequalities, which are certified using interval arithmetic. The extension from the Pompeiu--Schiffer problem is not formal. The earlier construction absorbs both boundary conditions into a single inverse-Laplacian equation. At the Dirichlet endpoint considered here, the nonzero Neumann datum forces the harmonic source modes to remain, producing a coupled interior--boundary system involving the full zero-Dirichlet inverse and its Neumann trace, together with a separate sign-recovery problem.
