AI 中文总结
该研究构造了一个十重对称共形映射生成的非圆形区域，作为平面Pompeiu问题与Schiffer猜想的计算机辅助反例，通过算子方程的后验收缩证明了其为对应方程的精确零点。
AI 中文摘要
平面Pompeiu问题起源于1929年，相关的Schiffer猜想是长期存在的刚性问题，将刚体运动积分变换、傅里叶零点集与超定Neumann特征值问题联系起来。我们构造了一个有界单连通非圆形区域Ω⊂ℝ²，其具有实解析Jordan边界，还构造了一个非常数函数u，使得在Ω内满足(Δ+k²)u=0，在∂Ω上满足u=1、∂_νu=0，其中k∈(31.967007261,31.967007293)。因此u是一个在边界上为常数的Neumann特征函数，Ω是Schiffer猜想的反例。格林恒等式还给出了对于所有ω∈𝕊¹，𝟙_Ω的傅里叶变换在kω处的值为0，故Ω不满足Pompeiu性质，也是有界单连通Lipschitz域的平面Pompeiu猜想的反例。我们将该区域构造为Ω=φ(𝔻)，其中φ是一个十重对称的共形映射，接近一个明确列出的301次多项式。在单位圆盘上，该解析问题转化为实系数空间上的三次算子方程F(g,p)=g+|p|²(1+Kg)=0，其中K以圆盘多项式基表示，是与零Dirichlet和Neumann迹兼容的范围内拉普拉斯算子的显式逆，且p=kφ'。圆盘多项式线性化系数的正性、K的精确界以及无限尾项的单调控制，在加权系数代数中建立了接近所列多项式的后验收缩，从而得到F的一个精确零点。
英文摘要
The planar Pompeiu problem, originating in 1929, and the associated Schiffer conjecture are long-standing rigidity questions linking rigid-motion integral transforms and Fourier zero sets to overdetermined Neumann eigenvalue problems. We construct a bounded simply connected noncircular domain $Ω\subset\mathbb{R}^2$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $(Δ+k^2)u=0$ in $Ω$, $u=1,\partial_νu=0$ on $\partialΩ$ for some $k\in(31.967007261,31.967007293)$. Thus $u$ is a Neumann eigenfunction which is constant on the boundary, and $Ω$ is a counterexample to Schiffer's conjecture. Green's identity also gives $\widehat{\mathbf 1_Ω}(kω)=0$ $(ω\in\mathbb S^1)$, so $Ω$ fails the Pompeiu property and is also a counterexample to the planar Pompeiu conjecture for bounded simply connected Lipschitz domains. We obtain the domain as $Ω=ϕ(\mathbb{D})$, where $ϕ$ is a ten-fold symmetric conformal map close to an explicitly listed polynomial of degree $301$. On the unit disc, the analytic problem becomes a cubic operator equation on real coefficient spaces, $F(g,p)=g+|p|^2(1+Kg)=0$, where $K$, expressed in a disk-polynomial basis, is an explicit inverse of the Laplacian on the range compatible with zero Dirichlet and Neumann traces, and $p=kϕ'$. Positivity of the disk-polynomial linearisation coefficients, sharp bounds for $K$, and monotone control of the infinite tails establish an a posteriori contraction near the listed polynomial in a weighted coefficient algebra, and hence an exact zero of $F$.
