维度三、四、六、八、十和十四中 Schiffer 与 Pompeiu 猜想的凸反例
Convex counterexamples to the Schiffer and Pompeiu conjectures in dimensions two to eighteen
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中文总结 AI 辅助
在多个维度构造凸非球域，其指示函数傅里叶变换在单位球面消失，否证 Schiffer 与 Pompeiu 猜想，方法结合共形映射、李代数约化与计算机辅助证明。
中文摘要 AI 辅助
在维度 $3,4,6,8,10,14$ 中，我们构造了具有实解析球面边界的、有界且凸的非球域。每个域都允许一个非平凡解 $\Delta u+u=0$，满足在边界上 $u=1$ 且 $\nabla u=0$。格林恒等式使得它们的指示函数傅里叶变换在单位球面上消失，从而在这些维度中即使在凸类内也否证了 Schiffer 和 Pompeiu 猜想。三维域是轴对称的，在后两个坐标上关于 $O(2)$ 不变，并关于第一个坐标反射不变，它是通过提升其子午截面的共形映射获得的。在维度 $n=2m+2$（$m=1,2,3,4,6$）中，这些域位于秩二的紧李代数 $\mathfrak{u}(2)$、$\mathfrak{so}(4)$、$\mathfrak{su}(3)$、$\mathfrak{so}(5)$ 和 $\mathfrak g_2$ 中。Harish-Chandra 的径向部分公式将这些情形归结为平面 Helmholtz 问题，而 Kostant 定理将凸性归结为 Cartan 截面。接近二进近似的精确解来自加权多项式系数空间中的计算机辅助压缩映射，对有限块使用区间界，对每个无穷尾部使用解析界。维度三中的第二个严格星形但非凸的例子作为独立结果保留。据我们所知，除了本工作的早期版本外，这些是维度至少为三的首个反例，也是任何维度中已知的首个凸反例。
英文摘要
We construct the first convex planar counterexample to the Schiffer and Pompeiu conjectures: a bounded strictly convex non-disc domain with real-analytic boundary carrying a nonconstant solution of $Δu+u=0$ with $u=1$ and $\partial_νu=0$ on the boundary. Together with the higher-dimensional constructions, this gives convex non-ball counterexamples in every dimension from two to eighteen, and in dimensions twenty and twenty-one, with real-analytic boundaries diffeomorphic to spheres and indicator Fourier transforms vanishing on the unit sphere. In dimensions $4$, $6$, $8$, $10$ and $14$, planar reductions through compact Lie algebras of rank two use Harish-Chandra's radial part formula and Kostant's convexity theorem. In dimensions $3$, $5$ and $7$, an axisymmetric formulation on the unit ball gives a quartic equation in weighted coefficient spaces. The exact two-sided inverse of the residual-corrected linearisation combines a Dirichlet Helmholtz inverse, a holomorphic boundary real-part problem and the material-derivative identity; it gives the planar Schiffer domain and, by one dimension-parametrised argument, the domains in dimensions $5$, $9$, $11$ to $13$, $15$ to $18$, $20$ and $21$. We also construct the first convex planar counterexample to Berenstein's conjecture: a strictly convex non-disc domain with real-analytic boundary carrying a sign-changing solution of $Δu+u=0$ with $u=0$ and $\partial_νu=1$ on the boundary. Planar existence for this problem is due to Colbrook, Sadeghi and Stepaniants. All existence proofs in this paper are computer-assisted.
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