计算机辅助证明正多边形在扭转刚度问题中的局部极大性
Computer-assisted local maximality of regular polygons for torsional rigidity
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- Faculty of Exact Sciences, Aurel Vlaicu University of Arad（阿拉德奥雷尔·弗拉伊库大学精确科学学院）
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中文总结 AI 辅助
本文通过有限元与区间算术，计算机辅助证明正多边形在扭转刚度除以面积平方的尺度不变泛函下，对5至25边均为严格局部极大值点。
中文摘要 AI 辅助
我们研究扭转刚度作为凸多边形标记顶点的函数。从分布的二阶形状导数出发，我们推导出关于顶点坐标的Hessian矩阵。在正多边形处，二面体对称性使得该矩阵在径向-切向坐标下成为分块循环矩阵，从而将其谱简化为二阶Hermitian矩阵的特征值。我们还推导了精确的二阶变分Galerkin恒等式以及有保证的泛函残差上界。有限元方法近似求解进入Hessian矩阵的偏微分方程解，FLINT/Arb提供认证所需的区间算术。在尺度不变设定下，我们精确认证了由相似性生成的四个零特征值以及$5\leq n\leq25$范围内的$2n-4$个严格负特征值。因此，该范围内的正多边形（模去相似性）是扭转刚度除以面积平方的严格局部极大值点。观测到的Hessian误差随网格尺寸近似二次方减小；证明该收敛速率所需的传递正则性作为猜想单独陈述。
英文摘要
We study torsional rigidity as a function of the labeled vertices of a convex polygon. Starting from the distributed second shape derivative, we derive the Hessian with respect to vertex coordinates. At a regular polygon, dihedral symmetry makes this matrix block circulant in radial-tangential coordinates, reducing its spectrum to the eigenvalues of Hermitian matrices of order two. We also derive an exact second-variation Galerkin identity and guaranteed functional residual majorants. Finite elements approximate the PDE solutions entering the Hessian, and FLINT/Arb provides the interval arithmetic needed for certification. In the scale-invariant setting, we certify exactly four zero eigenvalues generated by similarities and $2n-4$ strictly negative eigenvalues for $5\leq n\leq25$. The regular polygons in this range are therefore strict local maximizers, modulo similarities, of torsional rigidity divided by area squared. We prove broken regularity of the first and second material derivatives at each fixed regular polygon, and deduce an $O(h^2)$ error bound for the exact Galerkin Hessian and its eigenvalues on meshes fitted to the coarse fan.
