关于多面体域的瑞利 - 法伯 - 克拉恩不等式的数值研究
A Numerical Investigation of the Rayleigh$-$Faber$-$Krahn Inequality for Polyhedral Domains
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中文总结 AI 辅助
研究在规定面数的三维凸多面体中使拉普拉斯第一狄利克雷特征值最小化问题，通过支撑超平面参数化及拉格朗日框架求解，结果得出知名多面体或非规则最优形状，展现多种对称性。
中文摘要 AI 辅助
我们给出了在具有规定面数的三维凸多面体中，使拉普拉斯算子的第一狄利克雷特征值最小化问题的数值结果。此问题可视为经典瑞利 - 法伯 - 克拉恩不等式的多面体版本。通过用支撑超平面参数化多面体，该问题首先被重新表述为一个等效的有限维约束最小化问题。采用拉格朗日框架进行数值求解。在每个顶点恰好与\(d\)个面相邻的假设下，拉格朗日的灵敏度分析将\(d\)维多面体的面的有限维扰动与特征值和体积的无限维形状导数相结合。根据面数，数值结果产生了诸如柏拉图立体、棱柱和十四面体等知名多面体，或揭示了具有多种对称性的非规则最优形状。
英文摘要
We present numerical results for the problem of minimizing the first Dirichlet eigenvalue of the Laplacian among three-dimensional convex polyhedra with a prescribed number of facets. This problem can be viewed as a polyhedral version of the classical Rayleigh$-$Faber$-$Krahn inequality. Using a parameterization of polytopes by supporting hyperplanes, the problem is first reformulated as an equivalent finite-dimensional constrained minimization problem. A Lagrangian framework is employed for the numerical solution. Under the assumption that each vertex is incident to exactly $d$ facets, the sensitivity analysis of the Lagrangian combines finite-dimensional perturbations of the facets of $d$-dimensional polyhedra with infinite-dimensional shape derivatives of both the eigenvalue and the volume. Depending on the number of facets, the numerical results yield well-known polyhedra, such as Platonic solids, prisms, and the tetrakaidecahedron, or reveal nonregular optimal shapes exhibiting several symmetries.
