论文标题
用二维形状的二维形状的拉普拉斯特征值计算二面对称性
Computation of Laplacian eigenvalues of two-dimensional shapes with dihedral symmetry
论文作者
论文摘要
我们从数值上计算以任意精度算术的二二维形状的几种二维形状的Laplacian特征值。我们的方法基于具有域分解的特定解决方案的方法。我们对特征值$λ（n）$形状的渐近扩展特别感兴趣，其$ n $边缘为$λ（n）\ sim x \ sum x \ sum_ {k = 0}^{\ infty}^{\ infty} \ frac \ frac {c_k（x） \ infty $。以前，这种形式的膨胀仅以具有差异性边界条件的常规多边形而闻名，并且（令人惊讶的是）涉及Riemann Zeta值和单值多个Zeta值，这使得它们很有趣。我们为高阶$ C_K（x）$的闭合形式表达式提供了数值证据，并提供了更多的形状示例，这些形状可能是可能的（包括具有Neumann边界条件的常规多边形，常规的星形多边形和具有正弦边界的星形）。
We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We are particularly interested in asymptotic expansions of the eigenvalues $λ(n)$ of shapes with $n$ edges that are of the form $λ(n) \sim x\sum_{k=0}^{\infty} \frac{C_k(x)}{n^k}$ where $x$ is the limiting eigenvalue for $n\rightarrow \infty$. Expansions of this form have previously only been known for regular polygons with Dirichlet boundary condition and (quite surprisingly) involve Riemann zeta values and single-valued multiple zeta values, which makes them interesting to study. We provide numerical evidence for closed-form expressions of higher order $C_k(x)$ and give more examples of shapes for which such expansions are possible (including regular polygons with Neumann boundary condition, regular star polygons and star shapes with sinusoidal boundary).
