AI 中文总结
研究在穿孔闭流形上斯捷克洛夫特征值及相关函数向加权拉普拉斯 - 贝尔特拉米对应物的收敛速率，推导其高阶展开，确定二维和三维的校正尺度，通过辅助函数估计量化差异，弥合谱几何与点系统文献差距。
AI 中文摘要
我们建立了在由许多小测地球穿孔的闭流形上，斯捷克洛夫特征值和调和扩展特征函数向其加权拉普拉斯 - 贝尔特拉米对应物的最优收敛速率。孔的半径相对于它们的间距具有临界缩放，即每个孔的边界面积与其沃罗诺伊单元的加权体积平衡。然后我们推导了斯捷克洛夫特征值的高阶展开；在二维和三维中，我们确定了两个校正尺度。该展开由边界和体测度之间差异的不定库仑型能量控制，由极限算子的约化格林函数介导。证明依赖于对某些辅助函数的精确估计，我们引入这些函数以量化孔上的表面测度与其背景密度之间的差异测度。我们的论文旨在弥合谱几何中新兴文献与通过库仑型能量相互作用的点系统文献之间的差距。
英文摘要
We establish optimal convergence rates for Steklov eigenvalues and harmonically extended eigenfunctions toward their weighted Laplace--Beltrami counterparts on a closed manifold perforated by many small geodesic balls. The holes have radii that scale critically with respect to their spacing in the sense that the boundary area of each hole balances with the weighted volume of its Voronoi cell. We then derive a higher-order expansion of the Steklov eigenvalues; in dimensions two and three, we identify two correction scales. The expansion is governed by an indefinite Coulomb-type energy of the discrepancy between the boundary and bulk measures, mediated by the reduced Green function of the limiting operator. The proof relies on sharp estimates for certain auxiliary functions that we introduce in order to quantify the discrepancy measure between the surface measure on the holes and their background density. Our paper serves to bridge an emerging literature in spectral geometry with one on systems of points interacting via Coulomb-type energies.
