内半径约束下的形状优化与蜂窝结构的涌现
Shape optimization with inradius constraint and emergence of honeycomb structures
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中文总结 AI 辅助
本文研究固定内半径下最大化平均扭转刚度的形状优化问题，证明小半径球补集的最优域渐近呈蜂窝结构，并推广至Cheeger常数等变分能量。
中文摘要 AI 辅助
我们在固定内半径且补集由两两不相交、半径为 $\varepsilon$ 的球组成且相互距离有下界的 $\mathbb{R}^2$ 开子集中，最大化平均扭转刚度。我们证明，当 $\varepsilon$ 足够小时，最优值由补集呈现规则蜂窝结构的域序列渐近达到。通过 Delaunay 三角剖分，证明归结为对顶点处有圆形切口三角形域上的混合 Dirichlet--Neumann 问题的分析。我们的方法相当通用，可推广到其他变分能量，如 Cheeger 常数，对此我们还推导出比值 $\varepsilon/$ 内半径的显式界。
英文摘要
We maximize the average torsional rigidity among open subsets of $\mathbb{R}^2$ with fixed inradius whose complements consist of pairwise disjoint balls of fixed radius $\varepsilon$ with mutual distances bounded below. We prove that, provided $\varepsilon$ is sufficiently small, the optimal value is asymptotically attained by sequences of domains whose complements exhibit a regular honeycomb structure. The proof is reduced, via Delaunay triangulations, to the analysis of a mixed Dirichlet--Neumann problem on triangular domains with circular cutouts at their vertices. Our approach is quite general and extends to other variational energies, such as the Cheeger constant, for which we also derive an explicit bound of the ratio $\varepsilon/$ inradius.
