发表机构
University of Warwick; Università di Genova（华威大学; 热那亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将轻质结构最小柔度形状优化的松弛结果从各向同性二次能量密度推广到一般弹性张量及任意维度，完全解决了Bouchitté猜想。
AI 中文摘要
对于最小柔度形状优化问题，最近已有研究[《杜克数学杂志》172 (2023), 43--103]证明，当精确体积为\\(\e\\)的近似最优形状序列在\\(\e \todown 0\\)时收敛到一个极限广义形状，该极限形状由可能扩散的概率测度表示。该广义形状最小化所谓的轻质结构柔度泛函，正如Bouchitté所猜想的那样，该泛函用松弛被积函数表示，反映出最优形状必须仅涉及“低维”结构，如工程领域众所周知的Michell型桁架。然而，上述结果仅适用于最简单的各向同性二次能量密度，即Frobenius范数的平方，且仅适用于二维和三维情形。本研究将松弛结果推广到整个（各向同性或各向异性）弹性张量类别，并适用于任意维度，从而完全解决了Bouchitté猜想。由于先前的工作依赖于松弛被积函数的显式公式（归功于Allaire、Kohn和Strang），而这些公式在一般情况下不可用，因此需要若干新的论证和对原始方法的实质性修改。我们还显著简化和精简了原始论证的其他部分，并澄清了文献中的一些相关陈述。
英文摘要
For the minimum-compliance shape optimization problem it was established recently [\textit{Duke Math.\ J.} 172 (2023), 43--103] that a sequence of approximately optimal shapes of exact volume \(\e\) converges to a limiting generalized shape, which is represented by a possibly diffuse probability measure, as \(\e \todown 0\). This generalized shape minimizes the so-called light-structures compliance functional, which --- as conjectured by Bouchitté --- is expressed in terms of a relaxed integrand, reflecting that optimal shapes must only involve ``lower-dimensional'' structures, like the Michell-type trusses that are well known in the engineering domain. However, the aforementioned result only applies for the simplest isotropic quadratic energy density, that is, the squared Frobenius norm, in dimensions two and three. The present work extends the relaxation result to the entire class of (isotropic or anisotropic) elasticity tensors and to any dimension, thus solving Bouchitté's conjecture in full generality. Since the previous work relied on explicit formulas for the relaxed integrands (due to Allaire, Kohn, and Strang), which are not available in the general case, several new arguments and substantial modifications of the original approach are required. We also significantly simplify and streamline other parts of the original argument, in addition to clarifying some related statements in the literature.
Comments28 pages
