AI 中文总结
该研究针对体域和薄域，利用覆盖技术证明了特定权重下的加权几何刚性估计与Korn第一不等式，得到了薄域相关常数的最优尺度结果。
AI 中文摘要
本工作研究体域和薄域中的加权几何刚性估计与Korn第一不等式。我们考虑的权重是到区域边界部分的距离函数的非负幂。对于薄域的情况，距离取自该区域边界的薄面，且相关常数依赖于区域厚度；当中面包含平坦区域时，这些常数被证明在厚度趋近于零时具有最优尺度。我们采用了Acosta、Cejas和Duran在文献[\ref{bib: this http URL.}]中使用的一些覆盖技术来证明加权Poincaré不等式，后续Conti和Zwicknagl在文献[\ref{bib: this http URL.}]中使用这些技术证明了Lipschitz域中的加权Poincaré不等式和经典几何刚性不等式。然而，由于距离仅取自边界的一部分，覆盖部分变得更为精细，尤其是对于薄域，相关分析并不直接。
英文摘要
This work is concerned with weighted Geometric Rigidity Estimates and Korn's first inequalities in bulk and thin domains. We consider weights, that are a nonnegative power of the distance function to part of the boundary of the domain. For the case of thin domains, the distance is taken to be from the thin face of the domain boundary, and the constants depend on the domain thickness and when the mid-surface contains a flat region, the constants are proven to have optimal scaling as the thickness approaches zero. We employed some covering techniques utilized by Acosta, Cejas, and Duran in [\ref{bib:Aco.Cej.Dur.}] to prove a weighted Poincaré inequality, and later by Conti and Zwicknagl in [\ref{bib:Con.Zwi.}] to prove weighted Poincaré and classical Geometric Rigidity inequalities in Lipschitz domains. However, because the distance is taken to be only from part of the boundary, the covering parts become more delicate, especially for thin domains and the analysis becomes non-straightforward.
Comments25 pages. Most of the results were announced in the PhD thesis of Rodrigues in October, 2023
