AI 中文总结
该研究针对拉伸（变细）管上实阶$s>0$的受限Dirichlet拉普拉斯算子，分析其第一特征值的渐近行为，并推导缩放后算子的$G$-极限与对应二次型的$\u0393$-极限，涵盖分数阶与整数阶（多重调和算子）情形。
AI 中文摘要
我们研究$\boldsymbol{\rm R}^n\times\boldsymbol{\rm R}^k$的有界子集上（可能为分数阶的）Dirichlet拉普拉斯算子$\bigl(-\u0394_{n+k}\bigr)^s$（$s>0$）的谱性质与变分性质，这类子集在一个或多个方向上的尺度远大于其余方向。我们首先研究拉伸（或变细）管上第一特征值的渐近行为，特别涵盖整数$s\u22652$的情形，此时$\bigl(-\u0394_{n+k}\bigr)^s$退化为多重调和算子。随后我们计算了经缩放后的算子的$G$-极限，以及对应缩放二次型的$\u0393$-极限。
英文摘要
We study spectral and variational properties of the (possibly) fractional Dirichlet Laplacian $\left(-Δ_{n+k}\right)^s\!$, $s>0$, on bounded subsets of $\mathbb R^n\times\mathbb R^k$ whose extent in one or more directions becomes much larger than in the remaining ones. We first investigate the asymptotic behaviour of the first eigenvalue on stretching (or thinning) tubes, covering in particular the case of integers $s\geq 2$, for which $\left(-Δ_{n+k}\right)^s\!$ reduces to a polyharmonic operator. Then we compute the $G$-limit of the rescaled operators, and the $Γ$-limit of the corresponding rescaled quadratic forms.
