论文标题
通过两尺度切割和项目收敛的准晶体功能均质化
Homogenization of quasi-crystalline functionals via two-scale-cut-and-project convergence
论文作者
论文摘要
我们考虑与ferm \ begin {equination*} \ begin {Aligned} u_ \ varepsilon \ in l^p（ω; \ mathbb {r}^d）in l^p（ω; \ mathbb {r}^d） u_ \ varepsilon（x）\ big）\，dx，\ end {对齐} \ end {equation*}，其中$ u_ \ varepsilon $受恒定的线性偏差约束。基础复合材料的准晶体结构在对拉格朗日的第二个变量（$ f_r $）的依赖性中编码，并通过剪切和项目的方案进行建模，该方案将异质微结构解释为均质的高度较高空间的非均质子空间。我们分析的关键步骤是表征向量字段$ u_ \ u_ \ varepsilon $的序列序列的两尺度限制，它们位于给定常量的线性偏差偏差操作员的内核中，$ \ mathcal {a a} $，也就是说，$ \ nathcal calcal {a a} u}我们的结果提供了有关$ {\ rm \ Mathcal {a} = curl} $案例的文献中相关的结果的概括。
We consider a homogenization problem associated with quasi-crystalline multiple integrals of the form \begin{equation*} \begin{aligned} u_\varepsilon\in L^p(Ω;\mathbb{R}^d) \mapsto \int_Ωf_R\Big(x,\frac{x}{\varepsilon}, u_\varepsilon(x)\Big)\, dx, \end{aligned} \end{equation*} where $u_\varepsilon$ is subject to constant-coefficient linear partial differential constraints. The quasi-crystalline structure of the underlying composite is encoded in the dependence on the second variable of the Lagrangian, $f_R$, and is modeled via the cut-and-project scheme that interprets the heterogeneous microstructure to be homogenized as an irrational subspace of a higher-dimensional space. A key step in our analysis is the characterization of the quasi-crystalline two-scale limits of sequences of the vector fields $u_\varepsilon$ that are in the kernel of a given constant-coefficient linear partial differential operator, $\mathcal{A}$, that is, $\mathcal{A} u _\varepsilon =0$. Our results provide a generalization of related ones in the literature concerning the ${\rm \mathcal{A} =curl } $ case to more general differential operators $\mathcal{A}$ with constant coefficients, and without coercivity assumptions on the Lagrangian $f_R$.
