SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 1785-1817, January 2021.
We consider a homogenization problem associated with quasi-crystalline multiple integrals of the form $u_\varepsilon\in L^p(\Omega;\mathbbm{R}^d) \mapsto \int_\Omega f_R(x,\frac{x}{\varepsilon}, u_\varepsilon(x)), dx,$ where \(u_ǎrepsilon) 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_ǎrepsilon) that are in the kernel of a given constant-coefficient linear partial differential operator, (\mathcalA), that is, (\mathcalA u _ǎrepsilon =0). Our results provide a generalization of related ones in the literature concerning the (\mathcalA =curl ) case to more general differential operators (\mathcalA) with constant coefficients and without coercivity assumptions on the Lagrangian (f_R).

中文翻译：

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通过两步剪切和投影收敛实现准晶体功能的均质化

SIAM数学分析杂志，第53卷，第2期，第1785-1817页，2021年1月。
我们考虑与形式为$ u_ \ varepsilon \ in L ^ p（\ Omega; \ mathbbm {R} ^ d）\ mapsto \ int_ \ Omega f_R（x，\ frac {x } {\ varepsilon}，u_ \ varepsilon（x）），dx，$其中\（u_ǎrepsilon）受常数系数线性偏微分约束的约束。底层复合材料的准晶体结构根据拉格朗日第二变量（f_R）进行编码，并通过“切割并投影”方案建模，该方案将要均质化的异质微观结构解释为非均质的子空间。高维空间。我们分析中的关键步骤是表征矢量场（u_ǎrepsilon）序列的准晶体两尺度极限，这些极限在给定的恒定系数线性偏微分算子的核中，（\ mathcalA），即（\ mathcalA u_ǎrepsilon= 0）。我们的结果提供了有关（\ mathcalA = curl）情况的文献中的相关归纳为具有常数系数且没有拉格朗日（f_R）矫顽假设的更一般的微分算子（\ mathcalA）的概括。