基于双尺度切割投影收敛的准晶泛函均匀化
Homogenization of quasi-crystalline functionals via two-scale-cut-and-project convergence
摘要：
我们考虑一个均匀化问题与准晶多重积分的形式\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*}，其中$u_\varepsilon$是常系数线性偏微分约束。的准晶体结构的基础复合材料编码的拉格朗日，$f_R$的第二个变量的依赖，并通过切割和投影方案，解释异质微观结构被均匀化作为一个非理性的高维空间的子空间建模。在我们的分析中的一个关键步骤是准晶体的两个尺度的限制的向量场$u_\varepsilon$的序列是在一个给定的常数系数的线性偏微分算子，$\mathcal{A}$，即$\mathcal{A} u _\varepsilon =0$的内核的特征。我们的结果将文献中关于${\rm \mathcal{A} =curl } $情形的相关结果推广到更一般的常系数微分算子$\mathcal{A}$，并且不需要对拉格朗日量$f_R$作线性假设.
英文摘要：
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 $f_R$.
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