arXiv 2006.058562020-06math.AP

区域上局部周期椭圆问题的均匀化

Homogenization for locally periodic elliptic problems on a domain

Nikita N. Senik

摘要：

设$Ω$是$\mathbb R^d$中的Lipschitz域，$\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$是$Ω$上的强椭圆算子.我们假设$\varepsilon$是小的，并且函数$A$在第一个变量上是Lipschitz的，在第二个变量上是周期的，所以$\mathcal A^\varepsilon$的系数是局部周期的，并且是快速振荡的。给定$μ$的预解集，我们有兴趣在$L_p$上的算子拓扑中找到$(\mathcal A^\varepsilon-μ)^{-1}$和$\nabla(\mathcal A^\varepsilon-μ)^{-1}$的近似率，如$\varepsilon\to0$。众所周知，速率依赖于有效算子$\mathcal A^0$的正则性。证明了如果$(\mathcal A^0-μ)^{-1}$及其伴随是从$L_p(Ω)^n$到Lipschitz-Besov空间$Λ_p^{1+s}(Ω)^n$且$s\in(0,1]$有界的，则速率分别为$\varepsilon^s$和$\varepsilon^{s/p}$ .将所得结果应用于具有一致有界和$\operatorname{VMO}$系数的强椭圆算子的Dirichlet、Neumann和混合Dirichlet-Neumann问题.

英文摘要：

Let $Ω$ be a Lipschitz domain in $\mathbb R^d$, and let $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$ be a strongly elliptic operator on $Ω$. We suppose that $\varepsilon$ is small and the function $A$ is Lipschitz in the first variable and periodic in the second, so the coefficients of $\mathcal A^\varepsilon$ are locally periodic and rapidly oscillate. Given $μ$ in the resolvent set, we are interested in finding the rates of approximations, as $\varepsilon\to0$, for $(\mathcal A^\varepsilon-μ)^{-1}$ and $\nabla(\mathcal A^\varepsilon-μ)^{-1}$ in the operator topology on $L_p$ for suitable $p$. It is well-known that the rates depend on regularity of the effective operator $\mathcal A^0$. We prove that if $(\mathcal A^0-μ)^{-1}$ and its adjoint are bounded from $L_p(Ω)^n$ to the Lipschitz--Besov space $Λ_p^{1+s}(Ω)^n$ with $s\in(0,1]$, then the rates are, respectively, $\varepsilon^s$ and $\varepsilon^{s/p}$. The results are applied to the Dirichlet, Neumann and mixed Dirichlet--Neumann problems for strongly elliptic operators with uniformly bounded and $\operatorname{VMO}$ coefficients.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文