论文标题
在域上局部周期性椭圆问题的均质化
Homogenization for locally periodic elliptic problems on a domain
论文作者
论文摘要
令$ω$为$ \ mathbb r^d $中的Lipschitz域，让$ \ Mathcal a^\ varepsilon = - \ operatatorName {div} a（x，x，x，x/\ varepsilon）\ nabla $是$ω$上的椭圆形操作员。我们假设$ \ varepsilon $很小，函数$ a $是第一个变量中的Lipschitz，在第二个变量中是周期性的，因此$ \ Mathcal a^\ Varepsilon $的系数是本地周期性的，并且快速振荡。给定的$μ$在解决方案集合中，我们有兴趣找到近似值，例如$ \ varepsilon \ to0 $，对于$（\ Mathcal a^\ varepsilon-μ）^{ - 1} $ and $ \ nabla（\ nabla a^\ varepsil-y^\ varepsil-l _ $ 1} $ 1} $ 1} $ 1} $ 1} $ P $。众所周知，费率取决于有效运营商$ \ Mathcal a^0 $的规律性。我们证明，如果$（\ nathcal a^0-μ）^{ - 1} $且其伴随从$ l_p（ω）^n $到Lipschitz-besov space $ s $λ_p^{1+s}（ω） $ \ varepsilon^{s/p} $。
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.
