Let Ω be a C^1,s bounded domain (s > 1/2) in ℝ^d, and let \({{\cal A}^\varepsilon } = - {\rm{div}}\;A(x,x/\varepsilon )\nabla \) be a matrix elliptic operator on Ω with Dirichlet boundary condition. We suppose that ε is small and the function A is Lipschitz in the first variable and periodic in the second one, so the coefficients of \({{\cal A}^\varepsilon }\) are locally periodic. For μ in the resolvent set, we are interested in finding the rates of approximations, as ε → 0, for \({({{\cal A}^\varepsilon } - \mu {\rho ^\varepsilon })^{ - 1}}\) and \(\nabla {({{\cal A}^\varepsilon } - \mu {\rho ^\varepsilon })^{ - 1}}\) in the operator topology on L₂. Here ρ^ε(x)= ρ(x,x/ε) is a positive definite locally periodic function with ρ satisfying the same assumptions as A. Keeping track of the rate dependence on both ε and μ, we then proceed to similar questions for the solution to the initial boundary-value problem \({\rho ^\varepsilon }{\partial _t}{v_\varepsilon } = - {{\cal A}^\varepsilon }{v_\varepsilon }\).

中文翻译：

---

关于局部周期椭圆和抛物线算子的均质化

让Ω是Ç ^1，š有界域（小号> 1/2）中ℝ ^d，并让\（{{\ CAL A} ^ \ varepsilon} = - {\ RM {DIV}} \; A（X， x / \ varepsilon）\ nabla \）是具有Dirichlet边界条件的Ω上的矩阵椭圆算子。我们假设ε很小，函数A在第一个变量中是Lipschitz，在第二个变量中是周期性的，因此\（{{\ cal A} ^ \ varepsilon} \）的系数是局部周期性的。对于μ在解集，我们感兴趣的是发现近似的速率，ε →交通0，对于\（{（{{\ CAL A} ^ \ varepsilon} - \亩{\ RHO ^ \ varepsilon}）^ { -1}} \）和\（\ nabla {（{{\ cal A} ^ \ varepsilon}-\ mu {\ rho ^ \ varepsilon}} ^ {-1}} \）在L ₂上的运算符拓扑中。这里ρ ^ε（X）= ρ（X，X / ε）是正定局部周期性的函数ρ满足相同的假设甲。跟踪速率对ε和μ的依赖性，然后我们针对初始边界值问题\（{\ rho ^ \ varepsilon} {\ partial _t} {v_ \ varepsilon} =-{ {\ cal A} ^ \ varepsilon} {v_ \ varepsilon} \）。