论文标题
操作员估计Neumann Sieve问题
Operator estimates for Neumann sieve problem
论文作者
论文摘要
令$ω$为$ \ m athbb {r}^n $，$γ$中的一个域，是一种超平面相交的$ω$，$ \ varepsilon> 0 $是一个小参数，$ d_ {k，\ varepsilon} $，$ k = 1,2,2,3 \ dots $ $当$ \ varepsilon \至0 $时，孔的数量趋于无穷大，而其直径往往为零。令$ \ mathscr {a} _ \ varepsilon $为穿孔域中的neumann laplacian $ω__\ varepsilon =ω\setminusγ_\ varepsilon $，其中$γ_\ varepsilon = varepsilon = f varepsilon =γ\ setminus $ cup__ $ cup__ \ cup_k d _ （“筛子”）。众所周知，如果精心选择了孔的大小，$ \ mathscr {a} _ \ varepsilon $在$ω\setMinusγ$上以强烈的分辨率与laplacian收敛，则可能会受到所谓的$δ'$ - $γ$的条件。在目前的工作中，我们改善了这一结果：在相当普遍的假设下，我们根据$ l^2 \至l^2 $和$ l^2 \ to H^1 $运算符规范的孔的形状和位置的孔子和位置估算了估算。在后一种情况下，需要一个特殊的纠正器。
Let $Ω$ be a domain in $\mathbb{R}^n$, $Γ$ be a hyperplane intersecting $Ω$, $\varepsilon>0$ be a small parameter, and $D_{k,\varepsilon}$, $k=1,2,3\dots$ be a family of small "holes" in $Γ\capΩ$; when $\varepsilon \to 0$, the number of holes tends to infinity, while their diameters tends to zero. Let $\mathscr{A}_\varepsilon$ be the Neumann Laplacian in the perforated domain $Ω_\varepsilon=Ω\setminusΓ_\varepsilon$, where $Γ_\varepsilon=Γ\setminus (\cup_k D_{k,\varepsilon})$ ("sieve"). It is well-known that if the sizes of holes are carefully chosen, $\mathscr{A}_\varepsilon$ converges in the strong resolvent sense to the Laplacian on $Ω\setminusΓ$ subject to the so-called $δ'$-conditions on $Γ$. In the current work we improve this result: under rather general assumptions on the shapes and locations of the holes we derive estimates on the rate of convergence in terms of $L^2\to L^2$ and $L^2\to H^1$ operator norms; in the latter case a special corrector is required.
