论文标题
关于卷积类型非本地运算符均质化的操作员估计
On operator estimates in homogenization of non-local operators of convolution type
论文作者
论文摘要
本文研究有限的对称操作员$ {\ MATHBF {a}} _ \ varepsilon $ in $ l_2（\ MathBf {r}^d）$ with $$（{\ Mathbf {\ MathBf {a}} _ \ varepsilon U） \ int _ {\ mathbf {r}^d} a（（x -y）/\ varepsilon）μ（x/\ varepsilon，y/\ varepsilon）\ left（u（x）-U（x）-U（x）-U（y） - u（y）\ right）\，dy; $$这里$ \ varepsilon $是一个小的正参数。假定$ a（x）$是一种非负$ l_1（\ mathbf {r}^d）$函数，使得$ a（-x）= a（x）$和homments $ m_k = \ int _ {\ mathbf {\ mathbf {r} {r}^d}^d}^d} | x | x | x | x | x | x）还假定$μ（x，y）$是$ \ mathbf {z}^d $ - periodic均在$ x $和$ y $函数中，以至于$μ（x，y）=μ（y，x）$和$ 0 <μ__--\leqμ（x，x，x，x，x，y）我们的目标是研究分解$（{\ Mathbf {a}} _ \ Varepsilon + i）^{ - 1} $的极限行为，为$ \ varepsilon \ to0 $。我们表明，作为$ \ varepsilon \至0 $，操作员$（{\ MathBf {a}} _ \ varepsilon + i）^{ - 1} $在$ l_2（\ Mathbf {r}^d）$中收敛于运算符的norm中，以resolvent $（resolvent $（resolvent） i）^{ - 1} $的有效运算符$ {\ mathbf {a}}^0 $是二阶椭圆形差分操作员，具有$ {\ mathbf {a}}^0 = - \ operatateNAME {div div} g^0 \ nabla $的恒定系数。然后，我们获得了收敛速率的顺序估计。
The paper studies a bounded symmetric operator ${\mathbf{A}}_\varepsilon$ in $L_2(\mathbf{R}^d)$ with $$ ({\mathbf{A}}_\varepsilon u) (x) = \varepsilon^{-d-2} \int_{\mathbf{R}^d} a((x-y)/\varepsilon) μ(x/\varepsilon, y/\varepsilon) \left( u(x) - u(y) \right)\,dy; $$ here $\varepsilon$ is a small positive parameter. It is assumed that $a(x)$ is a non-negative $L_1(\mathbf{R}^d)$ function such that $a(-x)=a(x)$ and the moments $M_k =\int_{\mathbf{R}^d} |x|^k a(x)\,dx$, $k=1,2,3$, are finite. It is also assumed that $μ(x,y)$ is $\mathbf{Z}^d$-periodic both in $x$ and $y$ function such that $μ(x,y) = μ(y,x)$ and $0< μ_- \leq μ(x,y) \leq μ_+< \infty$. Our goal is to study the limit behaviour of the resolvent $({\mathbf{A}}_\varepsilon + I)^{-1}$, as $\varepsilon\to0$. We show that, as $\varepsilon \to 0$, the operator $({\mathbf{A}}_\varepsilon + I)^{-1}$ converges in the operator norm in $L_2(\mathbf{R}^d)$ to the resolvent $({\mathbf{A}}^0 + I)^{-1}$ of the effective operator ${\mathbf{A}}^0$ being a second order elliptic differential operator with constant coefficients of the form ${\mathbf{A}}^0= - \operatorname{div} g^0 \nabla$. We then obtain sharp in order estimates of the rate of convergence.
