Homogenization of nonlocal convolution-type operators: An approximation for the resolvent with corrector
Andrey Lvovich Piatnitski, Vladimir Anatolevich Sloushch, Tatiana Aleksandrovna Suslina, Elena Anatol'evna Zhizhina · EMS Press eBooks · 2025
In $L\_2(\mathbb{R}^d)$, we consider a selfadjoint bounded operator ${\mathbb A}\_\varepsilon$, $\varepsilon >0$, of the form $$ ({\mathbb A}\varepsilon u) (\mathbf{x}) = \varepsilon^{-d-2} \int{\mathbb{R}^d} a((\mathbf{x} - \mathbf{y})/\varepsilon) \mu(\mathbf{x}/\varepsilon, \mathbf{y}/\varepsilon) \bigl(u(\mathbf{x}) - u(\mathbf{y})\bigr), d\mathbf{y}. $$ It is assumed that $a(\mathbf{x})$ is a nonnegative function of class $L\_1(\mathbb{R}^d)$ such that $a(-\mathbf{x}) = a(\mathbf{x})$ and $\mu(\mathbf{x},\mathbf{y})$ is $\mathbb{Z}^d$-periodic in each variable and such that $\mu(\mathbf{x},\mathbf{y}) = \mu(\mathbf{y},\mathbf{x})$ and $0< \mu\_- \leqslant \mu(\mathbf{x},\mathbf{y}) \leqslant \mu\_+< \infty$. Moreover, it is assumed that the moments $M\_k (a)= \int\_{\mathbb{R}^d} \lvert \mathbf{x} \rvert^k a(\mathbf{x}),d\mathbf{x}$, $k=1,2,3,4$, are finite. We obtain an approximation of the resolvent $({\mathbb A}\_\varepsilon + I)^{-1}$ for small $\varepsilon$ in the operator norm on $L\_2(\mathbb{R}^d)$ with error of order $O(\varepsilon^2)$.