dorsal/arxiv
View SchemaHomogenization of L\'evy-type operators: operator estimates with correctors
| Authors | Andrey Piatnitski, Vladimir Sloushch, Tatiana Suslina, Elena Zhizhina |
|---|---|
| Categories | |
| ArXiv ID | 2601.06832vv2 |
| URL | https://arxiv.org/abs/2601.06832 |
| License | http://creativecommons.org/licenses/by/4.0/ |
Abstract
The goal of the paper is to study in $L_2(\R^d)$ a self-adjoint operator ${\mathbb A}_\eps$, $\eps >0$, of the form $$ ({\mathbb A}_\eps u) (\x) = \int_{\R^d} \mu(\x/\eps, \y/\eps) \frac{\left( u(\x) - u(\y) \right)}{|\x - \y|^{d+\alpha}}\,d\y $$ with $1< \alpha < 2$; here the function $\mu(\x,\y)$ is $\Z^d$-periodic in the both variables, satisfies the symmetry relation $\mu(\x,\y) = \mu(\y,\x)$ and the estimates $0< \mu_- \leqslant \mu(\x,\y) \leqslant \mu_+< \infty$. The rigorous definition of the operator ${\mathbb A}_\eps$ is given in terms of the corresponding quadratic form. In the previous work of the authors it was shown that the resolvent $({\mathbb A}_\eps + I)^{-1}$ converges, as $\eps\to0$, in the operator norm in $L_2(\mathbb R^d)$ to the resolvent of the effective operator $A^0$, and the estimate $\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} \| = O(\eps^{2-\alpha})$ holds. In the present work we achieve a more accurate approximation of the resolvent of ${\mathbb A}_\eps$ which takes into account the correctors. Namely, for $N\in\mathbb N$ such that $2-1/N < \alpha \le 2-1/(N+1)$, we obtain $$ \bigl\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} - \sum_{m=1}^N \eps^{m(2-\alpha)} \mathbb{K}_m \bigr\| = O(\eps). $$
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"abstract": "The goal of the paper is to study in $L_2(\\R^d)$ a self-adjoint operator ${\\mathbb A}_\\eps$, $\\eps \u003e0$, of the form $$ ({\\mathbb A}_\\eps u) (\\x) = \\int_{\\R^d} \\mu(\\x/\\eps, \\y/\\eps) \\frac{\\left( u(\\x) - u(\\y) \\right)}{|\\x - \\y|^{d+\\alpha}}\\,d\\y $$ with $1\u003c \\alpha \u003c 2$;\n here the function\n $\\mu(\\x,\\y)$ is $\\Z^d$-periodic in the both variables, satisfies the symmetry relation $\\mu(\\x,\\y) = \\mu(\\y,\\x)$ and\n the estimates $0\u003c \\mu_- \\leqslant \\mu(\\x,\\y) \\leqslant \\mu_+\u003c \\infty$. The rigorous definition of the operator ${\\mathbb A}_\\eps$ is given in terms of the corresponding quadratic form. In the previous work of the authors it was shown that the resolvent $({\\mathbb A}_\\eps + I)^{-1}$ converges, as $\\eps\\to0$, in the operator norm in $L_2(\\mathbb R^d)$ to the resolvent of the effective operator $A^0$, and the estimate $\\|({\\mathbb A}_\\eps + I)^{-1} - (\\A^0 + I)^{-1} \\| = O(\\eps^{2-\\alpha})$ holds. In the present work we achieve a more accurate approximation of the resolvent of ${\\mathbb A}_\\eps$ which takes into account the correctors. Namely, for $N\\in\\mathbb N$ such that $2-1/N \u003c \\alpha \\le 2-1/(N+1)$, we obtain $$ \\bigl\\|({\\mathbb A}_\\eps + I)^{-1} - (\\A^0 + I)^{-1} - \\sum_{m=1}^N \\eps^{m(2-\\alpha)} \\mathbb{K}_m \\bigr\\| = O(\\eps). $$",
"arxiv_id": "2601.06832",
"authors": [
"Andrey Piatnitski",
"Vladimir Sloushch",
"Tatiana Suslina",
"Elena Zhizhina"
],
"categories": [
"math.AP",
"math.FA"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Homogenization of L\\\u0027evy-type operators: operator estimates with correctors",
"url": "https://arxiv.org/abs/2601.06832",
"version": "v2"
},
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