Homogenization of non-symmetric convolution type operators
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| Format: | Preprint |
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2025
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| _version_ | 1866910995208732672 |
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| author | Piatnitski, Andrey Sloushch, Vladimir Suslina, Tatiana Zhizhina, Elena |
| author_facet | Piatnitski, Andrey Sloushch, Vladimir Suslina, Tatiana Zhizhina, Elena |
| contents | The paper studies homogenization problem for a bounded in $L_2(\mathbb R^d)$ convolution type operator ${\mathbb A}_\eps$, $\eps >0$, of the form $$ ({\mathbb A}_\eps u) (\x) = \eps^{-d-2} \int_{\R^d} a((\x-\y)/\eps) μ(\x/\eps, \y/\eps) \left( u(\x) - u(\y) \right)\,d\y. $$ It is assumed that $a(\x)$ is a non-negative function from $L_1(\R^d)$, and $μ(\x,\y)$ is a periodic in $\x$ and $\y$ function such that $0< μ_- \leqslant μ(\x,\y) \leqslant μ_+< \infty$. No symmetry assumption on $a(\cdot)$ and $μ(\cdot)$ is imposed, so the operator ${\mathbb A}_\eps$ need not be self-adjoint. Under the assumption that the moments $M_k = \int_{\R^d} |\x|^k a(\x)\,d\x$, $k=1,2,3$, are finite we obtain, for small $\eps>0$, sharp in order approximation of the resolvent $({\mathbb A}_\eps + I)^{-1}$ in the operator norm in $L_2(\mathbb R^d)$, the discrepancy being of order $O(\eps)$. The approximation is given by an operator of the form $({\mathbb A}^0 + \eps^{-1} \langle \boldsymbolα,\nabla \rangle + I)^{-1}$ multiplied on the right by a periodic function $q_0(\x/\eps)$; here ${\mathbb A}^0 = - \operatorname{div}g^0 \nabla$ is the effective operator, and $\boldsymbolα$ is a constant vector. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_07176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homogenization of non-symmetric convolution type operators Piatnitski, Andrey Sloushch, Vladimir Suslina, Tatiana Zhizhina, Elena Functional Analysis Analysis of PDEs The paper studies homogenization problem for a bounded in $L_2(\mathbb R^d)$ convolution type operator ${\mathbb A}_\eps$, $\eps >0$, of the form $$ ({\mathbb A}_\eps u) (\x) = \eps^{-d-2} \int_{\R^d} a((\x-\y)/\eps) μ(\x/\eps, \y/\eps) \left( u(\x) - u(\y) \right)\,d\y. $$ It is assumed that $a(\x)$ is a non-negative function from $L_1(\R^d)$, and $μ(\x,\y)$ is a periodic in $\x$ and $\y$ function such that $0< μ_- \leqslant μ(\x,\y) \leqslant μ_+< \infty$. No symmetry assumption on $a(\cdot)$ and $μ(\cdot)$ is imposed, so the operator ${\mathbb A}_\eps$ need not be self-adjoint. Under the assumption that the moments $M_k = \int_{\R^d} |\x|^k a(\x)\,d\x$, $k=1,2,3$, are finite we obtain, for small $\eps>0$, sharp in order approximation of the resolvent $({\mathbb A}_\eps + I)^{-1}$ in the operator norm in $L_2(\mathbb R^d)$, the discrepancy being of order $O(\eps)$. The approximation is given by an operator of the form $({\mathbb A}^0 + \eps^{-1} \langle \boldsymbolα,\nabla \rangle + I)^{-1}$ multiplied on the right by a periodic function $q_0(\x/\eps)$; here ${\mathbb A}^0 = - \operatorname{div}g^0 \nabla$ is the effective operator, and $\boldsymbolα$ is a constant vector. |
| title | Homogenization of non-symmetric convolution type operators |
| topic | Functional Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2506.07176 |