Abstract <p> In <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(L_2(\mathbb R^d)\)</EquationSource> </InlineEquation>, we consider a selfadjoint operator <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb A}_\varepsilon\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon >0\)</EquationSource> </InlineEquation>, of the form <Equation ID="Equi"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_Equi.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="306" /> </MediaObject> <EquationSource Format="TEX">\(({\mathbb A}_\varepsilon u) (\mathbf{x}) =\int_{\mathbb R^d} \mu\biggl(\frac{\mathbf{x}}{\varepsilon},\frac{\mathbf{y}}{\varepsilon}\biggr) \frac{(u(\mathbf{x}) -u(\mathbf{y}))}{| \mathbf{x}-\mathbf{y} |^{d+\alpha}}\,d \mathbf{y},\)</EquationSource> </Equation> where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(0< \alpha < 2\)</EquationSource> </InlineEquation>. It is assumed that a function <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu(\mathbf{x},\mathbf{y})\)</EquationSource> </InlineEquation> is bounded, positive definite, periodic in each variable, and is such that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="126" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu(\mathbf{x},\mathbf{y})=\mu(\mathbf{y},\mathbf{x})\)</EquationSource> </InlineEquation>. A rigorous definition of the operator <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb A}_\varepsilon\)</EquationSource> </InlineEquation> is given in terms of the corresponding quadratic form. It is proved that the resolvent <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(({\mathbb A}_\varepsilon+I)^{-1}\)</EquationSource> </InlineEquation> converges in the operator norm on <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(L_2(\mathbb R^d)\)</EquationSource> </InlineEquation> to the operator <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq10.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(({\mathbb A}^0+I)^{-1}\)</EquationSource> </InlineEquation> as <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon\to 0\)</EquationSource> </InlineEquation>. Here, <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq12.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb A}^0\)</EquationSource> </InlineEquation> is an effective operator of the same form with the constant coefficient <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq13.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu^0\)</EquationSource> </InlineEquation> equal to the mean value of <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu(\mathbf{x},\mathbf{y})\)</EquationSource> </InlineEquation>. We obtain an error estimate of order <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq15.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(\varepsilon^\alpha)\)</EquationSource> </InlineEquation> for <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq16.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(0< \alpha < 1\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq17.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="118" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(\varepsilon (1+| \operatorname{ln} \varepsilon|)^2)\)</EquationSource> </InlineEquation> for <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq18.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\( \alpha=1\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq19"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq19.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(\varepsilon^{2- \alpha})\)</EquationSource> </InlineEquation> for <InlineEquation ID="IEq20"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq20.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(1< \alpha < 2\)</EquationSource> </InlineEquation>. In the case where <InlineEquation ID="IEq21"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1180_Article_IEq20.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(1< \alpha < 2\)</EquationSource> </InlineEquation>, the result is refined by taking the correctors into account. </p>

Homogenization of the Lévy-type Operators

期刊论文
发表日期：

2025年9月1日

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Elena Zhizhina,
Andrey Piatnitski,
Vladimir Sloushch,
Tatiana Suslina

摘要

Abstract

In \(L_2(\mathbb R^d)\) , we consider a selfadjoint operator \({\mathbb A}_\varepsilon\) , \(\varepsilon >0\) , of the form \(({\mathbb A}_\varepsilon u) (\mathbf{x}) =\int_{\mathbb R^d} \mu\biggl(\frac{\mathbf{x}}{\varepsilon},\frac{\mathbf{y}}{\varepsilon}\biggr) \frac{(u(\mathbf{x}) -u(\mathbf{y}))}{| \mathbf{x}-\mathbf{y} |^{d+\alpha}}\,d \mathbf{y},\) where \(0< \alpha < 2\) . It is assumed that a function \(\mu(\mathbf{x},\mathbf{y})\) is bounded, positive definite, periodic in each variable, and is such that \(\mu(\mathbf{x},\mathbf{y})=\mu(\mathbf{y},\mathbf{x})\) . A rigorous definition of the operator \({\mathbb A}_\varepsilon\) is given in terms of the corresponding quadratic form. It is proved that the resolvent \(({\mathbb A}_\varepsilon+I)^{-1}\) converges in the operator norm on \(L_2(\mathbb R^d)\) to the operator \(({\mathbb A}^0+I)^{-1}\) as \(\varepsilon\to 0\) . Here, \({\mathbb A}^0\) is an effective operator of the same form with the constant coefficient \(\mu^0\) equal to the mean value of \(\mu(\mathbf{x},\mathbf{y})\) . We obtain an error estimate of order \(O(\varepsilon^\alpha)\) for \(0< \alpha < 1\) , \(O(\varepsilon (1+| \operatorname{ln} \varepsilon|)^2)\) for \( \alpha=1\) , and \(O(\varepsilon^{2- \alpha})\) for \(1< \alpha < 2\) . In the case where \(1< \alpha < 2\) , the result is refined by taking the correctors into account.