{"title":"非光滑情况下分数阶Dirichlet实现的Weyl渐近性","authors":"Gerd Grubb","doi":"10.7146/math.scand.a-138002","DOIUrl":null,"url":null,"abstract":"Let $P$ be a symmetric $2a$-order classical strongly elliptic pseudodifferential operator with \\emph{even} symbol $p(x,\\xi)$ on $\\mathbb{R}^n $ ($0<a<1$), for example a perturbation of $(-\\Delta )^a$. Let $\\Omega \\subset \\mathbb{R}^n$ be bounded, and let $P_D$ be the Dirichlet realization in $L_2(\\Omega)$ defined under the exterior condition $u=0$ in $\\mathbb{R}^n\\setminus\\Omega$. When $p(x,\\xi)$ and $\\Omega$ are $C^\\infty $, it is known that the eigenvalues $\\lambda_j$ (ordered in a nondecreasing sequence for $j\\to \\infty$) satisfy a Weyl asymptotic formula \\begin{equation*} \\lambda _j(P_{D})=C(P,\\Omega )j^{2a/n}+o(j^{2a/n}) \\text {for $j\\to \\infty $}, \\end{equation*} with $C(P,\\Omega)$ determined from the principal symbol of $P$. We now show that this result is valid for more general operators with a possibly nonsmooth $x$-dependence, over Lipschitz domains, and that it extends to $\\tilde P=P+P'+P”$, where $P'$ is an operator of order $<\\min\\{2a, a+\\frac 12\\}$ with certain mapping properties, and $P”$ is bounded in $L_2(\\Omega )$ (e.g. $P”=V(x)\\in L_\\infty(\\Omega)$). Also the regularity of eigenfunctions of $P_D$ is discussed.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases\",\"authors\":\"Gerd Grubb\",\"doi\":\"10.7146/math.scand.a-138002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $P$ be a symmetric $2a$-order classical strongly elliptic pseudodifferential operator with \\\\emph{even} symbol $p(x,\\\\xi)$ on $\\\\mathbb{R}^n $ ($0<a<1$), for example a perturbation of $(-\\\\Delta )^a$. Let $\\\\Omega \\\\subset \\\\mathbb{R}^n$ be bounded, and let $P_D$ be the Dirichlet realization in $L_2(\\\\Omega)$ defined under the exterior condition $u=0$ in $\\\\mathbb{R}^n\\\\setminus\\\\Omega$. When $p(x,\\\\xi)$ and $\\\\Omega$ are $C^\\\\infty $, it is known that the eigenvalues $\\\\lambda_j$ (ordered in a nondecreasing sequence for $j\\\\to \\\\infty$) satisfy a Weyl asymptotic formula \\\\begin{equation*} \\\\lambda _j(P_{D})=C(P,\\\\Omega )j^{2a/n}+o(j^{2a/n}) \\\\text {for $j\\\\to \\\\infty $}, \\\\end{equation*} with $C(P,\\\\Omega)$ determined from the principal symbol of $P$. We now show that this result is valid for more general operators with a possibly nonsmooth $x$-dependence, over Lipschitz domains, and that it extends to $\\\\tilde P=P+P'+P”$, where $P'$ is an operator of order $<\\\\min\\\\{2a, a+\\\\frac 12\\\\}$ with certain mapping properties, and $P”$ is bounded in $L_2(\\\\Omega )$ (e.g. $P”=V(x)\\\\in L_\\\\infty(\\\\Omega)$). Also the regularity of eigenfunctions of $P_D$ is discussed.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-10-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.7146/math.scand.a-138002\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7146/math.scand.a-138002","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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