{"title":"分数阶拉普拉斯算子及其他相关算子的Helmholtz解","authors":"Vincent Guan, M. Murugan, Juncheng Wei","doi":"10.1142/S021919972250016X","DOIUrl":null,"url":null,"abstract":"We show that the bounded solutions to the fractional Helmholtz equation, $(-\\Delta)^s u= u$ for $0<s<1$ in $\\mathbb{R}^n$, are given by the bounded solutions to the classical Helmholtz equation $(-\\Delta)u= u$ in $\\mathbb{R}^n$ for $n \\ge 2$ when $u$ is additionally assumed to be vanishing at $\\infty$. When $n=1$, we show that the bounded fractional Helmholtz solutions are again given by the classical solutions $A\\cos{x} + B\\sin{x}$. We show that this classification of fractional Helmholtz solutions extends for $1<s \\le 2$ and $s\\in \\mathbb{N}$ when $u \\in C^\\infty(\\mathbb{R}^n)$. Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation $\\psi(-\\Delta) u= \\psi(1)u$ in $\\mathbb{R}^n$, when $\\psi$ is complete Bernstein and certain regularity conditions are imposed on the associated weight $a(t)$.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"Helmholtz Solutions for the Fractional Laplacian and Other Related Operators\",\"authors\":\"Vincent Guan, M. Murugan, Juncheng Wei\",\"doi\":\"10.1142/S021919972250016X\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that the bounded solutions to the fractional Helmholtz equation, $(-\\\\Delta)^s u= u$ for $0<s<1$ in $\\\\mathbb{R}^n$, are given by the bounded solutions to the classical Helmholtz equation $(-\\\\Delta)u= u$ in $\\\\mathbb{R}^n$ for $n \\\\ge 2$ when $u$ is additionally assumed to be vanishing at $\\\\infty$. When $n=1$, we show that the bounded fractional Helmholtz solutions are again given by the classical solutions $A\\\\cos{x} + B\\\\sin{x}$. We show that this classification of fractional Helmholtz solutions extends for $1<s \\\\le 2$ and $s\\\\in \\\\mathbb{N}$ when $u \\\\in C^\\\\infty(\\\\mathbb{R}^n)$. Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation $\\\\psi(-\\\\Delta) u= \\\\psi(1)u$ in $\\\\mathbb{R}^n$, when $\\\\psi$ is complete Bernstein and certain regularity conditions are imposed on the associated weight $a(t)$.\",\"PeriodicalId\":50660,\"journal\":{\"name\":\"Communications in Contemporary Mathematics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2022-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Contemporary Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1142/S021919972250016X\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Contemporary Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/S021919972250016X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
期刊介绍：
With traditional boundaries between various specialized fields of mathematics becoming less and less visible, Communications in Contemporary Mathematics (CCM) presents the forefront of research in the fields of: Algebra, Analysis, Applied Mathematics, Dynamical Systems, Geometry, Mathematical Physics, Number Theory, Partial Differential Equations and Topology, among others. It provides a forum to stimulate interactions between different areas. Both original research papers and expository articles will be published.
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