加权Sobolev空间的一种新方法。

IF 0.8 4区 数学 Q2 MATHEMATICS
Monatshefte fur Mathematik Pub Date : 2025-01-01 Epub Date: 2025-01-09 DOI:10.1007/s00605-024-02044-z
Djameleddine Kebiche
{"title":"加权Sobolev空间的一种新方法。","authors":"Djameleddine Kebiche","doi":"10.1007/s00605-024-02044-z","DOIUrl":null,"url":null,"abstract":"<p><p>We present in this paper a new way to define weighted Sobolev spaces when the weight functions are arbitrary small. This new approach can replace the old one consisting in modifying the domain by removing the set of points where at least one of the weight functions is very small. The basic idea is to replace the distributional derivative with a new notion of weak derivative. In this way, non-locally integrable functions can be considered in these spaces. Indeed, assumptions under which a degenerate elliptic partial differential equation has a unique non-locally integrable solution are given. Tools like a Poincaré inequality and a trace operator are developed, and density results of smooth functions are established.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"206 4","pages":"893-920"},"PeriodicalIF":0.8000,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11909071/pdf/","citationCount":"0","resultStr":"{\"title\":\"A new approach to weighted Sobolev spaces.\",\"authors\":\"Djameleddine Kebiche\",\"doi\":\"10.1007/s00605-024-02044-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>We present in this paper a new way to define weighted Sobolev spaces when the weight functions are arbitrary small. This new approach can replace the old one consisting in modifying the domain by removing the set of points where at least one of the weight functions is very small. The basic idea is to replace the distributional derivative with a new notion of weak derivative. In this way, non-locally integrable functions can be considered in these spaces. Indeed, assumptions under which a degenerate elliptic partial differential equation has a unique non-locally integrable solution are given. Tools like a Poincaré inequality and a trace operator are developed, and density results of smooth functions are established.</p>\",\"PeriodicalId\":54737,\"journal\":{\"name\":\"Monatshefte fur Mathematik\",\"volume\":\"206 4\",\"pages\":\"893-920\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2025-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11909071/pdf/\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Monatshefte fur Mathematik\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00605-024-02044-z\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2025/1/9 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte fur Mathematik","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00605-024-02044-z","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/1/9 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

本文给出了一种定义权函数任意小时的加权Sobolev空间的新方法。这种新方法可以取代旧方法,即通过去除至少一个权函数非常小的点集来修改域。其基本思想是用弱导数的新概念代替分配导数。这样,在这些空间中就可以考虑非局部可积函数。实际上,给出了退化椭圆型偏微分方程具有唯一非局部可积解的假设。提出了庞卡罗不等式和迹算子等工具,建立了光滑函数的密度结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A new approach to weighted Sobolev spaces.

We present in this paper a new way to define weighted Sobolev spaces when the weight functions are arbitrary small. This new approach can replace the old one consisting in modifying the domain by removing the set of points where at least one of the weight functions is very small. The basic idea is to replace the distributional derivative with a new notion of weak derivative. In this way, non-locally integrable functions can be considered in these spaces. Indeed, assumptions under which a degenerate elliptic partial differential equation has a unique non-locally integrable solution are given. Tools like a Poincaré inequality and a trace operator are developed, and density results of smooth functions are established.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
1.60
自引率
11.10%
发文量
155
审稿时长
4-8 weeks
期刊介绍: The journal was founded in 1890 by G. v. Escherich and E. Weyr as "Monatshefte für Mathematik und Physik" and appeared with this title until 1944. Continued from 1948 on as "Monatshefte für Mathematik", its managing editors were L. Gegenbauer, F. Mertens, W. Wirtinger, H. Hahn, Ph. Furtwängler, J. Radon, K. Mayrhofer, N. Hofreiter, H. Reiter, K. Sigmund, J. Cigler. The journal is devoted to research in mathematics in its broadest sense. Over the years, it has attracted a remarkable cast of authors, ranging from G. Peano, and A. Tauber to P. Erdös and B. L. van der Waerden. The volumes of the Monatshefte contain historical achievements in analysis (L. Bieberbach, H. Hahn, E. Helly, R. Nevanlinna, J. Radon, F. Riesz, W. Wirtinger), topology (K. Menger, K. Kuratowski, L. Vietoris, K. Reidemeister), and number theory (F. Mertens, Ph. Furtwängler, E. Hlawka, E. Landau). It also published landmark contributions by physicists such as M. Planck and W. Heisenberg and by philosophers such as R. Carnap and F. Waismann. In particular, the journal played a seminal role in analyzing the foundations of mathematics (L. E. J. Brouwer, A. Tarski and K. Gödel). The journal publishes research papers of general interest in all areas of mathematics. Surveys of significant developments in the fields of pure and applied mathematics and mathematical physics may be occasionally included.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:604180095
Book学术官方微信