Elliptic operators in rough sets and the Dirichlet problem with boundary data in Hölder spaces

IF 1.7 2区 数学 Q1 MATHEMATICS Journal of Functional Analysis Pub Date : 2024-12-10 DOI:10.1016/j.jfa.2024.110801
Mingming Cao , Pablo Hidalgo-Palencia , José María Martell , Cruz Prisuelos-Arribas , Zihui Zhao
{"title":"Elliptic operators in rough sets and the Dirichlet problem with boundary data in Hölder spaces","authors":"Mingming Cao ,&nbsp;Pablo Hidalgo-Palencia ,&nbsp;José María Martell ,&nbsp;Cruz Prisuelos-Arribas ,&nbsp;Zihui Zhao","doi":"10.1016/j.jfa.2024.110801","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper we study the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in Hölder spaces. Our context is that of open sets <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, satisfying the capacity density condition, without any further topological assumptions. Our main result states that if Ω is either bounded, or unbounded with unbounded boundary, then the corresponding Dirichlet boundary value problem is well-posed; when Ω is unbounded with bounded boundary, we establish that solutions exist, but they fail to be unique in general. These results are optimal in the sense that solvability of the Dirichlet problem in Hölder spaces is shown to imply the capacity density condition.</div><div>As a consequence of the main result, we present a characterization of the Hölder spaces in terms of the boundary traces of solutions, and obtain well-posedness of several related Dirichlet boundary value problems.</div><div>All the results above are new even for 1-sided chord-arc domains, and can be extended to generalized Hölder spaces associated with a natural class of growth functions.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 5","pages":"Article 110801"},"PeriodicalIF":1.7000,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022123624004890","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

In this paper we study the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in Hölder spaces. Our context is that of open sets ΩRn+1, n2, satisfying the capacity density condition, without any further topological assumptions. Our main result states that if Ω is either bounded, or unbounded with unbounded boundary, then the corresponding Dirichlet boundary value problem is well-posed; when Ω is unbounded with bounded boundary, we establish that solutions exist, but they fail to be unique in general. These results are optimal in the sense that solvability of the Dirichlet problem in Hölder spaces is shown to imply the capacity density condition.
As a consequence of the main result, we present a characterization of the Hölder spaces in terms of the boundary traces of solutions, and obtain well-posedness of several related Dirichlet boundary value problems.
All the results above are new even for 1-sided chord-arc domains, and can be extended to generalized Hölder spaces associated with a natural class of growth functions.
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
求助全文
约1分钟内获得全文 去求助
来源期刊
CiteScore
3.20
自引率
5.90%
发文量
271
审稿时长
7.5 months
期刊介绍: The Journal of Functional Analysis presents original research papers in all scientific disciplines in which modern functional analysis plays a basic role. Articles by scientists in a variety of interdisciplinary areas are published. Research Areas Include: • Significant applications of functional analysis, including those to other areas of mathematics • New developments in functional analysis • Contributions to important problems in and challenges to functional analysis
期刊最新文献
Editorial Board Ricci-DeTurck flow from rough metrics and applications to scalar curvature problems A bridge connecting convex analysis and complex analysis and L2-estimate of d and ∂¯ A new characterization of the dissipation structure and the relaxation limit for the compressible Euler-Maxwell system Increasing resolution and instability for linear inverse scattering problems
×
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
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 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 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1