Observability of the heat equation from very small sets

A. Walton Green, Kévin Le Balc'h, Jérémy Martin, Marcu-Antone Orsoni
{"title":"Observability of the heat equation from very small sets","authors":"A. Walton Green, Kévin Le Balc'h, Jérémy Martin, Marcu-Antone Orsoni","doi":"arxiv-2407.20954","DOIUrl":null,"url":null,"abstract":"We consider the heat equation set on a bounded $C^1$ domain of $\\mathbb R^n$\nwith Dirichlet boundary conditions. The first purpose of this paper is to prove\nthat the heat equation is observable from any measurable set $\\omega$ with\npositive $(n-1+\\delta)$-Hausdorff content, for $\\delta >0$ arbitrary small. The\nproof relies on a new spectral estimate for linear combinations of Laplace\neigenfunctions, obtained via a Remez type inequality, and the use of the\nso-called Lebeau-Robbiano's method. Even if this observability result is sharp\nwith respect to the scale of Hausdorff dimension, our second goal is to\nconstruct families of sets $\\omega$ which have codimension greater than or\nequal to $1$ for which the heat equation remains observable.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"27 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.20954","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

We consider the heat equation set on a bounded $C^1$ domain of $\mathbb R^n$ with Dirichlet boundary conditions. The first purpose of this paper is to prove that the heat equation is observable from any measurable set $\omega$ with positive $(n-1+\delta)$-Hausdorff content, for $\delta >0$ arbitrary small. The proof relies on a new spectral estimate for linear combinations of Laplace eigenfunctions, obtained via a Remez type inequality, and the use of the so-called Lebeau-Robbiano's method. Even if this observability result is sharp with respect to the scale of Hausdorff dimension, our second goal is to construct families of sets $\omega$ which have codimension greater than or equal to $1$ for which the heat equation remains observable.
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
从极小集合观察热方程的可观察性
我们考虑的是热方程组在具有迪里希特边界条件的$\mathbb R^n$有界$C^1$域上的问题。本文的第一个目的是证明在任意小的 $\delta >0$ 条件下,热方程是可以从任何具有正 $(n-1+\delta)$-Hausdorff 内容的可测集合 $\omega$ 中观测到的。这一证明依赖于对拉普拉斯特征函数线性组合的一种新的谱估计,它是通过雷麦兹式不等式和所谓的勒博-罗比阿诺方法得到的。即使这个可观测性结果在豪斯多夫维度的尺度上是尖锐的,我们的第二个目标也是要构造出$\omega$集合的族,这些集合的codimension大于或等于$1$,对于这些集合,热方程仍然是可观测的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
自引率
0.00%
发文量
0
期刊最新文献
Holomorphic approximation by polynomials with exponents restricted to a convex cone The Denjoy-Wolff Theorem in simply connected domains Best approximations for the weighted combination of the Cauchy--Szegö kernel and its derivative in the mean $L^2$-vanishing theorem and a conjecture of Kollár Nevanlinna Theory on Complete Kähler Connected Sums With Non-parabolic Ends
×
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