{"title":"Approximate boundary controllability for parabolic equations with inverse square infinite potential wells","authors":"Arick Shao , Bruno Vergara","doi":"10.1016/j.na.2024.113624","DOIUrl":null,"url":null,"abstract":"<div><p>We consider heat operators on a bounded domain <span><math><mrow><mi>Ω</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>, with a critically singular potential diverging as the inverse square of the distance to <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span>. Although null boundary controllability for such operators was recently proved in all dimensions in Enciso et al. (2023) , it crucially assumed (i) <span><math><mi>Ω</mi></math></span> was convex, (ii) the control must be prescribed along all of <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span>, and (iii) the strength of the singular potential must be restricted to a particular subrange. In this article, we prove instead a definitive approximate boundary control result for these operators, in that we (i) do not assume convexity of <span><math><mi>Ω</mi></math></span>, (ii) allow for the control to be localized near any <span><math><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>∂</mi><mi>Ω</mi></mrow></math></span>, and (iii) treat the full range of strength parameters for the singular potential. Moreover, we lower the regularity required for <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span> and the lower-order coefficients. The key novelty is a local Carleman estimate near <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, with a carefully chosen weight that takes into account both the appropriate boundary conditions and the local geometry of <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span>.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0362546X24001433","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
We consider heat operators on a bounded domain , with a critically singular potential diverging as the inverse square of the distance to . Although null boundary controllability for such operators was recently proved in all dimensions in Enciso et al. (2023) , it crucially assumed (i) was convex, (ii) the control must be prescribed along all of , and (iii) the strength of the singular potential must be restricted to a particular subrange. In this article, we prove instead a definitive approximate boundary control result for these operators, in that we (i) do not assume convexity of , (ii) allow for the control to be localized near any , and (iii) treat the full range of strength parameters for the singular potential. Moreover, we lower the regularity required for and the lower-order coefficients. The key novelty is a local Carleman estimate near , with a carefully chosen weight that takes into account both the appropriate boundary conditions and the local geometry of .
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
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