{"title":"Unique continuation for the heat operator with potentials in weak spaces","authors":"Eunhee Jeong, Sanghyuk Lee, Jaehyeon Ryu","doi":"10.2140/apde.2024.17.2257","DOIUrl":null,"url":null,"abstract":"<p>We prove the strong unique continuation property for the differential inequality </p>\n<div><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n<mo>|</mo><mo stretchy=\"false\">(</mo><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub>\n<mo>+</mo> <mi mathvariant=\"normal\">Δ</mi><mo stretchy=\"false\">)</mo><mi>u</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo>|</mo><mo>≤</mo>\n<mi>V</mi>\n<mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo>|</mo><mi>u</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo>|</mo><mo>,</mo>\n</math>\n</div>\n<p> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi> </math> contained in weak spaces. In particular, we establish the strong unique continuation property for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi>\n<mo>∈</mo> <msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>∞</mi></mrow></msubsup><msubsup><mrow><mi>L</mi></mrow><mrow><mi>x</mi></mrow><mrow><mo stretchy=\"false\">[</mo><mi>t</mi><mo stretchy=\"false\">]</mo><mi>d</mi><mo>∕</mo><mn>2</mn><mo>,</mo><mi>∞</mi></mrow></msubsup></math>, which has been left open since the works of Escauriaza (2000) and Escauriaza and Vega (2001). Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis & PDE","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/apde.2024.17.2257","RegionNum":1,"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
We prove the strong unique continuation property for the differential inequality
with contained in weak spaces. In particular, we establish the strong unique continuation property for , which has been left open since the works of Escauriaza (2000) and Escauriaza and Vega (2001). Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces.
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