{"title":"弱空间中具有势的热算子的唯一延续","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":"37 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":\"37 1\",\"pages\":\"\"},\"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}","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
摘要
我们证明了微分不等式 |(∂t+ Δ)u(x,t)|≤V(x,t)|u(x,t)| 的强唯一连续性，其中 V 包含在弱空间中。特别是，我们建立了 V∈ Lt∞Lx[t]d∕2,∞ 的强唯一延续性质，这是自 Escauriaza (2000) 和 Escauriaza and Vega (2001) 的著作以来一直悬而未决的问题。我们的结果是洛伦兹空间中热算子的卡勒曼估计的结果。
Unique continuation for the heat operator with potentials in weak spaces
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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