{"title":"最优运输中自由边界的C2，α$C^{2，\\alpha}$正则性","authors":"Shibing Chen, Jiakun Liu, Xu-Jia Wang","doi":"10.1002/cpa.22151","DOIUrl":null,"url":null,"abstract":"<p>The regularity of the free boundary in optimal transportation is equivalent to that of the potential function along the free boundary. By establishing new geometric estimates of the free boundary and studying the second boundary value problem of the Monge-Ampère equation, we obtain the <math>\n <semantics>\n <msup>\n <mi>C</mi>\n <mrow>\n <mn>2</mn>\n <mo>,</mo>\n <mi>α</mi>\n </mrow>\n </msup>\n <annotation>$C^{2,\\alpha }$</annotation>\n </semantics></math> regularity of the potential function as well as that of the free boundary, thereby resolve an open problem raised by Caffarelli and McCann.</p>","PeriodicalId":3,"journal":{"name":"ACS Applied Electronic Materials","volume":null,"pages":null},"PeriodicalIF":4.3000,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"C\\n \\n 2\\n ,\\n α\\n \\n \\n $C^{2,\\\\alpha }$\\n regularity of free boundaries in optimal transportation\",\"authors\":\"Shibing Chen, Jiakun Liu, Xu-Jia Wang\",\"doi\":\"10.1002/cpa.22151\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The regularity of the free boundary in optimal transportation is equivalent to that of the potential function along the free boundary. By establishing new geometric estimates of the free boundary and studying the second boundary value problem of the Monge-Ampère equation, we obtain the <math>\\n <semantics>\\n <msup>\\n <mi>C</mi>\\n <mrow>\\n <mn>2</mn>\\n <mo>,</mo>\\n <mi>α</mi>\\n </mrow>\\n </msup>\\n <annotation>$C^{2,\\\\alpha }$</annotation>\\n </semantics></math> regularity of the potential function as well as that of the free boundary, thereby resolve an open problem raised by Caffarelli and McCann.</p>\",\"PeriodicalId\":3,\"journal\":{\"name\":\"ACS Applied Electronic Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.3000,\"publicationDate\":\"2023-09-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Electronic Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22151\",\"RegionNum\":3,\"RegionCategory\":\"材料科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ENGINEERING, ELECTRICAL & ELECTRONIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Electronic Materials","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22151","RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, ELECTRICAL & ELECTRONIC","Score":null,"Total":0}
C
2
,
α
$C^{2,\alpha }$
regularity of free boundaries in optimal transportation
The regularity of the free boundary in optimal transportation is equivalent to that of the potential function along the free boundary. By establishing new geometric estimates of the free boundary and studying the second boundary value problem of the Monge-Ampère equation, we obtain the regularity of the potential function as well as that of the free boundary, thereby resolve an open problem raised by Caffarelli and McCann.
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